finite element computation of unsteady viscous transonic ... - iitk.ac.in

FINITE ELEMENT COMPUTATION OF UNSTEADY VISCOUS TRANSONIC FLOWS PAST AIRFOILS S. Mittal Departmentof AerospaceEngineermg Indian .Instituteof Technology Kanpur, UP 208016, INDIA Abstract

In this article we present our results for computation of unsteady viscous transonic flows past stationary airfoils at various angles of attack. S'abilized finite element methods are employedto solve the co~pressible Navier-Stokes equations in their conservation law form. The non-linear equations resulting from the finite element discretizations are solvedusing the GeneralizedMinimal RESidual (GJvIRES) technique. Interesting flow P!lttems involving interactions betweenshock waves, boundary layers and shear layers are observedfor all the cases. It is observedthat the unsteadinessin the wake and the strength of the shocks.involved increase with an increase in the angle of attack. Keywords:

1

Finite-elements, Transonic, Viscous, Airtoil, Vortex-shedding.

Introd uction

Viscous transonic flows are associated with complex interactions between the boundary/shear layers and shock/expansion waves. In the transonic regime, the flows are quite sensitive to the freestream Reynolds and Mach numbers; the boundary/shear layer behaviour mainly depends on the Reynoldsnumber while the shock/e:Xpansionwavebehaviour .dependsmainly on the Mach number [1, 2]. A number of numerical studies have been devoted to the analysesof steady inviscid transonic flows [3, 4, .5, 6, 7, 8]. Fewer studies have been conducted.for unsteady, viscous flows [9, 10, 11, 12]. In this article, we present our results for the computation of unsteady viscous transonic flows past airfoils .at vaciousanglesof attack. We begin by reviewing the governing equations in Section 2. The equations are written in the conservation law form. Tire ~tabilized variational formulation of these equations in terms of the conservation variables is presented in Section 3. The SUPG (streamline-upwind/Petrov-Galerkin)' stabilization technique is employed to stabilize our computations against spurious numerical oscillations due to advection dominated flows. The SUPG technique was first introduced by Hughes and Brooks [13] for the advection-diffusion equation and for incompressibleflows. It was introduced in the context of inviscid compressibleflows by Tezduyar and Hughes [1, 14] and Hughes and Tezduyar [3]. In addition to the SUPG stabilizations we supplement our formulation with a shock-capturing term to provide stability of the computations in the presenceof discontinuities and large gradients in the flow. This idea, in the context of conservation variables, was demc;>nstrated by Le Beau and Tezduyar [6]. In the current work we employ the sameshock capturing operator as the one by Le Beau and Tezduyar [6] but with a modified coefficient to account for the unsteadinessin the flow. The effect of this modified term is demonstrated in [12] via a numerical example. In Section 4 we present our results for computation of flows past airfoils at various angles'.ofattack..

2

The -Governing Equations

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5

Conclusions

Results have been presented for the computation of unsteady, laminar, viscous transomc flows pit.o;t stationary airfoils at various anglesof attack. The computations involve long-time integratioll ()f the Navier-Stokesequations and demonstrate that the boundary conditions utilized are qmte ro'bust. The results show interesting flow patterns and a complex interaction between the boundary/shci~r liLycrs and shock/expansionwaves. The unsteadinessin the flow increaseswith an. increasein the aIlgle of attack. -

6

Acknow ledgment

Partial support for thjs work has come from the Aeronautical Dev(!loprncnt A~cncy, lll'g;>q.'3nH °m°.L nI 'uoJsnJ}Jp PUJ!t\SSo.tJon tj~J!t\ alliatjJS pnJA\dn lunoJsn,uuJp-J'IIl\lli \' 'S){oo.t[l .N'V pnu sJtj.'anH '11'{".L

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