Stochastically generated turbulence for wall bounded flows

An efficient stochastic method is applied to the problem of modelling unsteady turbulent velocity fluctuations within a turbulent flat plate boundary layer. White noise is spatially and temporarily convoluted by the time averaged Reynolds stress tensor and dissipation. The convolutions are conducted such that mean turbulent properties are reproduced and the spectral distribution of turbulent kinetic energy closely matches expected profiles. Wall bounded turbulent flow induces elevated levels of broadband aeroacoustic noise, especially near a sharp edge such as an airfoil trailing edge. In practice, the stochastic method of generating turbulent fluctuations may be used directly to produce turbulent noise sources for further aeroacoustic analysis, thus avoiding costly time dependent simulations of the Navier--Stokes equations. References C. Bailly and D. Juve. Numerical solution of acoustic propagation problems using linearized euler equations. AIAA Journal, 38(1):22--29, 2000. doi:10.2514/2.949 W. Bechara, C. Bailly, and Lafon. Stochastic approach to noise modeling for free turbulent flows. Journal of the Acoustical Society of America, 97(6):3518--3531, 1994. doi:10.2514/3.12008 M. Billson, L. Eriksson, and Davidson. Jet noise prediction using stochastic turbulence {M}odeling. 9th AIAA/CEAS Aeroacoustics Conference, (2003-3282), 2003. C. Blom, B. Verhaar, and Van Der Heijden. A linearized euler method based prediction of turbulence induced noise using time averaged flow properties. AIAA Journal, (A01-16882), 2001. R. Ewert. Slat noise trend prediction using {CAA} with stochastic sound sources from a random particle-mesh method ({RPM}). AIAA Journal, (2006-2667), 2006. R. Ewert. R{PM}-the fast random particle-mesh method to realize unsteady turbulent sound sources and velocity fields for {CAA} applications. AIAA Journal, (2007-3506), 2007. R. Ewert. Broadband slat noise prediction based on {CAA} and stochastic sound sources from a random particle-mesh ({RPM}) method. Computers and Fluids, 37(4):369--387, 2008. doi:10.1016/j.compfluid.2007.02.003 R. Ewert, C. Appel, and Dierke. R{ANS}/{CAA} based prediction of {NACA} 0012 broadband trailing edge noise and experimental validation. 15th AIAA/CEAS Aeroacoustics Conference, 2009. R. Ewert and W. Schroder. On the simulation of trailing edge noise with a hybrid {LES}/{APE} method. Journal of Sound and Vibration, 270:509--524, 2004. doi:10.1016/j.jsv.2003.09.047 R. Ewert and R. Edmunds. CAA slat noise studies applying stochastic sound source based solenoidal digital filters. 26th AIAA Aeroacoustics Conference, 2005. M. Karweit, P. Belanc-Benon, and Juve. Simulation of the propagation of an acoustic wave through a turbulent velocity field: a study of phase variance. Journal of the Acoustical Society of America, 89(1):52--62, 1991. doi:10.1121/1.400415 M. Klein, A. Sdiki, and Janicka. A digital filter based generation of inflow data for spatially developing direct numerical or large eddy simulations. Journal of Computational Physics, 186:652--665, 2003. doi:10.1016/S0021-9991(03)00090-1 R. H. Kraichnan. Diffusion by a random velocity field. The Physics of Fluids, 13:22--31, 1970. doi:10.1063/1.1692799 T. Lund, X. Wu, and Squires. Generation of turbulent inflow data for spatially developing boundary layer simulations. Journal of Computational Physics, 140:233--258, 1998. doi:10.1006/jcph.1998.5882 M. Mesbah. Flow noise prediction using the stochastic noise generation and radiation approach. Phd, Katholieke Universiteit Leuven, Faculteit Ingenieurswetenschappen, 2006. S. Chong, M. S. Perry, A. E. Henbest. A theoretical and experimental study of wall turbulence. Journal of Fluid Mechanics, 165:163--199, 1986. A. Smirnov, S. Shi, and Celik. Random flow generation technique for large eddy simulations and particle dynamics modeling. Journal of Fluids Engineering, 123:359--371, 2001. doi:10.1115/1.1369598 M. Snellen, L. Lier, and Golliard. Predictions of the flow induced noise for practical applications using the {SNGR} method. 10th International Congress on Sound and Vibration, 2003. P. R. Spalart. Direct simulation of a turbulent boundary layer up to Re$(\theta )=1410$. Journal of Fluid Mechanics, 187:61--98, 1988