Publications

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The frequencies of the bursting events associated with the streamwise coherent structures of spatially developing incompressible turbulent boundary layers were predicted using global numerical solution of the Orr-Sommerfeld and the vertical vorticity equations of hydrodynamic stability problems. The structures were modeled as wavelike disturbances associated with the turbulent mean flow. The global method developed here involves the use of second and fourth order accurate finite difference formula for the differential equations as well as the boundary conditions. An automated prediction tool, BURFIT, was developed. The predicted resonance frequencies were found to agree very well with previous results using a local shooting technique and measured data.

Galerkin methods are used in separable Hilbert spaces to construct and compute L{sup 2} [0,{pi}] solutions to large classes of differential equations. In this note a Galerkin method is used to construct series solutions of a nonhomogeneous Sturm-Liouville problem defined on [0,{pi}]. The series constructed are shown to converge to a specified du Bois-Reymond function f in L{sup 2} [0,{pi}]. It is then shown that the series solutions can be made to converge uniformly to the specified du Bois-Reymond function when averaged by the Ces{'a}ro-one summability method. Therefore, in the Ces{'a}ro-one sense, every continuous function f on [0,{pi}] is the uniform limit of solutions of nonhomogeneous Sturm-Liouville problems.

This SAND report summarizes the work completed for a Novel Project Research and Development LDRD project. In this research effort, new mathematical techniques from the theory of nonlinear generalized functions were applied to compute solutions of nonlinear hyperbolic field equations in nonconservative form. Nonconservative field equations contain products of generalized functions which are not defined in classical mathematics. Because of these products, traditional computational schemes are very difficult to apply and can produce erroneous numerical results. In the present work, existing first-order computational schemes based on results from the theory of nonlinear generalized functions were applied to simulate numerically two model problems cast in nonconservative form. From the results of these computational experiments, a higher-order Godunov scheme based on the piecewise parabolic method was proposed and tested. The numerical results obtained for the model problems are encouraging and suggest that the theory of nonlinear generalized functions provides a powerful tool for studying the complicated behavior of nonlinear hyperbolic field equations.