Kinetic energy of fluctuations

In order to derive the equation governing the kinetic energy of the fluctuations, it is required to multiply the time-dependent momentum equation by and is time-averaged by following the rules of time-averaging.

Subsequently subtraction of Equation (14.7) from the resulting equation yields the following equation for the turbulent energy budget. (The derivation will be carried out in one of the subsequent lectures).

Equation 15.1 states that the rate of change of turbulent kinetic energy (given on the left side) is due to the following contributions (in the same order as in the equation, on the right side):

     (a) pressure gradient work,
     (b) turbulent diffusion,
     (c) viscous diffusion,
    (d)  the production of turbulence and
    (e)  the rate of dissipation of turbulent kinetic energy.

Reconsider the production term in Equation (15.1). In cartesian coordinates, leading terms obtained on expansion are and . Since and have same sign, their product is positive and the related production term leads to growth of turbulence. In contrast to this, the order term is positive if , i.e., in expanding ducts and negative if , i.e., in ducts with contraction. As a matter of fact, the latter mechanism can destroy turbulence and result in relaminarization, a result not derivable from the eddy viscosity model.

The quantity in Equation (15.1) is the fluctuating rate of strain and is given by

(15.2)

In general, terms (a), (b) and (c) approach zero as we approach equilibrium and do not redistribute energy among the different points in a control volume of interest.