Center for TurbulenceResearch Proceedings of the Summer Program A new approachto turbulence mo deling By B Perot AND P Moin A new approach to Reynolds averaged turbulence mo deling is prop osed whichhas a computational cost comparable to two equation mo dels but a predictive capability approaching that of Reynolds stress transp ort mo dels This approach isolates the crucial information contained within the Reynolds stress tensor and solves trans p ort equations only for a set of reduced variables In this work direct numerical simulation DNS data is used to analyze the nature of these newly prop osed tur bulence quantities and the source terms which app ear in their resp ective transp ort equations The physical relevance of these quantities is discussed and some initial mo deling results for turbulentchannel ow are presented Intro duction Background Two equation turbulence mo dels such as the k mo del and its variants are widely used for industrial computations of complex ows The inadequacies of these mo dels are well known but they continue to retain favor b ecause they are robust and inexp ensive to implement The primary weakness of standard two equation mo dels is the Boussinesq eddy viscosityhyp othesis this constitutive relationship is often questionable in complex ows Algebraic Reynolds stress mo dels or nonlinear eddy viscosity mo dels assume a more complex nonlinear constitutive relation for the Reynolds stresses These mo dels are derived from the equilibrium form of the full Reynolds stress transp ort equations While they can signicantly improve the mo del p erformance under some conditions they also tend to b e less robust and usually require more iterations to converge Sp eziale The work of Lund Novikov on LES subgrid closure suggests that even in their most general form nonlinear eddy viscosity mo dels are fundamentally incapable of completely representing the Reynolds stresses Industrial interest in using full second moment closures the Reynolds stress transp ort equations is hamp ered by the fact that these equations are much more exp ensive to compute converge slowly and are susceptible to numerical instability In this work a turbulence mo del is explored which do es not require an assumed constitutive relation for the Reynolds stresses and may b e considerably cheap er to compute than standard second moment closures This approach is made p ossible by abandoning the Reynolds stresses as the primary turbulence quantityofinterest Aquasions Inc Canaan NH Center for Turbulence Research B Perot P Moin The averaged NavierStokes equations only require the divergence of the Reynolds stress tensor hence the Reynolds stress tensor carries twice as much information as required by the mean ow Moving to a minimal set of turbulence variables reduces the overall work by roughly half but intro duces a set of new turbulence variables which at this time are p o orly understo o d This pro ject attempts to use DNS data to b etter understand these new turbulence variables and their exact and mo deled transp ort equations Formulation The averaged NavierStokes equations take the following form for incompressible constantprop erty isothermal ow r u a u u ru rp r S rR b t where u is the mean velo city p is the mean pressure is the kinematic viscosity T S ru ru is twice the rateofstrain tensor and R is the Reynolds stress tensor The evolution of the Reynolds stress tensor is given by R T u rR r R P rT rq rq t where P is the pro duction term is the homogeneous dissipation rate tensor is the pressurestrain tensor T is the velo city triplecorrelation and q is the velo citypressure correlation The last four source terms on the righthand side must b e mo deled in order to close the system The pro duction term P is exactly represented in terms of the Reynolds stresses and the mean velo city gradients This is the standard description of the source terms but it is by no means unique and there are numerous other arrangements Note that turbulence eects in the mean momentum equation can b e represented by a b o dy force f rR One could construct transp ort equations for this b o dy force which has b een suggested byWu et al but mean momentum would no longer b e simply conserved Toguarantee momentum conservation the b o dy force is decomp osed using Helmholtz decomp osition into its soleno dal and dilatational parts f r r A constraint or gauge must b e imp osed on to make the decomp osition unique In this work we take r With this choice of gauge the relationship b etween and and the Reynolds stress tensor is given by r r rR a r r rR b Note that the choice of gauge inuences the value of but do es not aect how inuences the mean ow A new approa ch to turbulencemodeling Using these relationships transp ort equations for and can b e derived from the Reynolds stress transp ort equations u r r r q r rr rT Pr S a t u r r rq r rr r T Pr S b t These equations contain extra pro ductionlike source terms S and S whichcontain mean velo city gradients Note that the pro duction term is not an explicit function of and except under limited circumstances and in general must b e mo deled The inverse Laplacian r that app ears in these equations can b e thoughtofasan integral op erator Theoretical analysis Turbulent pressure Taking the divergence of Eq b the mean momentum equation gives the classic Poisson equation for pressure r p r u ru rrR Since this is a linear equation the pressure can b e split conceptually into two terms one can think of the mean pressure as b eing a sum of a mean ow pressure due to the rst term on the righthand side r P r u ru a mean and a turbulent pressure due to the second term on the righthand side r P r rR b tur b Given the denition of and assuming that is zero when there is no turbulence then it is clear that P For this reason will b e referred to as the tur b turbulent pressure This quantity is added to the mean pressure in the averaged momentum equation which results in P p b eing the eective pressure mean for the averaged equations The quantity P tends to vary more smo othly than mean p which aids the numerical solution of these equations For turbulentows with a single inhomogeneous direction the turbulent pressure can b e directly related to the Reynolds stresses In this limit Eq a b ecomes R where x is the direction of inhomogeneity This indicates that R for these typ es of ows Note that R is p ositive semidenite so is always greater than or equal to zero in this situation Positive is consistent with the picture of turbulence as a collection of random vortices with lower pressure cores emb edded in the mean ow It is not clear what the conditions for a negative turbulent pressure would b e if this condition is indeed p ossible B Perot P Moin Turbulent vorticity To understand the role of it is instructive to lo ok again at turbulentows that have a single inhomogeneous direction Under this restriction Eq b b ecomes R where x is the direction of inhomogeneityIf go es to zero i ik k when there is no turbulence then R or R and i ik k R These are the o diagonal or shear stress comp onents of the Reynolds stress tensor For twodimensional mean ows with two inhomogeneous ow directions only the third comp onentof is nonzero and Eq b b ecomes R R R R Since is resp onsible for vorticity generation it is appropriate that it b e aligned with the vorticityintwodimensionalows As a rst level of approximation it is not unreasonable to think of as representing the average vorticity of a collection of random vortices making up the turbulence and therefore will b e referred to as the turbulentvorticity For twodimensional ows with a single inhomogeneous direction R Note how the comp onents of reect the dimensionality of the problem while the mathematical expressions for these comp onents reects the degree of inhomogeneity Relationship with the eddy viscosity hypothesis The linear eddy viscosityhyp othesis for incompressible ows takes the form T R ru ru k I T where is the eddy viscosity I is the identity matrix and k is one half the trace T of the Reynolds stress tensor Taking the divergence of Eq and rearranging terms gives f r R r k u r r r uu rr T T T If the eddy viscosityvaries relatively slowly as is usually the case then the very last term involving the second derivative of the eddy viscosity will b e small and can b e neglected Under these circumstances the linear eddy viscosity mo del is equivalent to the following mo del k u r a T ru b T So to a rst approximation the turbulentvorticity should b e roughly equal to the mean vorticity times a p ositive eddy viscosity and the turbulent pressure should b e roughly equal to two thirds of the turbulent kinetic energy These results are entirely consistent with the ndings of the previous subsections A new approa ch to turbulencemodeling PHI PSI Figure Contours of turbulent pressure and negative turbulentvorticity for the separating b oundary layer of Na Moin Computational results Equations a and b relating the turbulent pressure and turbulentvorticity to the Reynolds stresses were used to calculate and from DNS data for two relatively complex twodimensional turbulentows a separating b oundary layer Na Moin and owover a backward facing step Le Moin The purp ose was to assess the b ehavior of these turbulence quantities in practical tur bulent situations and to provide a database of these quantities