Comparative Analysis of Numerical and Integral Methods for Calculating Turbulent Compressible Flow

The calculation of boundary layer parameters requires knowledge of the turbulent shear which, with our present incomplete knowledge of turbulence, must rely on empirical relationships. Another well-known problem is the non-linearity of the boundary layer equations. Early calculations rely on the well-known Von Karman integral equations but with the advent of the large digital computers, finite difference methods are presently commonly used. For calculation of the incompressible boundary layer there are two popularly used methods. A comprehensive comparison of this type of approximate method assuming different velocity profiles is given by Taylor. The second is a purely numerical method using finite difference approximation. Independent studies have been made by several authors, and the programs developed by Smith and Cebeci and Pletcher are quite general and can be used for conditions including heat and mass transfer. Despite this progress the existing methods cannot predict the point of flow separation reliably. Calculations for attached boundary layer or the calculations up to or near the separation point, however, can be done with a great degree of precision. For the case of the compressible turbulent boundary layer in an adverse pressure gradient, the calculation of the boundary layer development appears to be more uncertain. Mr. Rotta in a recent summary paper stated: \u27\u27With regard to tur­bulent boundary layers in an arbitrary pressure field of compressible f1ow, the same problems arise as in incompressible flow, however, with higher severity since Mach number and surface temperature come in as influencing factors. Several of the calculation methods, originally designed for incompressible boundary layers in pressure gradients, have been modified in or der to allow for the effects of Mach number and heat transfer. White has suggested categorizing compressible turbulent boundary layer approximations into five categories. They are: 1. Limiting Case Theories 2. Von Karman Integral Techniques 3. Compressibility Transformations 4. Finite Difference Approximations 5. Integral Relations Based on Wall Variables. The fifth method above is a new technique being developed by White, in which he extended his method to the case of the compressible boundary layer. White\u27s method is presently being modified to include axisymmetric flow problems. Studies up to now have indicated an accuracy and simplicity that may threaten to replace the now popular Von Kaiman\u27s integral methods. The results of Smith and Cebeci have indicated a fairly high accuracy. The main disadvantage to finite difference methods is that they require large computers to be effectively employed. The limiting case theories and compressibility transforms have been shown to be useful in restricted cases for rough approximations. Both methods must be used with caution and do not have as wide a range of applications as the other methods. The present study was initiated to find a reliable method to calculate compressible boundary layer parameters and determine boundary layer separation in choked inlets. The objective was to determine whether simplified methods of solution could be judged reliable enough for engineering purposes