Zero-Equation Models

In this model, no additional differential equation to needed to obtain the turbulent viscosity v_t , which is defined as a function of the mean flow. The Baldwin and Lomax (1978) is one such model based on the prandtl mixing-length theory. In this model, two different expressions are given for the turbulent viscosity.

• For the inner zone ,

(26.8)

where the mixing length l and the vorticity | ω | are given by

(26.9)

(26.10)

with A + = 26, and the von Kármán constant K=0.42.

• For the outer region

(26.11)

The factor F_ω decides the length scale depending upon whether the field point lies in the bounded zone or in the wake

(26.12)

where,

(26.13)

y_max is the value of y at which F(y) achieves its maximum value. The quantity U_dif is given by

(26.14)

The Klebanoff's Intermittency Function is expressed as

(26.15)

Here y is the distance normal to the wall, y_c the inner zone (viscous sublayer) thickness, and d the boundary layer thickness. The values of α , C_cp , C_Kleb , and C_wk are 0.0168, 1.6, 0.3 and 1 respectively. In many flows of industrial importance the zero-equation model as stated above or its variants find useful application. Maji and Biswas (1999) used a simpler zero equation model in order to predict complex flow in the spiral casing of a hydraulic turbine (Figure 26.1).

Figure 26.1: Secondary flow at different cross sections of a spiral for Re = 10⁶ (a)  
θ = 0° , (b) θ = 90° , (c) θ = 180°