ON THE APPROXIMATE AND NUMERICAL SOLUTION OF ORR-SOMMERFELD PROBLEMS

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Abstract
Two computer-oriented schemes for solution of problems in hydrodynamic stability theory are outlined. The first is a variational approach, which allows the eigenvalue problem to be reduced to an algebraic problem of matrix eigenvalue determination. Choosing relatively simple families of approximating functions, surprisingly accurate results can be obtained using only a few terms. Moreover, the matrix representation allows a portion of the eigenvalue spectrum to be found. The second scheme involves numerical integration, which is inherently difficult because of the high singularity of the Orr-Sommerfeld equation at large Reynolds number. Kaplan has suggested a method for extraction of rapidly growing solutions, and this idea has been used in a variety of recent calculations with remarkable success. The integrations are repeated with successively improved eigenvalues, starting from an initial guess. Experience has shown that the initial guess must be relatively good, and a few-term variational approximation provides a speedy means for selecting the initial value. Together the variational and integration scheme provide a powerful package for solution of linearized stability problems.