Turbulence in a wedge: the case of the mixing layerTurbulence in a wedge: the case of the mixing layer

TL;DR

This paper investigates turbulence in a wedge-shaped mixing layer, proposing a non-local closure model for turbulent stress based on Euler equations, and explores the resulting velocity field and pressure differences.

Contribution

It introduces a non-local integral equation approach for turbulence closure in wedge geometries, extending classical theories and satisfying realizability conditions.

Findings

Derived a nonlinear integral equation for the mean velocity field.

Showed the pressure difference contributes to lift on the splitter plate.

Demonstrated the model satisfies realizability conditions for turbulent stress.

Abstract

The ultimate goal of a sound theory of turbulence in fluids is to close in a rational way the Reynolds equations, namely to express the tensor of turbulent stress as a function of the time average of the velocity field. Based on the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations, it has been recently observed that the closure problem is strongly restricted, and that it implies that the turbulent stress is a non local function in space of the average velocity field, a kind of extension of classical Boussinesq theory of turbulent viscosity. This leads to rather complex nonlinear integral equation(s) for the time averaged velocity field. This one satisfies some symmetries of the Euler equations. Such symmetries were used by Prandtl and Landau to make various predictions about the shape of the turbulent domain in simple geometries. We explore specifically the case of mixing layer for which the average velocity field only depends on the angle in the wedge behind the splitter plate. This solution yields a pressure difference between the two sides of the splitter which contributes to the lift felt by the plate. Moreover, because of the structure of the equations for the turbulent stress, one can satisfy the Cauchy-Schwarz inequalities, also called the realizability conditions, for this turbulent stress. Such realizability conditions cannot be satisfied with a simple turbulent viscosity.