Eddy viscosity

It follows a mixing hypothesis, corresponding to molecular transport in ideal gases

(10.2)

where is the characteristic velocity scale of turbulence. The length scale quantifies the turbulent transport or mixing process and is traditionally called the mixing length. The symbol C is the model constant. The coefficient C should be expected to depend on the flow but the ambition is to develop general models in which C has a single value.

The simplest choice of a model for the eddy viscosity is to set = constant. This approach works well for self similar jets and wakes. These flows are statistically two dimensional and fully developed. This entails turbulence to be both in internal equilibrium and in equilibrium with the mean flow. A more general approach for self-similar jets and wakes to set and where is a measure of transverse variation of the mean streamwise velocity and is the breadth of the jet and wake.

The model for eddy viscosity can be completed by modeling the velocity scale as

(10.3)

This model for eddy viscosity and Reynolds stress is said to be “zero-equation” model. It does not require solution of a transport equation for the property of turbulence.

Generally, the characteristic velocity scale of the turbulence is estimated by

where

(10.4)

is the kinetic energy of turbulence. The model for eddy viscosity and Reynolds stress then requires solution of k -equation. The mixing length can be modeled by . The resulting k- l model is called “one equation” model of turbulence since it is based on a single transport equation, k -equation.

In complex flows it is not possible to identify a single unique or at every point in the flow. Many people use a modeled equation which also defines the length scale. This approach is known as “two-equation” model. The two-equation model is also known as k − model, where is the mean dissipation of the kinetic energy of turbulence. In this model the turbulent viscosity is estimated as

(10.5)

This implies,

(10.6)

Somewhat hand-waving, based on the eddy-viscosity model, with and , one can write (for fully developed turbulence)

(10.7)

where is the friction velocity, and is von Karman constant, = 0.4. The log profile agrees with the measurement in the zone.

(10.8)