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UBC Theses and Dissertations

Analysis of viscous flow stability by the finite element method Baldwin, John Frederick

Abstract

The stability of two dimensional viscous flow is studied by means of a Finite Element method. A small perturbation stream function is added to the stream function form of the Navier-Stokes equations. The linearized perturbation equation is then recast as a restricted variational principle and discretized using finite elements. The time dependance of the perturbation is taken as exp(-λt) which leads to an eigenvalue problem where the'real part of the eigenvalue indicates the stability and the imaginary part indicates the transient nature of the associated mode. The simplification to the Orr-Sommerfeld equation is made, and this problem is solved using cubic finite elements. The results are compared to those from other solution methods in the literature and the agreement is found to be excellent. The convergence is studied and as observed with other methods a very fine grid is needed to yield accurate results. The behaviour of the finite element method in this one dimensional problem is assessed to form a guideline for the two dimensional problem. The two dimensional problem is solved using 18 d.o.f. C¹ triangular elements. It is found that the Jacobian matrix used in the Newton-Raphson iteration for the steady laminar flow solution with the addition of a mass matrix forms the basis of the perturbation eigenvalue problem. Preliminary results for a two dimensional solution of the Poiseuille flow stability problem are presented. A comparison to the Orr-Sommerfeld results is drawn and it is noted that a grid much finer than allowed by present limitations is needed. The stability of recirculating flow in a square cavity is also studied. An unstable mode is observed but its critical Reynolds number varies with grid size, and it is doubtful if this corresponds to a physical instability. The continuum problem is real and unsymmetric, which suggests the possibility of complex eigenvalues. This is observed in the discrete spectrum. The complex eigenvalues are not thought to be spurious, however, since they are present at low Reynolds number (even at R = 0.001) and are seen to converge with grid refinement. This is in contrast to one dimensional advection diffusion problems where spurious complex eigenvalues in the discrete problem can lead to an excessively bumpy numerical approximation. The results indicate that the method works well but that it is very sensitive to grid refinement which is limited by the present computing methods and facilities. The main limitation is the matrix eigenvalue routine which does not take advantage of the banded nature of the finite element matrices and finds all of the eigenvalues though our interest lies only with the first few.

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