Eddy viscosity transport model based on elliptic relaxation approach

T HE modeling of near-wall turbulence with the k–" model usually has two major problems: An inclusion of a distance to thewall as an explicit parameter and an improper boundary condition for the dissipation rate ". The first problem makes the model inappropriate to simulate complexflows involvingmultiple surfaces. The second problem, as pointed out by Speziale et al. [1], has resulted in the use of a variety of derived boundary conditions that are asymptotically inconsistent or numerically stiff (e.g., zero normal derivatives of dissipation, links between dissipation, and derivatives of the turbulent kinetic energy, etc.). There are various proposals for the solution of both of these problems. On one hand, a number of low-Re number k–"models have been proposedwhich use empirical functions to account for near-wall turbulence without using the explicit wall distance as a parameter. And on the other hand, a dissipation equation " is replaced with ! "=k , k=" , and t t k=" to simplify the employment of the wall boundary conditions. This has resulted in models known as k–! (Wilcox [2]), k– (Speziale et al. [1]), t–k (Peng andDavidson [3]), etc. However, an integration of governing equations up to the wall has never been accepted for practical engineering applications, bearing in mind that computational meshes usually do not satisfy a strict criteria for the position of the cells next to the wall. This criteria is necessary so that the low-Reynolds number models can successfully describe the viscous sublayer and the buffer region. And although computing power continues to increase allowing computational meshes to be finer than ever, standard industrial applicationswill be done using the wall function approach. The universal solution is to provide the wall approach which combines the integration up to the wall with wall functions. Recently the various proposals, e.g., the automatic wall treatment (Esch andMenter [4]), compoundwall treatment (Popovac and Hanjalic [5]), etc., have provided an optimum solution for any computational mesh. The v–f of Durbin [6] has become increasingly popular as empirical damping functions are removed due to the employment of an additional velocity scale v derived by using an elliptic relaxation concept. However, the original model introduces the wall boundary condition for the elliptic relaxation function f proportional to 1=y (y is a wall distance) making computations more sensitive on very nearwall cells. Recently, two groups of authors, Hanjalic et al. [7] and Laurence et al. [8] made a similar, more robust formulation of the v–fmodel. They proposed an eddy viscosity model which solves a transport equation for the velocity scale ratio v=k instead of v. Therefore, the efficiency of the elliptic relaxation concept of Durbin [6] is kept which sensitizes v to the inviscid wall blocking effect, but the more robust wall boundary condition for the f equation is introduced, this time fwall is proportional to 1=y . Furthermore, the production of turbulence kinetic energy appears in the v=k equation while a dissipation rate " is in the v equation which is much more difficult to reproduce accurately in the near-wall layer. Hanjalic et al. [7] made even further simplifications by modifying the model’s constants to reduce the v=k equation to a source-sink diffusion form (see original reference). From the numerical aspects, solving v=k is also more robust as the maximum of this variable can be fixed on the physical limit equal to 2 [as k 0:5 u v w ]. It is also clear and these authors stated that as well that the –f or the v–f models are still inferior to the Reynolds-stress model or even some nonlinear k–" models and algebraic stress models for the calculations of particular flows and flow regions (e.g., three-dimensional flows with strong secondary motion, swirl or rotation, etc.), but much more accurate than the near-wall k–" models or similar two-equation models. At the same time, these models are more robust than any Reynolds-stress models or even two-equations nonlinear k–"models or algebraic stress models. Furthermore, these models represent an ideal choice when used in conjunction with the universal wall approach as for example proposed by Popovac and Hanjalic [5] related to their compound treatment of wall boundary conditions which combines the integration up to the wall with wall functions. The work presented here further simplifies the approach of Popovac and Hanjalic [5]. This is done by introducing the eddy viscosity transport equation instead of the dissipation rate equation [see the work of Peng and Davidson [3] (PD) who derived a twoequation t–k model]. Here, the elliptic relaxation approach is adopted for the k– ~ t– –f model and consequently, the compound wall treatment can be introduced very simply without modification for different variables.