Turbulence Modeling

At high free stream speeds external flows are likely to go through a transition from laminar to turbulence on the airfoil surface close to the leading edge. Depending on the value of the Reynolds number most of the flow on the airfoil becomes turbulent. The Reynolds decomposition technique applied to the Navier-Stokes equations results in new unknowns of the flow field called Reynolds stresses. These new unknowns introduce more unknowns than the existing equations which is called the closure problem of turbulence. In order to close the problem, the Reynolds stresses are empirically modeled in terms of the velocity gradients. All these models aim at finding the suitable value of turbulence viscosity qT applicable for different flow cases. The empirical turbulence models are in general based on the wind tunnel tests and some numerical verification. The simplest models of turbulence are the algebraic models. More complex models are based on differ­ential equations. Although so many models have been introduced, there has not been a satisfactory model developed to reflect the main characteristics of a tur­bulent flow. Now, we present the well known Baldwin-Lomax model which is used for the numerical solution of attached or separated, incompressible or com­pressible flows of aerodynamics. This model is a simple algebraic model which assumes the turbulent region to be composed of two different layers. Accordingly the turbulence viscosity reads

(lT)i; f°rz > Zc

(lT )o; forZ<Zc

Here, z is the normal distance to the surface, zc is the shortest distance where inner and outer viscosity values are equal. The inner viscosity value in terms of the mixing length I and the vorticity ю reads as

(lT )c = ql2 M Re and I = kz[1 — exp(—z+/A+)] (2-77a, b)

Here, к = 0.41 is the von Karman constant, A+ = 26 damping coefficient and z+ = zfXRe – The outer viscosity, on the other hand

(lT)dll KCcpFwFkl(z) ; Fw zmaksFmaks (2.78a, b)

Here, K = 0.0168 is the Clauser constant and Ccp = 1.6. Fmaks maximum of F(z) where zmaks is the z value at which Fmaks is found. For this purpose,

(2.79a, b)

Here, Ckl = 0.3 (Baldwin and Lomax 1978).

The research on turbulence models are of interest to many branches of fluid mechanics. The Baldwin-Lomax model is implemented for the aerodynamic applications of attached or separated flows considered here. More complex models based on the differential equation solutions are utilized even in commercial soft­wares of CFD together with the necessary documentations. Detailed information, scientific basis and their application areas for different turbulent models are pro­vided by Wilcox (1998).