A new low-Reynolds-number nonlinear two-equation turbulence model for complex ¯ows D.D. Apsley, M.A. Leschziner Department of Mechanical Engineering, UMIST, PO Box 88, Manchester, M60 1QD, UK Received 23 January 1997; accepted 13 August 1997 Abstract A new nonlinear, low-Reynolds-number k±e turbulence model is proposed. The stress±strain relationship is formed by successive iterative approximations to an algebraic Reynolds-stress model. Truncation of the process at the third iteration yields an explicit expression for the Reynolds stresses that is cubic in the mean velocity gradients and circumvents the singular behaviour that aicts the exact solution at large strains. Free coecients are calibrated ± as functions of y  ± by reference to direct numerical simulation (DNS) data for a channel ¯ow. By using the nonlinear stress±strain relationship, the sublayer behaviour of all turbulent stresses is reproduced. The extension to nonequilibrium conditions is achieved by sensitising the model coecients to strain and vorticity in- variants on the basis of formal relations derived from the algebraic Reynolds-stress model. The new model has been applied to a number of complex two dimensional (2-D) ¯ows, and its performance is compared to that of other linear and nonlinear eddy-vis- cosity closures. Ó 1998 Elsevier Science Inc. All rights reserved. Keywords: Turbulence modelling; Computational ¯uid dynamics; Nonlinear k±e model International Journal of Heat and Fluid Flow 19 (1998) 209±222 Notation a a ij  anisotropy tensor, Eq. (1) C coecient of linear term in nonlinear eddy-viscosity model (NLEVM) C l coecient in standard k±e model C e1 ; C e2 coecients in e equation D damping factor in e equation D=Dt @=@t U j @=@x j  derivative following the mean ¯ow f l damping factor in low Rey- nolds-number eddy-viscosity formula f P nonequilibrium factor in NLEVM coecients I identity matrix k turbulent kinetic energy l e  k 3=2 =e dissipation length P  1 2 P kk rate of production of turbulent kinetic energy P ij rate of production of u i u j q 1 ; q 2 ; q 3 coecients of quadratic terms in NLEVM R t  k 2 =me turbulent Reynolds number Re Reynolds number s s ij  dimensionless mean-strain ten- sor, Eq. (2) s dimensionless mean-strain in- variant, Eq. (5) U i U ; V ; W  mean-velocity components u i u; v; w turbulent-velocity components x i x; y ; z cartesian coordinates y n wall-normal distance y   y n k 1=2 =m dimensionless wall-normal dis- tance a; b; c coecients in algebraic Rey- nolds-stress model, Eq. (13) a; b; c values of ar; br; cr in log layer, Eq. (26) c 1 ; c 2 ; c 3 ; c 4 coecients of cubic terms in NLEVM e rate of dissipation of turbulent kinetic energy e w function with theoretical near- wall asymptotic behaviour of e m kinematic molecular viscosity m t eddy viscosity r k=e  @U i =@x j  2 q dimensionless shear parameter r k ; r e turbulent Prandtl numbers in k and e equations, respectively U ij pressure±strain correlation x x ij  dimensionless mean-vorticity tensor, Eq. (2) x dimensionless vorticity invari- ant, Eq. (5) 0142-727X/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved. PII: S 0 1 4 2 - 7 2 7 X ( 9 7 ) 1 0 0 0 7 - 8