Numerical investigation of laminar-turbulent transition in a flat-plate wake.

Abstract

Laminar-turbulent transition of high-deficit flat plate wakes is investigated by direct numerical simulations using the complete Naver-Stokes equations. The simulations are based on a spatial model so that both the base flow and the disturbance flow can develop in the downstream direction. The Navier-Stokes equations are used in a vorticity-velocity form and are solved using a combination of finite-difference and spectral approximations. Fourier series are used in the spanwise direction. Second-order finite-differences are used to approximate the spatial derivatives in the streamwise and transverse directions. For the temporal discretization, a combination of ADI, Crank-Nicolson, and Adams-Bashforth methods is employed. The discretized velocity equations are solved using fast Helmholtz solvers. Code validation is accomplished by comparison of the numerical results to both linear stability theory and to experiments. Calculations of two- and/or three-dimensional sinuous mode disturbances in the wake of a flat plate are undertaken. For calculations of two-dimensional disturbances, the wake is forced at an amplitude level so that nonlinear disturbance development may be observed. In addition, the forcing amplitude is varied in order to determine its effect on the disturbance behavior. To investigate the onset of three-dimensionality, the wake is forced with a small-amplitude three-dimensional disturbance and a larger amplitude two-dimensional disturbance. The two-dimensional forcing amplitude is varied in order to determine its influence on the three-dimensional flow field. Two-dimensional disturbances are observed to grow exponentially at small amplitude levels. At higher amplitude levels, nonlinear effects become important and the disturbances saturate. The saturation of the fundamental disturbance appears to be related to the stability characteristics of the mean flow. Larger forcing amplitudes result in the earlier onset of nonlinear effects and saturation. At large amplitude levels, a Karman vortex street pattern develops. When the wake is forced with both two- and three-dimensional disturbances, strong interactions between these disturbances is observed. The saturation of the two-dimensional disturbance causes the three-dimensional disturbance to saturate. However, this is followed by a resumption of strong three-dimensional growth that may be due to a secondary instability mechanism. Larger two-dimensional forcing amplitudes accelerate the saturation of the two-dimensional and three-dimensional disturbances as well as accelerate the resumption of strong three-dimensional growth. These interactions also result in complicated distributions of vorticity and in a significant increase in the wake width.