Numerical Computation of Unsteady Compressible Flow past a Plunging Aerofoil

Transcription

1 Journal of Applied Fluid Mechanics, Vol. 5, No. 4, pp , Available online at ISSN , EISSN Numerical Computation of Unstead Compressible Flow past a Plunging Aerofoil K. S. Kumar and S. V. Sajjan Computational and Theoretical Fluid Dnamics Division, Council of Scientific and Industrial Research, National Aerospace Laboratories, Bangalore , India Corresponding Author sivakumar@nal.res.in (Received June 7, 2010; accepted October 9, 2011) ABSTRACT Unstead Renolds-averaged Navier-Stokes (RANS) computations are presented for subsonic and transonic flow past a plunging NACA 64A010 aerofoil. The Implicit RANS solver used for obtaining the time-accurate solution is based on finite volume nodal point spatial discretization scheme with dual time stepping. Results for the subsonic and transonic cases compare well with the eperimental data, thus demonstrating the capabilit of the solver to provide useful unstead pressure data for aeroelastic analsis. Kewords: Viscous flow, RANS solver, Implicit method, Heaving aerofoil, Nodal point scheme. NOMENCLATURE A, B, R, S Jacobian matrices C d drag coefficient C p surface pressure coefficient C l lift coefficient C m moment coefficient F, G inviscid flu vectors M free stream Mach number Pr Prandtl number Pr t turbulent Prandtl number Re free stream Renolds number U Vector of conserved variables U free stream velocit U n solution vector at time level n V, W viscous flu vectors c aerofoil chord e energ non-dimensional amplitude in plunge ( 0 / c ) h ij area of quadrilateral non-dimensional reduced frequenc (c / 2U ) n time level p pressure t phsical time 1. INTRODUCTION The Navier-Stokes equations constitute the basic mathematical model for numerical simulation of compressible flows. The permit the analsis of comple flow phenomena such as shock/boundar laer interaction, flow separation, wake flow, hsteresis, periodic vorte shedding and so on. Since the t * non-dimensional time ( U t / c ) u, v velocit components, Cartesian coordinates 0 amplitude of plunge or heave (t) instant plunge distance of the aerofoil non-dimensional instant plunge distance of the aerofoil boundar curve control volume surrounding the nodal point (i, j) of the curvilinear grid t real or phsical time step α o amplitude of pitching oscillation α m mean angle of attack γ ratio of specific heats λ,µ viscosit coefficients µ free stream viscosit coefficient µ t turbulent viscosit coefficient ρ densit ρ free stream densit non-dimensional angular frequenc mathematical nature of the governing equations changes from elliptic in subsonic region to hperbolic in supersonic region, compressible flow computations require sophisticated numerical techniques and enormous computational effort. Due to rapid advances achieved in numerical methods as well as computer technolog, there has been considerable progress in the development of efficient Navier-Stokes solvers in the

2 last two decades. Several finite difference and finite volume time stepping schemes have been applied successfull to a variet of two and three-dimensional fluid flow problems. Even though stead flow computations have attracted much attention, the focus has slowl shifted to unstead flow computations in recent ears [1]. Unstead flows are encountered in man aerospace applications and prediction of unstead air loads plas a vital role in aircraft and helicopter design (Mabe, 1999; McCroske 1982; McCroske, W. J. 1988). Since wind tunnel testing of unstead flow situations is difficult and epensive, computational studies of wing stall and flutter, buffeting, gust response, dnamic stall, blade-vorte interaction of helicopter rotors etc. can provide important design data. Understanding these interesting unstead flow phenomena through Navier- Stokes computations therefore presents a significant challenge for computational fluid dnamics. The Implicit Renolds-averaged Navier-Stokes Solver (IMPRANS) for simulating unstead compressible flows over aerofoils and wings has been emploed for solving the unstead RANS equations on a moving mesh in an inertial frame of reference Vimala Dutta et al. (2003) The solver is based on an implicit finite volume nodal point scheme wherein a control volume is formed b joining the centroids of the neighbouring cells around a nodal point in the computational domain. The efficienc of the solver for making time-accurate computations is enhanced b implementing an implicit dual time stepping procedure. In this approach, an equivalent pseudo stead state problem is solved at each real time step using local time stepping. The algebraic edd viscosit model due to Baldwin and Loma is used for turbulence closure Baldwin et al. (1978). This solver has been etensivel validated for computing unstead flow past pitching aerofoils and wings Sharanappa et al. (2008) helicopter rotor blades Vimala Dutta et al. 2005; Sharanappa et al. 2006) and wind turbines Dutta et al. (2007). Plunging or heaving is a vertical displacement of the aerofoil perpendicular to the free stream direction. Most of the previous computational studies that have been published in literature are on plunging aerofoils for low speed flows, of interest to MAVs (Sarkar et al. 2006; Viieru et al. 2007; Andro et al. 2009; Ashraf et al. 2007). The present work is one of onl a few to consider for unstead flow past plunging aerofoil at subsonic and transonic Mach numbers. 2. IMPRANS SOLVER The solver is based on an implicit finite volume nodal point spatial discretization scheme with dual time stepping. Inviscid flu vectors are calculated b using the flow variables at the si neighbouring points of heahedral volume. Turbulence closure is achieved through the algebraic edd viscosit model of Baldwin and Loma. 2.1 Governing Equations The two-dimensional Renolds averaged Navier-Stokes equations for a moving domain can be written in nondimensional conservative form as U F G V W (1) t where ( u t ) ( v t ) u U u( u t ) p F u( v t ) (2) G v v( u t ) v( v t ) p e, e( u t ) pu, e( v t ) pv V = V 1U,U 1 Re v( u W = W 1 +V 2 U,U v ) u( u U u v v u 2u v ) 2uu U, + W 2 0 U,U T 2 1 M 0 u v 1 u v 2 v Re T u ( u v ) v ( u v ) 2vv 2 1 M Pr Pr Here and are the Cartesian coordinates and t is the time variable; t and t are the Cartesian velocit components of the moving domain. For a fied domain, the grid speeds t and t are zero. U is the vector of conserved variables; F and G are inviscid flu vectors and V and W are viscous flu vectors. The primitive variables are densit ρ, velocit components u, v in the and directions, pressure p, temperature T and total energ e per unit volume. The non-dimensional variables used in the above equations have been obtained b using the following free stream values as reference quantities: ρ (densit), U (velocit), µ (viscosit), ρ U 2 (pressure), T (temperature), and so on. Some characteristic length such as chord c of an aerofoil is chosen as the length scale. (3) (4) M and Re are the free stream Mach number and Renolds number respectivel; γ is the ratio of specific heats and Pr is the prandtl number. In addition, the viscosit coefficients λ and µ are related b the Stokes relation 3λ 2μ 0 (5) and the Sutherland s law of viscosit is given b 3/ 2 T C1 T C 2 (6) For turbulent flows, the laminar viscosit coefficient µ is replaced b, µ+µ t and µ / Pr is replaced b (µ / Pr) + (µ t / Pr t ); the turbulent viscosit coefficient µ t and the turbulent Prandtl number Pr t are provided b a 132

3 turbulence model. Finall the sstem is closed using the perfect gas equation of state in non-dimensional form as T P (7) M 2 The Euler equations for inviscid flow are obtained from the Navier-Stokes equations b setting 1/ Re Computational Method There are essentiall two tpes of algorithms to solve unstead compressible Renolds-averaged Navier- Stokes equations, namel, eplicit and implicit. In eplicit schemes, the dependent variables at an new time-level are epressed in terms of variables at previous time steps and no matri inversion is involved. Eplicit determination of the dependent variables is not possible in implicit schemes and the difference equations at all grid points have to be solved simultaneousl. In eplicit methods, the time step imposed b numeric is much smaller than the time step that is required to resolve the phsical unsteadiness of the flow problem. Even though eplicit schemes are eas to implement, the need a ver large number of time steps. Since convergence acceleration techniques like local time stepping used for stead flows destro time accurac and unstead flow computations often need to be continued for a long time, the total CPU time requirement becomes ver large. Implicit time discretization, on the other hand, allows much larger time steps and hence can be emploed for time-accurate computations of flows of practical interest within reasonable CPU time. The computational work per time step to construct and solve the resulting matri equations is, of course, larger than in an eplicit scheme. The use of integral forms of Euler/Navier-Stokes equations leads to the finite volume approach whereas the divergence form ields the finite difference approach. The main advantage of the finite volume method is the capabilit to handle arbitrar geometr through direct discretization in the phsical space. In cell-centered finite volume schemes, the dependent variables are associated with the centre of the cell in the computational mesh. In cell-verte or nodal point schemes, the flow quantities are specified at cell vertices rather than at cell centre s. With a view to implement the cell-verte finite volume approach to implicit schemes, an implicit finite volume nodal point schemes has been developed at National Aerospace Laboratories (NAL). The numerical scheme for solving the two-dimensional Navier-Stokes equations governing the viscous compressible flow over aerofoils has been derived b using Euler s time differencing formula with nodal point discretization. Certain basic ideas from implicit finite difference scheme of Beam et al. (1978) and Steger (1978),the nodal point schemes of Ni (1982) and Hall (1985), the Runge kutta timestepping scheme of Jameson et al. (1981) and cellcentered schemes due to Hollanders et al. (1980) Hollanders et al. (1985) have been combined efficientl in the present method. Appling Euler s implicit time differencing formula n1 n n1 U 2 U U t O t t to the governing Eq. (1), we obtain n 1 n U t F V G W 0 (8) (9) Here U n = U (t) = U (n t) is the solution vector at time level n and U n = (U n+1 - U n ) is the change in U n over time step t. In order to facilitate the finite volume formulation, the above equations are written in the integral form as U n dd t 1 1 F -V n d - G -W n d (10) 0 where Ω is an two-dimensional flow domain and Γ is the boundar curve. In the nodal point finite volume approach (Ni 1982 and Hall 1985), the flow variables are associated with each mesh point of the grid and the integral conservative equations are applied to each control volume obtained b joining the centroids of the four neighbouring cells of a nodal point. Fig. 1(a) shows a tpical control volume Ω ij formed b joining the centres a, b, c, d of the four cells surrounding a nodal point (i, j) of a bod fitted curvilinear grid, i, j being the spatial indices along curvilinear coordinate directions ξ, η. Application of nodal point spatial discretization to Eq. (10) leads to the following equations for the computational cell Ω ij U n h t n 1 n 1 F V d G W d 0 (11) ij ij ij where h ij is the area of quadrilateral abcd and the integral refers to a contour integration around the boundar Γ ij of the cell Ω ij in anticlockwise direction. The flues are calculated across the four sides of the control volume abcd. Linearising the changes in flu vectors using Talor s series epansions in time and assuming locall constant transport properties, Eq. (11), can be simplified to n t n U A R n U n d B n S n U n d ij h ij ij ij t n n F V d G W d h ij ij ij Here A, B, R and S are the Jacobian matrices given b F G V W A, B, R and S U U U U 2.3 Moving Aerofoil s 1 2 (12) (13) For simulating unstead flow over moving aerofoils, the governing equations are solved in an inertial frame of reference. The flow field is discretized b emploing a C-grid which is fied to the aerofoil so that the computational grid translates and rotates rigidl with the aerofoil. For an grid point z = + i in the moving domain fied to the aerofoil which is shown in Fig. 1(b). ( z z ) e c i Constant;... z zc i ( z z c ); z z c ( i )( z zc ) (14) 133

4 The nodal points (, ) and the grid speeds ( t, t ) at an particular instant are computed from the prescribed aerofoil motion, thus eliminating the need for multiple grid generation. Along the moving bod surface, conditions for zero relative velocit and adiabatic wall are imposed. The stead solution at initial incidence is used as the starting solution. first grid spacing on the aerofoil surface is c in the direction normal to the aerofoil surface. The grid points are properl clustered near the leading and trailing edges and the wall. The close-up view of the grid is shown in Fig. 2(b) Fig. 2(b). Close-up view of the grid Fig. 1(a). A tpical control volume Fig. 1(b). Moving aerofoil configuration 3. GRID GENERATION Fig. 2(a).C- Grid around the aerofoil The structured C-grid with a grid size of (streamwise normal) is generated around the NACA 64A010 aerofoil, as shown in Fig. 2(a). Here, 167 points are distributed on the aerofoil surface and 41 points in the wake region. The outer boundar is located 30 chords awa from the aerofoil surface. The 4. RESULTS The computation is carried out for two-dimensional unstead compressible viscous flow over a plunging aerofoil at subsonic and transonic Mach numbers, where sinusoidal plunging or heaving motion of the aerofoil is given b (t) = 0 cos (t). Here 0 is the amplitude in plunge, is the frequenc of oscillation, c is the aerofoil chord length and U is the free stream velocit. When all the parameters are non-dimensionalized with respect to c and U, the time, amplitude in plunge and reduced frequenc can be defined as t* = U t / c, ha = 0 / c and k = c / 2U. The non-dimensional displacement of the input motion is given b the epression, (t*) = ha cos (2k t*). The time-accurate results are presented for flow around the aerofoil undergoing small amplitude heave motion corresponding to the eperiments of Davis et al. (1980) The input parameters used for the present computations are listed in Table 1 for the NACA 64A010 aerofoil. The unstead variation of non-dimensional plunge displacement of the aerofoil, with non-dimensional time (t * ) for five ccles of heaving oscillation is shown in Fig. 3. The first harmonic components of unstead surface pressure data have been evaluated through a Fourierseries representation of the surface pressure coefficient C p. The mean values and the Fourier coefficients corresponding to real and imaginar parts of the first harmonics are plotted in Fig. 4. The mean and fundamental frequenc pressure data show good agreement between computations and the eperiments of Davis and Malcolm Davis et al. (1980).The presence of shocks on both the surfaces of the aerofoil and their strength and location are well predicted in the transonic flow regime. 134

5 for one complete ccle at approimatel equal time intervals are shown in Fig. 7 and Fig. 8 respectivel. The Mach number contours are plotted in Fig. 9 and Fig.10 for both the cases respectivel. The Mach number contour plots and the surface pressure distribution plots show that in transonic case, there is a strong shock on the upper surface that weakens when it reaches the upstream position during upward stroke of the aerofoil similarl there is a strong shock on the lower surface that becomes weak when it reaches upstream position during downward stroke of the aerofoil. In both cases, plunging motion of the aerofoil creates asmmetr in the flow field even for smmetric aerofoils. This leads to different pressure fields on the upper and lower surfaces and increases the lift compared to the static aerofoils. Fig. 3. Unstead variation of heave amplitude with time on plunging NACA 64A010 aerofoil for subsonic and transonic cases. Fig. 5. Time histories of C l, C d and C m for subsonic (left) and transonic (right) cases. Fig. 4. Unstead surface pressure distribution for subsonic (left) and transonic (right) cases. (Eperiment Davis et al.(1980): Upper; Δ Lower Computation: Upper; Lower) The time histories of C l, C d and C m for five ccles of heaving motion are plotted in Fig. 5, for both the cases. The aerodnamic loads attain a periodic behavior after four ccles of oscillations. The unstead variation of lift and moment coefficients with heave distance is shown in Fig. 6, for both subsonic and transonic cases. The computed loops of the aerodnamic coefficients clearl demonstrate the hsteretic propert eisting between the up-stroke and down-stroke motion of the aerofoil. The unstead surface pressure coefficient distribution on plunging aerofoil for subsonic and transonic cases Fig. 6.Unstead variations of lift and moment coefficients with heave distance for subsonic (left) and transonic (right) cases 135

6 Fig. 9. Mach number contours during one complete ccle for subsonic case. Fig. 7. Unstead surface pressure distribution during one complete ccle for subsonic case. (--- Upper surface, Lower surface) Fig. 10. Mach number contours during one complete ccle for transonic case. Table 1 Unstead aerodnamic input flow conditions used for plunging NACA 64A010 aerofoil Fig. 8. Unstead surface pressure distribution during one complete ccle for transonic case. (--- Upper surface, Lower surface) Input Subsonic case Transonic case parameter M Re m k=c/2u h a

7 5. CONCLUSION The applicabilit of the Implicit Renolds-averaged Navier-Stokes Solver (IMPRANS) for aeroelastic studies is illustrated through time accurate computations for two-dimensional subsonic and transonic flow past a plunging aerofoil. The unstead aerodnamic data obtained for NACA 64A010 aerofoil is validated with the available eperimental results. The plunging motion of the aerofoil creates asmmetr in the flow field even for smmetric aerofoil's. This leads to different pressure fields on the upper and lower surfaces and increases the lift compared to the static aerofoils. In transonic case, there is a strong shock on the upper surface that weakens when it reaches the upstream position during upward stroke of the aerofoil similarl there is a strong shock on the lower surface that becomes weak when it reaches upstream position during downward stroke of the aerofoil. ACKNOWLEDGEMENTS The authors gratefull acknowledge Dr. P. K. Dutta and Dr. Vimala Dutta for their constant encouragement and support during the work. REFERENCES Unstead aerodnamics (1996, Jul). Procee-dings Two da Conference Roal Aero-nautical Societ, London. Mabe, D. G. (1999, Januar). Unstead Aerodnamics: Restrospect and prospect. Aeronautical Journal, 103(1019), Review Paper No. 003, McCroske, W. J. (1982). Unstead airfoils. Ann. Rev. Fluid Mech., Vol. 14, McCroske, W. J. (1988, March). Some Rotorcraft Applications of Computational Fluid Dnamics. NASA TM Vimala Dutta, P. K., Dutta and Sharanappa (2003, March). An Implicit RANS Solver for Unstead Compressible Flow Compu-tations. In Proceedings of the Seminar on State of the Art and Future Trends of CFD at NAL, NAL SP Baldwin, B. S. and H. Loma, (1978). Thin Laer approimation and algebraic model for separate turbulent flows, AIAA Paper No Sharanappa V., Sajjan, Vimala Dutta and P. K. Dutta (2008, August). Numerical Simu-lation of flow over pitching bodies using an implicit Renoldsaveraged Navier-Stokes solver. In Procedings of 12 th Asian congress of Fluid Mechanics, Daejeon, Korea. Vimala Dutta, Sharanappa and P. K. Dutta (2005, August). Navier-Stokes Compu-tations for a Helicopter Rotor Blade in Hover. In Proceedings Eighth Annual CFD Smposium, CFD Division of Aeronautical Societ of India, Bangalore, CP 18. Sharanappa V., Sajjan, Vimala Dutta and P. K. Dutta (2006 August). Viscous Unstead Flow Around a Helicopter Rotor Blade in Forward Flight. In Proceedings 9 th Annual CFD smposium, CFD Division of Aero-nautical Societ of India, Bangalore. Dutta, P. K., Vimala Dutta and Sharanappa V., Sajjan (2007, November). RANS Compu-tation of Flow Past Wind Turbine Blades. In Proceedings of 7 th Asian Computational Fluid Dnamics Conference, Bangalore, 26 th 30 th, (Invited Paper). Sunetra Sarkar and Karthik Venkatraman (2006). Numerical Simulation of in-compressible viscous flow past heaving airfoil. International Journal for Numerical Methods in fluids, Vol. 51, Jian Tang Dragos Viieru and Wei Sh (2007, Januar). Effects of Renolds Number and Flapping Kinematics on Hovering Aero-dnamics. 45th AIAA Aerospace Sciences Meeting and Ehibit, Reno, Nevada. Jean-Yves Andro and Laurent Jacquin (2009). Frequenc effects on the aerodnamic mechanisms of a heaving airfoil in a forward flight configuration. Aerospace Science and Technolog 13, Ashraf, M. A., J. C. S Lai and J. Young, (2007, December). Numerical Analsis of Flapping Wing Aerodnamics. 16 th Australasian Fluid Mechanics Conference, Crown Plaza, Gold Coast, Australia. Beam, R. M. and R. F. Warming, (1978, April). An Implicit Factored Scheme for the Comp-ressible Navier-Stokes Equations. AIAAJ, 16( 4), Steger, J. L. (1978). Implicit Finite Difference Simulation of flow about Arbitrar Geo-metries with application to airfoils. AIAAJ, Vol. 16, pp Ni, R. H. (1982). A Multiple Grid Scheme for Solving the Euler Equations. AIAAJ, Vol. 20, Hall, M. G. (1985, April). Cell Verte Multigrid Scheme for Solution of the Euler Equations. RAE- TM-Aero 2029, In Proceedings Conf. on Numerical methods for fluid dnamics, Jameson, A. W., Schmidt and E. Turkel (1981). Numerical Solution of Euler Equations b Finite Volume Methods Using Runge Kutta Time Stepping Schemes. AIAA Paper

8 Hollanders, H. and H. Viviand, (1980). The Numerical Treatment of Compressible High Renolds Number Flows. Computational Fluid Dnamics, Vol. 2, Kollmann, W.,(ed), Hemisphere Publishing Corporation, Hollanders, H., A. Lerat and R. Perret, (1985, November). Three-Dimensional Calculation of Transonic Viscous Flows b an Implicit method. AIAAJ, 23(11) Davis, S. S. and G. Malcolm (1980, August). Eperimental Unstead Aerodnamics of Conventional and Supercritical Aerofoils. NASA TM