where κ is the Karman constant and C is a constant. A large number of experiments in incompressible boundary layers proved the existence of the log-layer, with κ ~ 0.41 and C ~ 5.0 [7]. A typical turbulent boundary layer is shown in Fig. 1↑, where this layer is clearly visible. However, it also appears that the hypothesis does not hold anymore in the viscous sub-layer and buffer layer, very close to the wall (y⁺ < 30). In order to tackle this problem, Van Driest (1956) suggested the introduction of a damping function in the mixing length [9]. where A⁺₀ = 26. Despite some physical argument regarding the Reynolds stress behaviour at the wall, the function was introduced because of its good fit with experimental data. From the previous graph, it can also be seen that Prandtl’s initial definition of the mixing length does not hold in the outer layer (roughly y ⁄ δ > 0.1). In this region, Clauser (1956) suggested that the eddy viscosity should be given the following form [4]: with U_e being the external velocity, δ^* the boundary layer displacement thickness and α a closure coefficient. The last major improvement upon Prandtl’s initial model is the introduction of an intermittency function. Indeed, the flow in the outer region is not always turbulent. It fluctuates between a laminar and a turbulent state. Hence, Klebanoff (1955) suggested that the outer eddy viscosity should be multiplied by the following function [6].

The Cebeci-Smith model (1974) is the direct application of the concepts described previously [3]. The eddy viscosity is given by where y_m is the distance from the wall for which ν_Ti = ν_To. The two different eddy viscosities are computed using the following equations.

where the closure coefficients are κ = 0.40, α = 0.0168 and A⁺ = 26⎛⎝1 + y(dP ⁄ dx)/(ρu²_τ)⎞⎠^{− 1 ⁄ 2}. Notice that the constant A⁺₀ introduced by Van Driest was modified to take the effect of pressure gradient into account. The closure coefficients are typically determined on a few test cases, as shown in the example section.

There are a multitude of algebraic models. One of the most famous is probably the Baldwin-Lomax model (1978) [1]. The later resembles in many points to the Cebeci-Smith model, but some of its modeling problems have been tackled. In particular, the outer eddy viscosity is based on parameters that are sometimes not readily available, such as the displacement thickness δ^* or the external velocity U_e. The Baldwin-Lomax model modifies the formulation of the eddy viscosity to remove these parameters from the equation. The model is therefore based on local properties only. Both models provide very similar results for boundary layers, but Baldwin-Lomax turns out to be more reliable for separated flows, when δ^* becomes negative.