Analysis of terms

13.1.4 Analysis of terms

The diffusion of energy

is due to molecular diffusion, described by the term:

turbulent diffusion, described by the term:

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(13.22) The two last terms are unclosed. Here we face with the famous problem noted

.ı jk u 0 0 i 0 Cı ij u k /p

first by Friedman and Keller (1924): Effort to derive the equations for the second order moments u 0 i 0 u k results in the necessity of determination of new

unclosed terms including third order moments u 0 u 0 u i 0 j k . Using the method proposed by Friedman and Keller in 1924 it is possible to derive equations for moments of arbitrary order. However, the equation for the m th order will contain unclosed moments of the m C 1 th order. Impossibility of obtaining of a closed system of equations for a finite number of moments, known as the Friedman-Keller problem is a direct consequence of the nonlin- earity of the Navier Stokes equations.

The also unclosed term

describes the redistribution of the energy between different tensor compo- nents u 0 i u 0 k caused by correlation between the stresses and pressure fluctua- tions.

The term

is responsible for the energy generation, i.e. the transport of the energy transfer from averaged (mean) flow to oscillating flow (fluctuations). And, finally,

is the dissipation. This unclosed term is responsible for the transformation of the turbulent kinetic energy into the inner energy of the flow. RSM model based on equations (13.14) is used to determine the Reynolds stresses from the transport equations. It is not based on the Boussinesq approach and takes the anisotropy of stresses into account. This model is the best one among RANS models.

Equations of the k – ε Model