The structure of turbulent shear flow

Abstract

A theoretical investigation is made of the mixing layer between two streams. The work is divided into four sections. The first involves the solution of the mean problem of laminar and turbulent mixing. The equations of motion are written in terms of a similarity variable. An eddy viscosity hypothesis is made to describe the shear stresses. The similarity equations for both laminar and turbulent problems are solved numerically by an iterative scheme.
The second section examines the stability of the mixing layer. The Orr-Sommerfeld equation of hydrodynamic stability is solved numerically. Doth cases of spatial and temporal amplification are examined. The shear layer is shown to be unstable to both spatially and temporally growing disturbances at all Reynolds numbers. A correction is made for the divergence of the mean flow and leads to a value of critical Reynolds number of 12.3.
The third section presents a model for the turbulent mixing layer. A set of partial differential equations describing the flow are obtained. A Fourier transform technique is employed to reduce this set to an ordinary differential equation for the fluctuating flow field. The homogeneous form of this equation is solved numerically. The resulting predictions of fluctuating velocity, pressure and their correlations are compared with measured values. The agreement is good in certain cases and this serves as a guide to components of the flow governed by non-linear processes.
The final section examines the non-linear growth of the mixing layer through transition from laminar to turbulent flow. A set of integral momentum and energy equations are written in terms of a number of shape parameters of mean and fluctuating flow. The amplitude of disturbances is shown to grow rapidly and reach a limiting value. A sudden growth of the mixing layer thickness
through transition is also predicted. Comparison is made with experimental results.

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