On the interactions between large-scale structure and fine-grained turbulence in a free shear flow. III. A numerical solution

Abstract

We present the results of a numerical computation of the interactions between the horizontally periodic monochromatic component of a large-scale coherent structure and the fine-grained turbulence in a mixing layer. The dependent variable characterizing the coherent structure is thetotalflow quantity which includes both the Reynolds-averaged (here the horizontal average) mean flow and the large-scale coherent structure, this being distinguished from the fine-grained turbulence by a conditional average linked to the periodicity of the coherent structure. The dependent variables characterizing the fine-grained turbulence are the conditionally averaged turbulent stresses, which are themselvestotalquantities comprising the Reynoldsaveraged mean stresses and the coherent-structure induced or modulated stresses. The transport equations for such conditionally averaged quantities are identical in form to the Reynolds-mean equations. Closure approximations for the Reynolds-mean stresses due to Launderet al. (1975) are directly extended to the conditionally averaged stresses. These approximations include the pressure-velocity strain redistribution which accounts for rapid distortions due to the large-scale coherent structure, a simple diffusional approximation to the triple correlations and an approximate transport equation for the rate of viscous dissipation. The large-scale structure is two dimensional with its vorticity axis perpendicular to the mean flow direction, the fine-grained turbulence being three dimensional. The initial conditions are made self-consistent by numerically solving the horizontally homogeneous, time-dependent, Reynolds-mean problem in the absence of coherent structure until the mean Reynolds stresses and viscous dissipation become self-similar as the shear layer growth rate becomes constant. The solution to the Rayleigh equation with this self-similar mean velocity profile is then obtained. The wavenumber mode which corresponds to the most-amplified case is thensuddenlyimposed, together with the Reynolds mean velocity, as the initial condition for the large-scale structure. It is argued that, since the coherent structure-induced stresses and viscous dissipation function develop in finite time, the initial conditions for the conditionally averaged turbulence quantities are precisely those for the initial self-similar Reynolds-mean quantities, consistent with the sudden imposition of a large-scale structure upon the flow. The structural details are presented in terms of the time evolution of the conditionally averaged streamfunction and vorticity contours. While the streamline patterns resemble Kelvin’s cats’ eyes, the vorticity patterns display large non-uniformities within the cats’ eyes; thus it is not possible to construct an analytical nonlinear critical layer theory for this problem consistent with the numerical results. The physical interpretation is presented, after Reynolds (horizontal) averaging, in terms of the time evolution of global energy-exchange mechanisms between the Reynolds-averaged mean flow, the large-scale coherent structure and the fine-grained turbulence. The mean flow loses energy to both components of the oscillations, the large-scale structure gains energy from the mean flow and loses energy to the fine-grained turbulence, and the fine-grained turbulence gains energy from both the mean flow and the large-scale structure and loses kinetic energy through viscous dissipation of the small eddies. The possibility that the large-scale structure transfers energy back to the mean flow (like ‘damped’ disturbances in hydrodynamic stability interpretations) is also shown. Owing to the nonlinear interaction between the mean flow and large-scale structure, through the action of the large-scale Reynolds stresses, the global energy of the large-scale structure first increases and then decays with time. Such a growth and decay reflects the inability, at some time, of the largescale to effectively transfer energy from the mean flow and to subsequently transfer energy back to the mean flow through the action of the large-scale Reynolds stresses. The simultaneous evolution of the global fine-grained turbulent energy exhibits an increase from its initial self-preservation level to a final, higher one, in response to the new mean flow condition caused by the evolution of the large-scale structure. This physical picture of the global energy evolution is consistent with previous much-simplified analyses of the temporal problem (Liu & Merkine 1976a) and approximately consistent with results for the spatial problem as well (Alper & Liu 1978). The structural details of the direct production mechanisms of the conditionally averaged horizontal and vertical contributions to the fine-grained turbulence energy from the two dimensional, large-scale structure are shown in terms of contour plots for the cats’ eyes regions; of these, the dominant production mechanisms essentially resemble the contours of the horizontal and vertical contributions to the turbulence energy. Since there is no direct production mechanism for the spanwise contribution to the turbulence energy, this contribution must arise entirely from the pressure-velocity strain redistribution mechanism obtained from the closure approximation. The structural details of this redistribution mechanism essentially resemble those of the spanwise contribution to the turbulence energy. Certain apsects of available experimental observations are interpreted in terms of the present considerations. Finally, the importance of well-controlled experiments to further study the large-scale coherent structures in free turbulent shear flows is re-emphasized.