Issue1. Vol.1 (April, 2013) - IIT Mandi

More documents

Recommendations

Info

ESSENT Society for Collaborative Research and Innovation, IIT MandiAmongst the most popular methods that havebeen used quite frequently in CFD is the finitedifference method. Here the basic methodologyinvolves discretizing the problem domain bysetting up a grid and then approximating thederivatives appearing in the governing equationsby difference quotients at each grid point. Suchapproximation yields a system of algebraicequations which can then be solved with somematrix algorithm. Over the years, the secondorder central difference schemes, because oftheir easy and straight-forwardness inapplication, have for quite some time been apopular choice for discrete approximation ofpartial differential equations. Such methods areknown to yield quite good results on reasonablemeshes if the solution is well behaved. But forcertain problems such as convection dominatedflows; the solution may exhibit oscillatorybehaviour if the mesh is not sufficiently refined.However mesh refinement inevitably brings inadditional points into the system resulting in anincreased system size and consequently morememory and CPU time are required to solvesuch problems on a computer. Againdiscretisation on a non-compact stencil increasesthe band width of the coefficient matrix. Theseultimately result in increased arithmeticoperations. Thus neither a lower order accuratemethod on a fine mesh nor a higher orderaccurate one on a non-compact stencil would becomputationally cost effective. Therefore thereis a need for the development of schemes whichare higher order accurate and also compact atthe same time.Higher Order Compact (HOC) Scheme:Of late, the Higher Order Compact (HOC) finitedifference schemes came up for the computationof incompressible viscous flows to overcomethe above mentioned problems of a second orderschemes. There exist several mechanismsthrough which finite difference schemes canachieve higher-order compactness. One of themis based on Taylor series expansion. Kalita et al.[1] first developed this type of HOC scheme onrectangular non-uniform grids for the steady 2Dconvection–diffusion equation with variablecoefficients without any transformation. It wasbased on the Taylor series expansion of acontinuous function at a particular point for twodifferent step lengths and approximation of thederivatives appearing in the 2D convection–diffusion equation on a non-uniform stencil. Theoriginal PDE was then used again to replace thederivative terms appearing in the finitedifference approximations, resulting in a higherorder scheme on a compact stencil of ninepoints. Later on Ray and Kalita [2] developed atransformation-free HOC scheme for steadystateconvection diffusion problems on nonuniformpolar grids and then extended that ideafor time dependent cases [3]. The higher orderaccuracy of the HOC methods combined withthe compactness of the difference stencils yieldhighly accurate numerical solutions on relativelycoarser grids in various real life problems.Applications of HOC scheme till date:Viscous flow past a bluff body has long beenappreciated owing mainly to its theoretical andpractical implications. Vorticity is generatedclose to the surface because of the no slipboundary conditions. The coupling between theunsteady wakes of body surface with theirmotion is important in structural, offshore andthermal power engineering applications. As anapplication, this has to be taken care of whileusing a drill with some flammable fluid say inan oil well. The drill bit usually cylindrical inshape is subjected to inline, cross flow rotationsor oscillations which is the case of flow past acylinder. The cylinder motion appears to triggera distinctive vortex shedding pattern whichtogether with the wake structure gives rise toenhanced unsteadiness, pressure fluctuations,structural vibrations and noise in addition toincreased heat and mass transfer. This alsoproduces a change in the phase of vortexinduced forces on the body, and can affect thesign of mechanical energy transfer between themoving body and the flow. Attempts to findways to control or lessen the unfavourablebehaviour associated with the formation andshedding of the vortices remains to be one of themost intriguing problems in computational fluid48 ESSENT|Issue1|Vol1
ESSENT Society for Collaborative Research and Innovation, IIT Mandidynamics. Since most of the bluff bodies used inengineering and other applications are circularcylinders, they warranted this in-depthtreatment.Application of HOC scheme to RotationallyOscillating circular Cylinder:Recent works done by Kalita and Ray [3] andRay [4] have already ascertained the accuracyand reliability of the HOC scheme tocharacterize the diagnostics of flows behind astationary and a rotating cylinder. Excellentqualitative and quantitative agreements arefound in each case. Our approach is anextension of above scheme to rotationaloscillations.Fig 1(a): Vorticity contours for αm = 2.0, Sf = 0.2 and Re= 200 where dashed and solid lines represent negative andpositive vortices respectively.Fig 1(c): Phase DiagramThe oscillatory rotation of the cylinderintroduces the boundary conditions in terms ofnon-dimensional speed α ,which can be denotedby α = α m sin 2πS f t, where S f is the forcedoscillation frequency, α m is the dimensionlesspeak rotation rate and t is the non-dimensionaltime. Fig: 1(a) depicts the instantaneousvorticity contours plotted by using our schemein a fixed frame at a low Reynolds number Re =200 after few complete oscillations. Positive andnegative vortices are seen shed from both sidesof the cylinder forming a vortex street. Anotherunavoidable consequence of an object movingthrough a fluid is the parallel and perpendicularforce acting on a body specified as Drag andLift forces respectively. Sinusoidal variation oflift and drag coefficient curves in Fig: 1(b) atalmost fixed amplitude shows that vortices shedfrom both sides of the cylinder are almost ofsame size and strength. Phase diagram in Fig:1(c) also predicts the periodic nature of thevortex shedding.Fig 1(b): Lift and Drag coefficientsFig 3(a): Streamlines for Re = 500 by the present scheme.49 ESSENT|Issue1|Vol1
Page 2 and 3: ESSENT Society for Collaborative Re
Page 88: ESSENT Society for Collaborative Re

Issue1. Vol.1 (April, 2013) - IIT Mandi

Create successful ePaper yourself

Delete template?

Save as template?