Prandtl's Mixing Length Theory

It proposes a way to express Turbulent Viscosity through dimensional analysis in order to solve the the Closure problem in RANS

The Kinematic Turbulent Viscosity ($v_t$) in the Boussinesq Assumption

$$v_t\propto {\mathrm{m}}^2/\mathrm{s}$$

Through dimensional analysis

$$v_t\propto U_{\text{turb }}{l}_m$$

Where,

• $U_{\text{turb }}\to$ Characteristic Velocity
• $l_m\to$ Mixing Length

Similarly fluctuating velocity can be written in terms of Mean Strain

$$u^{\prime }\sim l_m\frac{\partial \bar{u}}{\partial y}$$

Plugging into the Reynolds Stress Tensor

$$-\overline{u_i^{\prime }{u}_j^{\prime }}=l_m^2\left|\frac{\partial {\bar{u}}_j}{\partial y}\right|\frac{\partial {\bar{u}}_i}{\partial y}$$

Note: $j$ here is the vertical component and it cannot be negative

Using the Boussinesq Hypothesis

$$v_t\left(\frac{\partial {\bar{u}}_i}{\partial x_j}\right)=l_m^2\left|\frac{\partial {\bar{u}}_j}{\partial y}\right|\frac{\partial {\bar{u}}_i}{\partial y}$$ $$\boxed{v_t=l_m^2\left|\frac{\partial \bar{u}}{\partial y}\right|}$$

Thus we have an equation to calculate Eddy Viscosity

Note: The Mixing length ($l_m^2$) is selected base on the kind of flow.