STEADY, UNSTEADY AND LINEAR STABILITY OF FLOW PAST AN ELLIPTIC CYLINDER S.J.D. D'ALESSIO
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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 4, Number 4, Fall 1096 STEADY, UNSTEADY AND LINEAR STABILITY OF FLOW PAST AN ELLIPTIC CYLINDER S.J.D. D'ALESSIO ABSTRACT. Diecud in thin work is the twedimensional steady state and unsteady flow of a viscous incompressible fluid past an inclined elliptic cylinder. Numerical solutions to the Navier-Stokes equations have been obtained for moderate Reynolds numbers, R, up to 100 and for various inclinations, a. It is observed that for low R the limiting unsteady problem agrees well with the steady state results while for 20 < R < 30 the unsteady flow succumbs to vortex shedding. Good agreement in the steady state results is found with existing results for R = 40. Also included is a linear stability analysis. The minimum critical Reynolds number at which the flow becomes unstable due to asymmetric disturbances has been estimated to lie in the range 35 < & < 40 and to occur for an inclination in the interval 40 < a, < 50. Similarly, for the circular cylinder & was estimated to be between loo < Re < Introduction. Considered in this paper is the twedimensional flow of a viscous incompressible fluid past an inclined elliptic cylinder. This represents a fundamental problem in fluid mechanics since it serves as a model problem for bodies of other shapes having more complex flows. The difficulties encountered in solving this problem can most certainly be expected when dealing with other shapes. The problem of flow past cylindrical bodies contains all the essential features common to situations encountered in practice and industry. These features include an unbounded flow domain, the implementation of boundary conditions on the surface and at large distances, and the different length scales present in the boundary layer and far wake. For these reasons and others this problem has received numerous theoretical, numerical and experimental investigations and hence is well understood. The present work differs from previous studies in several respects. Firstly, we are interested in both the steady state and unsteady cases and have carried out a detailed investigation into the comparisons and differences between these problems. A similar study concerned with Accepted for publication by the editors on October 15, Copyright Rocky Mountain Mathemstice Consortium
2 342 S.J.D. D'ALESSIO steady and unsteady flow past a rotating circular cylinder has been reported by Badr, Dennis and Young [5]. Also, numerical solutions to the steady Navier-Stokes equations have been obtained for Reynolds numbers up to R = 100. At the moment, published numerical solutions for Reynolds numbers up to R = 20 can be found in D'Alessio and Dennis [I] while Dennis and Young [ll] have extended this to R = 40. The main point of their paper is to make a detailed study of how separation occurs with R and inclination for the range 5 5 R Here, detailed solutions are presented for Reynolds numbers in the range 40 < R < 70 and for inclinations 0" 5 a 5 90" where a is the angle that the major axis of the ellipse makes with the oncoming uniform flow. Also, we present results for the case R = 100 and o = 45'. Lastly, the present work includes a linear stability analysis of the steady state results with respect to asymmetric disturbances. The stability of the flow past a circular cylinder due to forced disturbances is discussed by Pate:l [16] while Alfonsi and Giorgini [3] report on the nonlinear perturbation of vortex shedding from a circular cylinder. In addition, there have been several experimental studies into the stability of the wake of the circular cylinder. These include the works of Coutanceau and Bouard 18, 91, Gerrard [13], Taneda [19, 201 and Tritton [21, 221 to mention a few. Jackson [15] studied the onset of vortex shedding in the flow past circular and elliptic cylinders, triangular prisms and flat plates by means of finite-element simulations where the transition from steady to periodic flow was marked by a Hopf bifurcation. This Hopf bifurcation was located by working with an extended set of steady state equations. The paper is organized as follows. In the following section we present the governing equations in a stream function/vorticity formulation for both the steady state and unsteady cases and introduce a coordinate system which is convenient for an elliptic cylinder. Section 3 is devoted to the linear stability analysis. Then, in Section 4 we discuss numerical techniques used to solve the steady and unsteady problems as well as offer a method for solving the stability problem. In describing the numerical schemes the difficulties associated with solving the steady and unsteady equations will be pointed out and ways of overcoming these inherent difficulties will be proposed. Following this, in Section 5, we present and discuss the numerical results obtained. Lastly, we include a summary in the concluding section.
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