The Tollmien-Schlichting instability of laminar viscous flows

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1981
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Virginia Polytechnic Institute and State University
Abstract

In this work, an analysis of the Tollmien-Schlichting instability of laminar viscous flows is presented. For two-dimensional incompressible flow past a flat plate with porous suction strips, we use linear triple-deck, closed-form solutions for the mean flow to do a linear, parallel, spatial stability analysis. We develop a simple linear optimization scheme to determine the number, spacing, and mass-flow rate through the strips and conclude, surprisingly, that suction should be concentrated near the Branch I neutral point of the stability curve.

We then verify the results of our optimization scheme with experimental data. We find that the theory correctly predicts the experimental results and conclude that the optimization scheme is reliable enough to replace the experiment as a tool in designing efficient strips configurations in so far as two-dimensional, incompressible flows are concerned.

For axisymmetric incompressible flow past a body with porous strips, we develop linear triple deck, closed-form for the mean-flow quantities, solutions which account for upstream influence. These solutions are linear superpositions of the flow past the body without suction plus the perturbations due to the suction strips. The flow past the suctionless body is calculated using the Transition Analysis Program System (TAPS).

Using these linear triple deck, closed-form solutions we then develop a simple linear optimization scheme to determine number, spacing, and mass flow rate through the strips on an axisymmetric body. At present, we are finishing the development of and documentation for a computer code for official distribution that will interface with TAPS and suggest efficient configurations using our theory.

For compressible three-dimensional flow, we use the method of multiple scales to formulate the three-dimensional stability problem and determine the partial-differential equations governing variations of the amplitude and complex wavenumbers. We then propose a method for following one specific wave along its trajectory to ascertain the characteristics of the most unstable disturbance. Numerical results using the flow over the X-21 wing as calculated from the Kaups-Cebeci code will be published when they become available.

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