An implicit finite volume nodal point scheme has been developed for solving the two-dimensional compressible Navier-Stokes equations. The numerical scheme is evolved by efficiently combining the basic ideas of the implicit finite-difference scheme of Beam and Warming (1978) with those of nodal point schemes due to Hall (1985) and Ni (1982). The 2-D Navier-Stokes solver is implemented for steady, laminar/turbulent flows past airfoils by using C-type grids. Turbulence closure is achieved by employing the algebraic eddy-viscosity model of Baldwin and Lomax (1978). Results are presented for the NACA-0012 and RAE-2822 airfoil sections. Comparison of the aerodynamic coefficients with experimental results for the different test cases presented here establishes the validity and efficiency of the method

Dutta, Vimala

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National Aerospace Laboratories Institutional Repository

Ail implicit finite volume nodal point scheme for the solutionof two-dimensional compressible Navier-Stokes equationsVimala DuttaComputational and Theoretical Fluid Dynamics DivisionNational Aerospace Laboratories, Bangalore 560 017, INDIAAbstract: An implicit finite volume nodal point scheme has been developed for solv-ing the two-dimensional compressible Navier-Stokes equations. The numerical schemeis evolved by efficiently combining the basic ideas of the implicit finite-differencescheme of Beam and Warming (1978) with those of nodal point schemes due to Hall(1985) and Ni (1982). The 2-D Navier-Stokes solver is implemented for steady, lami-nar/turbulent flows past airfoils by using C-type grids. Turbulence closure is achievedby employing the algebraic eddy-viscosity model of Baldwin and Lomax (1978). Re-sults are presented for the NACA-0012 and RAE-2822 airfoil sections. Comparisonof the aerodynamic coefficients with experimental results for the different test casespresented here establishes the validity and efficiency of the method.1. IntroductionDue to rapid advances achieved in numerical methods as well as computer technology,there has been considerable progress in the development of Navier-Stokes solvers in thepast decade. Several finite-difference and finite-volume time stepping schemes have beensucessfully applied to a variety of two- and three-dimensional fluid flow problems.The main advantage of the finite-volume method is the capability of handling arbitrary.geometries through direct discretisation in the physical space. This approach has theflexibility of modifying the arrangement of control volumes and the evaluation of fluxesthrough the control surfaces. In the cell-centred finite-volume scheme, the dependentvariables are associated with the centre of the cell in the computational mesh. In cell-vertex or nodal point schemes, the flow quantities are specified at cell vertices rather thanat cell centres or as cell averages.With a view to implement the cell-vertex finite-volume approach to implicit schemes,a new implicit finite-volume nodal point scheme for the solution of 2-D Navier-Stokesequations is presented here. The numerical scheme has been developed by combiningthe basic ideas of nodal point schemes due to Hall (1985) and Ni (1982) with those ofimplicit finite-difference scheme of Beam and Warming (1978). Since implicit schemes havesignificantly larger stability bounds than explicit methods, fewer time steps are needed tocompute unsteady flow over a given interval of time or to converge to a steady state,2. Finite volume formulationIn order to facilitate a finite-volume formulation, the two-dimensional Navier-Stokes equa-tions for unsteady compressible flow are written in integral form after applying Euler'simplicit time-differencing formula. The computational domain is partitioned into a finiteVimala DuttThe integral conservative equations are applied to each control volume formed by joininjthe centres of the neighbouring cells of a nodal point.The line integrals occurring in the resulting equations are evaluated by summing u\the contributions due to the flux terms over the four edges of the computational cell. Th<terms containing inviscid flux vectors are calculated by using the flow variables at the fou:neighbouring points. The derivatives in viscous flux terms are evaluated by using Taylor':series expansion around the location at which the gradients are needed.Second- and fourth-order dissipation terms are added to ensure convergence and tcsuppress oscillations near shock waves. The turbulence model of Baldwin and Loma>(1978) is employed for the computation of eddy viscosity in turbulent Hows.3, Compressible viscous flow around airfoilsThe implicit finite-volume nodal point scheme thus developed is applied here for steadycompressible viscous flow around airfoils. The computational domain considered for theflow past an airfoil is bounded by the airfoil surface, an outer far field boundary and awake cut extending from the trailing edge of the airfoil. C-type grids are generated usingan algebraic grid generation technique (Jain 1983). At the body .surface, no-slip conditionfor velocity and adiabatic wall condition for temperature are imposed. Along the branchcut in the wake the conservative variables are obtained by averaging their values on eitheraide of the cut. At the outer boundary, viscous effects are neglected and non-reflectingfarfield boundary conditions are constructed using the theory of dmraeterislics for locallyone-dimensional flow normal to the boundary.4. ResultsComputations have been carried out for steady laminar/turbulent flows past the NACA-0012 and RAE-2822 airfoil sections.Figure 1 shows the Cp distribution, convergence history, streamlines, pressure andMach contours for laminar subsonic flow past NACA 0012 airfoil at M,x, = 0.8, a = 10°,Re = 500 and CFL=50. The lift, drag and moment coefficients compare well with thoseobtained by other methods (Miiller 1986, Miiller and Rizzi 1986, Chakrabartty 1987).Since it is an implicit method it has been possible to achieve fast convergence. Thestreamlines show a separation region with separation and reattacbment [joints at x/c =0.39, x/c = 0.98 respectively, which match well with those (x/c = 0.37, x/c = 0.97)predicted by Miiller (1986). The pressure and Mach contours are also very similar tothose obtained by Miiller (1986). Figure 2 presents the surface pressure distributions forhigh Reynolds number flows over the same airfoil. Good agreement with the experimentalresults (Thibert et d. 1979) is observed for the subsonic Mach number case with A/TO = 0.5,a = 1.77% £eoo = 2.91 x 106. In the transonic flow case with M^ = 0.799, a = 2.26°,He^ - 9x10 , the shock location is predicted farther downstream compared to experiment(Harris 1981)._ Results for transonic turbulent flow around RAE-2822 airfoil section are presented in*igs. d and 4 Pressure distribution, shock location, upper surface skin friction distributionanl ^ I**6 Dynamic coefficients are well predicted in the first test case, viz., MTO = 0.73,!L~H f n 1 * 1 1 ? ™ ? ' i , tllough no separation is reported in the experimentalresults (Cook et al 1979), the skin friction distribution shows a small separation nearJhe Sfrf f% I thie se?S? ^  cfe' x '-c '> M*> = °'75> « = 2.81% Re'** 6.2 x 106 ,lorn th^ ,vn ' sk°1d\alld /o6 ^°W downstream of the shock wave are slightly differenthowevt XT* f (C>°^ f al< 1979)- The l'PPer skin friction distribution,nowever, indicates separation at the foot of the shock wave from x/c - 0 64 to We = 0 67in agreement with experimental r«nlf.« P,^ *t!.. i - V01? f /f : ~ ()M lo x'c - u > 0 (An implicit finite volume nodal point scheme for 2-d compressible N-S equations 53not remain separated upto the trailing edge, there is another separated flow region closeto the trailing edge.5. ConclusionsAn implicit 2-D Navier Stokes solver has been developed for steady laminar/turbulentcompressible flow past airfoils by employing a cell-vertex finite-volume approach. The2-D Navier-Stokes equations governing the flow are solved by an implicit, finite-volumenodal point scheme. The numerical algorithm is derived by combining the basic ideas ofthe implicit finite difference technique of Beam and Warming (1978) with those of nodalpoint schemes due to Hall (1985) and Ni (1982). Results presented for NACA-0012 andRAE-2822 airfoil sections show good agreement with experimental results. The implicitnature of the numerical scheme permits the use of larger time steps and hence faster rateof convergence to steady state solution is possible.AcknowledgementThe author is thankful to Prof. R. Narasimha for his keen interest and encouragementin this indigenous effort. She thanks Dr. S.S. Desai, Dr. M.D. Deshpande and Dr.P.Ramamoorthy for their support and encouragement at different stages of the present work.She is grateful to Dr.S.K. Chakrabartty for providing the grid generation program andturbulence model routine and also for allowing her to continue this work. Her heartfeltthanks are also due to her dear friend Mrs. K. Dhanalakshmi for the graphics routines inparticular, and computational assistance in general.Baldwin, B.S. and Lomax, H. (1978), "Thin layer approximation and algebraic model forseparated turbulent flows," AIAA Paper No. 78-257.Beam, R.M. and Warming, R.F. (1978), "An implicit factored scheme for the compressibleNavier-Stokes equations," AIAA «/., 16(4), pp.393-402.Chakrabartty, S.K. (1987), "Numerical solution of two-dimensional Navier-Stokes equa-tions by finite volume method," DFVLR-IB-129-87/23.Cook, P.H., McDonald, M.A., and Firmin, M.C.P. (1979) "Airfoil R.AE 2822 - pressuredistributions and boundary layer and wake measurements," AGARD AR.-138.Hall, M. G. (1985), "Cell vertex multigrid scheme for solution of Euler equations," RAE-TM-Aero-2029, Proc. Conf. on Numerical Methods for Fluid Dynamics, UK, pp.303-345.Harris, C.D. (1981), "Two dimensional aerodynamic characteristics of the NACA 0012airfoil in the Langley 8-foot transonic pressure tunnel," NASA TM-81927.Jain, R.K. (1983), "Grid generation about an airfoil by an algebraic equation method,"DFVLR-IB-129-83/40.Miiller, B. (1986), "Navier-Stokes solution of laminar transonic flow over a NACA 0012airfoil," FFA Report-140.Muller, B., and Rizzi, A. (1986), "Runge-Kutta finite volume simulation of laminar tran-sonic flow over a NACA 0012 airfoil using the Navier-Stokes equations," FFA TN-1986-60.Ni, R.H. (1982), "A multiple grid scheme for solving the Euler equations," AIAA J., 20,1565-1571.54 Vimala DuttaThibert, J.J., Grandjacques, M., and Ohman, L.H. (1979), "NACA 0012 airfoil," AGARDAR-138.CL •CDpCDfCDCM (le)Present0.41190.14120.11380.2550-0.1111Miiller0.41990.14110.12210.26320.1145Muller &Rizzi0.43270.14550.12390.26940.1199Chakrabartty0.47110.13650.14400.2805-0.1140200 400 600 800 1000°-° 0.2 0.4 0.6 0.8 1.0x/c(i) Streamlines(ii) Pressure contoursAn implicit finite volume nodal point scheme for 2-d compressible N-S equations 57-1.0--0.5-0.0-0.5-1.0-1.5CLCDCMCDfPresent0.198910.006710.004470.00535Expt.0.1950.0070.0011— Present Computationo Experiment0.0I0.2I0.4 0.6IT0.8I1.0x/cFigure 2a. Surface pressure distribution for a NACA-0012 airfoil.(M,, = 0.57 a = 1.77°, Re = 2.91 x 106, Grid = 165 x 61)-1.5 HCLCDCMCDfPresent0.373470.03617-0.018540.00514Expt.0.390.0331-0.02Present Computationo Experimentoo-1J5-1.0-030.0CLCDCMCDfPresent0.776750.01871-0.093920.00564Expt.0.8030.0168-0.099Present computationo Experimentx/c"3-H2-JI-Ir"o.o 0.2— Present computationo Experiment0.4 0.6x/c0.8Figure 3. Surface pressure and skin-friction distributions for an RAE-2822 airfoil.(Mw = 0.73, a = 2:79°, Re = 6.5 x 1Q6, Grid = 247 x 65)1.0-15 H-1.0-J-QS'-ACLCDCMCDfPresent0.766540.02708-0.1060.00541Expt.0.7430.0242-0.1065H4-13-JU1-x/cPresent computationo Experimentx/cFigure 4. Surface pressure and skin-friction distributions for an RAE-2822 airfoil.(M. = 0.75, a = 2.81°, Re = 6.2 x 106, Grid = 247 x 65)I0)S3oa.cog-ia>toCn1.0 -J 1-Present computationo Experiment0-o.o oa o.4 0,6x/cFigure 3. Surface pressure and skin-friction distributions for an RAE-2822 airfoil.(M^ = 0,73, a = 2.79°, Re = 6.5 x 10* Grid - 247 x 65)0.8sjs-15 H-1.0--05'-U05-1.0-1.50.0 0.2CLCDCMCDfPresent0.766540.02708-0.1060.00541Expt.0.7430.0242-0.1060.4I0.6\0.8I1.0X/C4-oT~lX2-1 —0-0.0I0.2— Present computationo ExperimentI0.4I0.6x/cFigure 4. Surface pressure and skin-friction distributions for an RAE-2822 airfoil,(Mw = 0.75, a = 2.81°, Re = 6.2 x 106, Grid = 247 x 65)0.8 1.0th£3c2CDboa.Qb"03S-CoCOPresent computationo Experiment1-0-0.0 0.2 0.4x/cFigure 3. Surface pressure and skin-friction distributions for an RAE-2822 airfoil.(M« = °-73/ a = 2.79°, Re = 6.5 x 1G6, Grid = 247 x 65)0.8 1.0-1.5 H-1.0--0.5-0.0-0.0CLCDCMCDfPresent0.766540.02708-0.1060.00541Expt.0.7430.0242-0.106X/C5H4-U 2-I0.0I0.2— Present computationo Experiment10.4I0.6x/cFigure 4. Surface pressure and skin-friction distributions for an RAE-2822 airfoil.(M^ = 0.75, a = 2.81 °, Re = 6.2 x 106, Grid = 247 x 65)I0.8I1.0£3O0.ens-CDCr?CO

An implicit finite volume nodal point scheme for the solution of two-dimensional compressible navier-stokes equations

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An implicit finite volume nodal point scheme for the solution of two-dimensional compressible navier-stokes equations

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