Similarity law in the developed turbulent boundary layer

Аuthors

Ha L. V.

Moscow Institute of Physics and Technology (State University) (MIPT), 16, Gagarin St., Zhukovsky,140180, Russia

e-mail: halevan@mail.ru

Abstract

Developed turbulent boundary layer contains organized vortex structures, which determine various physical properties of the flow. Recent experimental and numerical studies confirm the existence of coherent structures. The flow in the developed turbulent boundary layer is well described by Navier-Stokes system of equations. However, solution of these nonlinear equations presents severe difficulties. One of possible alternative approaches to solve these equations describing the developed turbulent boundary layer is a waveguide model, where a hypothesis for the equality by order of tangential stresses at the side and maximum value of the least damping mode of Tollmien-Schlichting waves decrement value. This hypothesis can be expressed by the equation , which determines the change of the thickness of the boundary layer downstream. Here δ^** is pulse thickness loss; and — damping mode of spectral problem least eigenvalue for the Orr-Sommerfeld equation at the turbulent longitudinal velocity profile in the boundary layer; τ_w — the value of tangential stresses at the side; a — constant determined from the comparison with experimental data.

To do it we, firstly, solve the spectral problem at non-uniform grid for Tollmien-Schlichting equation for vertical component of speed. Here we used numerical integration to transfer differential equation to difference form of second order of accuracy. Such approximation gives us standard matrix, which is solved in «Mathematica» package. Then we will find the δ^**(x) dependence, inserting it into the formula, and obtain a. This hypothesis allows separate out the coherent structure and close the moment equations system, determining the average coherent pulsations dynamics and stochastic component spectral density. This work is aimed at the test of this hypothesis.

Keywords:

incompressible viscous fluid, turbulent boundary layer, Orr-Sommerfeld equation, the wave of the least damping mode, non-uniform grid

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