Scaling of turbulent kinetic energy and dissipation in turbulent wall-bounded flows

A new scaling is developed for the turbulent kinetic energy (TKE) and its dissipation in a turbulent wall-bounded flow. In the traditional dimensional analysis of wall-bounded turbulence, the control parameters for the near-wall region are the kinematic viscosity v and the wall shear stress, resulting in the friction velocity u(tau) as the characteristic velocity scale and the viscous length scale nu/u(tau )as the characteristic length scale. Although the mean streamwise velocity scales well with the friction velocity, the TKE, and, in particular, the TKE dissipation does not scale with the friction velocity. In the present paper, a new dimensional analysis is performed to identify a proper scaling for the TKE and its dissipation. The control parameters in the near-wall region are the kinematic viscosity and the TKE dissipation at the wall is an element of(k,w). The new inner velocity scale for the TKE budget def equation is the Kolmogorov wall velocity u(is an element of) = (def )(nu is an element of(k,w))(1/4), and the proper length scale is the Kolmogorov wall length nu/u(is an element of). The profiles of the TKE and its dissipation in the near-wall region collapse well under the new scaling, and the TKE profiles in the outer layer also scale well with the new scaling. However, the TKE peak value k(max) does not scale with the Kolmogorov wall velocity u(is an element of) or the friction velocity u(tau) but with a mixed scale. One mixed scale is u(tau)U(infinity) first proposed by DeGraaff and Eaton [J. Fluid Mech. 422, 319 (2000)], where U-infinity is the mean velocity in the free stream or at the channel centerline. A new mixed scale developed in this paper for k(max) is nu is an element of(k,w)/ u(tau)(2)and justification is provided by adding a control parameter to the new dimensional analysis.