1. The Spanwise Perturbation of Two-Dimensional Boundary Layers. 2. The Turbulent Rayleigh Problem. 3. The Propagation of Free Turbulence in a Mean Shear Flow

Citation

Crow, Steven Collins (1966) 1. The Spanwise Perturbation of Two-Dimensional Boundary Layers. 2. The Turbulent Rayleigh Problem. 3. The Propagation of Free Turbulence in a Mean Shear Flow. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/75J2-GM72. https://resolver.caltech.edu/CaltechETD:etd-11172005-144841

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. 1. The Spanwise Perturbation of Two-Dimensional Boundary Layers. Large spanwise variations of boundary-layer thickness have recently been found in wind tunnels designed to maintain two-dimensional flow. Bradshaw argues that these variations are caused by minute deflections of the free-stream flow rather than an intrinsic boundary-layer instability. The effect of a small, periodic transverse flow on a flat-plate boundary layer is studied in this chapter. The transverse flow is found to produce spanwise thickness variations whose amplitude increases linearly with distance downstream. 2. The Turbulent Rayleigh Problem. Rayleigh flow is the non-steady motion of fluid above a flat plate accelerated suddenly into motion. Laminar Rayleigh flow is closely analogous to laminar boundary-layer flow but does not involve the analytical difficulty of non-linear convection. In this chapter, turbulent Rayleigh flow is studied to illuminate physical ideas used recently in boundary-layer theory. Boundary layers have nearly similar profiles for certain rates of pressure change. The Rayleigh problem is shown to have a class of exactly similar solutions. Townsend's energy balance argument for the wall layer and Clauser's constant eddy viscosity assumption for the outer layer are adapted to the Rayleigh problem to fix the relation between shear and stress. The resulting non-linear, ordinary differential equation of motion is solved exactly for constant wall stress, analogous to zero pressure gradient in the boundary-layer problem, and for zero wall stress, analogous to continuously separating flow. Finally, the boundary-layer equations are expanded in powers of the skin friction parameter [...], and the zeroth order problem is shown to be identical to the Rayleigh problem. The turbulent Rayleigh problem is not merely an analogy, but is a rational approximation to the turbulent boundary-layer problem. 3. The Propagation of Free Turbulence in a Mean Shear Flow This chapter begins with the assumption that the propagation of turbulence through a rapidly shearing flow depends primarily on random stretching of mean vorticity. The Reynolds stress [...] acting on a mean flow [...] in the x direction is computed from the linearized equations of motion. Turbulence homogeneous in x, z and concentrated near y = 0 was expected to catalyze the growth of turbulence further out by stretching mean vorticity, but [...] is found to become steady as [...]. As far as Reynolds stress is a measure of turbulent intensity, random stretching of mean vorticity alone cannot yield steadily propagating turbulence. The problem is simplified by assuming that all flow properties are independent of x. Eddy motion in the y, z plane is then independent of the x momentum it transports, and the mean speed U(y,t) is diffused passively. The equations of motion are partially linearized by neglecting convection of eddies in the y, z plane, and wave equations for [...] and U(y,t) are derived. The solutions are worthless, however, for large times. Turbulence artificially steady in the y, z plane forces the mean speed gradient steadily to zero. In a real flow the eddies disperse as fast as U diffuses. Numerical experiments are designed to find how quickly concentrated vortex columns parallel to x disperse over the y, z plane and how effectively they diffuse U. It is shown that unless a lower limit on the distance between any two vortices is imposed, computational errors can dominate the solution no matter how small a time increment is used. Vortices which approach closely must be united. Uniting vortices during the computations is justified by finding a capture cross section for two vortices interacting in a strain field. The experiments confirm the result that columnar eddies disperse as fast as they transport momentum.

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