METHOD OF LINES FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN THE STREAM-FUNCTION FORMULATION

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1 TASKQUARTERLY9No1,17 35 METHOD OF LINES FOR THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN THE STREAM-FUNCTION FORMULATION ZBIGNIEW KOSMA Institute of Applied Mechanics, Radom University of Technology, Krasickiego 54, Radom, Poland (Received 3 October 2003; revised manuscript received 29 July 2004) Abstract: The aim of this paper is to simulate the laminar motion of viscous incompressible fluid and the transition between the laminar and the turbulent state in simply connected domains. The developed numerical algorithms are based on the solution of an initial-boundary value problem for the full incompressible Navier-Stokes equations, written in the form of a fourth-order equation for the stream function. The spatial derivatives and the boundary conditions are discretized on uniform grids by means of sixth-order compact schemes together with fourth-order finite-difference formulas, while the continuity of the time variable is preserved. The resulting system of ordinary differential equations has been integrated using the backward-differentiation predictor-corrector method. The efficiency of the numerical algorithms is demonstrated by solving two problems of viscous liquid plane flows in a square driven cavity and a backward-facing step. Calculations for the cavity flow configurationhavebeenobtainedforreynoldsnumbersrangingfromre=100tore=30000on uniform50 50and grids.Calculationsforthebackward-facingstephavebeenmadefor Re 3000withchannellengths,L,withintherange10 30,on30L 30uniformgrids.Thecomputed stream-function contours and velocity fields have been compared with numerical results reported in the literature. Keywords: Navier-Stokes equation, stream-function formulation, method of lines, compact schemes, driven cavity problem, backward-facing step flow 1. Navier-Stokes equations in the stream-function form The problem of determination of the unsteady plane motion of viscous incompressible fluid can be formulated mathematically as an initial-boundary value problem for the fourth-order equation which governs the stream-function distribution[1, 2]: 2 ψ t + 2 ψ x ψ y 2 ψ ψ y x = 1 Re 4 ψ, (1) tq109b-e/17 11 V 2005 BOP s.c.,

2 18 Z. Kosma where Re denotes the Reynolds number, and 2 is the Laplace operator. The governing Equation(1) was transformed into a dimensionless form with appropriate normalizations. The simplicity of formulation of Equation(1) lies in the fact that the continuity equation is satisfied automatically, and then the boundary conditions for the streamfunction in simply connected domains are explicitly specified by given velocity distributions. A disadvantage of this formulation is the high order of Equation(1). The velocity components are defined as: u= ψ y, v= ψ x, (2) andvorticity,ω= v/ x u/ y,satisfiestheequationω= 2 ψ. 2. Numerical approach Approximations of all the derivatives with respect to spatial independent variables x, y occuring in Equation(1) have been performed using compact finite difference schemes[3], defined as generalizations of the classical Padé schemes. The results of the theory of these schemes, applied to an auxiliary discrete function,ϕ k,ofarealvariableξ k : ϕ k =ϕ(ξ k ), ξ k =ξ 0 +kh (k=0,1,...,n), (3) yield the following sixth-order tridiagonal approximations of the first derivatives: 1 3 ϕ k 1+ϕ k+ 1 3 ϕ k+1= 14 9 ϕ k+1 ϕ k ϕ k+2 ϕ k 2, 2h 9 4h (k=2,3,...,n 2), (4) and a similar three-point formula for the second derivatives: 2 11 ϕ k 1+ϕ k ϕ k+1= ϕ k+1 2ϕ k +ϕ k 1 h ϕ k+2 2ϕ k +ϕ k h 2, (k=2,3,...,n 2). (5) In practice, the non-periodic boundary formulation for the first derivatives is given by the following third- and fourth-order relations: ϕ 0+2ϕ 1= 1 ( 5 h 2 ϕ 0+2ϕ ) 2 ϕ 2 (k=0), 1 4 ϕ 0+ϕ ϕ 2= 3 ) (ϕ 2 ϕ 0 (k=1), 4h 1 4 ϕ N 2+ϕ N ϕ N= 3 ) (ϕ N ϕ N 2 (k=n 1), 4h 2ϕ N 1+ϕ N= 1 ( 1 h 2 ϕ N 2 2ϕ N ) 2 ϕ N (k=n), (6) tq109b-e/18 11 V 2005 BOP s.c.,

3 Method of Lines for the Incompressible Navier-Stokes Equations while the non-periodic boundary formulation for the second derivatives is taken in the following form: ϕ 0+11ϕ 1= 1 ( ) h 2 13ϕ 0 27ϕ 1 +15ϕ 2 ϕ 3 (k=0), 1 10 ϕ 0+ϕ ϕ 2= 6 ( ) 5h 2 ϕ 2 2ϕ 1 +ϕ 0 (k=1), 1 10 ϕ N 2+ϕ N ϕ N= 5 ( ) (7) 6h 2 ϕ N 2 2ϕ N 1 +ϕ N (k=n 1), 11ϕ N 1+ϕ N= 1 ( ) h 2 ϕ N 3 +15ϕ N 2 27ϕ N 1 +13ϕ N (k=n). Asetofformulasfordeterminingthespatialderivativesofthe 2 ψfunction canbeobtaineddirectlyfromequations(4) (7)byreplacingϕ k with( 2 ψ) k. In order to solve the initial-boundary value problem for Equation(1), we cover therelevantquadraticorrectangulardomaininthex yplanewithagridsystem defined by: x=x 0 +kh (k=0,1,...,k), y=y 0 +lh (l=0,1,...,l), wherehisthesamespacingofthegridinthexandydirections. Usingcentraldifferences[4],theLaplacianoperator 2 inthenon-stationary term 2 ψ/ tcanbereplacedwithitsfinitedifferenceapproximations: ( 2 ψ ) i,j = 1 12h 2 ( ψ i 2,j ψ i+2,j ψ i,j 2 ψ i,j ψ i 1,j +16ψ i+1,j +16ψ i,j 1 +16ψ i,j+1 60ψ i,j )+O(h 4 ), at interior grid points i=2,3,...,k 2, j=2,3,...,l 2. Systems of ordinary differential equations can thus be obtained in these nodes of the assumed uniform computational grid(8): (8) (9) a i 2,j ψ i 2,j +a i+2,j ψ i+2,j +a i,j 2 ψ i,j 2 +a i,j+2 ψ i,j+2 + +a i 1,j ψ i 1,j +a i+1,j ψ i+1,j +a i,j 1 ψ i,j 1 +a i,j+1 ψ i,j+1 +a i,j ψ i,j = f(ψ k,l ), (10) where the dots indicate first derivatives with respect to time, being the only continuous independentvariable,andcoefficientsa i,j areconstant.theright-handside, f(ψ k,l ), contains approximated values of the discretized convective and diffusive terms of Equation(1),whileindicesk,laremeshpoints(8). Thelackingequationsalongthei=1,i=K 1,j=1andj=L 1linescanbe obtained from boundary conditions involving a constant and specified stream-function and its normal derivatives on the boundaries. Based on the following finite-differences expressions[4]: ϕ 0= 1 ) ( 25ϕ 0 +48ϕ 1 36ϕ 2 +16ϕ 3 3ϕ 4 +O(h 4 ), 12h ϕ N= 1 ) (3ϕ N 4 16ϕ N 3 +36ϕ N 2 48ϕ N 1 +25ϕ N +O(h 4 ), 12h (11) tq109b-e/19 11 V 2005 BOP s.c.,

4 20 Z. Kosma we can find additional relationships: ψ 1,j = 3 4 ψ 2,j 1 3 ψ 3,j ψ 4,j, ψ K 1,j = 3 4 ψ K 2,j 1 3 ψ K 3,j ψ K 4,j, ψ i,1 = 3 4 ψ i,2 1 3 ψ i, ψ i,4, ψ i,l 1 = 3 4 ψ i,l ψ i,l ψ i,l 4. Hence we can obtain fourth-order accurate difference approximations to ( 2 ψ ) i,j withmeshpointsvaluesofi=2,3,k 3,K 2,j=2,3,L 3,L 2, and the corresponding Equations(10) at these points, for example: 32 3 ψ 3, ψ 2,3 36 ψ 2,2 = f(ψ k,l ), 61 4 ψ 2, ψ 4,2 ψ 5, ( 3 ψ 3, ) ψ 3,2 = f(ψ k,l ), ψ 2, ψ 4,3 ψ 5, ψ 3, ψ 3,4 ψ 3, ψ 3,3 = f(ψ k,l ). The modified system of linear algebraic equations(10) with coefficients determined from Equations(9) and(13) for the time derivatives can be easily solved by means of the over-relaxation Gauss-Seidel method[4] and leads to a system of ordinary differential equations in the Cauchy form(a method of lines approach): (12) (13) ψ i,j =f(ψ k,l ). (14) The obtained initial value problem for the system of ordinary differential equations(14) is integrated using the two-step backward-differentiation predictorcorrector method[5, 6]. Considering an initial value problem for an unknown function ϕ(t): for this particular case becomes: predictor corrector dϕ =f(t,ϕ), ϕ=ϕ(t), dt ϕ(0)=ϕ 0, t [0,T], (15) ϕ (0) n+1 =ϕ n 1+2 tf(t n,ϕ n ), (16) [ ] ϕ (q+1) n+1 =1 (4ϕ n ϕ n 1 )+2 tf(t n+1,ϕ (q) n+1 3 ) (q=0,1,...). (17) BDF-methods are currently in common use for stiff equations. The right-hand sides in Equations(16) and(17) are calculated only once at each step of integration, which allows one to decrease the calculation time. The 2-step BDF-formula(17) is stable in all the left conformal half-planes of the corresponding characteristic equation. Another numerical scheme for incompressible viscous flow, formulated as Equation(1) for the stream-function, was proposed in[7]. In this paper the biharmonic tq109b-e/20 11 V 2005 BOP s.c.,

5 Method of Lines for the Incompressible Navier-Stokes Equations equation is discretized with a compact scheme and the advection of vorticity is implemented with a high-resolution central scheme. 3. The driven cavity problem 3.1. Computational domain and boundary conditions Letusconsiderasthefirstexampleofthedevelopedmethodthedrivencavity flowshowninfigure1,containingwaterincontactwithasteadyairflowoverits surface. In this case, the stream function, ψ, has to fulfil the following boundary conditions: ψ(0,y)=ψ(1,y)=ψ(x,0)=ψ(x,1)=0, ψ x (0,y)=ψ x (1,y)=ψ y (x,0)=0, ψ y (x,1)=1. (18) Figure 1. The driven cavity problem: geometry and boundary conditions Cavity flows have been frequently employed to test the accuracy of Navier- Stokes solvers[7 46]. Numerical simulation of viscous incompressible fluid motion in a square cavity is not only important technologically, it is also of great scientific interest as it displays most fluid mechanical phenomena in a simple geometrical setting. Multiple regions of re-circulation, non-uniqueness, transition and turbulence occurnaturallyandcanbestudiedinthesamedomain Numerical results Thequadraticdomaininthex-yplane(Figure1)wascoveredbyagridsystem, N N,definedbyx=ih,y=jh(i,j=0,1,...,N),whereh=1/N. Severalpreliminarytestshadbeendone[47 49]on50 50and grids accordingtothealgorithmwithsecond-andfourth-orderapproximationsof 2 ψ in the time-dependent term in connection with fourth-order approximations of the spatial derivatives derived from the theory of cubic spline functions. The proposed method seemed to be quite promising as an incompressible Navier-Stokes solver for laminar as well as turbulent flows, taking into account that the transition from laminar toturbulentflowsoccursaboutre 8000[38,41]. tq109b-e/21 11 V 2005 BOP s.c.,

6 22 Z. Kosma The computed streamlines on the and grids obtained according tothepresentalgorithmforreynoldsnumbersrangingfromre=100tore=30000 are presented in Figures 2 6. Converged solutions for Re = 100 with the assumption of ψ 0 =0wereobtainedandthenusedastheinitialconditionsforthecaseofRe=400, andsoforth.thetimestepsweretakenas t=10 3 forthe50 50gridand t=10 4 forthe grid,whileiterationsforthesolutionofthesystemof Equations(10) and the backward-differentiation corrector(17) were repeated until an accuracyof[ , ]wasachieved.theprocessofintegrationwasterminated afterarrivingatthesteadystate(re 10000),definedas: max ( ) Re ψ i,j δ, δ [1 10 9, ]. (19) The effect of the spatial grid s size on the accuracy of the time-dependent solution has also been studied. It was possible to compute steady solutions for Reynoldsnumbersupto10000,buttheresultsonthe50 50gridatRe>1000 differed considerably from results reported in the literature[7 45]. Steady states werereachedafterabout timesteps.Asconfirmedin[15],anunstable solutionatre=30000wasalsofoundonthe gridandinthiscasethe computations were performed until a specified number of iterations(equal to ) was reached. Figures2 6showhowflowsinthecavitydependontheReynoldsnumber.The flow configuration is characterized by the locations of the centres and sizes of the main vortex and the secondary vortices. The primary vortex moves down to the centre of the cavity as the Reynolds number increases. It is getting stronger and its location becomes virtually invariant for Re 5000[15]. Small eddies develop in the vicinity of thetwolowercornersandintheupperrightcorner,andtheircentresalsomovevery slowly towards the cavity s centre with the increase of Re. The interaction of these eddying motions with the mean shear-driven flow eventually leads to a turbulence, as smalleddiesareunstableandthestrengthoftheprimaryeddyatthecentreofthe cavityvariesintime.allthesecondaryvorticesonthe gridforthesteady state solutions are shown in Figures 7 12, where velocity vectors are plotted with vector length proportional to the magnitude of velocity Discussion of results The driven square cavity flow is a good benchmark problem as it offers a deceptively simple model on which numerical techniques may be examined and very accurate numerical results are available for comparison. The computed streamfunction contours and distributions of velocity components, appearance of vortices in the flow field and the observed locations of primary, secondary and additional corner vortices are compatible with the numerical results reported in the literature [7 46]. Especially the stream-function contours are graphically comparable with thewell-knownfiguresobtainedbyghiaetal.[9].the gridisfoundto be good enough to capture the flow details including the tertiary vortices up to Re= Figures show the computed profiles concerning the u velocity component ontheverticalcentrelineofthecavity(x=0.5),thevvelocitycomponentalong the horizontal centreline of the cavity(y = 0.5) and their comparisons with the tq109b-e/22 11 V 2005 BOP s.c.,

7 Method of Lines for the Incompressible Navier-Stokes Equations Figure 2. Stream-function contours: Re = 100 and 400, grid correspondingdatafromghiaetal.[9].itisworthnotingthatghiaetal.used a meshatRe 3200anda meshatRe 5000.Althoughweused 50 50and meshes,theaccuracyofthesolutionsappearsalmostthesame asthoseinthereferencedpaper.forthereynoldsnumbersof7500and10000the present method yields slightly higher extremal values of the velocity components since itisdifficulttoresolvetheverythinboundarylayerwithauniformgrid,although therateofthisthinningisveryslowforre The results indicate that with an increase of the Reynolds number the effect of viscosity is confined to a thin layer close to the solid boundaries. Another effect is that the local curvature of the resulting mean velocity profiles increases. In order to obtain additional detailed local flow characteristics for high Reynolds numbers(re 10000),veryfineuniformornon-uniformmeshesshouldbeused [9,13,15,16,40,42].In[49]gridshavebeenclusterednearthewallsusingalgebraic stretching functions. 4. Backward-facing step flow 4.1. Computational domain and boundary conditions The second case considered is the motion of viscous liquid in a rectilinear twodimensional backward-facing step(flow over a sudden expansion), the flow geometry ofwhichisshowninfigure21.thedownstreamchannelwasdefinedtohaveunit height.thestepheightandtheheightoftheinletregionarethesame.thelength of the computational domain, L, was taken from the range and increased with the Reynolds number. The co-ordinate system for describing locations in the channeliscentredinthetopcorneranditsaxesareparalleltothechannelsides. Aparabolicvelocityprofileu(y)=24y(0.5 y)for0 y 0.5attheinletandno-slip conditions on solid walls are prescribed. This produces a maximum inflow velocity of u max =1.5andanaverageinflowvelocityofu ave =1.0.Attheoutlet,theimposed boundary conditions also assumed a parallel flow and depended on the flow model. A parabolic velocity profile is also used for laminar flow, while for laminar-turbulent flowthe ψ/ yderivativeviaanextrapolationformulaiscalculatedand 2 ψ/ y 2 =0 tq109b-e/23 11 V 2005 BOP s.c.,

8 24 Z. Kosma Figure 3. Stream-function contours: Re = 1000 and 3200, grid Figure 4. Stream-function contours: Re = 1000 and 3200, grid Figure 5. Stream-function contours: Re = 5000 and 7500, grid tq109b-e/24 11 V 2005 BOP s.c.,

9 Method of Lines for the Incompressible Navier-Stokes Equations Figure6.Stream-functioncontours:Re=10000and30000, grid Figure 7. Velocity distribution in the bottom left-hand corner: Re = 3200 and 5000 Figure8.Velocitydistributioninthebottomleft-handcorner:Re=7500and10000 tq109b-e/25 11 V 2005 BOP s.c.,

10 26 Z. Kosma Figure 9. Velocity distribution in the bottom right-hand corner: Re = 400 and 1000 Figure 10. Velocity distribution in the bottom right-hand corner: Re = 3200 and 5000 Figure 11. Velocity distribution in the bottom right-hand corner: Re = 7500 and tq109b-e/26 11 V 2005 BOP s.c.,

11 Method of Lines for the Incompressible Navier-Stokes Equations Figure12.Velocitydistributionintheupperrightcorner:Re=3200,5000,7500and10000 is postulated. Hence, boundary conditions for laminar flow correspond to those defined in the literature[19, 35, 49 73] and can be expressed in the following manner: ψ=0fory= 0.5,x [0,L], ψ=0forx=0,y [ 0.5,0], ψ=2y 2 (3 4y) forx=0,y [0,0.5], ψ=0.25+(3 4y 2 )y/4forx=l,y [ 0.5,0.5], ψ=0.5fory=0.5,x [0,L]. (20) Figure 21. Geometry of the backward-facing step problem Results of computations of viscous incompressible flows in a backward-facing geometry have been reported by many of authors[19, 28, 35, 49 73]. This simple configuration involves a few re-circulating flow regions(see Figure 22) and vortexshedding phenomena, experimentally studied by Armaly et al.[50] for the flow of air. Unfortunately, the three-dimensional effects in the channel become significant for Re > 400, which makes comparison of the measurements with the two-dimensional simulations less than satisfactory. However, the laminar(re < 1200), transitional (1200<Re<6600)andturbulent(Re>6600)zonesoftheflowcanbeclearly identified from the measurements. tq109b-e/27 11 V 2005 BOP s.c.,

12 28 Z. Kosma Figure 13. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=400,the50 50grid Figure 14. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=1000,the50 50grid Figure 15. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=3200,the50 50grid Figure 16. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=1000,the grid tq109b-e/28 11 V 2005 BOP s.c.,

13 Method of Lines for the Incompressible Navier-Stokes Equations Figure 17. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=3200,the grid Figure 18. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=5000,the grid Figure 19. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=7500,the grid Figure 20. Distributions of the x and y components of velocity on the centrelines ofthecavityatre=10000,the grid tq109b-e/29 11 V 2005 BOP s.c.,

14 30 Z. Kosma Figure 22. Re-circulation zones in the backward-facing step geometry 4.2. Numerical results The assumed rectangular computational domain(figure 21) on a uniform CartesianN Mgrid(8)canbediscretizedasfollows: x i =ih, y j = 0.5+jh, (21) whereh=1/mistheidenticalgridsizeinthexandydirections,andindicesi,j, (0 i N,0 j M)arerelatedtothexandydirections,respectively. By using the presented algorithm, computations were undertaken on the 30L 30grid(Equation(21))andforchannellengths,L,withintherange10 30, withtimesteps t= Re=200, t= Re=400and t= Re 600.Iterationsrangingfrom10000to30000werenecessarytoachievefor laminarflows(re 1200)withaccuracy(19)withintherange[1 10 8, ],while keeping the other calculation parameters same as in Section 3.2. Steady state solutions ofthisproblemwereobtainedforreynoldsnumbervaluesof200,400,600,800,1200 and 2000(see Figures 23 28). Calculations according to the algorithm with second-order discretizations of the 2 ψ/ ttermandcubicsplinefunctionapproximationsofthespatialderivativescan befoundin[49].thecomputationswereperformedon30l 30and40L 40grids, L=20andL=25,forReynoldsnumbersRe Comments on the results The proposed algorithms have been proved to be also applicable as an incompressible Navier-Stokes solver for the numerical simulation of laminar and transitional motion of viscous incompressible fluids over a backward-facing step. The computed separation and reattachment points are in very good agreement with the numerical results reported in the literature. For example, the lengths of the re-circulation Figure 23. Stream-function contours for Re = 200 on the grid tq109b-e/30 11 V 2005 BOP s.c.,

15 Method of Lines for the Incompressible Navier-Stokes Equations Figure 24. Stream-function contours for Re = 400 on the grid Figure 25. Stream-function contours for Re = 600 on the grid Figure 26. Stream-function contours for Re = 800 on the grid Figure 27. Stream-function contours for Re = 1200 on the grid tq109b-e/31 11 V 2005 BOP s.c.,

16 32 Z. Kosma Figure 28. Stream-function contours for Re = 2000 on the grid Figure 29. Stream-function contours for Re = 3000 on the grid, laminar inlet velocity profile regions(figure22)obtainedwiththepresentmethodforre=800arecomparedin Table 1 with those reported by other authors. Inallthepapers[50 73]computationshavebeenperformedforthe100 Re 800laminarregimeonly,sinceithasbeenestablishedthattheflowissteadyfor inlet Reynolds numbers up to Re = 800. The present calculations have shown that theproblemconvergestoasteadystatesolutionforre=1200(asalreadyconfirmed by experimental results[50] and numerical calculations[49]) and for Re = The calculations have indicated that flows over a backward-facing step become unsteady and unstable for Reynolds numbers Re > 2000(see Figure 29). Therefore, the choice of proper boundary conditions for the inlet and outlet velocity profiles arises as another important problem. The influence of inflow and outflow boundary conditions has been studied by assuming experimentally determined turbulent velocity profiles in the inlet aswellasintheoutletchannel[62];theresultsareshowninfigure30.atre=3000, for both kinds of velocity profiles, the transient flow behind the backward-facing step is composed of successive eddies generated along the lower and upper walls. tq109b-e/32 11 V 2005 BOP s.c.,

17 Method of Lines for the Incompressible Navier-Stokes Equations Table 1. Comparison of predicted separation and reattachment points for Re = 800 Authors Grid x 1 x 4 x 5 Gartling[51] Pentaris et al.[52] Pappou, Tsangaris[54] Barton[55] Barton[57] Keskar, Lyn[58] Domański, Kosma[66] Kosma[49] Kosma[49] Present calculations(figure 26) Figure 30. Stream-function contours for Re = 3000 on the grid, turbulent inlet and outlet velocity profiles 5. Concluding remarks The developed computational algorithms for the solution of incompressible flow of viscous fluids substantially improve the previous versions proposed in papers [47 49].Theauthorisnotawareofanyothersimilarmethodforthesolutionof incompressible viscous flows; only in[7] a central difference scheme for a pure streamfunction formulation of incompressible viscous flow was introduced. In their present form, the algorithms are applicable to numerical simulation of both laminar and turbulent motion as a solution of incompressible Navier-Stokes equations written in the form of a fourth-order equation for the stream function. The method of lines is adopted, the essence of which lies in discretizing all the spatial derivatives, while preserving the continuity of the time variable. The computed stream-function contours and distributions of velocity components in a wind-driven cavity and a rectilinear backward-facing step fit well the numerical results presented in numerous previously cited references. Because of their two-dimensionality, the present simulations are tq109b-e/33 11 V 2005 BOP s.c.,

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20 36 TASKQUARTERLY9No1 tq109b-e/36 11 V 2005 BOP s.c.,