Utah State University DigitalCommons@USU All Graduate Theses and Dissertations Graduate Studies 5-1965 A Study of the Structure of Shear Turbulence in Free Surface Flows Maddineni Venkateswara Rao Utah State University Follow this and additional works at: https://digitalcommons.usu.edu/etd Part of the Engineering Commons Recommended Citation Rao, Maddineni Venkateswara, "A Study of the Structure of Shear Turbulence in Free Surface Flows" (1965). All Graduate Theses and Dissertations. 1460. https://digitalcommons.usu.edu/etd/1460 This Thesis is brought to you for free and open access by the Graduate Studies at DigitalCommons@USU. It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of DigitalCommons@USU. For more information, please contact digitalcommons@usu.edu. Logan. Utah Illiiliiiluliiiill 3 9060 01399 8348 .-A STUDY OF THE STRUCTURE OF SHEAR TURBULENCE Approved: IN FREE SURFACE FLOWS by Maddineni Venkateswara Rao A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Fluid Mechanics Return To: Major Professor " UTAH WATER RESEARCH LABORATORY UTAII STATE UNIVERSITY LOGAN , UTMf 84322 "'"D'iJlrFM' Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 1965 ACKNOWLEDGMENTS This investigation could not have been a,ccomplished successfully without the cooperation and help of many faculty members and employees of the College of Engineering. At the outset the author wishes to single out the help rendered by Dr. C. G. Clyde, under whose supervision the research has been carried out. The author likes to take this opportunity to express his deepest sense of gratitude to him for his constant encouragement and constructive criti-cism in the course of the work, for his generosity in sparing considerable time to discuss every phase of the work, for his guidance in designing the equipment and valuable consultation in debugging the experimental equipment, and finally for his active interest in providing an environment conduci ve to research by acquiring neces sary equipment in a short time . . The author extends his sincere thanks to the committee members Dr. A. A. Bishop, Dr. G. H. Flammer, Dr. G. Z. Watters, and Dr. R. K. Watkins for their advice and painstaking review of this dissertation. Special thanks are due Dr. G. H. Flammer for permitting the author to conduct tests in the flume belonging to his project and to Professor L. S. Cole for permission to make use of the facilities of the anechoic chamber and to borrow various instruments from the stores of his department. i Since re thanks are offered to Professor C. N. Merkley for his help in constructing the tranducer probe, to Mr. Pat Buller and Professor D. G. Chadwick for their help in debugging the experimental equipment, and to my friend and colleague Mr. M. A. Abaza -_ for extending his helping hand many a time ungrudgingly. Thanks are due to -Profes sor Elliot Rich of the Civil Engineering Department and Dr. V. E. Hansen of the Utah Water Research Laboratory for making availQble necessary funds. Finally the author acknowledges with gratitude the financial aid supplied by the U. S. A. I. D. (TCM) scholarship and the study leave granted by the Madhya Pradesh Government, India. Maddineni Venkateswara Rao ii TABLE OF CONTENTS Page ACKNOWLEDGMENTS TABLE OF CONTENTS LIST OF TABLES i iii v LIST OF FIGURES v Chapter I. II. III. IV. INTRODUCTION . EQUATIONS OF MOTION AND ENERGY FOR TURBULENT FLOW IN TWO-DIMENSIONAL OPEN CHANNELS. STATISTICAL THEORY OF TURBULENCE. Statistical Specification of Turbulent Fields. Intensity of Turbulence Double-velocity Correlation Function Scales of Turbulence . Turbulence-Energy Spectral-Density Function Kolmogoroff ' s Theory of Local Isotropy . Spectral Laws of Locally Isotropic Turbulence Si milarity hypothesis Physical transfer the'ories ' Homogeneous Shear Turbulence Relationships Between Statistical Quantities of Turbulent Shear Flows EXPERIMENTAL APPARATUS AND PROCEDURE 1 8 16 19 21 23 29 32 38 41 41 42 47 53 57 Description of Apparatus 57 Flow system. 57 Tranducer probe . 62 Electronic instrumentation 65 DISA random signal indicator and correlator . 65 Spectrum analyzer 69 Data indicator and recorder 69 Variable time delay network 71 Auxilliary instruments 71 iii Chapter . Page V. VI. VII. VIII. Details of Procedure Measurement of average velocity. Measurement of autocorrelation coefficient Measurement of spatial correlation coefficients Measurement of longitudinal microscale of turbulence Measurement of turbulence-energy spectrum and intensity of turbulence DATA PROCESSING . ReI ati ve Intensity of Turbulence . One-Dimensional Energy Spectrum Function Longitudinal Average Macroscale of Turbulence. Longitudinal Micros cale of Turbulence RESULTS AND DISCUSSION . Distribution of Relative Intensity of Turbulence One-dimensional Energy Spectrum Function Energy spectra in the outer region of flow Energy spectra near the bed . Dissipation spectra Test of simplified form of Heisenberg's equation Autocorrelation Curves Macroscale of Turbulence Microscale of Turbulence Correlation of Turbulence Properties CONCLUSIONS SUGGESTIONS FOR FURTHER RESEARCH 72 72 73 74 75 77 79 79 81 83 84 86 86 94 95 107 109 116 120 125 132 142 LITERATURE CITED 145 149 152 160 167 LIST OF SYMBOLS AND DEFINITIONS APPENDIXES A. Theory of operation of the tranducer probe . 168 B. Calibration of the tranducer probe 171 Frequency response 171 Sensiti vity 174 C. Calibration of the spectrum analyzer . 180 D. Block diagram of the internal circuit of the random signal indicator and correlator 183 iv Table 1. UST OF TABLES Comparison of the values of Kolmogoroff's universal constant (1.' evaluated by different investigators 2. Values of the dimensionless quantity in equation (90) at various relative depths, Reynolds numbers, and Figure 1. 2. 3. 4 . 5. 6. relati ve roughnes ses UST OF FIGURES Experimental flume--plan and sectional elevation Flume-bed roughness meshes Experimental flume and staging--general view . Design details of the tranducer probe . Photograph of the final shape of the tranducer probe. A general view of the instrumentation set-up 7. Block diagram of the tranducer probe and associated electronic instrumentation 8. 9. 10. 11. 12. Typical oscilloscope pictures of turbulent total-head fluctuations at RN =.7.25 x 10 4 --;- Series At. Distribution of relative intensity of turbulence as a function of relative depth with RN as a parameter--Series A. . . . . . . - . . • Distribution of relative intensity of turbulence as a function of relative depth with RN as a parameter--Series B . . . . . . . . . . Distribution of relative intensity of turbulence as a function of relative depth with RN as a parameter--Series C. . . . . . . . . . Distribution of relative intensity of turbulence as a function of relative depth with relative roughness as a parameter at RN = 7.25 x 104 v Page 115 143 Page 58 60 60 63 66 66 67 70 88 89 90 91 13. 14. 15. 16. 17. 18. 19. 20, 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. Distribution of relative intensity of turbulence as a function of relative depth with relative roughness as a parameter at RN == 8. 30 x 104 .......... . 4 One-dimensional energy spectra at RN == 8. 30 x 10 --Series A . . . . . . . . . . . . . . . . . . . . . 4 One-dimensional energy spectra at RN == 7. 25 x 10 --Series A , . . . . . . . . . . . . . . . . . . . . 4 One-dimensional energy spectra at RN == 5.22 x 10 --Series A . . . . . . . . . . . . . . . . . . . " . 4 One-dimensional energy spectra at R == 8. 3 x 10 --Series B . . . . . . . . . . . . ~ . . . . . . 4 One-dimensional energy spectra at RN == 7. 25 x 10 --Series B . . . . . . . . . . . . . . . . . . . . . 4 One-dimensional energy spectra at RN == 5. 22 x 10 --Series B . . . . . . . . . . . . . . . . . . . . . 4 One-dimensional energy spectra at RN == 8. 3 x 10 --Series C . . . . . . . . . . . . . . . . . . . . . 4 One-dimensional energy spectra at RN == 7. 25 x 10 --Series C . . . . . . . . . . . . . . . . . . . . . 4 One-dimensional energy spectra at RN == 5.22 x 10 --Series C . . . . . . . . . . . . . . . . . . Effect of depth of flow on transition wave number 4 One-dimensional dissipation spectra at RN == 8. 3 x 10 --Series A . , 4 One-dimensional dissipation spectra at RN == 8. 3 x 10 --Series C . . . . . . . . . . . . 4 One-dimensional dissipation spectra at RN == 7.25 x 10 --Series C . . . . 4 One-dimensional dissipation spectra at RN == 5.22 x 10 --Series C . . . . . . . . . . . . Test of simplified form of Heisenberg's equation at RN == 8.3 x 10 4-- Series C . . • . Test of simplified form 'of Heisenberg's equation at RN == 8. 3 x 1 04 - - S eri e sA. . . . . . . . . . . Autocorrelation curve at RN == 8.3 x 104 at a relative depth x 2(Fi == O. 179 -- Seri~s C . . . . . . . vi 92 95 96 97 98 99 100 101 102 103 108 III 112 113 114 118 119 121 31. 32. 4 Autocorrelation curve at RN = 8. 3 x 10 at a relative depth x 2/h = O. 284 -- Series C . . . . . . Autocorrelation curve at RN = 5.22 x 104 at a relative depth x2/h = O. 137 -- Series C . . . . . . . . . . . 33. Autocorrelation curve at RN = 5.22 x 104 at a relative 34 . 35. depth x 2/h = O. 242 . . Variation of L Ih versus x 2/h at different values of Xl RN and h/ks for the case of fully-developed turbulent flow .. Variation of L Ih versus x 2/h in transition regime of xl 122 123 124 127 flow with RN as a parameter. . . . . . . . . . . 128 Variation of X. as a function of relative depth x 2/h . ~ 1. . 4 with h/ks as a parameter at RN = 8.30 x JO . 36 . 133 37. Variation of X. as a function of relative depth x 2/h Xl with h/ks as a parameter at RN = 7.25 x J 04 134 38. Variation of X. as a function of relative depth x 2/h Xl - . 4 with h/ks as a parameter at RN = 5. 22 x 10 135 39. Variation of X. as a function of relative depth x2/h Xl with RN as a parameter -- Series A .... 138 40. Variation of X. as a function of relative depth x 2/h Xl with ~ as a parameter -- Series B ..... 139 41. Variation of X. as a function of relative depth x 2/h Xl with R as a parameter -- Series C . . . . N 140 vii 42. 43. 44. 45. 46. 47. 48. Experimental set-up for calibration of the tranducer probe at low-irequency range ... . . . . . . . . . Low-frequency response of tranducer ·probe No.2. Block diagram of the experimental set-up for calibration of the tranducer probe in anechoic chamber. . . . . .' . . . . . . . . . . . . Pressure sensitivity of the tranducer probe No. 2 Pressure sensitivity of the tranducer probe No. 3 Response of spectrum analyzer Block diagram of the internal circuit of the random- . signal indicator and correlator ........ . viii 172 173 176 178 179 182 183 CHAPTER I INTRODUCTION Turbulence is a familiar phenomenon which gives rise to complicated problems in many branches of engineering. Hinze (1959. p. 1-2) has set forth the following definition for turbulence: II Turbulent fluid motion is an irregular condition of flow in which the various quantities show a random variation in time p nd space coordinates, so that statistically distinct average values can be discerned. II Osborne Reynolds (1894) was the first to introduce the notion of statistical mean values into the study of turbulence. He visualized tur-bulent flow as the sum of mean and eddying motion. By introducing this sum of mean velocity and fluctuating velocity into the Navier-Stokes equations and with the aid of the continuity equation, he derived equations giving re-lationships between the various components of the fluctuating velocity. It was soon realized that before any further results could be obtained from a theoretical analysis of Reynold;s equations of motion, a mechanism had to be postulated for the ihteraction of fluctuating v~locity components at dif-ferent points in the turbulent field. Consequently, three decades after Reynold's: work, phenomenological theories of turbulence, such as the mo-mentum-transfer theory of Prandtl (1926), the vorticity transport theory of Taylor (1932) and the similarity theory of Karman (1930) were introduced. Not only are they based on unrealistic physical models, but they do not 2 furnish any information beyond temporal-mean velocity distribution. A complete theory of turbulence should describe the mechanism of production of turbulence, its convection, diffusion, distribution, and eventual dissipation for any given boundary conditions. A step towards this obj ecti ve was undertaken by G. 1. Taylor (1921, 1935, 1938) when he proposed a theory based on the statistical treatment of the random fluctuations of the properties of the turbulent velocity field as suming a stationary, homogeneous, isotropic turbulence . He introduced various concepts such as turbulence intensity, Lagrangian correlation co-efficient, Eulerian time and double-velocity space-correlation coefficients, different scales of turbulence, and one-dimensional energY-6pectrum function. These parameters would constitute what has come to be called the structure of turbulence. He further · showed that the .knowledge of the turbulence intensity and the Lagrangian correlation function alone would be sufficient to determine the laws of diffusi.on, independent of any physical model for mixing. Renewed interest was created among research workers in this field by his pioneering efforts; and subsequently many theoretical investigations have been made by Karman and Howarth (1938), Robertson (1940), Batchelor (1948), Kampe de Fe'riat (1948), and others. Progress in hot-wire annemometry and electronic instrumentation has made it pos-Sible to check the validity of the theory in flows of air in wind tunnels just downstream from screens which generate a homogeneous and near-isotropic turbulence. The most notable studies made in this connection are thoseof Dryden (1938) and Simmons and Salter (1938). 3 Although the concept of isotropy is essential for mathematical sim-plicity, it is not encountered in actuality. Nevertheless, the Fourier a~nalysis of the equations of motion leads to some important clues pertaining to the mechanism of the production and dissipation of turb~lent energy. From a Fourier transformation of the Reynolds equations there arise three distinct terms: (1) a term describing the production of turbulence and its convection and diffusion by the mean motion, (2) a term describing the interaction of eddies characterized by different wave numbers and the transfer of turbulent energy from one eddy to another of different wave number, and (3) a term describing the dissipation of turbulent energy by viscosity. From the form of these terms it can be inferred that the production term is important in the low wave-number region, the viscous dis sipation term is important in the high wave-number region and the second term accounts for the transfer of energy between wave numbers throughout the spectrum. Next the question arises regarding whether or not the transfer of turbulent energy and its dissipation by viscosity at high wave numbers is independent of the manner in which the turbulence is generated. An answer to this is found in Kolmogoroff ' s (1941) theory of local isotropy which states that the fine scale structure (high wave-number components) j, of actual turbulent shear flows at high Reynolds numbers is isotropic. The idea of local isotropy was put forward independently by Onsager (1945), and Obukoff (1941). Kolmogoroff derived spectral laws for the distribution of turbulent energy with wa ve number in the region of local isotropy or 4 equilibrium range. Various spectral laws for the equilibrium range have been put forward by Heisenberg (1948), Obukoff (1941), Karman (1948), Chandrasekhar (1949), Kov.e.sznay (1948), and Tachen (1953) to bring the theory into closer agr'eement with experimental results. As in the case of isotropic turbulence, dynamical equations either for the correlation function or for the spectrum function could not yet be developed for turbulent shear flows due to formidable mathematical dif-ficulties. As a matter of fact, very little has been accomplished in the analytical study of the statistical theory of self-maintained shear turbu-lence such as occurs in fully-developed turbulent flow in pipes and open channels, To overcome these mathematical difficulties, perhaps new physical hypotheses might be required. Meanwhile, however, considerable insight into the mechanics of turbulent shear flows could be obtained by extensive experimental studies of the distribution of the intensity of tur-bulence, macr~, and microscales of turbulence, double-velocity correlations and energy spectrums, since these parameters constitute the structure of turbulent shear flows as well. Furthermore, such studies might lead to the formulation of new physical hypotheses concerning the actual mechanism of turbulence in shear flows. The fact that experimental methods and .d theoretical analysis are always complementary to each other in the search for understanding inthe ,physical sciences lends credence to this approach. Recently many experimental investigations have been made in turbulent shear flows. In the case of turbulent flow in a pipe, Romano 5 (1954) measured the energy-spectrum function, Brookshire (1961) measured intensity and microscale, and Jordan (1963) studied the distribution of microscale and macroscale. The investigations of Laufer (1951, 1954) on the structure of turbulent shear flows in pipes and two-dimensional channels are the first and the most exhaustive. All of the above studies were made in air flows using hot-wire anemometers as sensors. But no such systematic investigations have been made in the past into the struc-ture of turbulent shear flows either in smooth or rough open channels, presumably because of a lack of a reliable liquid-turbulence measuring probe with the necessary sensitivity, response , · and strength. Some partial attempts have been made by Ippen (1957) and Clyde (1961), who used di-aphragm types of pressure tranducers whose maximum frequency responses were low. In order to overcome this deficiency of experimental data on the structure of turbulent shear flows in open channels, this proj ect has been undertaken. For this investigation, a total-head tube with a barium titanate ceramic tranducer, as the sensor, was constructed, similar to the one developed by Eagleson et al. (1959). The device has a sufficiently high, as well as low, frequency response. Such a total-head probe was believed to have clear advantage over other types for measurements in unclear water even though it has the serious disadvantage of measuring only one of the three components of the fluctuating velocity vector. The objectives of this research are: 1. To study the turbulence-energy distribution as a function of relative position and Reynolds number. 2. To determine the existence of local isotropy in the wave-number space and the applicability of the spectral laws of locally.i.sotropic turbulence. 6 3. To determine whether or not the pattern of the distribution of relative intensity of turbulence versus relative depth in the case of a rough boundary differs from the published results for smooth-boundary flows. 4. To determine autocorrelation functions and obtain therefrom the distribution of the mean macroscale of turbulence as a function of relative depth and Reynolds number. 5. To determine the distribution of microscale as a function of relati ve position and Reynolds number. 6. To make an attempt to obtain some relationships between the statistical quantities describing nonisotropic flows. Literature pertinent to the structure of turbulence is examined crit-ically as a basis for this experimental investigation. Also thebretidal studies concerning the shape of turbulence energy spectrum are reviewed and compared with the shape determined by experiment. A concluding discussion will point out the contributions of this re-Search as well as the limitatiors imposed by the instrumentation used. In 7 the light of the experience gained during thi~ research, some suggestions for further investigations are offered in the closing chapter. CHAPTER II EQUATIONS OF MOTION AND ENERGY FOR TURBULENT FLOW IN TWO-DIMENSIONAL OPEN CHANNELS Osborne Reynolds (1894) was the first to make systematic ex-8 perimental and theoretical investigations of turbulent flow. He assumed that the Navier-Stokes equations describe the turbulent motion at every instant. He also suggested that the instantaneous velocity and pressure could each be expressed as the sum of their time average and fluctuation, i. e. : U. = U. + u. i = 1, 2, 3 (1 ) 1 1 1 where T 1 S U . (t) dt U. = 1 2T 1 -T T 1 S u. (t)dt = 0 u. = 1 2T 1 -T Similarly P = P + p (2) Reynolds substituted equations (1) and (2) into the Navier-Stokes equations for steady incompressible fluid flow and took time averages in the resulting equations term by term and obtained the following equations for the mean flow: au, 1 P U, Q J oX, J = a2 u, _a_ (P + -yh) + ~ 1 ax, ax, ax , 1 J J - p ' a u ,u, 1 J ax, J 9 (3) These equations differ from the Navier-8tokes equations only in the pres-ence of additional terms involving velocity fluctuations. The term p u,u, 1 J is usually referred to as the Reynolds-stress tensor. The Reynolds equations (3) could not be solved, as there are ten unknowns with only four equations including the continuity equation. The Reynolds stresses must be known before there is any hope of solving the equations of motion for turbulent flow. In steady unidirectional flow in an open channel, ...,. U 1 (x2) U ' = 1 U 2 = 0 U 3 = 0 In fully-developed uniform turbulent flow the means of the squares of fluctuating velocities and means of the products of fluctuating velocities are functions of x 2 alone. Therefore, equation (3) becomes (P + -yh) = (4) Equation (5) reduces to - - 2 P + ')Ih + P u 2 = F (xl) = e Equation (4) becomes At x 2 = 0, - ex 2 u l = u 2 = 0 and T = T o d Ul (T 0 - e x 2) = iJ. dx - P ul u2 2 This is the equation of motion for turbulent flow in an open channel. 10 (5) (6) (7) Multiplication of the equation (7) by the mean-velocity gradient leads to (7a) This is the mean-energy balance equation. The term at the left represents the ra.te per unit volume at which mechanical energy is lost by the me~n 11 flow; the first term at the right is the rate of energy dis sipation by the direct action of viscosity, and the second is the rate of turbulence pro-duction. Klebanoff (1954) compared the production of turbulent energy and viscous dissipation for the region near the wall (0 < y* < 30) in his turbulent boundary layer studies and concluded that approximately 40 per cent of the energy extracted from the mean flow is directly dis-sipated by viscosity and, of the remaining 60 per cent which is converted into turbulent energy, 30 per cent is produced in the same region. From this it is evident that, in order to obtain a complete picture of turbulent-energy balance, conditions near and within the laminar sublayer have to be known. This is a very difficult task from the experimental point of view. Further insight into the mechanism of turbulent action results from a study of energy equations which describe the energy balance for the turbulence at a given sect ion. These equations can be derived by utilization of the Reynolds technique of substituting mean and fluctuating components for the instantaneous values of velocity and pressure and by following his rules for taking averages. For more details reference may be made to books by Einze (1959), Rouse (1959), and Townsend (1956). Multiplying the Navier-Stokes equations by U,' the following 1 equation is obtained: au, 1 P U,U, a 1 J x, J = a2u. u. a (P + 'Yh) + tJ. U : 1 1 ax. 1 ax. ax. 1 J J (8) 12 upon rearrangement, 1 a (u. u. u.) a (yh) a (pu.) 1 1 J U. 1 P = - + 2 ax. 1 ax. ax. J 1 1 2 1 a (u. u.) au. au. 1 1 1 1 ~ - ~ ax. 2 ax. ax. ox. J J J J (9)" Introduction of the velocity and pres sure perturbations = u 3 -p = p + p into equations (9) and utilization of the fact that the mean and fluctuating velocities are functions of x 2 only, the following turbulence energy equation is obtained: d UI d u1 u2 - 1 d 222 p u I u2 dX2 + P U1 dX2 +-p dX2 u2 (uI + u 2 + u 3) 2 d d2 U - (pua) ,+ ~ 111 1 = ,),S 0 U 1 + dX2 2 dX2 d 2 au au . 1 2 2 2 f'.( a/X ax.l) 2 ~- (u1 + u2 + u3 ) dx2 J J ' (10) 13 In light of equation (4), equation (10) can be simplified to dU1 2 2 2 (~)2 d [ u1 + u 2 + u 3 ) + U2pJ T = 'pu 2 ( + dX2 dX2 2 1-1 dx ' 2 1 d 2 (U~ 2 2) f1(Ui ] G~) 2 1-1- + u 2 + u3 + 2 ax. ax. (11 ) dX2 J . J Von Karman (1937) obtained this form for the case of turbulent flow between parallel plates in connection with the discussion of non- . isotropic flow in terms of the statistical theory of turbulence. The terms from left to right ha ve the following physical meaning: The first term corresponds to the rate of production of turbulent energy per unit volume by the action of the shearing stresses at a point. The second term represents the rate of convective diffusion by turbulence of the total turbulence energy. The third term stands for the rate of dissipation of energy by the action of the viscous shear stresses. The fourth term is the rate of energy diffusion by viscous forces. The fifth term expresses the rate of turbulent-energy dissipation into heat by the action of vis-cosity. The diffusion terms vanish upon integration of equation (11) over the space. This means that the diffusion process acts only to transfer energy from one place to another and the energy is neither created nor destroyed , nor is there any net exchange in energy with another direction of diffusion. 14 In the absence of a statistical theory of shear turbulence, exten-sive experimental studies were made by Laufer (1~ 51, 1954), Klebanoff and Diehl (1951), Klebanoff (1954), and Schubauer (1954) to sketch out ade-quately the spatial budget of energy in nonhom.ogeneous shear flows which occur in turbulent boundary layers in pipes and channel flows. The basis used was that formed by the equations of the type (7a) and (11), incor-porating the ideas of the Reynolds number, similarity, and self-preservation. For a high Reynolds number flow it was shown by Laufer (1951) that the second and third terms on the right s ide of equation (11) may be neglected. Hence, the equation (11) reduces to d = The term on the left side of the equation can be obtained directly from the measurements of T and from the mean-velocity profile. The determination of the dissipative derivatives is the most difficult task because of the im-portance of the region near the wall. Since only five terms (first four and seventh) in the dissipation function can be measured in the vicinity of the wall, the remaining four were estimated by Laufer (1954) and Klebanoff (1954) using isotropic relationships .' From the known distributions of turbulence-energy production and dissipation, the distribution of diffusion of energy can easily be calculated from equation (12). Because of the limitations of the piezoelect ric probe, no attempt could be made to measure the distribut ions of the various terms in equation (1 2) in this investigation. 1 5 For later use, the salient points in the conclusions of the studies of Laufer and Klebanoff are summarized briefly: The flow section can be divided into three regions; the wall or inner region, intermediate region, and the outer region, or fully turbulent region. In the inner region various energy rates, production, diffus ion, and dissipation are of e c ual impor-tance and show 'CI sharp maximum, within y* < 30. Klebanoff speculated that in a small region near the wall (y* < 20) the rate of dissipation is greater than the rate of production and the deficiency is compensated by an inward flow of energy toward the wall from the outer region, because of the action of the pressure forces. In the outer region the production is negligible, and the gain of energy resulting from the flux of kinetic energy by turbulent diffusion from the wall region compensates for the loss by dissipation. In the intermediate region (the major part of the section), turbulence-energy balance is furnished by production and dissipation only. If these conclusions were correct, 85 per cent of the total dissipation (viscous and turbulent) takes place in y* < 30, signifying the importance of the region near the wall. This qualitative picture of turbulence-energy balance could be vindicated only with detailed measurements in the wall region by improved instrumentation, even though the above conclusions are quite plausible. Some of the terms in the equation (1\2) will be used later to analyze the statistical characteristics of nonisotropic turbulent flow. 16 CHAPTER III STATISTICAL THEORY OF TURBULENCE The phenomenological theories of turbulence proved to be fruitful to describe momentum, mass, and heat transfer processes, not so much because they describe correctly the mechanism underlying turbulent trans-port processes, but because they have resulted in useful and practical semi-empirical relationships" Neither Prandtl nor Taylor discusse 's in detail the mechanism by which the lumps of fluid adapt their transferable property to their new environments. Due to pressure fluctuations, the momentum of each lump of fluid does not remain constant during its travel over a distance equal to the mixing length as Prandtl assumed in his theory. The constanc¥ of vorticity during the travel of the lump over a distance as assumed by Taylor is true only in the two-dimensional case, and the extension of the theory to three-dimensions is too involved to be of much practical use. Even though von Karman was successful in sug-gesting a similarity theory of turbulence independent of any physical model, his theory is restricted to the low~requency components of tur-bulence only because he assumed that the turbulence mechanism is independent of viscosity except in the laminar sublayer. The statistical theory of turbulence introduced by Taylor (1921, 1935) describes in a statistical way the character and properties of turbulence such as the intensity of turbulence, the scale of turbulence, and the eddy~iffusion coefficient by means of correlation coefficients and the energy spectrum of turbulence without speculating as to the mechanism of turbulence itself. Before treating the se parameters in greater detail, some discussion about how the behavior of a turbulent flow field could be represented mathematically and what kind of simplifying assumptions may be made is desirable. 17 There are two distinct ways of representing a turbulent flow field, namely Eulerian and Lagrangian. In the Eulerian representation, the var-iation of statistical parameters at one or two fixed points in the turbulent flow field is considered. In the Lagrangian description, attention is focused on a single fluid particle and the variation of statistical para-meters with time as the fluid particle travels through the flow field is considered. Although the Lagrangian description has certain advantages over the Eulerian description in the development of theory of turbulent diffusion in particular, it has a s~rfous disadvantage in that the experimental deter-mination of Lagrangian statistical quantities can only be obtained through a tedious process of data reduction, whereas Eulerian quantities can be determined directly by the modern electronic instruments. The statistical theory of real turbulent flows (shear flows) is very complicated and mathematically intractable. Therefore, in the develop-ment of the theory, Taylor and subsequent investigators considered for 18 mathematical simplicity an idealized homogeneous and isotropic turbulent flow in the absence of mean motion. Homogeneous turbulence is character-ized by the invariance of the statistical characteristics for any translation of the coordinate system in the space occupied by the fluid ~ If, in addition to homogeneity, the mean value of any function of the turbulent velocity components and their space derivatives is invariant with respect to rotation and reflection of the coordinate system, the turbulent .fi'eld is isotropic. Thus, from isotropy it follows that Const = 0 (13) Isotropic turbulence is an idealized flow situation that can be realized even approximately only under controlled laboratory conditions. Homo-geneity permits the sampling point to be chosen arbitrarily. Most of the real flows are nonisotropic; consequently, it is extremely difficult to realize homogeneity even in one direction, except in axisymmetric flow. For the more general case of nonisotropic turbulence, (14) Even though the statistical theory of nonisotropic or turbulent shear 19 flows has not been developed yet, the various statistical parameters and concepts applicable to isotropic turbulence are equally applicable to turbulent shear flows as well. In turbulence theory, the fluid motion is assumed to consist of the superposition upon a mean flow of eddies covering a range of sizes with distinguishable upper and lower limits. The upper size limit of the eddies is usually determined by and is of the order of magnitude of the dimension of the containing apparatus. The lower limit is determined by viscosity effects and decreases with increase in average velocity, other conditions remaining the same. As the eddy becomes smaller, the velocity gradient in the eddybe.ccnnes lar.g~·r; and thus the viscous shear becomes larger. This increase in viscous shear counteracts the eddying motion, thus further limiting the size of the smaller eddies. Even in the limit for moderate flow velocities, for example, air flows at speeds below about 200 ft/ sec and water flows up to still higher speeds if there is no cavitation (Corrison, 1961), the size of the smallest eddies is much greater than the mean free path of molecules so that fluid flow could still be treated as a continuum flow. Statistical Specification of Turbulent Fields The velocity vector u (x, t) in a turbulent flow field is a random function of position and time w!l0se values conform to probability dis-tributions which are determined for any particular problem by the boundary 20 conditions and initial conditions. The values of I u l at various positions and instants of time are connected by means of joint-probability distribu-tion functions as a consequence of the equations of motion and continuity. As pointed out by Townsend (1956) and Batchelor (1953), for the detailed statistical description of a turbulent field, a knowledge of the complete system of j oint-probability distributions of the values of the velocity lUi at I n l points of the space at the appropriate value of I t l is required, since it is supposed that the statistical properties of the turbulence field at different val ues of I t l are uniquely related by the equations of momen-tum and continuity. From the viewpoint of the experimenter, the above statement is an admission that it will never be possible to describe com-pletely the statistical properties of even the simplest turbulent flow. Nevertheless, some progress has been made in the development of the theory of turbulence using correlation and spectrum functions even though these functions give an incomplete description of turbulent motion. As Townsend points out, The central role that the correlation and spectrum functions play in current theory and experiment is due to their comparative simplicity and convenience of measurement. The correlation function is one of the infinite set of integral moments of the basic joint-probability distribution function, and attempts to use the correlation function as a complete description amount mathematically, to making a hypothesis concerning the nature of the complete probability distribution function and, physically, to making statements about the eddy structures that exist in fully developed turbulence. (Townsend, 1956, p. 12.) 21 Intensity of Turbulence Since the fluctuating component of velocity is a property of vital interest in the statistical treatment of turbulence, an average value of this fluctuating velocity which is non-zero must be defined. An average value of I u l that has received wide use is the root-mean-square. The reason for this popularity is that the root-mean-square (a measure of the spread of the probability-density curve of the random variable) of I u l is easy to measure with existing experimental equipment and the mean square of I u l occurs quite naturally in many of the equations governing the turbulent flow. The root-mean-square of the fluctuating velocity is also known as the intensity of turbulence and may be defined mathematically, = 1,2,3 (15) This definition of turbulence intensity was first introduced by Dryden and Kuethe (1930). It should be noted that the average defined above is a statistical or ensemble average. From the viewpoint of the statistical theory of turbulence, it is the ensemble average that should be used; but it is usually not possible to measure the ensemble av~rage experimentally. For example, if the ensemble average value of the fluctuating velocity in a channel at a given time is to be measured, it would require that the 22 averaging process be made in an infinite number of identical channels in which the physical conditions are the same with the component of fluc-tuating velocity being measured at the same time in all channels. This is, of course, experimentally impossible. The impossibility of measur-ing ensemble averages has led to the extensive use of time average that could easily be measured by conventional measuring instruments. There-fore, it is necessary to prove that a time average is identical with a statistical average for the experimental field (ergodic theorem). Un-fortunately, no ergodic theorem has yet been proved in fluid dynamics, as pOinted out by Pai (1957). Since the experimental field of turbulence is such that the velocity at a fixed point is a stationary random function of time, applicabilit'y of ergodicity could be assumed reasonably; and thereby time average and statistical average are considered identical here as is customary in the turbulence literature. So time average of square root of variance may be written as: Ut = ( U~) 1/2 = [~t SLl~~ dt ] 1/2 The relative intensity of turbulence is defined as J~ u. 1 U. 1 i. e. u l i U. 1 i=1,2,3 (16) (1 7) 23 It should be noted that the turbulence intensity characterizes only the magnitude, but not the frequency, of the turbulence fluctuations. Double-Velocity Correlation Function The statistical theory of turbulence is not concerned with the joint-probability distribution functions themselves, but rather with the correlations between various velocity components. A large part of the theory is concerned with the properties of the covariance of the velocities at the two points or the double-velocity correlation. Von Karman and Howarth (1938) introduced the correlation-tensor function as a general-ization of the particular correlation coefficients discussed by Taylor (1935). The velocity-correlation tensor for two points separated by space -+ vector r is -+ R .. (r) 1J = u. (~ u. (; + ?) 1 J It should be noted that since physical quantities under a mean-value (18) sign are taken at the same instant, explicit reference to time is omitted. -Von Karman first pointed out that R .. (r) forms a tensor 'of the second 1J order. Under the assumption of isotropic turbulence in the absence of mean flow, von Karman and Howarth considered the special case where the line connecting the two points is parallel to one of the cartesian Coordinate axes and showed that all the components but two of the 24 correlation tensor vanish. /The two nonvanishing components are the longitudinal-and lateral-velocity correlations which depend only on the distance Irt between the two points. The longitudinal-velocity correla-tion function and lateral-velocity correlation function first introduced by von Karman and Howarth (1938) are given by 2 u f(r) = (19) .... -r = (r, 0, 0) 2 -u g (r) = R22 (r ) (20) These investigators also showed that f(r) and g(r) are not independent, but are related by the equation g(r) = f(r) + r2 df(r) dr which follows from the continuity condition. The correlation tensor (21 ) -R .. (r) can thus be expressed in terms of a single scalar function, either 1J f(r) or g (r). The simpler two-point triple-velocity correlation is given by (22) Von Karman and Howarth (1938) showed that this forms a tensor of the third order, and under conditions of isotropy all but three of the compon-ents of the triple-correlation tensor vanish. The nonvanishing components 25 h (r), k (r) , and q (r) are given by (J)3/2 h{r) -= T 22, 1 (r) (23) (J)3/2 k{r) - -= T 11, 1 (r) r = (r, 0, 0) (24) (-//2 u 2 q (r) = -T 21, 1 (r) (25) For incompressible flow, from the continuity condition it follows that h (r), k (r), and q (r) are related by the relations k{r) = 2h{r) q (r) = - h (r) - ~ ah{r) Br (26) (27) -Thus it follows from the above relations that T .. k{r.) can be expressed 1J, in terms of a single scalar function. Using the above relations and the Navier-Stokes equations of motion, von Karman and Howarth (1938) obtained the following dynamical equation for the propagation of the correlation function u 2 f (r): ~ - 3/2 » 2f~ + 2 (u2) Bt\'u j 2 = 2 vU Ira2f + i £L] L8r2 r ar (28) This equation is the fundamental equation for studying the dynamical behavior of a homogeneous and isotropic turbulence. The only ca.se for 26 which the equation (28) can be said to have been completely solved is for turbulence in the final period of decay where the inertial term h(r) (the triple correlation) can be neglected. A satisfactory solution applic-able to the maj or portion of the time history of turbulence has not been obtained. 'I'he chief difficulty lies in obtaining a determinate set of dynamical equations. From the momentum and continuity equations and from equations invol ving moments (correlation functions) of the fluctuating velocity, com-ponents of any order can be constructed. Each n-order equation so obtained involves the moments of order n + 1 as a direct consequence of the nonlinearity of the Navier-Stokes equations. The various methods of approximation proposed to make the infinite set of dynamical equations finite can be divided into two broad classes. In the first class, models of dynamical processes are postulated on physical grounds. The examples are the theories of Kolmogoroff (1941) and Heisenberg (1948), etc., which are treated later. The second class consists of schemes for systematic analytical approximations. So far, two different approaches have been taken in the second class. One of them consists of ignoring moments of 'n + I' order in equations for I n' order moments. This method was used by Deissler (1958, 1960) for the study of the decay of homogeneous turbulence at earlier times or at higher Reynolds numbers. It is not known whether this approximation yields satisfactory results for interesting cases of the very large Reynolds number. 27 Another approach is to introduce the hypothesis that the fourth-order moments of the distribution of simultaneous velocity components are related to second-order moments as they would be for a normal prob-' ability distribution. Fourth-order moments can then be expressed in terms of second-order moments, and the set of moment equations is closed. proudman and Reid (1954) and Tatsumi (1957) have extensively discussed the decay process of isotropic three-dimensional turbulence on the basis of this hypothesis. The dynamical equations for two-dimensional iso-tropic turbulence that follow from this hypothesis were derived by Ogura (19 62a). Ogura (1962a, b) sol ved numerically the dynamical equations for two- and three-dimensional turbulence for the whole time history of turbulence. The most distinct feature revealed by his calculation is that the energy spectrum becomes negative for medium-sized eddies. He tentatively concluded that the generation of unphysical negative energy is 111.0st likely the ' consequence of the quasi-normality hypothesis. Loitsianskii (1945) took from each term of equation (28) the fourth moment and showed that (29) is an invariant under the assumptions that and Lim r- 00 The integral (29) is usually referred to as Loitsianskii's invariant. The interpretation given by Loitsianskii is that the integral gives the total amount of disturbance introduced by the turbulence generating system. Besides the correlation coefficients f (r) and g (r), the auto-correlation coefficient RE (T) between the values of u l at the same 1 point but different time stand t + T is important and is given by 28 (30) The correlation coefficient is a measure of the relatedness of the variable ul at the two times t and t + T As T tends to be zero, RE (T) 1 tends to be equal to one. As T . recomes larger, RE approaches 1 zero. This means that ul (t) and ul (t + T) are no longer correlated but they may not be independent. From a plot of RE (T) vs T, two 1 inferences could be drawn. First the rapidity with which the relatedness is lost with time delay is shown and, second, detection of a periodic signal hidden in the random velocity fluctuation is indicated by peaks in the plot at intervals of I T I equal to the period of the periodic signal. 29 Scales of Turbulence i The concept of a scale representative of a particular turbulence under consideration was used by Prandtl (1926) and Taylor (1915) in the mixing-length theories. However, their definition of the scale was based on the assumption that the fluid elements act like the molecules in a gas. Hence the scales so defined are fictitious and lack in physical reality. This fact led Taylor (1935) to inlcC!1duce a number of s.cales which charac-terize various statistical aspects of turbulence. The macroscale and microsc ale gained widest acceptance. Taylor set forth the following definition for the scale of turbulence in the Eulerian system: It is clear that whatever we may mean by the diameter of an eddy a high degree of correlation must exist between the velocities at two points which are close together when com-pared with this diameter. On the other hand, the correlation is likely to be splall between the velocity at two points sit-uated many ed<;iy diameters apart. If, therefore, we imagine that the corre}ation R~2 between the values of u1 at two points distaI)t J ,.x2 apart in the direction of the x2 coordinate axis has been', cJ.~lermined for various values of x2 we may plot a curve of RX2 against x :2 and ~his curve will represent, from the statistical point of 'view, the distribution of u1 along the x 2 axis. (Taylor, 1935, p. 426.) With these words he defined 00 (31 ) The length L is considered as a possible definition of the average x2 size of the eddies and is termed the integral scale or macroscale of 30 turbulence. Correspondingly, a length L may be defined by the xl relation: 00 (32) where R is the correlation coefficient between the values of the xl component ul at two points separated by distance xl in the direction of xl-coordinate axis. Another macroscale using the autocorrelation coefficient may be defined as 00 o (33) Taylor (19 3 5) showed that R can be expanded in a series such that x 2 2 4 2 I X " (:U1 ) I x 2 ( a ~l) 2 _ . R I )'2 = - --- + --x 2 2 ! 2 4 ! 2 . x . . aX2 ul 2 u l . If the above series is truncated at the second term then I R = I - 2.' x 2 . Hence, the curvature of the R versus x 2 curve at x = 0 is a x 2 2 ( measure of (34) (35) This relationship can be expressed as ( ~)2 = aX 2 . 2 2 u 1 Taylor intorduced the Eulerian microscale of turbulence, )., defined as = Lim x -0 2 31 (36) (37) 2 Thus,)., is a measure of the rpdius of curvature of the R curve at x 2 The general expression for the mean rate of turbulent-energy dissipation per unit mass (Goldstein, 1938, p. 222) is: au 2 au 2 2 E = V [ 1 ) 2 (_2) + 2 (au3 ) 2 (axl + aX2 aX3 (38) which reduces to E = 7.5 (39) 32 in case of incompressible isotropic turbulence, as first proved by Taylor (1935). Combination of equations (36), (37), and (39) gives E = 15 (40) The IX. I value may now be regarded as a measure of the diameters of the smallest eddies which are responsible for the dissipation of energy. For this reason, X. is sometimes called dissipation scale or length. The theory proposed by Taylor does not predict the distribution of the scale of turbulence or the distribution of the intensity of turbulence for any type of flow. In fact there are no theories that will predict the distribution of the scale of turbulence. Therefore, experimental data must be relied upon to determine how the scale of turbu-Ience is distrib-uted across a channel. Laufer (1951) has determined the distribution of the scale of turbulence in a two-dimensional channel flow of air, but there is no data available to show the distribution of the scale of turbulence in an open channel (either smooth or rough). Turbulence Energy Spectral Density Function A more de'tailed character of turbulence can be obtained by con-sidering the distribution of the total kinetic energy of turbulence among eddies of different sizes, i. e., turbulence spectrum. It is convenient to consider the energy variation with either frequency or wave number. 33 Taylor (193 8) introduced the concept of spectrum into turbulence. Taylor's one-dimensional energy spectrum El (n) may be defined as 00 S E-\ (n) dn = u': or o 00 (41) 2 . El (n}dn is the contribution to ul over the frequencIes between nand n + dn. The normalized one-dimensional energy spectrum F 1 (n) or F 1 (k1) may be defined as 00 00 S PI (n)dn = 1 or S PI (kI)dki = 1 o 0 (42) Taylor (193 8) showed that F 1 (n) and the normalized longitudinal correla-tion function f(xl} are Fourier cosine transforms of each other, i. e. , (43) and F 1 (n) (44) From Equations (32) and (44), when n = 0 (45) Therefore, the value of L can be obtained from the intercept of the xl F 1 (n) curve at n = 0 . 34 It could also be shown that the normalized autocorrelation function ~1 (T) and F 1 (n) are Fourier cosine transforms of each other, i. e. 00 RE (T) = S F 1 (n) Cos (2lT nT)tln 1 (46) 00 F 1 (n) = 4 S RE (T) COS(2lT nT)~ T o 1 (47) Spectrum function is more intuitive than that of correlation function, and one of the advantages of working with the spectrum function is that various relations can be deduced without the restrictive assumptions of isotropy; the field need only be homogeneous. In view of the three-dimensional character of turbulence, it is obvious that the energy-spectrum function must also have a three-dimensional character. Therefore, ft should be noted that Taylor's spectrum function is a one-dimensional section of what is in fact the three-dimensional spectrum function. It should also be pointed out that even though the one-dimensional spectrum function is very con-venient for experimental measurements, it is not too convenient for theoretical study. Heisenberg (1948), therefore, introduced a three-dimensional-space spectrum function E (k); while Batchelor (1953) and Kampe de F6riet (1940) introduced a spectrum tensor E . . (k). For this 1) purpose the relation between correlation tensor R .. (r) and energy-1) spectrum tensor E .. (k), is to be studied. They are related by the 1) following three-dimensional Fourier transform pair: 35 -. ik r -= R .. (;) S E .. Ck) dk 1J 1J e (48) --... S ike -E .. (k) = 1/8TT3 R .. (7.) d r 1J 1J e (49) where k is a wave-number vector and dr: = drl ' dr-2 ' dr-3 and dk = dkl' dk 2' dk3 . Physical significance can be attached to the quantity E .. (k) by substituting ; = 0 in equation (48), 1J R .. (0) = u. (~ u. (~ 1J 1 J = S E .. (k) d k 1J (50) Tn~refore, -E .. (k) 1J represents a density in wave-number space of contri-butions to u, (~) u. (~). Thus it describes how the energy associated with 1 J each velocity component is distributed over the various wave.numbers ~n a harmonic resolution of the velocity field. It is this physical significance -that makes E .. (k) the most important single quantity describing the fielc;l 1J of turbulence. Correlation and spectrum functions of a single scalar variable can ... -be obtained by averaging R .. (r) and E .. (k) over all directions of the 1J 1J - -vector arguments rand k: T .. (r) = 1 S R .. (r) d A (r) 1J 4TTr2 1J (51 ) and 36 'M. . (k) = S E .. (k) d A (k) 1) 1) (52) where d A is an element of area on a sphere with a radius equal to the magnitude of the vector. These scalars are related by Fourier Sine ' transforms from equations (50) and (52) 00 1/2 R .. (0) 11 = 1/2 u. (;Z) u. (~ 1 1 =S 1/2 \]£. (k) d k 11 (53) o The quantity 1/2 'i (k) is the contribution to the turbulent kinetic energy in the wa ve .. number range k to k + dk and will be called E (k) , the three-dimensional energy spectral-density function of Heisenberg and hence E (k) = 1/2 'lI .. (k) 11 (54) 00 S E (k) d k = 1/2 u. 6Z) u. (~ 1 1 (55) o Batchelor (1953) has shown that for the case of isotropic turbulence E1 (k1) is related to E (k) by 00 k 2 ill) E1 (k1) = 1/2 S (1 --t ) dk k k k ' 1 (56) or E(k) (57) 37 Equation (56) can be used to transform the quantity E (k) established from theory to a quantity E1 (k1) that can be measured in the laboratory, if the turbulent field established in the laboratory is isotropic. Conversely, equation (57) can be used to transform laboratory data into a form that can shed light on the mechanism of energy transfer. The dynamic equation of energy spectrum can be obtained by taking a Fourier transform of the von Karman- Howarth equation (28) as shown by Lin (1947). a~ E (k) = W(k) (58) or k k a~ S E(k) dk o = S W(k) dk o (59) The left-hand side of this equation represents the change of total kinetic energy of turbulence. The first term on the right represents the interaction of eddies of different wave-numbers, thereby transferring energy by inertial effects to or from the eddies in the wave number region 0 to k and, therefore, W(k) is referred to as the transfer-spectrum function. The second term on the right can be shown to represent the dissipation of turbulence energy into heat. The weighting factor k 2 in the integral in equation (59) clearly shows that the dissipation of energy is practically all associated with high wave-number components which contain c;i neg-ligible amount of energy. This deduction has a very important bearing on later developments of the theory of energy spectrum. 38 Knowledge of the spectrum function E (k) is sufficient to des cribe the turbulent field completely; and, therefore, determination of E (k) is one of the basic problems in the study of turbulence, To determine E (k) either equation (58) or (59) may be used. The complete solution of equation (58) could not be obtained as the form of the transfer-spectrum function W(k) is unknown. There have been two approaches to overcome this difficulty. In the first approach-similarity-equation (58) is discarded and some conclusions about the functional form of ::8 (k) are drawn from more basic physical principles, namely dimensional considerations. The contributions of Kolmogoroff (1941), On'Sager (1945), von Karman and Lin (1949) belong to this approach . In the second approach, (physical transfer theories) an explicit functional form for W(k) is assumed which .makes possible the solution of the dynamical equation (58). This second approach was used by Obukoff (1941), Kovasznay (1948), Heisenberg (1948), Chandrasekhar (1949), Tachen (1953), and von Karman (1948). Among the two approaches, only those studies concerning the determination of the energy-spectrum function in the equilibrium range will be briefly considered in a later section. Kolmogoroffi s Theory of Local Isotropy The following model for turbulent-energy production and dissipation at high Reynolds numbers was suggested by Kolmogoroff (1941): The mech-anism of turbulence production from the instability of the mean flow is 39 determined by the boundary conditions. Pulsations of small wave number produced from the instability of the mean flow give rise to pulsations of higher wave number because of the inertial forces, and these in turn break down into pulsations of much higher wave number. Turbulence energy cascades from the region of low wave numbers to a region of high wave numbers. This cascading of turbulent energy continues until wave num-bers of pulsations are so high that not all the energy is transferred to the next higher order, but part is dissipated into heat by viscous forces. Kolmogoroff reasoned that, at high Reynolds numbers, the range between wave numbers of pulsations charact eristic of mean flow and those dis-sipating all their energy by viscosity is large. In this case, the region of pulsations of high wave numbers should be far enough removed in the cascading process from the production region to be independent of the production mechanism. Therefore, in any turbulent field at high Reynolds number, a domain Q can be defined which is small enough that it will possess local isotropy. The domain Q will contain pulsations of high wave number describing a motion that is isotropic and steady-state. The fundamental quantities upon which the structure of the motion in the domain Q may depend are the mean dissipation of energy per unit time per unit mass of the fluid E {which determines the intensity of the energy flow in the cascading phenomenon) and the kinematic viscosity I I V • These considerations led Kolmogoroff to make his first hypothesis: .. At sufficiently high Reynolds number there is a range of high wave 40 numbers where the turbulence is statistically in equilibrium and the probability distributions defining the turbulence are uniquely determined by the quantities, e and v." (Hinze, 1959, p . 184 . ) When this equilibrium range is sufficiently wide, it is further argued that the lower wave-number components in the equilibrium range will contribute so little to the total viscous dissipation that a subrange will exist in which the properties will be det ermined solely by E • This concept provides Kolmogoroff's second hypothesis: .. At the lower wave number end of the equilibrium range there is an inertial subrange in which the probability distribution defining turbulence is uniquely determined bYe. II (Hinze, 1959, P. 186.) The concept of local isotropy has exerted profound influence and served as the starting point for many later investigations. One of the most important problems of turbulent shear flows is the estimation of the rate of turbulence-energy dissipation. The theoretical equation for the rate of turbulence-energy dis-sipation (second term on the right side of the equations (12) ) per unit volume of fluid could not be availed of for practical computations as all the nine terms contained in it cannot be measured. But for isotropic turbulence, the expression for the rate of turbulence-energy dissipation per unit mass given by equation (40) is very simple to make use of as it is fairly easy to measure velocity fluctuation and the associated micro-scale. Hence if the existence of local isotropy in the domain of turbulent 41 shear flows is proved, usage of the expression for I E I in isotropic turbulence for estimation of the rate of energy dissipation in noniso-tropic turbulence would be justified. Besides, the concept of local isotropy facilitates the application of the results of studies in isotropic turbulence to the phenomena in actual turbulence (nonisotropic turbulence) that are determined mainly by the fine scale structure where local isotropy prevails. Spectral Laws of Locally Isotropic Turbulence Similarity hypothesis The concept of Kolmogoroff and his two hypotheses can be used to determine the energY-ciistribution function E (k) in locally-isotropic tur-bulence. According to Kilmogoroff' s first hypothesis, the structure of the equilibrium range at the end of high wa ve numbers is determined by £ and v. If the structure is expressed in terms of the energy-spectrum function E (k), similarity considerations lead to a spectrum function of the form (60) 3 1/4 where 11 = (v / E) and cp is a universal dimensionless function of 11k . From Kolmogoroff' s second hypothesis at the high Reynolds number equation (60) must reduce to a form independent of viscosity. From similarity conditions it follows that 42 E(k) = a € 2/3 k -5/3 (61 ) where a is an absolute constant. This is called ' lmogoroff's spectrum law. From equations (56) and (61), the longitudinal one-dimensional spectrum function takes the form 18 2/3 -5/3 E 1 (k 1 ) = 55 ,u E k 1 , 2/3 k -5/3 = a E 1 (62) where 18 a' = 55 a . Laboratory data obtained at high Reynolds numbers should, there-fore, give a straight line with slope - 5/3 when plotted on logarithmic paper if KOlmogoroff's hypotheses are correct. Physical transfer theories The basic assumption made in the development of physical-transfer theories is that W(k) is determined by E (k) only. The specific form of relationship between W(k) and E(k) depends on the particular mechanism of energy transfer considered. Among the physical-transfer theories the most successful one is the eddy-viscosity transfer theory due to Heisenberg (1948). While considering the rate of dissipation of energy by eddies with wa ve numbers les s than a particular k, Heisenberg distinguished between the energy directly dissipated in the form of molecular motion and thermal energy and the energy transferred in the form of kinetic energy to all eddies with wave numbers exceeding the specified k. He argued that 43 the mechanism governing the transfer of energy is similar to the one governing the viscous dissipation provided the molecular viscosity is replaced by a suitable eddy viscosity. Therefore, the expression for the energy transfer is k 5 W(k) dk = o k - 2 v t (k) S k 2 E (k) d k o (63) Heisenberg further assumed that vt(k) can be expressed as an integral over the spectrum in the form where kH is a numerical constant of order unity. Substitution of equation (63) in equation (59) leads to For the equilibrium range, k 00 .-£. 5 E(k) d k~ .-£.5 E(k) d k = - E at at o 0 k = - 2 v 5 k 2 E (k) d k o Combination _of' equat{ons .(65) 'and. (66) results in 00 k E = 2 [ v ~ kH 5 JE~~) d k ] 5 k 2 E(k) d k k 0 (64) (65) (66) (67) 44 Bass (1949) and, independently, Chandrasekhar (1949) obtained an exact solution of (67) in the form 2/3 E(k) = (98~-) -5/3 k 3 [ 8 v 1 + --3 k2 H E 4 J- 4/ 3 k ' (68) In the region where the effect of viscosity is negligible, equation (68) reduces to E(k) = (~)2/3 k -5/3 9kH (69) which is identical with Kolmogoroff's spectrum law in the inertial subrange. For large values of 'k' where viscosity plays an important part equation (68) reduces to ( E kH ) 2 -7 E(k) = -2- k ·2v ' (70) Equations (69) and (70) can be transformed to equations for the longitudinal one-dimensional spectrum function by substituting in equation (56) to give for the low wavenumber region and the high wave-number region, respectively, E1 (k1) 9 (~) 2/3 -5/3 = k1 55 9kH (71 ) and E1 (k1) _1 (~)2 -7 = k1 63 2v2 (72) 45 By substitution of equation (68) into equation (56) the longitudinal one-dimensional spectrum function for the entire region is obtained. 00 -4/3 00 E1 (k1) = 1/2 S Mk -8/3 [ 1 +_ P k 4 ] dk - 1/2 k~ S Mk -14/3 where kl k1 -4/3 [1 + Pk 4 ] d k p = 8 2/3 and M = (_e) 9kH (73) (74) Equation (73) can only be sol ved over the entire range by graphical inte-gration once P and M have been determined. Kovasznay (1948) postulated that the contribution to W(k) comes from a narrow wavGflumber band in the immediate vicinity of k. Since, under these circumstances, W(k) can only be a function of E (k) and k, it follows from dimensional arguments that k S W(k) dk = o - a [E(k) ] 3/2 k 5/2 (75) where a is an absolute constant. Substituting (75) in (59) he sol ved for E (k) and obtained 2 1/3 5/3 [ 2/3 r.:d \4/3 ] 2 E (k) = ( ~1 0 ) k - . 1 - T \.. ) (76) 46 where kd = [ e/v3 ] 114 Obukhoff (1941) assumed that the energy transfer across the wave numer ~_k~ is analogous to the energy transfer from the main motion to the turbulent motion through the turbulent shear stresses; and it follows, therefore, k k 00 S W(k) dk = o a [2 S k 2 E (k) d k ] 1/2 S E (k) d k o k (77) where a is an absolute constant. Substituting (77) in equation (59) he obtained a solution for E(k) h ' h d k- 5/3 , h' '1 b w IC re uces to const. x In t e Inertia su range. Von Karman (1948) visualized that the tLansfer function W(k) may be interpreted as the difference between the energy supplied by the eddies wi th wa ve numbers les s than 'k' and the energy transferred to the smaller eddies with wave numbers greater than 'k'. On this basis he defined k 00 W (k) = 2 a [E f3 k 'Y S E i3' k 'Y' d k - E i3' k 'Y' S E i3 k 'Y d k ] o k (78) which includes, as special cases, the assumptions made by Obukoff, Kovasznay, and Heisenberg and obtained E (k) = E (k '/21 7/6 e __ where k is a function of time. e (k/k ) 4 e (79) 47 Homogeneous Shear Turbulence Most of the research in turbulence so far has dealt with the case of infinite homogeneous isotropic turbulence fields. Extensions to specific isotropic turbulence fields were made by Batchelor (1946) and Chandrasekhar. (1950 a, b). They studied axisymmetric turbulence, a field of turbulence with const ant mean velocity where the average value of any function of the velocities and of their deri vati ves is invariant with respect to rotations and reflections about a fixed axis. However, in many practical problems in fluid mechanics such as in turbulent boundary layers, pipe flow, and channel flows, the fields of turbulence are nonhomogeneous with mean velocity gradients. No theoretical studies of nonhomogeneous turbulent shear flows have been made to date. But in the recent years some theoretical studies of homogeneous turbulence with a uniform trans verse velocity gradient have been made by Tachen (1953), Burgers and Mitchner (1953), Reis (1952), Deissler (1961, 1962), Fox (1964), and Lumley (1964). Even though the homogeneous turbulent shear flow is a hypothetical one, it will be a better representation of the actual turbulence than the one considered before. The development of the theory of homogeneous shear turbulence has often followed the ideas developed in the theory of homogeneous isotropic turbulence. Analogous to Heisenberg ~ theory, there exist the nearly identical theories of Tachen (1953), Reis (1952), and Burgers and Mitchner (1953). Analogous to the work of Proudman and Reid (1954), 48 there is the work of Craya (1957). DeisslerJ ~ (1961) work in shear turbulence follows similar lines to his ·work in homogeneous decaying turbulence. The following treatment briefly outlines the significant results of these studies of homogeneous shear turbulence. Most of the investigators of homogeneous shear turbulence obtained general two-point correlation equations by use of the Na vier-Stokes equations. After taking the Fourier transform of each term in the correlation equations and introducing suitable substitutions, the invest-igators obtained the following dynamic equation for the energy spectrum: k d U k .2.. S E (k, t) d k + d~ S E (k, t) at x 2 o 0 k k = S F(k, t) d k - 2 v S E(k, t)k2dk o 0 (80) In the case of stationary shear flow, the equation for the energy spectrum is k - 00 k dU • = 2 v S E (k) k 2 d k - dx 1 S E (k) d k - S F (k) d k o 2 k 0 (81 ) On the basis of this eq uation, a few solutions for E(k) were obtained that are valid in the range of large wave numbers. In obtaining a solution of the equation (81), Tachen (1953) con-dU sidered two cases: 1. The vorticity -d 1 of the main motion is small x 2 compared with the vorticity of the turbulence in the wa ve-number range under consideration. 2. The vorticity of the main motion is comparable 49 to the vorticity of the turbulence in the wave~umber range under consider-ation. Utilizing the concepts of Boussinesq and Heisenberg, he expressed the second term (representing the interaction between the mean flow and its fluctuating part) and the third term (representing turbulent transfer) on the right hand side of the equation (81) in terms of the energy-spectrum function E (k). Then he sol ved the equation (81 )for not too large II k" and for v'ery "large " "'k" and obtained for case one E (k) ex:: k - 5/3 (82) and (83) These spectral laws were obtained by Heisenberg for the nonviscous sub-range and viscous subrange in isotropic turbulence respectively. Burgers and Mitchner (1953) obtained the same solution as (82) assuming that the second term on the right side of equation (81) can be neglected. In case two, Tachen argued that there will be violent interactions between the vorticities of the two motions and violent resonance may occur. It is reasonable to expect that it will occur in the non-viscous subrange where II k" is not large. As the result of resonance, the pro-dUction function predominates the viscous dissipation function and eddy transfer function; hence the first and third terms on the right side of the equation (81) were neglected by him to obtain a solution for E (k) E (k) ex: k -1 as (84) 50 The corresponding longitudinal one-dimensional spectrum function takes the form (85) Tachen further showed that strong interaction between the main motion and the turbulence can occur only in the u~ component of turbulence as the turbulence production by the main motion occurs only in that component in homogeneous parallel shear flow. . -1 Hence II k II spectral law can be expected only for the energy spectrum of the longitudinal component of turbulence. Reis (1952) obtained a solution of equation (81) assuming Heisen-berg's expression for the turbulent-transfer term, and making a hypothesis on the form of the term for the interaction between main flow and turbulence. For large wave numbers, his solution is similar to the one obtained by Bass and Chandrasekhar (refer to equation 68) for the case of isotropic turbulence. Thus k -5/3 and k -7 spectral laws are again obtained by him. However, from his analysis he found a fundamental difference between isotropic and shear turbulence in the equilibrium range: The greater the mean velocity gradient, the larger will be the amount of energy in the equilibrium range, whereas in isotropic turbulence the amount of energy in this range is negligible and the range of validity of the k -5/3 law will extend further to higher wave-numbers. These results may be attributed to this: As the mean-velocity gradient increases, the characteristic time of the mean flow decreases; and it will approach the reaction times of the small eddies, resulting in a greater interaction between the mean flow and the small eddies in the equilibrium range. Deissler' s (1961) analysis is based on the general two-point equations for the velocity correlations and for the pressure-velocity correlations. He assumed the turbulence to be weak enough for triple correlations to be negligible compared to double ones. He derived the corresponding equation for the rate of change of the spectral tensor Eij . For the case of initially isotropic turbulence, solutions for E22 , E12 , E .. components of E .. were obtained by Deissler and solutions 11 1J 51 for the remaining components were obtained by Fox (1964). They utilized these solutions to obtain a complete description of the redistribution of turbulent energy u. u. among its directional components 1 1 the effects of mean transverse velocity gradient as the turbulence decays. Their study leads to the following description of the sequence of turbulent-energy processes. The turbulent-€nergy production by the mean-velocity 2 gradient occurs only in the u l component of the energy and predominantly · in the low wave-;1umber or larg~ddy region. The pressure forces which depend on the velocity gradient transfer the energy into other components. In doing this, the pressure forces increase the anisotropy of the turbulence; and, in particular, they oppose local isotropy in the high wave-number region. But the triple correlations which are neglected by them can also affect the pressure forces at high Reynolds numbers and may tend to 52 increase isotropy. Hence, their results cannot be taken as definitive, but they may serve as a warning to use caution in assuming local isotropy in boundary-layer flows especially very near to boundaries. The two approaches above do not provide the detailed mechanism of the transfer of energy from one wa ve number to another. Improvement in the study can be made by considering not only the equations for the two-point double correlations, but also the equations for three-point trip~e correlations. Closure of this system of equations can be made by the application of the quasi-gaussian hypothesis. Craya (1957) adopted this procedure and derived the dynamical equations for the energy-spectral tensors. These equations are too complicated to obtain any solution. Lumley (19 64) derived spectrap-energy equations for wall turbulence, that is, turbulence which obeys similarity relations and in which the only characteristic length is the distance to the wall. He made use of those equations for qualitative study of the spectral budget of energy. The picture of spectra1-energy budget obtained as a, result of his study was summarized by Lumley and Panofsky in lucid terms and is quoted here: Energy removed from the mean motion by working of the Reynolds stresses against gradients of the mean velocity is fed in at lower wave numbers. The spectrum is anisotropic here. Distortion of the eddies by the mean strain rate serves to transfer energy from the lower to the higher end of the anisotropic part; inertig forces or vortex stretching, due to the fluctuating velocity gradients serve to transfer energy from the low wav~number anisotropic range to higher wave numbers. As the wave number increases, under the influence of pressure forces, the spectrum becomes more isotropic and feeding falls off. For sufficiently large Reynolds number, there is a range of wave numbers in which no significant feeding or dissipation is taking place, only inertial transfer; finally, at sufficiently high wave numbers, dissipation takes place. The spectrum in the ranges in which only inertial transfer and dissipation take place may be regarded as isotropic. (Lumley and Panofsky, 1964, p. 82.) Relationships Between Statistical Quantities of Turb~lent Shear Flows In the previous sections, several statistical quantities which constitute the structure of turbulent shear flows have been considered. Not only do they depend on the position of the sampling point and the Reynolds number. of flow, but they must be related to the representative 53 mean values of flow as well. If they could be related to easily measurable mean quantities such as the relative depth of flow, the resulting relation-ships could form a basis for the application of the statistical theory of turbulence to practical problems. The rate of energy dissipation per unit volume in turbulent shear flows may be expressed on the basis of the analogy with the case of isotropic turbulent flow, as E o = const. p v Provided that ~ and u~ are proportional to ui the expres sion for turbulence energy per unit volume is given by (86) 54 E = 1/2 p ( u~ + u; + u~)= Canst. p u~ (87) This turbulent energy is dissipated by the action of viscosity in the time 00 of decay defined by S RL (T) dT. where RL (T) is the Lagrangian o VI . VI correlation coefficient of a turbulent fluid particle velocities at an interval of time T. The time of decay is related to the Eulerian system such .hat L xI a-u l 1 The rate of turbulent-energy dissipation is given by E I == Const. (88) (89) A combination of equations "(86) and (89) leads to the following relation~ == Const (90) The experiments of Laufer (:19 S1, 1954) proved that the dissipation of turbulence is nearly balanced with the production rate of turbulence in x 2 the range of approximately (0 < h ~ 1/2) and with the diffusion rate . in the outer region (0. 8 < Y/h ~ 1). In addition, the turbulence intensities referred to the friction-velocity u* are functions of x Ih. On the basis of this conclusion, the macroscale of turbulence relative to the flow depth 55 can be expressed as a function of relative depth as indicated by Hino (:1~ 60). The rate of production of energy in steady unidirectional flow is given by T (91) The rate of dissipation of turbulent energy can be expressed as (92) Reference to equation (11) reveals that the rate of diffusion of turbulent energy is given by (the first term on right~aI)d side). C ( 2 + 2 2) d u2 '~I ;2 + u3 ] = dx~ 3 2 pu* d u' u 1 2 = C 2 -h d(X2/h) 3 u* (93) In light of the isotropic character in the .outer region, the diffusion rate as well as the dis sipation rate should nearly be constant in this range; and, therefore, equation (93) can be rewritten as Qiffus'ion rate = h (94) 56 If equation (92) is equated to equations (91) and (94) o < x 2!h < 0.5 0.8 < x 2/ h < 1 (95) This relation is analogous to the experimental result, obtained by Nikuradse (Schlichting, 1960, p. 51}) from his measurements on artificially roughened pipes in that the variation of the mixing length with the wall distance (i!R) is independent of Reynolds number and wall roughness and x is a function of R2 only, provided the flow is fully-developed turbulent flow. The validity of the above approximate relationship will be test ed in Chapter VI, making use of the experimental data obtained. 57 CHAPTER IV EXPERIMENTAL APPARATUS AND PROCEDURE Description of Apparatus Flow system The flow system used in this study ~ a flume which is 3 feet wide, 2 feet deep, and 24 feet in length (see Figure 1). The slope of the flume is adj ustable from 0 to 4 per cent. At the head end of the flume a head tank equipped with a diffusion partition receives the discharge from the reservoir directly through a 1. 0 foot-diameter pipe which is provided with a regulating valve near the head tank. Discharges up to 5 c. f. s. were possible and were measured by an electromagnetic flow-meter. The entrance to the experimental flume was provided with a rounded inlet which was designed to provide a smooth transition from the relatively quiet conditions in the head tank to the flow of the main channel. At the entrance to the flume, a sluice gate wa s provided for regulating the flow in the flume in addition to the valve on the supply pipeline. In the en-trance to the flume, a series of screens were placed across the head tank for suppressing surface waves and eliminating secondary currents. At high flows, secondary rollers appeared in the entrance section. These were suppressed by manipulating the sluice gate. During the investigation, there was no evidence of the persistence of such disturbances if ever -1-- 4 .0 -f t P.::~~..rz:be 2·0 './ ,/ '- P\"'Obe,s I I 8-0 I 2.D-4-t / + ' "WI tr I t " I i 2;0 f , I ./ 3·0 J l ~ I -=!----JI:;;+t -_.# W&f ~" --M ~ --d~~ ~ Figure 1 . Exp8rimental flume _0- plan and sectional elevation I " 3~o ~ooden bloc __ " ,"xj ~z , .. -+-:-- - 0 s 8 t ,. .. '. '.--.. _.'l)~ 21J . - ~ II I I • 1I1~ T 3 I /1 .. _ ----·---~l ~ , II 62. ~ ~l ' ~ f (J1 co present beyond a few feet from the entrance to the flume. As further insurance to guarantee homogeneity, the sampling point was chosen at a distance of 15 feet and 7. 25 inches from the entrance to the flume. The discharge from the flume was allow~d to fall freely into a channel which drained into Logan River. Since the water depths were normally small and the flume bottom extremely rough, the influence of the draw-down curve was not felt more than 6 feet up-stream of the end of the fl ume. 59 The bottom of the flume was covered with expanded metal screens in order to provide a uniformly rough surface for the generation of turbu-lence. These screens are illustrated in Figure 2 and ha ve diamond shaped openings. The minor diagonals of the meshes measure 13/8 inch and 5/16 inch while the respective major diagonals measure 27/8 inch and 5/8 inch. The screens were attached to the bottom of the flume using marsh adhesive and Silastic RTV-731. Care was taken to eliminate bulges or dips in the height of the meshes and to prevent unusual wave patterns which might interfere with the statistical homogeneity in the turbulence in the axial direction. The simplest nonisotropic turbulent flow is the axisymmetric flow in a pipe in which the turbulent field is nonhomogeneous in one direction only. In the case of uniform turbulent flow in the flume, the turbulent field is nonhomogeneous in two directions., namely;, perpendicular to the mean flow and transverse to the mean flow. As the ratio of the breadth 60 Series A Series C Figure 2. Flume bed roughness meshes. Figure 3. ExperiPlental flume and staging--General view. 61 of the flume to the maximum depth of flow worked within the investigation is more than 12, it could be reasonably assumed that the flow, u-sed 'in the flume was essentially two-dimensional. The simple geometry of two-dimensional channel flow allows an integration of Reynolds equations; and the turbulent shearing stress can then be related directly to the shearing stres s on the bottom, which in turn can be determined from the slope of the mean-velocity profile at the bottom d the flume. Another advantage of choosing fully-developed channel flow as an experimental flow system in this study is that, in contrast with the flow behind screen grids, the rre an-tlow conditions are steady. Furthermore, no decay of mean or fluctuating quantities exists in the direction of the flow because the turbulence generated by the bottom roughness of the flume was in equi-librium with turbulence decay. The position of the sampling point in the transverse direction was chosen in the center of the flume. The Reynolds number of the flow was calculated using depth of flow as the characteristic length. The depth of flow was measured from the effective bed which was arbitrarily chosen at a height equal to half of the thicknesstof the mesh measured from the bed of the flume. The location of the test section ranged from 12 feet 0 inches to 19 feet 0 inches downstream from the entrance of the flume. The variation of depth of flow in that reach wa s not more than 5 per cent. In order to isolate the tranducer from the oscillations of the flume, a separate staging acros s the flume was erected; and the t randucer probe was mounted on the frame, which was resting on the staging, which rested on the floor. The det ails are shown in Figure 3. Tranducer probe A radially-polarized hollow cylindrical barium titanate ceramic, 62 1/16 inch in. outside diameter and 0.033 inch in inside diameter and 1/2 inch in length purchased from Gulton Industries, Inc., Metuchen, New Jersey, was used as the pick-up. It was supplied with a very thin waterproof coating of silicone varnish. It was provided with electrical leads already soldered to the inner and outer faces. The inside lead is 32 gauge stranded, insulated copper wire with outside diameter not greater than O. 030 inch and length 18 inches. The lead on the outer face is a gold foil tab 1/16 inch wide, 1 inch long, and O. 0015 inch thick. The details of the various parts of the probe are shown in Figure 4. The following steps were followed in its construction: The rear portion of the tip connector was tinned with sil ver solder, and while it was still hot the connector was screwed tightly into a 1/8-inch stainless-steel tube. This provided a good physical and electrical connection between the two parts. The tranducer was then glued to the exposed end of the tip connect0t.' with a very thin layer of hard drying II Epoxy glue. II The inner lead of the tranducer was carried back through the tube to a II Glenite C5-RllI miniature cable connector. The gold foil tab was fastened to the flat on the tip connector with a small amount of silver ~ ~ " (Y) J C\J r 12/1 • I 2-3/8' .\ Ii . ~ VEr' O.D. sto;nles~ Glennite c5-R\ steel c.onnector Miniature Cdble FRONT VJEw A~ CSUP'POrt Arl7) Connecier ~ n~ PLAN VrEw '"TIP Connector-A..+ Solder- \/8·~ l/8' erdoS'S \/8r o. D.)( .055* J.o. ~stainless steel 1U.be ~/d RadiLtS Gold ledf' solder 5ECT. A.-A T{p Cov& ~ .v } See l1P . Section ~ y- Kerr E/osr,c en lr \ \:. '( .. \\ .1:. \:~. \\ \\ \\ i \\ 'K \ \\ \ \ h\ \\ \\ { '\: It \\ \, \ \ \ \ \\ \\ \?t 0 0\11 ~r~ss Le~ ~ \: \: \. .. • B UY T'P SECTION Figure 4. Design details of the tranducer probe. Ie 4 G ul ton BOri um Iltdnate ce.rdMic.. Transdl.lcer en w conducting paint. The tip connector and the tube thus form the ground lead from the ceramic and also provide electrical shielding for the inner lead. 64 Then the brass diaphragm was punched from O. 001 inch shim stock and very carefully soldered to the tip cover. The outer surface of the ceramic was then coated with a layer of II Kerr' Syringe Elasticon. II Its purpose is to fill the space between the ceramic and tip cover, thus giving lateral support to the ceramic element and at the same time pro-viding a certain degree of vibration isolation. Before the elasticon could set, the tip cover was slipped over the tranducer and pushed onto the tip conf-1ector. The' length of the tip cover had been previously adj usted such that when pushed on as far as possible a slight deflection of the dia-phragm by the ceramic was effected. This ensured good contact between the diaphragm and the ceramic. The j oint between the tip cover and the main portion of the tube was then sealed with epoxy cement. This com-pleted the assembly of the tip section. To provipe additional strength to the tube, a brass strip was soldered along the forward edge of the main portion of the tube. The final shape of the tranducer probe is shown in Figure 5 along with ceramic crystal. Since the barium titanate crystal has a very low capacitance, the external cable must also have a low capacitance in order to achieve high Sensitivity. For this reason Amphenol No. 621-685 shielded cable was used to connect the tranducer electrodes and the II Glenite C5-Rl miniature 65 cable connect or" at the other end. In order to increase the low-frequency response of the tranducer probe, a Glenite cathode follower Model F-407 with an input impedance of at least 100 megohms was connected to the miniature cable connector. A Glenite C22-5 low noise cable was used to connect the cathode follower and the input provided on the power panel of the po~er supply to the cathode follower. Electronic instrumentation DISA random signal indicator and correlator. Figures 6 and 7 show the electronic system used to measure the response of the tranducer probe to the turbulent fluctuations. The particulars of each one of the units are furnished below. The DISA random-signal indicator and correlator type 55AO 6 shown in Figure 6 could be used to measure the two root-m-ean-square voltages of the two input signals, "A" and II B", the time derivative of signal "A" as well as the root-mean-square value of the sum and difference of the signals "A" and "B" irrespective of their wave form. The principle of its operation would be clear by looking at the block diagram of the internal circuit shown in Appendix D. The input signals A and B are applied to identical preamplifiers via separate calibrated attenuators. The am-plified signals are applied to the output terminals A and B and to a summing amplifier whose output provides the amplified A + B signal. Preamplifiers A and B consist of the amplifier proper followed by an isolating amplifier. The isolating amplifier in amplifier B is also 66 Figure 5. Photograph of the final shape of the tranducer probe. Figure 6. A general view of the in~trum~ntation set-up. Tranducer Glenite ~ektronix " probe (Pie- . cathode pre-:zd>electric : -- follower .... ~mplifier crystal) F-407 trype 122 AD-YU Disa random rype 802 J -r signal indic.ator Dl=>lr:l 1 inc> and correlator Type 55A06 . iI'randucer . Glenite Tektronix , , probe (Pie- cathode -pre- ,,, .-amplifier - ., zoelectric follower crystal) F-407 Type 122 n Quan-Tech Dumont 766 Model 303 Fairchild Spectrum oscilloscope Analyzer - with camera) ~-Figure 7. Block diagram of the tranducer probe and associated electronic instrumentation. I Q) '-l 69 Spectrum analyzer. The energy spectrum was obtained with the aid of wave.and noise-spectrum analyzer Model 303 of Quan-Tech which ana-lyzes a.n input .. signal present in the frequency range of 30 cycles to 100 K. C. It is provided with four selectable constant band widths of 10, 30, 100 and 1000 C. P. S. Band-widths of 10 and 30 cycles enable accurate measurement in the low-frequency range. All measurements were made with 30 C. P. S. band-width as it was found to result in easier reading of . the analyzer output. It is also provided with two selectable time constants . 01 sec (fast) and 1 sec (slow). Slow position was used throughout the measurements since it made the reading of the instrument easier. The R. M. S. voltmeter is calibrated in various scales as selected by the milli-volt full scale switch. The instrument is equipped with two input terminals Xl and Xl 0, and it '.is'.also provided w"ith ' aL meter multiplier which is to be set to a range higher than the signals to be measured. In the measurements reported here, input X 1 and meter multiplier 10 were used. To obtain an absolute voltage measurement the reading in milli-volts should be multiplied by 10 with gain control full clockwise. Data indicator and recorder. Visual observation of the signal and noise at various points in the circuit was made with Dumont Type 766 tran-sistorized dual trace oscilloscope. It was provided with a Dumont Type 450A polaroid camera which was used for photographing the signal traces on the oscilloscope screen. Some typical oscilloscope pictures of total-head fluctuations are shown in Figure 8. Figure 8. 70 Scale: 1 volt per centimeter 10 milliseconds per centimet er Typical oscilloscope pictures of turbulent total head fluctuatiDns at RN = 7.25 x 104 -- Series A. 71 Variable time-d.elay network. Type 802 J Decade variable delay-line supplied by AD-YU Electronics, Inc., was used in conjunction with the correlator to obtain autocorrelations. It is capable of delaying the signal from 21-lS to 200,000 I-lS. It is provided with four switches; and when all of them are set at maximum, total delay is obtained. The cutoff frequency . at 3 db varies with the amount of delay cor 'fesponding to the maximum setting of each one of the switches. Care was taken to have impedance matching at the input and output terminals of the delay line. The signal source of the delaY]¥1e was input .1.1 All of correlator. As the source impedance was higher than the input im-pedance of delay line, a variable resistor was connected parallel to the input terminal of the delay unit. The output terminal of the delay line was connected to input .11 BI.' of correlator with a resistor of 2200n connected parallel. As the input voltage requirement of the correlator is low, it was not found necessary to amplify the output of the qeHry line' before the signal was fed to input JI B" of Itbe-Correlator. Auxiliary instruments. The Glenite cathode follower Model F-407 was used for opupling the tranducer probe with the input of the Tektronix preamplifier Type 122 in order to achieve maximum power transfer by match-ing the high output impedance of the probe with the high input impedance of the cathode follower probe which was kept very near the tranducer probe to minimize the shunt capacitance for maximum sensitivity. The output of the cathode-follower probe was connected to the input on the power panel 72 with a low~oise Glenite C22-5 cable. 8ingle-ended output on the power panel was connected to the input of the preamplifier. Most of the time the preamplifier gain was set at 1000. The low-frequency and high-frequency response selectors on the preamplifiers were set at 8 c. p. s.-and 10K. C., respectively, as it was found expedient from the consider-ations of the range Gfrrequencies of fluctuations present in the flow system and the noise contributed by selected band width of frequency components. All the connections between the units were made with Belden type 8254 RG-62/U and Amphenol No. 621-685 shielded cables with 80-235 or BNC connectors in order to minimize noise. Preamplifiers were provided with a separate power supply. All the units had power suppl:iea 'Jr:om~ r-eg-ulated power source. Extreme precautions were taken to ground the measuring system properly to eliminate, as much as possible, the 60-cycle noise and to reduce the noise as much as pos sible from various other source s. Details of procedure Measurement of average velocity. In order to determine the rates of energy dissipation and mean-friction velocity in the experimental chan-nel, it was es sential to determine the slope of the total-energy line (8 ) e within the test section. To obtain '8 " mean-velocity distributions at e the two ends of the test section were to be measured. Besides, at all locations where measurements were made with the tranducer probe for 73 intensity of turbulence, energy-spectrum, microscale, and autocorrelations, local mean velocities were also to be determined. Dwy'er pit 0t : tubes or total-head tubes were used to meas ure mean velocities. These tubes have an outside diameter of 1/8 inch and aninside diameter of O. 047 inch. Literature concerning the effect of turbulence on total-head .tube readings was reviewed by, Hinze (1959); but in the absence of an accurate procedure to account for the effect, it was neglected in this study. In order to compute the local mean velocity at a point, it is essen-tial to know the difference between the total and static heads. The inner and outer tubings of the total-head tube were connected to an inclined man-ometer board. Water was used as the manometer fluid. For every position of total-head tube at least five to ten minutes were allowed for manometer liquid levels to reach steadiness before readings were taken. For calculat-ing mean-velocity distributions, at least 15 to 20 measurements were taken at each section .. Meas urement of a.u1ocorrelation coefficient. The signal II All from the Tektronix amplifier output terminal was connected to the input II All of the correlator. The input terminal of the external variable time--delay network wa s connected to the input II All of the correlator. The signal from the output terminal of the time~elay network, , designated signal II BII was connected to input II BII of the correlator. Input II All and input II BII of the correlator were connected to the oscilloscope. With zero time delay, both signals II A" and II B" on the . 74 oscilloscope trace were identically the same. To measure the correlation coefficient, the instantaneous sum and difference of the two time-dependent signals II All and II 811 were first derived. The ratio of the root-mean-square of the sum and difference signals was then measured by means of .an f. _m. s. ratio-meter. It could be shown that the measured ratio is an unique fun-ction of the correlation coefficient if the r. m. s. value of the signals II All and II BII had been made equal. This equalization was performed in the variable-gain pre-amplifiers. Then from the precalibrated scale for the correlation coefficient, the value of the correlation coefficient was obtained. At zero time delay, the value of the correlation coefficient was found to be nearly one, as it should be. The same procedure was repeated for increasing time delays until the delayed signal II BII was attenuated to such an extent that the stgnal ttace was distorted considerably. Measurement of spatial correlation coefficients Besides the autocotrelation coefficient, attempts were made to measure R .. , Xl ul and R ' . by .simultaneous meas urement of the x3 ul turbulent fluctuations at two different points located along xl' x 2' and X3 axes using two tranducer probes No. 3 and No.2. The signals from the two probes were amplified by two Tektronix preamplifiers, and the output terminals of the preamplifiers were connected to the two inputs II A" and II BII of the correlator. To measure R _, one probe was placed very closely behind the x lu1 other; and it was found that the probe located downstream was affected by 75 the wake of the other. To overcome this difficulty too downstream probe was placed slightly to one side of the other with a longitudinal spacing of 1/8 inch. The correlation coefficient obtained was O. 5. At a distance of 1/4 inch the correlation became negative; and at distances more than that, correlation remained negative. Similarlj) to measure R ,the two probes were placed adjacent x 3u1 to each other with zero spacing; and the correlation coefficient found was O. 6. When the spacing was increased to 3/8 inch, the correlation became negative; and at increas eq-" spacings the correlation coefficient remained negative. Simila{ readings occurred with the measurement of R x 2u1 As sufficient data could not be obtained for plotting correlation curves for each one of these spatial correlation coefficients, no further measurements were made. It was concluded that much finer tranducer probes, such as a hot-film anemmometer, were needed to measure, successfully, spatial correlations. Measurement of longitudinal microscale of turbulence A method suggested by Townsend (1947) was used to measure X-xI It has been found experimentally by Townsend that the mean square of the time deri vati ve agrees closely with the mean square of the space deri vati ve in the direction of the mean motion,; that is ( ~)2 ~ U2(~)2 at 1 aXI (96) 76 The microscale X. is defined by equations (36) and C3 7), namelYt xl _1_ ..L '\ 2 = 2 I\. 2ul xl (97) ( aUl )2 ( aUl )2 There is a definite relationship between aX2 and aXl as shown by TaY,lor (1935), such that (98) Combining equations (96), (97), and (98) 1 1 1 . aUl 2 ~ ..-- ( at) X. 2 2 ~ xl u l Ul (99) or 2 X. = Ul ul xl CaUl 2 " at) (loa) The differentiater, incorporated in the t.andom-signal indicator and correlatoT. (See Figure 6), provides the product of the time derivative of the amplifier output signal and a selective time constant itT. It The RMS ratiometer of the correlator now measures the root-mean-square value of the ratio "r" of the input signal from the amplifier output terminal to its time deri v "ati ve multiplied by T • Then the numerical value of the microscale is computed from the following relation: ~ = Ul . ,.. . r xl Measurement of turbulence-energy spectrum and intensity of turbulence (101) Every time the spectrum analyzer was used for making energy-spectrum measurements, nulling was done and the calibration of the r. m. s. voltmeter of the analyzer was checked. The details of the calibration of the analyzer are outlined in Appendix C. Measurements of energy spectrum and turbulence intensity were 77 made in the following four steps: (1) measurement of the Fourier compon-ent of turbulence intensity, (2) measurement of corresponding noise output, (3) measurement of total output for turbulence intensity, (4) measurement of corresponding noise output. In steps 2 and 4, the probe was sealed from contact with pressure fluctuations in the flow. In step 1, the signal from the amplifier output terminal was fed into the input terminal of the analyzer. The frequency dial was I tuned to 20 c. p. s. and the reading was taken from the r. m. s. voltmeter of the analyzer when the output stayed steady at least for 5 minutes. This procedure was repeated at different tunings of the frequency dial until the output of the analyzer was negligibly small. Exactly the same data-taking procedure was used in step 2. In steps 3 and 4, the spectrum analyzer was by-passed, and total outputs for turbulence intensity and noise were measured with the r. m. s. voltmeter of. . the :~taildom-s.lg.na-r inciTcator and · correlator. 78 The entire above procedure was repeated for every position of the probe for a particular flow condition. 79 CHAPTER V DATA PROCESSING Sample calculations are presented to illustrate the procedure used in processing of the experimental data to evaluate the relative intensity of turbulence, one-dimensional energy spectral-depsity function, longitudinal average macroscale of turbulence, and longitudinal microscale of turbulence. Relative Intensity of Turbulence The total signal VT obtained with the uncovered tranducer probe consists of real turbulence si9nal VR. and noise VN . arising from the structural vibration of the probe, accoustical noise in the surroundings, thermal noise, microphonics, etc. (102) Squaring both sides of the equation (102) and taking time averages, (103) Since the turbulence signal and the noise signal are two independent random signals, they are not correlated and hence (104) 80 Therefore, equation (103) reduces to (105) Equation (105) provides a convenient method of resolving the true signal from the noise. The following example for the calculation of the intensity of turbulence is based on the readings of the tranducer probe at a relative height of x 2/ h = O. 065 in the run of Series B at RN = 8. 3 x 104. V'T = 1. 40 volt at a gain of 1000 VN = O. 52 volt at a gain of 1000 As shown in Appendix B, the sensitivity of tranducer probe No. 3 is 14 .. 0 millivolts per p. s. i. of pressure ,change. This is equal to 6.066 milli-volts per foot of head change. Including 1000 times amplification, it becomes 6.066 volts per foot of head change. Designating the r. m. s. voltmeter reading due to true turbulence signal as Vk' the above sensi-tivity combined with equation (2) in Appendix A gives U1 6. 066 - u l = VI g 1 R (106) Using equation (105) and the above values of VI T and VN, the corresponding value of Vk is given by V R = 1. 30 volt and U1 = 9. 75 ft/sec substituting. the above values in equation (106), 6. 066 32. 2 x 9. 75 ui = 1. 30 u' = 1 1.30 1.838 = 0.707 ft/sec One-Dimensional Energy Spectral Density Function From equation (3) in Appendix A, 2 2 C2 tf 1 d VR = d u1 2 g or 2 2 2 d u1 = g d VR 2~2 C U1 Similarly, 2 2 2--d u1 (n) = g d V R (n) C2~ The wave number is related to the frequency by the expression, 81 (107) (108) (109) (110) 82 (Ill) Equation (109) may also be written in the form, where :::: 2 9 (112) 2 A VR (k1) :::: mean square of the output of the wave analyzer in volts resulting from turbulence fluctuations contained within the band width provided by the analyzer at wave number 1<1 corresponding to the frequency ·s iet on the wave analyzer :::: turbulence energy of fluctuations contained in the band width at kl corresponding to frequency set on the wave analyzer. Dividing equa tion n 12) by eq ua tion (Ill), 2 2 -g A VR (k1) U1 C2~ x 2 iT A n 2 2 g A VR (kl ) C 2U 2iT An 1 The selected nominal band width :::: 30 c. p. s. (113) Actual band width established by the analyzer (see Appendix C) = 59 . 0 c. p. s. The common factor in equation (113) 2 = --g'-----2 C 2TT~n = = o. 076 6.0662 :x 2 TT x 59.0 The following example for the calculation of El (k1) is based upon the spectrum measurements at x 2/ h = O. 065 in the run of Series B at RN 83 = 8. 3 x 10.4 at nominal frequency 160 c. p. s. set on the wave analyzer. The actual frequency is identically the same as the nominal frequency (See Appendix C). The local mean velocity U 1 = 7. 53 ftl sec. The corresponding wa ve number from equation (110) is I k == 2 TT X 1 60 == 133. 5 ft- 1 1 7. 53 From measurements, A v~ (kl ) == 0.0507 volts 2 . Substituting the above values in equation (113) one obtains = 0.076 x 0.0507 7. 53 == 7. 6 x: 5. 0 7 :: 1 0 - 4 7.53 -4 3 2 == 5.10 X. 10 ft Isec Longitudinal Average Macroscale of Turbulence The average macroscale of turbulence is given by the equation (33) 00 (,-) d,-Frenkiel (1958) suggested that when the correlation curve is such that the negative correlation coefficients exist, it will then be necessary to measure the coefficients RE (T) up to values of T sufficiently 1 large to allow determination of L with sufficient accuracy. There-xl fore, equation (33) may be rewritten (114) where T represents the smallest value of T at which the auto-c correlation curve intersects the axis of T • The area under the autocorrelation curve up to the intersection point was obtained with a planimeter. At x 2/ h = 0.179 for run of Series B at RN TC S RE (T) d T = 4. 32 milliseconds o 1 Ul = 9.05 ft/sec. L ~ 9.05 x 30.5 x. 4.32 = 1.19 cm. xl Longitudinal Microscale of Turbulence From equation (101) }... = Ul . T • r xl 4 = 8.3 x 10 84 85 0.179 in run of Series B at RN = 8.3 x 1.04 Ul = 9.05 ft/sec T = O. 5 milliseconds r = 2. 00 A. = 9. 05 x. 30. 5 x O. 5 x 2. 0 = O. 2 7 6 em . xl CHAPTER VI RESULTS AND DISCUSSION Distribution of Relative Intensity of Turbulence For all the series of tests, the boundary-layer growth in the flume has been calculated using the empirical equation of Halbronn (1954), and it i~ found that at the sampling section the boundary-layer thickness already reached the depth of flow. It is· als:o found that the flow in the channel corresponded to fully rough-wall condition (u*k ;, s.. v > 70) for all the runs with the artificial roughness on the bed. The most serious difficulty encountered in performing tests in open-channel 86 flow is to keep the relative roughness constant. It should be noted that the relative roughness in an open-channel, unlike the case of pipe flow, varies with the depth of flow. Hence, variation in the Reynolds number of flow keeping the relative roughness constant has to be attained by varying the slope to increase the velocity. But with the expanded metal mesh glued to the bed of the flume, this could not be achieved while keeping the flow uniform. Thus, attempts to keep the depth of flow constant for all the runs in a series were dropped as it was considered that attainment of uniform-flow conditions was a more important objective. Therefore, even though the relative roughness would be a more pertinent parameter, the absolute roughness size I k' • s (Nikuradse ' s equivalent 87 sand-grain roughness), which remains constant for the runs in a particular series, is chosen in place of relative roughness. The distribution of relative intensities of turbulence in the open-channel is shown in Figures 9 to 11 for three different series (i. e., for three different bed-roughness conditions) at three Reynolds numbers. Fig-ures 9 and 11 refer to fully-deyeloped turbulent-flow conditions while Figure 10 corresponds to transition regime of flow. Not more than one measurement could be taken in the region of the law of the wall or the constant-stress layer whose thickness varies from 1/5 to 1/7 of the thickness of the boundary- -layer or the depth of flow in this case. All three figures show that the relative intensity of turbulence decreases over the entire depth of flow as the Reynolds number increases. This result is in agreement with Laufer's (1951) measurements in air flow in a two-dimensional, smooth-surfaced channel. It could also be seen from these figures that the relative intensity of turbulence increases from the outer region of flow towards the bed of the channel. This in-creasf3 is much more pronounced near the bed in the two dases with artificial roughnes s on the bed of the channel (Figures 9 and 11) indicat-ing that the effect of roughness i,s more prominent near the bed than in the region far from the bed. In all cases, there is a tendency for the relati ve intensity of turbulence to approach a constant value towards the surface indicating that the influence of the variation of the Reynolds number is negligible near the surface. From Laufer's (1951) measurements, Q) .~ o. 1 ~ (tj ...... Q) q:: I~"" "-El o 0 Q (!J8,,8 ~ G Series A 4 9---RN = 8.3 x 10 .e.--!{N = 7. 2 5 x 1 04 13 - - ~N = 5. 22 x 1 0 4 8 Q 88 ~.... 0 ~ ____________________________ ~ ______________ ~ __ ~ o o. 2 0.4 0.6 0.8 1.0 x 2/h -relative depth Figure 9. Distribution of relative intensity of turbulence as a function of relative depth with RN as Q parameter -- Series A. o 'I!J Series B e--~ = 8.30 x 104 N 89 4 C-""~N = 5.22 x 10 o~~ ______ ~ ____________________________________ ~ o 0.2 0.4 0.6 0.8 1. 0 x2/h -relative depth Figure· 1 O. Oistribution of relative intensity of turbulence as a function of relative depth with ".RN :as ·a: p.arameter~.;; Series . ·B .. ··· . .', Q) () c Q) ...... :j 0.4 ~ 0.3 +-' 4-1 o ~ +-' ..... til C Q) O. 2 ~ ..... Q) > ..... +-' to ...... Q) 0:: . I O. 1 I P""" "-- ...... :j o A 8 e & 8 c:.> b" a 0 ~ ~ <:> Series C 4 (:) - - RN = 8. 30 x .10 4 A -- RN = ? 25 x 10 4 B - - . RN = 4. 06 x 10 I:] ~ G G> & 11 (!J 0 e ~O\--------~--------------------------------------~ 0.4 0.6 0.8 1.0 x 2/h - relati ve depth Figure 11. Distribution of relative intensity of turbulence as a function of relative depth with RN as a parameter--Series C. 90 Q) () c Q) .-1 ::1 ..0 0.4 ~ 0.3 -4-1 4-1 o >. -4-1 -..-I U) C Q) ~ 0.2 -..-I A o <:> <:> (g---h/k = 3.58 s 4---fi/k = 1.95 s Q 91 o.~ __ ~ __ ~~ __ ~ __ ~ __ ~ __ ~~ __ ~ __ ~ __ ~ __ ~ o o. 6 o. 8 1 :", 0 x 2/h - relati ve depth Figure 12: Distribution of relative intensity of turbulence as a function of relative depth with relative roughness a s a parameter at RN = 7. 25 x 104. 0.4 0) () r:: -0) ,....j ::s ..0 ~ O. 3 ::s +-J 4-l 0 >. +-J -r-I (J) r:: 0) 0.2 +-J r:: -r-I 0) > -r-I +-J co ,....j 0) 0::: '..-1 0. 1 I::> "'-,...-1 ::s 0 o 0 o A 0 -O. 2 0.4 4 RN = 8.30 x 10 G> - ---h/k = 3. 84 s &- ---h/k = 2.14 s EJ- ---h/k = 97.0 s O~ 6 O. 8 x2/h -relative depth 92 1.0 Figure 13. Distribution of relative intensity of turbulence as a function of relative depth with relative roughness as a parameter at RN = 8. 30 x 104. :t 93 it is known that the relative intensity of turbulence reaches a maximum value at the edge of the laminar sublayer in two-dimensional channel flow over a smooth surface. From the results presented in Figures 9 to 11, it appears that in the case of rough open-channel flow in which the viscous sublayer is completely disrupted at u*k: / > 55, the maximum value of s v relative intensity of turbulence is reached somewhere in the constant.· stress · ·layer, i. e., in the region where the production of turbulence is maximum. It may be pointed out that the variation in relative roughness in Figures 9 and 11 for series A and C is so small that perhaps h/ks may be considered constant. The distribution of the relative intensities of turbulence as a fun-ction of the relative depth with the relative roughness as parameter at two Reynolds numbers· is shown in Figures 12 and 13. It should be strongly emphasized that the two roughnesses are not geometrically similar, and it is assumed that the cumulative effect of each roughness mesh can be represented by the calculated equivalent sand-grain roughness size "]<1 :". S Even though the variation in the values of h/k is not high, it appears s from the figure~ that the relative intensity of turbulence increases over the entire depth with the decrease in the values of h/k . The trend of s the variation in the relative intensity of turbulence as a function of the depth of flow is not affected by the roughness at all even though its influence on the rate of variation of ul /U 1 near the bed is quite appreciable. 94 One-Dimensional Energy-Spectrum Function .§pergy spectra in the outer region of flow The one-dimensional energy spectra of the longitudinal turbulence-velocity component in the turbulent shear flow in an open channel in the outer region and near the bed are presented in Figures 14 through 22. The spectra are all reasonabl V: well fitted by a - 5/3 power law in k1 over a considerable range as predicted by Kolmogoroff' s theory (See Chapter III and equation (62)). This implies that the existence of the region of locally-isotropic turbulence independent of viscosity (inertial subrange) therefore is plausible. It should be strongly emphasized that the straight line fitting of the energy-spectra data having a slope of - 5/3 is not by itself an irrefutable evidence of the existence of the inertial subrange. There are a few other tests which the energy-spectra data should satisfy besides, before one could conclude the presence of local isotropy in wave-number space. The shear spectrum is a direct t est of local isotropy (See Hinze, 1959, p. 502). Corrison (1949) used the technique of measuring the var-iation of shear correlation coefficient with frequency to verify local isotropy in free-turbulent shear flow in a round jet. In this case the monotonic de-crease of the shear correlation coefficient with frequency is a clear indication of the presence of local isotropy. Another method of checking local isotropy is to derive the spectrum function E1 (k1) from the measured E2 (k1) using the relation for isotropic turbulence given by Hinze (1959) -3 10 o Series A 4 RN = 8 . 30 x 10 ti/k = 3.84 s x2/ti = 0.065 x'2/ti = O. 716 95 (SlOpe (-5/3) / Slope (-7) I 10-7 ~------~~~~~----~--------------------~--~ 1 0 ,Figure 14 . -1 k -Ft 10 3 1 4 One-dimensional energy spectra at RN = 8. 30 x 10 --Series A -3 10 o o Series A 4 RN = 7.25 x 10 h/k = 3.58 s 96 (SlOpe (-7) 10-7~ __________ .. __ ~~ __ ~ __ ~~~ .... ~~~ ____ ~. 10 Figure 15. -1 k - Ft 1 4 One-dimensional energy spectra at RN = 7. 25 x 10 --Series A. o Series A 4 RN = 5.22 x 10 h/k = 3.39 s 97 . (Slope (-7) 0--- x2/fi = 0 . 716 10-7~------------~~~--~--~~~~~----~--~ 10 2 103 10 Figure 16. -1 k , - Ft .1. 4 One-dimensional energy spectra at RN = 5 . 22 x 10 --Series A. -3 10 -4 10 .-J -6 10 \ o .. /lope (-5/3) Series B 4 RN = 8.3 x 10 h/k = 97.0 s 0--- x2/h = 0.065 0--- x 2/h = 0.705 -1 k - Ft 1 "/lope (-7) 98 ~"igure 1 7 . 4 One-dimensional energy spectra at RN = 8. 3 x 10 --Series B o o o I Slope (-5/3) o o o Series B 4 RN = 7.25 x 10 ii/k = 137. 0 s 10- 6 (;)--- x 2/ii = 0.06 B--- x 2/h = O. 710 10-7~ ________ ~~~~ ____ ~~~~~~~ __ ~~~~ 10 -1 kl - Ft 99 ' ? igure 18. 4 One-dimensional energy spectra at ~ = 7. 25 x 10 --Series B. -3 10 -4 10 ~- () 2.4 1.6 0.8 o <:> o Series A 4 RN = 8. 30 x 10 x 2/h = 0.716 O'~~ ________________ m-________ ~ ____ ~~ ____________________________ ~ ______ ~ __ __ o 100 200 300 400 -1 k - Ft 500 600 1 Figure 24. One dimensional dissipation spectra at RN = 8. 3 x 104 -- Series A. 700 800 I-' I-' I-' 3. '2 N U ' Q) 2.4 Cf) " ~ ~..-.... ........ ~ -~r-I1. 6 N r-I ~ O. 8 o o Figure 25. o <:> o I (!) 100 200 300 <:) 400 500 Series C 4 RN = 8.30 x 10 x 2/h = 0.705 h/k = 2.14 s 600 k - Ft-1 1 : One-dimensional dissipation spectra at RN = 8. 3 x 10 4_-Series C. 700 800 ~<:: ...... ...... l'V 4.0 3. 2 I 0 N U Q) I / Cf.) "-~ 2. 4 J7..L.4 N;~ 1.6l f O. 8 o o·~· " ~ o 100 o 0 200 300 "'-~ 400 500 -1 kl - Ft Series C RN= 7.25 x 10 x 2/h = 0.710 ii/k = 1.95 s <:> 600 700 Figure 26. One-dimensional dis sipation spectra at RN = 7. 25 x 104 --Series C. 4 800 ...... ...... w N C) (]) CI.) " +-J ~........ ~ ........ ~ N ........ ~ 2. 4 1.6 0-. o 0 o O. 8 o o Series C 4 RN = 5.22 x 10 x 2/ii = 0.716 ii/k = 1.470 s Q " :;:x. 1.... -!. __ :=:s:.az:::::. CL»f! ! o 100 200 300 400 500 600 Figure 27. k' - Ft-1 1 One-dimensional dissipation spectra at RN = 5 ~ 22 x 104_-Series C. 700 800 900 ...... ...... ~ i i 5: Table 1. Comparison of the values of Kolmogoroff l s universal constant a' evaluated by different investigators. Author Grant et al. (1962) Gibson and Schwartz (1963) Laufer (1954) Present work Flow Tidal channel Grid turbulence Pipe Rough open channel 4 8.3 x 10 a 0.1 1.44 0.472 2.70 0.885 1.96 0.642 3.24 1.006 The values quoted above are a verage ones. The table indicates that the value of a' varies by a factor of 2. If one bears in mind the different conditions (such as type of flow, grossly different kinematic dis-sipation rates varying by a factor of one hundred or more) under which the data of Table 1 were obtained and the different equipment used and assoc-iated experimental difficulties, the four values of 0.1 reported above are perhaps reasonably consistent and appear to bear out the Kolmogoroff theory for an inertial subrange. In the light of this conclusion, as wel~ as from various other considerations discussed in the previous two sub-sections, it is believed that there is sufficient evidence for the validity of Kolmogoroff l s theory of local isotropy in the outer region of fully-developed turbulent shear flow in the rough open chann~l. It may be of interest to mention the importance of the knowledge of the validity of Kdmogoroff's theory of local isotropy for a particular region in a specified field of turbulent shear flow and approximate value of a' . 116 An observation of E1 (k1) at one value of k1 permits one to evaluate E, the most important single measurement in a turbulent field. Average Reynolds stress and eddy viscosity can be calculated from 'EI and mean-velocity distribution provided production and dissipation of energy balance, a hypothesis quite well confirmed for 'regions not too close to boundary_ Test of simplified form of Heisenberg's equation Of all the physical transfer theories that are considered in Chapter III, Heisenberg's theory received the best attention as it is based on a more sound physical basis. Hence his theory has been singled out to be tested by the data presented in Figures 14 to 22. In the region of universal equilibrium range, the turbulence energy distribu-tion function should be a function of E , v, and k. Beyond the respec-tive transition wave numbers, the spectral data of runs in Series A and C in the outer reg.ibn of flow should be representable by one equation. The difficulty of transforming the equation of Heisenberg (equation (68)) to the corresponding equation for the longitudinal one-dimensional spectrum function is pointed out in Chapter III. The two extreme cases resulting from this equation are given by 'equations (69) and (70). The corresponding longitudinal one-dimensional spectrum equations are given by (71) and (72) . The simplified form of Heisenberg's equation for one-dimensional spectrum function is given by TIT = 9 (~!..]/3 k -5/3[1 (9 x 63 )31-4.; ~ ~ 3 . k 4 J-4/3 E 1 (k1) 55 9 k I 1 + 55 3 1 H (118) This has the form of equa~ion (68) with the extremes given by equations (71) and (72). This simplified Heisenberg\ s equation will be testec;i to verify whether this could represent the entire region of universal equilibrium range. Equation (118) could be written as (119) By rearranging equation (119) 1 + 2.73 k 4 2 1 E kH (120) The plot of the left-hand side of equation (120) against k{ should give a straight line with an intercept of unity. Such sample plots are shown in Figures 28 and 29. It is seen from these figures that the straight line correlation predicted by equation (120) is obtained. Hence it could be considered that the transition region between inertial subrange and the viscous subrange is correctly represented by the simplified form of Heisenberg's equation. 118 10.0 - o 8.0 6. 0 4. 0 2.0 Series C 4 RN = 8.30 10 x2/h = O. 705 Elk = 2.140 s o ~ ____________________________________ ~ o 1;.0 2. 0 3. 0 4. 0 k~ x lOll Figure 28. Test of simplified form of Heisenberg's equation at RN = 8. 3 x 1 04 - - S eri esC. o Series A 4 RN = 8.30 x 10 x2/h = 0.716 il/k = 3. 840 s 119 O~ ________________ ~ ____________________________ _ o 1.0 2. 0 3. 0 4.0 5. "0 kl x 1011 Figure 29. Test of simplified form of Heisenberg's equation at . 4 RN = 8. 3 x 1 0 - - Se rie s A. 12.0 Autocorrelation Curves Sample autocorrelation curves are shown in Figures 30 to 33. They are primarily used for calculating the average size of macroscale of turbulence as explained in Chapter V. Besides this use, a few in-ferences could also be drawn from these curves. At zero time delay the value of the autocorrelation coefficient is almost one (allowing for the error in the reading due to the inertia of the indicator needle). As the time delay increases, the autocorrelation coefficient reduces rapidly until it becomes- zero. As the time delay further increases, the A. C. C. becomes negative indicating that the fluctuating quantities are negatively correlated. At long time delay the A. C. C. assumes again positive values as shown in Figures 30 to 33, indicating that the fluctuating quantities are positively correlated at the corresponding time delays. At further increase in time delay, the A. C. C. again assumes negative values. The measurement of A. C. C. had to be terminated at the time delays noted on the curves as the delayed signal was completely distorted; this is due to the characteristic performance of the variable time-delay net-work whose cutoff frequency at 3db band-width point decreases as the time delay increases. It should be mentioned, however, that this feature of time-delay network does not materially affect the meas urement of the autocorrelation function as the high frequencies make little contribution to auto co.r.r~ifrtti.k:lUH function at large time delays. +-' C Q) -~ () -M ~ ~ Q) o {) c o -M +-' CO ...-1 Q) s.... s.... o {) o +-' ~ ....... J:, r-I J:,L.:I 0::: 0 . 8 o. 6 0. 4 0. 2 o -0 . 2 r-o 12 24 Series C . 4 RN = 8.30 x 10 x 2/h = 0 . 179 T = 9. 6 milliseconds m 36 · 48 <:> 60 T - milliseconds o Q 72 84 96 Figure 30 . Autocorrelation curve at RN = 8. 3 x 104 at a relative depth x 2/h = O. 179 -- Series C __ .L ___ ~_._.J. 108 120 "-r. aA ...... ~ ~ Q) ...... . ~ 0.8 'H 'H Q) o u ~ o ...... 4-J ro Q) 0 . . 6 s.... s.... o u o 4-J ~ ~ .J::, ......-t j:.L..:I 0::: 0 . 4 0 .. 2 o -0.2 o 10 2G Series C 4 RN = 8.30 x 10 x2/h = 0.284 T = 14 r 0 milliseconds m 30 40 50 T - milliseconds 60 70 4 . -~igure 31. Autocorrelation curve at RN = 8. 3 x 10 at a relative depth x 2/h = 0 , 284 -- Series ..,.-. . o 80 90 .~ C . ~ +-' ~ Q) • ..-i 0 • ..-i 4-1 4-l Q) 0 0 ~ 0 • ..-i +-' cO .....-j Q) !..-: ~, 0 0 0 +-' ::J ~ J::, r-I ~ 0::: 1.0 O. 8 O. 6 0.4 O. 2 o Series C 4 RN = 5. 22 x 10 x 2/ii = 0.137 T =.: 15. 0 milliseconds m -O u 2, I ! . ! o 10 20 30 40 50 50 70 80 T -milliseconds Figure 32 Autocorrelation curve at RN = 5. 22 x 10 4 at a relative dep t h x 2/h. = O. 137 -- Series C . ? ;.i r-~ '. 1.0 +-' !:: Q) '0 0.8 ....... 4-: '+-1 Q) o o !:: o ....... ~ 0.6 ~ 0.12 8 0 o - - RN = 8. 30 x 1 04; Elk s = 3. 84 4 -6 - - RN = 5. 22 x 1. 0 ; h/k s = 3. 39 EB - - RN -- 8. 30 x ). 04; ti/k s = 2. 14 4 A- - RN = 5. 22 x 1 0 -; h/k = 1. 84 s (J) ED i 0 ~ CD 0 • 0 0 e 127 o~ ______________________ ~ ____________ ~ Figure 34. 0.2 Variation of L xl o. 6 0.8 versus different values of ~ and h/ks for the case of fully developed turbulent flow. o. 60 0.48 0.36 o o 0.24 0.12 -o o Series B 4 0-- RN=8.30xlO 4 $ - - RN = 5. 22 x 10 128 O~--~------------------------~--~------------~ o O. 2 0.4 O. 6 O. 8 1.0 Figure 35. Variation of L versus x 2/h in transition regime x 1 of flOTN with RN as a parameter. 129 slight scatter due to errors from various source s, ~ t' could be concluded from this figure that all the points fall on the same curve. Thus the variation of the macroscale with the depth of flow ( LXI /h)' in the case of fully-developed turbulent flow, is a function of the relative depth of flow (x2/ii) only and is independent of the Reynolds number and relative roughnes s of the bed of the channel. This is a significant result . The same conclusion has been proved in the last section of Chapter III, using the results of Laufer's (1951, 1954) studies of turbu-lence energy balance and adopting an equation for the rate of turbulence energy dissipation in shear flow similar to the one applicable to isotropic turbulent flow. The result is given in the equation (95). It may be inter-esting to compare the above result with an analogous conclusion reached by Nikuradse from his measurements on pipes artificially roughened with the sand of different grain size. That is that the variation of the mixing length with the wall distance (l/R) is independent of the Reynolds num-ber and wall roughnesses and is a function of only x 2/ R . Reference may be made to Schlichting (1960) for Nikuradse' s findings. But it should be noticed that the macroscale tends to decrease as the water surface is approached, whereas the mixing length 111" increases with increasing distance from the wall indefinitely. The variation of the values of L with the relative depth of x 1 / h flow in the transition regime of flow is shown in Figure 35. Figure 34 is in direct contrast with the Figure 35, as it should be. It is evident from 130 this Figure 35 that L is a function of the Reynolds number and, xl i ii perhaps, relative roughness. It is also clear that at lower Reynolds numbers the values of L are much higher. This indicates that x l / h the turbulence has a larger scale at lower Reynolds numbers as might be expected from ideas on the origin of turbulence. As there is much similarity among open-channel flows and boundary-layer flows and pipe flows, an effort is made to explain the trend of the variation of the macro-scale over the depth of flow making use of the results of turbulence energy balance studies of Klebanoff (1954) and Laufer (1954) in boundary-layer flow and pipe flow respect-i vely. Reference may be made to Chapter II, where the salient features of the results of their studies are presented briefly. From the out er region of flow in the open-channel where turbulence production is negligible and diffusion is predominant, the macroscale of turbulence increases until the intermediate region is reached. In this region the rate of turbulence energy production is significant, and the energy bal-ance is provided by production and dissipation only. This increase in macroscale in the intermediate region may be attributed to the increase in energy production. But as the bed of the channel is approached, even though the energy production becomes markedly predominant, the viscous effects tend to break the larger eddies into smaller ones. The viscous action increases rapidly in the wall region (y* < 30), which accounts for the sharp decrease in the macroscale of turbulence. It may be pointed t '31 out that in most of the runs only one measurement of the macroscale could be taken in the wall region. From this analysis it appears that the region of maximum macroscale is where the effect of viscosity is small and turbulence generation is significant enough. It should be pOinted out that the macroscale measured in this investigation is only average macroscale at different positions over the depth. In order to measure the actual macroscale at any position, it is necessary to obtain the longitudinal spatial correlation curve. As explained in Chapter IV, efforts to obtain R (xl) were not success-xlul ful with the tranducer probes used. Difficulties in getting R (xl) xlul curve even with hot-wire or film ', anemometer are pOinted out by Hinze (1959). For this reason, perhaps, Laufer (1951) in his detailed inves-tigation of turbulent flow in a two-dimensional channel measured only the distributions of L x 2 and derived the distribution of L , xl assuming that L xl is twice the value of L x3 which is strictly true only in isotropic turbulence. Even though Jordan (1963) made an intensive study of the distribution of the scale of turbulence in a pipe, he confined his study only to L . x2 Hence the results of the distribution of the average macroscale of turbulence obtained in this ,study can be compared only to Laufer's work with caution. He obtained distribution of L in a two-dimensional xl channel, 5 inches wide and 60 inches tall . The 5-inch width was directed along the x 2 axis, His Reynolds numbers were based on the 132 half-width of the channel and the maximum velocity. The trend of the variation of L acros s the width of his channel and over the depth in xl the open channel is significantly similar. The maximum value of L xl in his channel occurs at x2/d ~ O. 5 (d corresponds to half width), whereas in this work it occurs approximately at x 2/h = O. 54. It 'could also be noted th~t at corresponding Reynolds numbers in the transition regime , k u-* ( 5 < ~ < 70) the ratio of the maximum value of LXI tq character-istic linear dimension (in his case a 5-inch width) is nearly the same in a two-dimensional channel as well as in an open channel, irrespective of the medium of fluid. It should also be noted that the maximum value of L in the fully-developed turbulent regime is less than in the transition xl regime, even if the ~ in both cases were the same, suggesting that roughness decreases the value of L xl Microscale of Turbulence The distribution of the longitudinal micros cale of turbulenceJ also called dissipation length ~ ~ as a function of the relative depth of flow ~. with relative roughnes s as a parameter at three Reynolds numbers, is presented in Figures 36 to 38. The trend of the variation of ~ seems xl to be unaffected by the relative roughness. The dis sipation length ~ xl increases from the water surface until it attains a maximum value, and thereafter it decreases rapidly as the bed of the channel is further ap-proached. In the case of higher values of h/k , the microscale of s ......... Ul 6 o -0.8 o. 6 x~ 0.4 ,< o. 2 o 0G A o Figure 36. o o o [J 0AD~~6 o A A ~ 0.2 0.4 ' 0 . 6 i xn/h L. (:) A 8 A O. 8 4 RN = 8.30 x 10 o ---h/k = 97.0 s /!l- --h/k = 3. 84 s El---ii/k == 2.14 s 1.0 Variation of A. as a function of relative depth x 2/h with il/k xl s as a parameter with RN = 8.30 x 104. I-' W W ---00 6 0 -........ ~ ,< o. 6 I <:> o 0 O. 4 ~ G> L <:) AlA & 0.21- ~ GJ G Q A " A 0 0 A GJ G A [:J EJ A 0 4 RN = 7.25 x 10 G---h/k :::: 1.37 s 8---h/k = 3.58 s rn -- -h/k = 1.94 s o I ! ! ! ! I o 0.2 0.4 0.6 0.8 1.0 Figure 37. x /h 2 Variation of ~ as a function of relative depth Xl - 4 h/ks as a parameter at RN = 7. 25 x "10 . x 2/h with ~ (.V ~ "~ 0.6 ---(!) ~ _ 0.4 [ 0 0 &, -6 6 8 'x ~ 0.2 I- £ [!) t:J GI ~ (!) <:) <:> 8 G EI El 6 e I:::. A \ -~ 4 RN = 5.22 x 10 0--- ,h/k" = 72.0 s 8--- Ji./k = 3. 39 s [!]--- h/k = 1. 84 s o ~I ________________________ ~ __ ~ ____ ~ __ ~ ________ ~ o Figure 38. O. 2 " O. 4 x 2/h O. 6 O. 8 1.0 Variation of A. as a function of relative depth x 2/h with ii/k xl s parameter at RN = 5.22 x 10 4 as a I-' W (J1 136 turbulence X. reaches a maximum value nearer to the wall than for the xl cases with lower values of h/k for all the three Reynolds numbers. It s could also be inferred that the microscale of turbulence X. decreases Xl considerably with the decrease in the values of ii/k . These results s could be explained in terms of what is known about turbulence energy balance ,in boundary layer flows from the studies of Klebanoff (See Chapter lI) and the relation between the dissipation length X. ,the ~l rate of energy dis sipation and the mean square value of the velocity fluctuation that could be considered to be similar to the equation (40). In the outer region of flow near the surface, the p.roduction of , turbulence is negligible and whatever amount of energy is diffused from ~ I the wall region into this region is dis sipated. As E ex: 2' X. X. Xl in the Xl 2 outer region is small due to small values of ul and E. In the wall region both production and dissipation reach maximum values and are of equal importance except in a very thin layer near the wall (i. e., y* < 20). No specific quantitative correlation is known between the production of turbulence and turbulent velocity fluctuation ul . It is suspected that the mean square velocity fluctuation at a point is a few orders less in magnitude than the rate of generation of turbulence. Hence it follows 2 ul from E ex: -- that X. should reduce in magnitude as the wall is X. 2 Xl Xl approached. In the intermediate region both production and dis sipation of turbulence are considerable but are of equal magnitude~ Hence X. Xl 137 has to have maximum value somewhere in the intermediate region. Not only are there no turbulence-energy-balance studies in open channel flows, but there are also no similar studies even in turbulent boundary-layer flows past rough surfaces. Hence it is very difficult to speculate on the relative magnitudes of production, diffusion and dissipation of turbulence in differ~nt zones of rough open channels. Since the value of ~ is bound to increase for decreasing values of h/k , it is inferred that s ~ is smaller over the entire section, for smaller values of h/k, at xl s all the three Reynolds numbers. It is suspected that the intermediate region is displaced a little towards the surface by the presence of bed roughness which explains"l'perhaps~why the maximum value of ~ occurs xl farther from the wall at low values of h/k . s In Figures 39 to 41 the variation of the longitudinal microscale of turbulence as a function of the relative depth from the bed is presented with Reynolds number as a parameter at three different relative roughnesses. It should be pointed out again that the relative roughness could not be kept absolutely constant for flows at different Reynolds numbers but the variation in h/k for Series A and C is small. It indicates from Figure 39 that s the dis sipation length ~ remains constant over the relative depth of xl flow for the range of Reynolds numbers 5.22 x :10 4 to 8.30 x 10 4 . But for the little scatter either due to the nonconstancy in h/k or ex-s perimental errors, one arrives at the same conclusion from the Figures 40 and 41. The only explanation that could be offered is that the range --. Ui S () '-'" ~ ~ ,,< O. 6 O. 4 O. 2 o G~ ~0'tJ & 0[!) rP AG] % I:] 0Q [J Q' (3 A a:> o Series A 4 O---RN = 5. 22 x 10 4 ~- - -RN = 7. 25 x 10 4 o --""RN = 8. 30 x 10 O~, __________ ~ ______________ ~ __ ~ ____ ~ __________ ~ __ ~ o 0.2 0.4 O. 6 0.8 1.0 x 2/h Figure 39. Variation of A. as a function of relative depth x 2/h with xl RN as a parameter -- Series A. ~ w co -U) s o '-' ~ X ,< o. 6 0.4 0.2 <:) ~ 8~ AA 13 ~ ~blA I!J t:l " ~ ~ <:> o 86 A CD l!:l B Series B 4 0---RN = 5. 22 x 10 4 A- - -R = 7. 25 x 10 N 4 [!J- - -R -- 8. 3 0 x 10 N " o-~ ____________________________________________________ ~ o Figure 40. x Iii 2 Variation of A. as a function of relative depth x2/h with xl RN as a parameter -- Series B. ~ W <.0 -Ul e u ........ ........ X ,< o. 6 0.4 o. 2 m G> 8 E 8 ~QA ~ [3 o (:)' A o m A GJA o o • ~ , o o. 2 0.4 O. 6 O. 8 x 2/h Series C 4 0---R = 5 22 x 10 N . 4 ~--~=725xl0 N . 4 [:]--1\ = 8.30 x 10 N " 1.0 Figure 41. Variation of A. as a function of relative depth x 2/h with xl RN as a parameter -- Series C. ......... ~ o 141 of Reynolds numbers investigated is too small to show marked distinct trends. Similar distributions of the dissipation length were obtained by Laufer (1951) in a 5-inch wide, two-dimensional channel and by Jordan (1963) and Brookshire (1961) in a 5-inch diameter pipe with the maximum occurring at different distances from the wall. The ranges of values of the dissipation length X. from Laufer's two-dime nsional channel in-xl vestigation and Jordan's and Brookshire's investigations in a pipe are 0.198 cms to 0.685 cms and 0.127 cms to 0.456 cms respectively. The range of values of X. from this investigation is 0 . 120 cms to xl O. 51 cms. It may be m~ntioned that Laufer's values were obtained at lower Reynolds numbers, and it might explain the reason for his slightly higher values of X. • The range of Reynolds numbers in this investigation Xl [La. comparable to that of Jordan and Brookshire. It should be strongly emphasized that the tranducer probe cannot be accurately sensitive to eddies smaller than the diameter of the barium titanate crystal (1/16 inch). Hence the actual lower limit of X. could Xl not be ascertained accurately but only the average value of the lower limit of X. is obtained. The important influence of roughness is to Xl extend further the lower limit of X. Xl 142 Correlation of Turbulence Properties In the last section of Chapter III, two important relations (equations (90) and (95)) connecting the statistical properties of turbu-lence are presented, and it is indicated that such relationships may be of som~ use in solving practical problems. The left-hand side of the equation (90) is actually a dimensionless grouping of the three statis-tical characteristics of turbulence and fluid property. This dimension-less grouping should have a constant value at different positions over the depth of flow at any given value ~ ' of Reynolds number: and relative roughness. In Table 2, the calculated values of t.. .. e equation (90) are presented for four different Reynolds numbers and relative roughnesses . The values of the expression in the last three columns are reasonably constant if one keeps in mind the experimental errors involved in deter-mining especially the values of L and A. and the approximate xl Xl assumptions made in deriving that equation. In the first two columns the values are spread more, and the cause could be attributed to experimental errors only. Regarding e quation (95), it has already been proved experimentally (See Figure 34) that L is a function of relative depth of flow only in Xl Iii the case of fully-developed turbulent flow. Laufer (1954) showed that ui/u* is a function of relative depth only in the outer region of flow. Table 2. Values of the dimensionless quantity in equation (90) at various relative depths, Reynolds numbers, a.nd relative roughnesses . - - ~- - - --- - -Series A Series B Series B Series B \ Series C 4 4 4 .- 4 RN = 5.22 x 10 RN = 8. 3 x 10 RN = 5.22 x 10 RN = 4. 06 x 10 ~ = 8.3xl0 Elk = 3.39 Elk = 97.0 Fi/k = 72.0 Fi/k = 65. 70 Elk = 2.14 s s s s s ~ 2 I ~ 2 u' ~ 2 u l ~ 2 u"i ~2 u' u1 x 2 ~ x 2 xl 1 x 2 ~1 x 2 xl 1 x 2 xl 1 h vL Fi vL 11 vL h vL Fi vL xl xl xl xl xl 0.137 78. 40 0. 179 84. 30 O. 084 159.0 O. 065 73. 0 0.179 64. 80 0. 242 121.0 0.284 181.50 0.242 134.0 0. 242 97.40 O. 284 79 . 50 0. 452 108.0 O. 344 172 . 50 0.284 122. 0 O. 344 94 . 50 O. 390 78. 60 04 452 117.0 O. 530 72 . 60 O. 663 76.80 O. 705 64. 70 O. 530 110.8 O. 546 61 . 60 0. 716 94 . 0 0.716 103 . 60 O. 705 47 . 40 ---- . ------------- _ . 4 ~ ~ w .;' 144 These :.experimental results could form a basis for the approximate estimation of L ,A. ,ul' and perhaps E for a given boundary shear xl xl flow problem. The mean flow quantities such as Reynolds number of flow, shear velocity, relative roughness, and relative depth can easily be estimated. If the shear flow problem, for example, is a flow in an energy dissipator, then at " any relative depth of flow (preferably in the outer region of flow) the corresponding value of L can be found from xl the Figure 34, since the flow in energy dissipators is usually fully-developed turbulent flow. At the same relative depth, u:s'fmg; Laufer's plot, the value of ui ca n be obtained from the known value of the shear velocity. If the tabulated values of the expression on the left-hand side of the equation (gO) are available for extensive range of Reynolds numbers and relative roughnesses, the value corresponding to the known values of RN and ii/k s for the problem at hand can be evaluated by interpolation. u ll are already known, the value of A. can be determined. If xl the value of E is required in the outer region of flow, where local isotropy can be expected to exist, it could be evaluated using thl- equation (40) as the knowledge of every quantity in that equation can be determined as explained above. 145 CHAPTER VII CONCLUSIONS From the experimental investigation and the subsequent inter-pretation and discussion of results, the following conclusions can be made concerning the structure of shear turbulence in rough open channels. 1. The relative intensity of turbulence over the entire depth of flow decreases with increase in RN at a constant bed rough-ness and increases with decrease in the value of ii/k at a s constant value of RN . It seems to attain a maximum value in the constant stress layer where the production of turbulence is also maximum. 2. The influence of roughness on the relative intensity of tur-bulence is markedly pronounced near the bed, and the trend in the variation of ul / U 1 is not affected by roughness. 3. In the outer region of fully-developed turbulent flow in the rough open channel, the wave number at which the transition between locally isotropic turbulence and anisotropic turbulence occurs, is independent of the Reynolds' number, but varies inversely \vith the depth of flow. This is in complete agree-ment with the picture of the mechanism of the generation, transfer, and dissipation of turbulent energy as proposed by 146 Kolmogoroff. It has been verified that both the inertial and viscous subranges of the universal equilibrium range (local isotropy) can be expected to exist in the energy spectrum of turbulence in the wave number space. 4. The longitudinal one-dimensional energy spectrum function (equation 118) for the entire region of universal equilibrium range, obtained by combining equations (9 9) and (70) for the two extremes (inertial subrange and viscous subrange), a simplified form of Heisenberg equation (68), describes sat-isfactorily the transition between inertial and viscous subranges. 5. It is concluded that local isotropy cannot exist in the wave number space in turbulent shear flow in the open channel near the bed due to the presence of steep mean velocity gradients. 6. The contribution to turbulence energy in the entire wa ve num-ber range increases as the bed of the channel is approached. 7. The concept of local isotropy is inadequate for obtaining the distribution of the rate of dissipation of turbulent energy in turbulent shear flow even though it could be made use of in the outer region to evaluate the local rate of energy dis sipation. 147 8. The rapidity with which the autocorrelation falls off increases with decreasing values of the relative depth measured from the bed at the sampling locations . Thus the correlation decays faster near the bed. 9. The longitudinal average macroscale of turbulence decreases IO. as the bed of the channel is approached and it increases with increasing distance from the bed until a maximum value is reached; and thereafter it begins to decrease as the water surface is approached. The macroscale is maximum in the intermediate region where the viscous action is negligible and turbulence generation is considerable; it is approximately located at x 2/ h = O. 54. In the case of fully-developed turbulent flow, L /ii xl is a function of relative depth of flow (x2/ h) only, and independent of Reynolds' number and relative roughness of the bed of the channel. In the transition regime of flow, L Ih is a function x . 1. of Reynolds' number as well as relative roughness and the tur-bulence has larger scale (L ) at lower Reynolds numbers. xl 11. The longitudinal microscale of turbulence, A. ,increases from . ~l . the liquid surface until it attains a maximum value at a relative distance from the bed around x 2/ h = O. 55 and then it decreases rapidly as the bed of the channel is further approached. The position of the maximum value of A shifts towards the xl 12. surface as the relative roughness ii/k decreases. The s influence of roughnes s on the shape of the distribution of ~ is negligible . Xl At a constant value of RN , the value of ~ decreases Xl 148 over the entire depth of flow as the relative roughness ii/k s decreases. Then ~ remains constant over the relative Xl depth of flow for the range of Reynolds numbers in this in-vestigation at a constant relative roughnes s ii/k . s I 3. Equation (90) is proved to be valid for a set of values of RN and ii/k . A procedure for estimating important statistical s characteristics of turbulence such as L ,~ ,uI' and ~ Xl Xl in fully-developed turbulent shear flow in channels on the basis of the knowledge of mean flow quantities like RN, u*' h/ks' and x 2/ii is suggested. I 4. The presence of artificial roughness on the bed of the channel has an indirect influence on the structure of turbulence in open-channel flows by transforming the flow into fully-developed turbulent flow at a lower Reynolds number. 1 5. Due to the acoustical properties of the tranducer probe, it is considered that it is not ideally suited for measuring auto-correlation function and s cales of turbulence. 149 CHAPTER VIII SUGGESTIONS FOR FURTHER RESEARCH As pOinted out in Chapters IV and VI, due to the limitations of the tranducer probe, complete information on t he structure of shear turbulence in open-channels could not be obtained. On the basis of the experience gained in this investigation, it is believed that improvements can be made in organizing measurement of data . Besides, prior to this work, there have been very few turbulence measurements in open channels and no measurements in rough open channels so far as the author knows . Hence, it is felt that there is sufficent need to pursue this study much more inten-s~ely as well as extensively. With this view in mind, a few suggestions are offered below to extend the present author's research with the hope that they may be of some use to someone else. 1. Measuremen't$ should be made of relative intensity of turbulence over a wider range of values of RN ·with h0s as a parameter and over a wider range of h/ks with RN as a parameter, keeping the geometrical similarity of the roughness constant, 2. Attempts should be made to make measurements within the constant stres'S.· layer extending to the laminar sublayer possibly using a device such as the hot-film a nemometer with highly viscous fluid and uniform sand-grain roughness 150 on the bed. At present no mean or turbulence energy balance studies have been made either in the vicinity of a rough wall or in open channels. This study may also be used to sub-stantiate or refute the speculation that turbulence energy moves from the outer region to the inner region by the action of pressure forces which compensate for greater dissipation within the region of y* < 20. 3. It is necessary to use a device such as the hot-film annemom-eter (which can measure lateral velocity fluctuations) if further turbulence measurements are to yield a more complete picture of the structure of shear turbulence. a. As no conclusive studies have been made yet about the validity of Kolmogoroff's concept of local isotropy for spectra of lateral velocity fluctuations, it will be of great interest to obtain cross-spectra. Besides, such data can be made use of in conj unction with equation (115) in applying a further check on the existence of local isotropy in spectra of longitudinal velocity fluctuations. Furthermore, measurements of spectra of shear stress and shear correlation coefficient can be made and can also be used as additional checks on the occurrence of local isotropy in spectra of ul' i Sl b. Measurements of s pace correlations and their associated macroscale s and microscales could not be made in this study. It ma y be pointed out that measurements of Rx (xl) ul and associated L have not been made in any kind of x l wall shear fl ow. No turbulence study is complete with-out mea s uring the s e quantities. 4. It may be intere s ting to measure the di s t r ibution of A. over xl the relative depth taking h/k as a parameter over a range s of constant values of RN and to repeat the same taking RN as a parameter over a range of constant values of h/k , s keeping the geometry of bed roughness similar. S. It will be useful to chec k the validity of the equation (90) over a wider range of RN and il./k s . 6. The possibility of replacing the Lagrangian correlation with Eulerian space- time correlation is worth investigating. If such a substitution were possible, it paves the way for the direct determination of the Lagrangian correlation coefficient and associated mean scale s of the eddies participating in the diffusion process . Thi s simplif ied procedure for the determin-ation of the Lagrang ian s cale s of eddie s could be used to determine a fun ctional ex pression for the diffusion coefficient in turbulent shear fl ow on the bas i s of Kolmogoroff's theory and to evaluate the ex tent of influence of the relative roughness of the bed on the d iffusion coeffi cient. LITERATURE CITED Bass, J. 1949. II Sur les bases mathematiques de la theorie de la turbulence d' Hei senberg. II Comptes Rendus "des Seances De l' Academie des Sciences, Paris. 228:228. Batchelor, G. K. 1946. II The theory ofaxisymetric turbulence. " Proceedings Royal Society, Series A. 186:480. 1 52 Batchelor, G. K. 1948. II Energy decay and self-preserving correlation functions in isotropic turbulence. II Quarterly of Applied Mathematics. 6:97. Batchelor, G. K. 1953. II The theory of homogeneous turbulence. II Cambridge University Press, Cambridge, Great Britain. 197 p. Brookshire, W. A. 1961. II A study of the structure of turbulent shear flow in pipes. II Unp-ublished Ph. D. dissertation in Chemical Engineering. Louisiana State University, Baton Rouge, Louisi'ana. Burgers, J. N. and M. Mitchner. 1953. II On homogeneous nonisotropic turbulence connected with a mean motion ha ving a constant velocity gradient. II Proceedings Kon. Ned. A Kad. Van. Wet. B., 56(3) and 56(4). (Original not seen; abstracted in J. O. Hinze. 1959. II Turbulence" McGraw-Hill Book Company, Inc., New York. 586 p.) Chandrasekhar, S. turbulence. II A. 200:20. 1949. II On Heisenberg's elementary theory of Proceedings of the Royal Society of London, Series Chandrasekhar, S. 1950a. II The theory of axisymmetric turbulence. II Philosophical Transactions of the Royal Society of London, Series A. 242:557. Chandrasekhar, S. 19 SOb. II The decay of axisymmetric turbulence. II Proceedings of the Royal Society of London, Series A. 203: 358. Clyde, Calvin G. 1961. II Fluctuations of total head near a smooth wall in a turbulent open channel flow. II Unpublished Ph. D. dissertation in civil engineering. University of California, Berkeley, California. Corrison, S. 1949. II An experimental verification of local isotropy. II Journal of Aeronautical Science. 16: 7 57. Corrison, S. 1961. II Turbulent flow. II American Scientist. 49: (3); 300-325. Craya, A. 195 7a. II Sur les Correlations triples entrois points en turbulence homogene. II qomptes Rendus des Scienaes De 1'1 Academie des ScienGes, Paris. 244:560-562. Craya, A. 195 7b. II Une methode d' investigation des corr~lations doubles pour la turbulence homage' ne en pre s ence de vitesses moyennes. II Comptes Rendus des Scienc.e:s De L' Academie des S'ciences" Paris. 244:1448-1450. Craya, A. 1957 c. II Sur les equations de la turbulence homoge' ne 153 en presence d' un champ de vitesse moyenne. II Comptes Rendus des Seances De L'Academie desSt:ienc-e,s" Paris. 244:847-848. Craya, A. 195 7d. II Sur la structure des correlations triples en turbu-lence homoge' ne associee a' unchamp de vitesses moyennes. II Comptes Rendus des ScienC'8s De L" Academie des Science~, Paris. 244:1609-1611. Deissler, R. G. ' 1958. II On the decay of homogeneous turbulence before the final period. II The Physics of Fluids, 1 (2): 111-121. Deis sler, R. G. 1960. II A theory of decaying homogeneous turbulence. II The Physics of Fluids. 3(2):176-187. Deis sler, R. G. 1961. II Effects of inhomogeneity and of shear flow in weak turbulent fields. II The Physics of Fluids. 4(10):1187-1198. Deissler, R. G. 1962. II Pressure fluctuations in a weak turbulent field with a uniform transverse velocity gradient. II The Physics of Fluids. 5(9):1124-1125. Dryden, H. L. and A. M. Kuethe. 1929. II Effect of turbulence in wind tunnel measurements. II National Advisory Committee for Aeronautics, Report Number 342. Dryden, H. L. 1938. II Turbulence investigations at the National Bureau of Standards. II Proceedings of the Fifth International Congress of Applied Mechanics. p. 362. 154 Eagleson, P. S., C. J. Huval, and F. E. Perkins . 1961 . II Turbulence in the early wake of a fixed flat plate. II Massachusetts Institute of Technology Hydrod ynamics Laboratory, Technical Report No. 46. Fox, J. 1964. II Velocity Correlations in weak turbulent shear flow. II The Physics of Fluids . 7(4):562-564. Frenkiel, F. N. 1958. II Statistical study of turbulence-spectral functions and correlation coefficients . II National Advisory Committee for Aeronautics, Technical Memorandum 1436. Gibson, M. M. 1962 . II Spectrum of turbulence at high Reynold s number. II Nature. 195:1281-1283. Gibson, M. M. 1963. II Spectra of turbulence in a round jet. II Journal of Fluid Mechanics . . Part II. 15:161-173. Gibson, C. H. and W. H. Schwarz. 1963 . II The universal equilibrium spectra of turbulent velocity and scalar fields. II Journal of Fluid Mechanics . 16: 365-384. Goldstein, S. 1938. II Modern developments in fluid dynamics. II Oxford University Press, London. Volume 1. Grant, H. L., A. Moilliet, and R. W. Stewart. 1959. II A spectrum of turbulence at very high Reynolds number. II Nature. 184:808-810. Grant, H. L., R. W. Stewart, and A. Moilliet. 1962. II Turbulence spectra from a tidal channel. II Journal of Fluid Mechanics. 12:241-263. Halbronn. 1954. Discussion of II Turbulent boundary layer on steep slopes" by W. J. Bauer. Transactions of the American Society of Civil Engineers . 119:1238. Heisenberg, W. 1948. II On the theory of statistical and isotropic turbulence. II Proceedings of the Royal Society of London. Series A. 195:402-406. Heisenberg, W. 1958. II Zur Statistischen theorie der turbulence. II Z. Phys. Vol. 124. pp. 628 - 657. (1948). English translation available as National Advisory Committee for Aeronautics, Technical Memorandum 1431 (1958). 155. Hino, M. 1959. Dis cuss ion of paper, H Eddy d iffusion in homogeneous turbulence. 01 by OrIob, G. T. Proceedings of the American Society of Civil Engineers. 85:HY9. Hinze, J. O. 1959. II Turbulence!! McGra w- Hill Book Company, Inc., New York . 586 p. Hueter, T. F. and R. H. Bolt. 1955. H Sonies . II John Wiley and Sons, Inc., New York. 456 p . Ippen, A. T., R. S. Tonkin, and F. Raichlen. 1955 . II Turbulence m 0.1 U1). Ippen et ale (1955) estimated an error of 11 percent in determination of u l I when ul /U 1 = O. 1 and an error of 22 percent when ul!U1 = 0.2 . If VI is the tranducer output in volts due to HI acting on the R probe, and C is the calibration constant in volts per foot of head ·u u l "- VI C 1 1 = R g (4) Hence g VI u l R = 1 C U1 (5) This equation was used for calculating u l I from the measurements of the r. m. s . voltmeter of the random signal indicator and correlator to which the signal from the amplifier output terminal was fed. 1.70 Care should be taken in interpreting the turbulence data obtained with the tranducer probe because not only the probe responds indiscrimi-nately to pressure fluctuations of ' acoustical origin, but also to free or forced yibrations. Besides, the natural frequency of the probe itself might interfere with the energy_spectrum measurements in indicating an abrupt peak at the natural frequency irrespective of the turbulence characteristics being measured. As the s pectrum measurements were made in this study at frequencies much below the resonant frequency (80 K. C. ), peaks were not observed in the spectrum measurements , i 71 Appendix B Calibration of the tranducer probe Frequency response. The details pertaining to the computation of the low frequency relative response of the tranducer probe can be found in the publications of Perkins and Eagleson (1959) and Rueter and Bolt (1955). The experimental set up used for the determination of fow frequency re-sponse of the probe is shown in Figure 42. At a constant amplitude a series of tests was run at frequencies from 2 to 16. 7 c. p. s. It was found that by 14 c. p. s. the sensitivity of the probe had reached a con-stant limiting value. This value was assigned to represent a relative response of 1. 0 and the relative response at lower frequencies was computed from the experimental data. The plot of relative response against frequency is shown in Figure 43. The estimated low frequency response curve is superimposed on the measured points. The agreement between estimated and measured low frequency response of the probe is seen to be good. This result indicates that the probe may be used to measure fluctuations of as low frequency as 4 c. p. s. with sufficient accuracy. The high frequency response of the probe in water could not be determined due to the limitation imposed by the mechanical system to generate high frequency pressure fluctuations in the pressure chamber. Neubert (1963) pointed out that the useful frequency range of a pd.ezo-elect ric tranducer is one-fifth of the frequency interval between zero and Glenite cathode follower F-407 Tektronix preamplffi.er Type 122 Tektroriix preamplifier ...-----.~_--II Type 122 Disa random signal indi-cator and correlator Type 55A06 Dumont 766 Fairchild oscillos cope \.\lith camera) Figure 42. Experimental set-up for calibration of the tranducer probe at low frequency range . ....... 'l r-v J i I I I _\ _ _ __ I t-- -~ ! I ~ I i o _I ~ O. 6 1 -----> Q) Q) :> -r-! +oJ co r-4 ID ~ P:: O• 4 o. 2LI -1-01 i l , I . ~ 1 1 o 2. 0 4. 0 6. 0 8. 0 10. 0 12. 0 Frequency in cycles per second Figure 43. Low frequency response of tranducer probe . number 2. 14.0 16. 0 ...... "-l W 174 the lowest resonant frequency if a magnification error of 4 per cent in its output is tolerated. The lowest resonant frequency of th~ tranducer in water is 80, 000 c. p. s. Therefore, its useful frequency range is much above the frequencies of turbulent fluctuations which the probe was re-quired to measure. However, the high frequency response of the probe was measured in air in an anechoic chamber up to 1000 c. p. s. The response was found to be flat up to that frequency. Sensitivity. As the stiffness of the tip diaphragm of the probe is extremely small, its effect on the attenuation of the stres s which is actually induced in the ceramic crystal can easily be neglected. Using the physical properties of the crystal furnished by the manufacturer and neglecting the stiffness of the diaphragm, the sensitivity of the probe was calculated to be 19. 15 millivolts per p. s. i. Unlike the frequency response, the dynamic sensitivity of the tranducer is only a function of its physical properties such as the relative dielectric constant of the crystal material, the absolute dielectric constant of the free space between its faces, Young's modulus of elasticity and the piezoelectric coefficient. Thus it makes no difference in what kind of fluid medium the sensitivity of the probe is determined. As the pressures can be generated at very high frequencies by ordinary loud speakers and these pressures can be accurately measured in an anechoic chamber with standard laboratory microphones, it was decided to determine the sensitivity of the probe in the anechoic chamber of the electrical engineering department. 175 The block diagram of the experimental arrangement used in the anechoic chamber is shown in Figure 44. The standard precision micro-phone and the probe were mounted before the speaker system at the same distance and elevation side by side. At a particular frequency a sinus-oidally varying sound pressure wave was generated using audiogenerator and speaker system. The signals from both the microphone and probe were fed to the inputs of the two channel oscilloscope for visual inspec-tion of 1:he response of the probe and microphone. The output of the microphone was measured with vacuum tube voltmeter and the output of the ,tranducer probe was measured with r. m. s. voltmeter of the random signal indicator and correlator which was used in turbulence measurements. At a constant frequency the amplitude was varied over a range cor-responding to preselected readings of microphone output expressed in db of pressure level and the corresponding outputs of the probe were measured simultaneously. The same procedure was repeated by isolating the probe from contact with the pressure fluctuations in order to measure the noise due to hum and possible structural vibration. The two steps outlined above were repeated at different frequencies to make sure the high frequency response of the probe is flat. From the readings of the microphone output, absolute values of the pressure changes acted on the probe were calculated in p. s. i. Correspono.ing actual outputs of the probe, after allowance for noise, were calculated in millivolts at a preamplifier gain of 100. Plots of General radio beat frequency Audio-gene~ator l!..Y.Q.e 13043 --A ~ewlett- \ r--~ackard lVacuum tubE Ee:.clh kit 1YQll. m~ter .-. Audio- Model 400~ amplifier Mode 1 .4A:rQ 'I' II Preamplifier Dumont Tektronix 766 ~tec Lancing ~ Type 122 Fairchild speaker system HMikel--· r--- scope Model N800 A r HrobeL -- 'DISA random . :.t Tektronix signal indi4ator Glenite cathode ~ Preamplifie - and correlator ~ follower - Type 122 Type 55A06 r-407 I Figure 44. Block diagram of the experimental set-up for calibration of the tranduc-er probe in anechoic chamber. I-' -.......] (j) 177 pressure change in p. s. i. against probe output in millivolts at a gain of 100 are shown in Figures 45 and 46 for probes 2 and 3. The sensi-tivity of the probes 2 and 3 are respectively 16. 5 millivolts and 14.0 millivolts per p. s. i. pressure change respectively. Their estimated sensitivity is 19.15 millivolts per p. s. i. pressure change. The dif-ference -between the estimated and measured sensitivity of the probes is small compared to what Perkins and Eagleson (1959) obtained and this improvement is believed to have been achieved by avoiding depolarization of the ceramic crystal by heat generated in using silver soldering to seal the j oint between the tip cover and main portion of the tube. The small difference, however, could be accounted for neglecting the stiffness of the tip diaphragm and the discrepancy between the actual dimensions and properties of the ceramic crystal and the values used in computation as furnished by the manufacturer. The sensitivity of the probes was also determined in the water medium using the experimental setup shown in Figure 42, and the results compared well with the values obtained in air medium as expected. --. M " o 4-1 o 178 8.0· .. --------~------~--------~---------.--------~--------~ 7. 0 ___ ------t------+------.i-- ----+------+----"-'JIII~ 6. 0 (Jl5. O~---------+------+-J - ,-i j:! ::s j:! -,-i -ro4. O.---------~----------~----------+-~~----~-----------+----------~ ~ j:! -,-i Q) 01 j:! 10 "8 3. 0 Q) ~ ::s (Jl (Jl Q) ~ ~ 2. O~--------~----~~~----------+----------+----------r---------~ 1. 0 Oc-______________________________________ ----------------~ o 2. 0 4. 0 6. 0 8. 0 10 . 0 12.0 Probe output in millivolts Figure 45. Pressure sensitivity of the tranducer probe No.2. 179 8. 0 / I I( 0 I 7. 6. -M I I 0 -l . If . ~ , I 0 .--t ~ 0 Ul 5. +oJ -.-4 s:: :J 0 5V . I s:: -.-4 -H j , Cf) 4. ~ s:: -.-4 Q) 0'1 s:: 10 3 . ..c:: 0 Q) \.., :J Ul Ul Q) ~ ~ 2. 0 ~ V 57 ~~J ~ 1/ ~ 0 ( .. J~ 1/ --0 I ~ 7 / u v ) I 1. --_. V , 0 I ~ o 2. 0 4. 0 6. 0 9. 0 10.0 Probe output in millivolts Figure 46. Pressure sensitivity of the tranducer probe No.3 . Appendix C Calibration of the spectrum analyzer The harmonic wave analyzer used to measure the spectrum of turbulence was a Quan-Tech Model 303 . To calibrate the frequency 180 dial of the wave analyzer, Hewlett-Packard's Electronic Counter Model 5228 was used as a reference. After nulling was done by adjusting the zero beat following the steps outlined in the operation manual a signal from the audiooscillator was fed to the inputs of the analyzer and the electronic counter. Then the frequency dial of the analyzer was adj usted until the analyzer output was a maximum as meas ured with the r ~ . m .. $ .. voltmeter of the analyzer. The frequency corresponding to the maximum output of the analyzer and the frequency read by the electronic counter were noted. This procedure was repeated for different frequencies. In all cases the frequencies read on the frequency dial of the analyzer were identically the same as the frequencies read on the electronic counter indicating that the frequency dial of the analyzer is 'in complete adj ustment. The ~ r. :m. s. voltmeter in the wa ve analyzer was calibrated by a similar procedure. A signal from the audiooscillator was fed to the wave analyzer input with the analyzer frequency dial set to give maximum out-put. The 7analyzer output was measured with the vacuum-tube voltmeter. This procedure was repeated for different values of the input voltage and 181 frequency. It was found that the readings of the r.. m .. S ·,. voltmeter and the vacuum-tube voltmeter were identically the same indfcating the 'r, ar .. s .. voltmeter of the analyzer reads accurately enough. Spectrum measurements were made with the wave analyzer with a fixed nominal band width of 30 c . p. s. The e ffective band width is defined ,as the rectangular band width having the same area as the area under the experimentally determined band shape. To determine the band shape, a signal from the audiooscillator was fed to the input of the analyzer with the analyzer frequency dial set to a preselected value; say 500 c. p. s. The gain control of the analyzer was adjusted until the r .. in .. s. voltmeter read a maximum which was then taken as a reference value. Then the frequency dial of the analyzer was adjusted such that the r .. m; .. $. voltmeter read O. 8, O. 6 . .. O. 01 times the reference value. The corresponding frequencies were read on the electronic counter. A plot of the output of the analyzer versus the frequency was drawn and shown in Figure 47. A rectangle having the same area as the area under the curve is fitted and the width of the rectangle,or in other words, the effective band width was found to be 59. 00 c. p. s . -~---r---t----I,---~--1--- r - ~ __ I I ...... - I -I-T---t-+~~-l~It-. ____ (~I I • t - -1-r--t-l~-----i---1-J ~l I ~ - ~ ---i - I 1--~---t---r~1---+---~-t--_J1l r~J ' _. - - Ir--+---~~ ,_ ~ Q - - __ _ . I _ v ~ .J ·c .~ )--.. - f-- -I-- . - _.+ ' ~ ~ , ' '" _ 5 - -- ---- --'-- I I ' - -, J U -. -~ I -l- '" l r--t----t--+~-. ~ 7 ~ L rI-j-r~+-_~~v~- ~-r~\~~~--L-1--+------1-.. ~ v~ \ ~ -I /'tT- ~ -~ ~ _ I ,I '.- '-0.. I I I I ~----I ~ 340 - 380 420 460 500 540 Frequencies in Cycles per Second Figure 47. Response curve of spectrum analyzer,.. 580 620 660 I-' 00 N INPU SIGNAL VARIABLE ----~ CALIBRATED CAIN . ISOLATION r ATTENUATOR r-- ~ AMPLIFIER - 100 V AMPliFIER 10 m INPU SIGNAL VARIABLE 4,... , CALIBRATED GAIN H .... SIGN ~ " ~ ATTENUATOR INVERTER ~ "-- lOOV AMPLIFIER 10m ~ ISOLATION AMPLIFIER H SUMMING AMPLIFIER .. I" " A-SUMMING H AMPliFIER ] ~ d T-d DIFFEREN H -TIATOR VOLTMETER AO INPUT SELECTOR BO A+B A-BG TdA O dt BALO 1 TdA dt OUTPUT AMPliFIER • I OUTPUT AMPLIFIER II B031 .... , ___ ~ I A-B 04 : I I I BALO 1 ~ A A03 A+B04 1: 2: RATIOMETER SELECTOR OUTPUT AMPLIFIER III BAL A TdA dt 3: . A .B 4: A+B A-B AND CORR. COEFF. Figure 48. Block diagram of the internal circuit of the random Signal indicator and correlator SQUARING AND AVERAGING NETWORK I SQUARING AND AVERAGING NElWORK II SQUARING AND AVERAGING NETWORK III RMS VOLTMETER RMS VOL 1ME-TER ~ 't1' CD· ::1 . Or ..... x-; ~ t:: ...... ex> w

A turbulence description of couette flow

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A turbulence description of couette flow

Abstract

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