Eigenvalues of the Squire equation for laminar and developed turbulent boundary layers
DOI: 10.34759/trd-2020-112-5
Аuthors
Moscow Institute of Physics and Technology (National Research University), 9, Institutskiy per., Dolgoprudny, Moscow region, 141701, Russia
e-mail: selim.rs@phystech.edu
Abstract
The stability of the eigenvalue problem for two-dimensional laminar and turbulent external flow over a flat plate was numerically studied using the theory of time linear stability. That is, the classical as well as the efficient approach are considered in detail for the eigenvalue problem: two different methods are studied for deducing the spectrum of eigenvalues, namely the finite difference method and the collocation method based on basic functions. The first approach of the physical model discretizing leads to algebraic equations with large matrices that are difficult to solve efficiently, while the second one creates matrices that are usually complete and have a large number of conditions. This problem is being discussed here in the Appendix to the Squire equation, which describes laminar and turbulent boundary layers. The average velocity profile of laminar boundary layers is obtained numerically. The dispersion ratio as a function of the wave number α and other flow parameters for the problem (such as, the Reynolds number) is being defined for two different velocity profiles. The algorithm is realized in Mathematica, and the calculated eigenvalues are being compared between the two different methods.
The linear stability of a small class of engineering problems can be studied by solving the Orr-Sommerfeld equation. The most famous examples of this are the Blasius Boundary layer and the plane of the Poiseuille flow. While the plane flow of the Poiseuille is strictly parallel, in the first case, an irrational argument relative to the parallel mean flow must be called in, in such a way that the system of stability equations, obtained by substituting small wave-like perturbations in the Navier-Stokes equations and linearization of the Blasius profile, will be reduced to the Orr-Sommerfeld equation. A numerical method for solving the Orr-Sommerfeld spectral equation in a two-dimensional boundary layer (β = 0) was studied by the Chebyshev collocation method for laminar and boundary layers. To obtain the spectrum of eigenvalues of the Squire equation for the background field of a developed turbulent boundary layer, a collocation method (pseudospectral) based on Chebyshev polynomials was used. The technique was debugged on the profile of Blasius and Mucker. It is obvious that an increase in the number of Chebyshev polynomials has a significant impact on the accuracy of determining eigenvalues for the Blasius and Miskeg profiles. The impact of Chebyshev polynomial degrees on the accuracy of determining the real and imaginary parts of the eigenvalues of the first mode for laminar and turbulent boundary conditions was considered.
Keywords:
incompressible viscous liquid, turbulent boundary layer, Squire's equation, collocation method, Chebyshev polynomialsReferences
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