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1、Turbulence, Heat and Mass Transfer 4 ? ? ? ? ? ? !?“$#?% martin.kuntz , robin.langtry Abstract This document describes the current formulation of the SST turbulence models, as well as a number of model enhancements. The model enhancements cover a modified near wall treatment of the equations, which
2、allows for a more flexible grid generation process and a zonal DES formulation, which reduces the problem of grid induced separation for industrial flow simulations. Results for a complete aircraft configuration with and without engine nacelles will be shown. 1. Introduction The starting point for t
3、he development of the SST 1,2 model was the need for the accurate prediction of aeronautics flows with strong adverse pressure gradients and separation. Over decades, the available turbulence models had consistently failed to compute these flows. In particular, the otherwise popular k- 3 model was n
4、ot able to capture the proper behaviour of turbulent boundary layers up to separation 4. The Johnson-King model 5 was the first formulation, which allowed the accurate prediction of separated airfoil flows. Unfortunately, the model was not easily extensible to modern three-dimensional Navier-Stokes
5、codes due to its algebraic formulation. The k- model is substantially more accurate than k- in the near wall layers, and has therefore been successful for flows with moderate adverse pressure gradients, but failes for flows with pressure induced separation 1. In addition the -equation shows a strong
6、 sensitivity to the values of in the freestream outside the boundary layer 6. The freestream sensitivity has largely prevented the -equation from replacing the -equation as the standard scale-equation in turbulence modelling, despite its superior performance in the near wall region. This was one of
7、the main motivations for the development of the zonal BSL and SST models. The zonal formulation is based on blending functions, which ensure a proper selection of the k- and k- zones without user interaction. The main additional complexity in the model formulation compared to standard models lies in
8、 the necessity to compute the distance from the wall, which is required in the blending functions. This is achieved by the solution of a Poisson equation and is therefore compatible with modern CFD codes. The SST model was originally used for aeronautics applications, but has since made its way into
9、 most industrial, commercial and many research codes. This is in agreement with the present authors experience that the need for accurate computations of flows with pressure- induced separation goes far beyond aerodynamics. The SST model has greatly benefited from the strength of the underlying turb
10、ulence models. In particular, the accurate and robust near wall formulation of the Wilcox model has substantially contributed to its industrial Turbulence, Heat and Mass Transfer 4 usefulness. As well, all the model additions developed by Wilcox for rough walls and surface mass injection etc. can be
11、 used with minor modifications 7. While the original model formulation has largely stayed unchanged from the formulation given in 1 (small modifications see bellow), there have been several areas of improvement carried out within the CFX codes. Robustness opimisation have brought the model to the sa
12、me level of convergence as the standard k- model with wall functions. An improved near- wall formulation has reduced the near wall grid resolution requirements, which has resulted in a substantial improvement for industrial heat transfer predictions 8. Finally, the zonal formulation of the model has
13、 been beneficial in the formulation of an industrial Detached Eddy Simulation (DES) model. A large number of model validation studies and applications can be found on the internet. 2. SST Model Formulation In this section, the complete formulation of the SST model is given, with the limited number o
14、f modifications highlighted. ()() () + += + i tk i k i i x k x kP x k 4 t k * (1) ()() ()() ii w i t ii i xx k F xx S x U t + + += + 1 12 21 22 Where the blending function F1 is defined by: = 4 2 2 2* 1 4500 maxmintanh yCD k , y , 5 k F k (2) with = 10 2 10, 1 2max ii kw xx k CD and y is the distanc
15、e to the nearest wall. F1 is equal to zero away from the surface (k- model), and switches over to one inside the boundary layer (k- model). The turbulent eddy viscosity is defined as follows: () 21 1 ,maxFSa ka t = (3) Where S is the invariant measure of the strain rate and F2 is a second blending function defined by: = 2 2* 2 5002 maxtanh y , 6 k F (4) A production limiter is used in the SST model to prevent the build-up of turbulence in s
