More on Budget Equation

Equation (15.3) implies that in this situation, the rate of production of turbulent energy by Reynolds stresses equals the rate of viscous dissipation. The most obvious deduction is that the turbulence is produced only where the gradient of mean velocity is different from zero. Production is most prominent in the wakes of bluff objects and adjacent to the solid walls. Generally, wake formation takes place due to the separation from a solid body. Therefore a wall is essential, directly or indirectly, for the production of turbulence.

Townsend (1949) measured the various terms of the energy balance equation at the section in the wake of a circular cylinder. Here is given by , where x is the distance of the measurement point from the origin, a is the distance of the virtual source of wake from the from the origin and D is the diameter of the cylinder. In Figure (15.1) the contributions made by various terms in the wake along the normal direction have been shown. Near the centre, production is negligible. The production is maximum at . At , diffusion and convection are negligible and the production equals the local dissipation. This is the only point in the wake zone where Prandtl's mixing length hypothesis can be applied with a high degree of accuracy. Outside the wake, the diffusion of kinetic energy, production and dissipation are small. Convection in the axial direction takes up energy which is supplied through the equalization of pressure across the wake by the pressure-velocity correlation term .

Figure 15.1  Energy Balance in the wake of a
                                  Circular Cylinder(after Townsend, 1949)