Nonequilibrium scalings of turbulent wakes

The most basic and arguably most important property that any theory or model of turbulence must predict is the mean flow profile. A model or theory of turbulence which can do this for a wide range of turbulent flows on the basis of only few fundamental and robust assumptions is still lacking. However, in the case of canonical boundary-free turbulent shear flows such as turbulent jets, wakes, and mixing layers, one can predict how the mean velocity difference and the size of the mean flow profile evolve with streamwise distance on the basis of two cornerstone assumptions [1]. These two assumptions may not be sufficient for a complete mean flow profile prediction, but they do lead to some of its most important aspects. The first assumptions is self-preservation of one-point turbulence statistics and the second is the scaling of the turbulence dissipation rate (see [1–3]). The high Reynolds number scaling of the turbulence dissipation rate traditionally used is the one consistent with the Richardson-Kolmogorov equilibrium cascade. The resulting streamwise developments (power laws of streamwise distance) of the mean velocity difference and the size of the mean flow profile are in many classical textbooks (e.g., [1,2,4]) for many canonical boundary-free turbulent shear flows. Recently, however, a new high Reynolds number dissipation law has been found in decaying and in forced periodic turbulence [5] and in near-field grid-generated decaying turbulence for many different types of grids (see [6]). The region where this nonequilibrium dissipation law holds can be long, depending on the turbulence-generating grid. Before proceeding to the implications of this nonequilibrium dissipation law for mean flow profiles, we recall the reasons (see [6]) why this dissipation law is termed “nonequilibrium.” In a Richardson-Kolmogorov equilibrium cascade of turbulent kinetic energy the turbulent dissipation rate instantaneously equals the rate with which energy is fed into the cascade at the large energycontaining scales. This rate scales with the kinetic energy and size of these large scales at the instant considered and is independent of viscosity. Hence the turbulence dissipation must scale in the same way. Any different scaling of the turbulence dissipation must therefore characterize an instantaneous imbalance between dissipation and the rate with which energy is fed into the cascade at the large scales and it must therefore be a nonequilibrium scaling. This argument has a mathematical analog in terms of the generalized Karman-Howarth equation which can be found in Ref. [6] and which is based on setting the unsteady term in this equation to zero (for equilibrium). Of course, there are also spatial inhomogeneity terms in this equation which are negligible if small enough scales are considered in flows which are not extremely inhomogeneous (i.e., if inertial range scales are smaller than the length scale characterizing statistical inhomogeneity). Goto and Vassilicos [7] considered the Lin equation, which is the Fourier equivalent of the Karman-Howarth equation when the turbulence is statistically homogeneous, and ran direct numerical simulations (DNS) of periodic turbulence to confirm the above. The Lin equation holds in this context and the DNS of [7] did indeed confirm that the turbulence dissipation and the large-scale energy flux (which in their context corresponds to the rate at which energy is fed into the cascade at the large scales) scale differently. The reason for this difference lies in the cascade time-lag which is why instantaneous equilibrium is absent.