Calculation of NACA 0012 Airfoil through Roe s Scheme Method

Transcription

1 International Conference Recent treads in Engineering & Technology ICRET 2014 Feb 13-14, 2014 Batam Indonesia Calculation of NACA 0012 Airfoil through Roe s Scheme Method Mohd Faizal bin Che Mohd Husin, Dr. Ir. Bambang Basuno, and Dr. Zamri bin Omar Abstract The present work performs the Roe s scheme in solving Euler equation, applied to the solution of flow over 2D airfoil NACA 0012 and compares to experimental data and XFOIL software. As one of the approximation solvers for Riemann scheme, Roe s scheme is extended to second order through MUSCL scheme. MUSCL scheme has been applied in order to eliminate the spurious oscillation due to shock wave presence. In this work, flow is treated as compressible with Mach number around 0.3 to 0.8 and Reynolds number is set at 3 million. Results show a good potential have been made by present method especially at low angle of attack and low Mach number. To improve the accuracy of solution, present study proposed viscous effect should be included where viscosity plays a major role in determination of aerodynamic characteristics particularly for high speed aerodynamic. Keywords Inviscid flow, Euler solver, Roe s scheme, and MUSCL scheme. S I. INTRODUCTION INCE early 20 th century, aerodynamics progressively evolved in a wide range and become very interesting subject in engineering and mathematic. Study on flow behavior for single airfoil is neither new nor primitive subject in aerodynamics. Since Eastman and Abbott 1, investigation has been made on various types of NACA airfoil through experiment within variable-density wind tunnel. Consequently, Eastman made another attempt which more systematic experimentation with relating a number of N.A.C.A airfoil in a wide range of Reynolds number 2. In other occasions, experimental data on single airfoil also been carried out by Wenzinger 3, Eastman and William 4, Abbott, Von Doenhoeff, and Stivers 5, Moyers 6, Ferri 7, and many others. Numerical solution on Navier Stokes equation gave major influence in aerodynamics analysis. As example, an investigation was made by Korn on shock-free transonic around airfoil by applying numerical method in solving linear partial differential equations 8. Subsequently, many works was made by researchers to examine the capability of solution methods. Lax, Roe, McCormack, Godunov, Ritchmeyer and Mohd Faizal bin Che Mohd Husin, is a PHD student of University Tun Hussein Onn Malaysia, Johor, Malaysia corresponding author s phone: ; marchupagecat@gmail.com. Dr. Ir. Bambang Basuno, is a Senior Lecturer at Department of Aeronautic, University Tun Hussein Onn Malaysia, Johor, Malaysia bambangb@uthm.edu.my. Dr. Zamri bin Omar, is a Senior Lecturer at Department of Aeronautic, University Tun Hussein Onn Malaysia, Johor, Malaysia zamri@uthm.edu.my. Rusanov who invented applicable schemes frequently gained attention by succeeding researchers. Taylor explained some of favorite schemes thoroughly in dealing with such serious difficulties of aerodynamic problem 9. This work engaged with a wide scope of aerodynamics properties such as subsonic, transonic and supersonic speed, viscous and inviscid, compressibility effect, high Reynolds numbers and various approach of solution namely potential flow, Euler solver and Navier stokes solution. Behind these advances, experimental approach exhibited similar improvement as computational one since the technology of wind tunnel experienced successive modification. Gregory and Wilby 10 on their study aerodynamics characteristics of airfoil NPL9615 and NACA 0012 provided a complete data set for these airfoils at various Mach number of subsonic flow. Consequently, Gregory made another effort with O Riley on NACA 0012 which including effect of upper surface roughness 11. Another excellent accomplishment has been done by Harris 12 via experiment on two-dimensional NACA 0012 within Langley 8-foot wind tunnel where measurement is conducted at subsonic speed and relatively high Reynolds number. Experiments on NACA 0012 were also made by Langley 13, Nash, Quincey, Callinan 14and Murdin15. All those experimental results show uncertainty difference of each other and it caused difficulty to validate CFD results. Jameson carried on aerodynamics discovery with his work on airfoils through numerical potential flow solution 16. This work offers the solution of flow at sonic Mach number and also implements artificial viscosity as a shockwave treatment. The great work by Jameson on Euler methods can be found in 17 by solving Euler equation with finite volume methods. Those methods were solved by Runge-Kutta time stepping schemes. Engaged with time stepping schemes, accuracy of solution was improved and the stability region can be extended. The latest work, which dealing with Euler equation is by Arias et. al18. In this research finite volume has been simulated for a flow over airfoil NACA 0012 by using Jameson, MacCormack, Shue, and TVD schemes. This work presented two computer codes where both approach implement finite volume methods to solve Euler equations. First code namely ITA works on two-dimensional structured grid and it possess the capacity to work with three different schemes: i the Jameson scheme using a five stage Runge- Kutta time integration; ii the MacCormack scheme, based upon the predictor and corrector strategy to advance in time; 51

2 International Conference Recent treads in Engineering & Technology ICRET 2014 Feb 13-14, 2014 Batam Indonesia data have been taken from Harris in 21 as suggested in 22. Another comparison has been made with aerodynamics software namely XFOIL by Drela where represents a combination of panel method and global Newton method. Resolution that offered by XFOIL mainly applicable for low Reynolds number case, while at high Reynolds number case, software exhibit inconsistence results. A brief description of Euler solution and computer code is shown in the next section. iii and finally the Shu scheme, which uses a variation of the Jameson time integration, in order to better capture of shock waves. Another effort that related to present work is explained by Maciel 19 which demonstrated several high resolution of TVD s scheme to be dealt with two-dimension aerodynamic problem. There are six schemes employed here namely Roe s, Van Leer vector splitting, Harten-Yee, Yee-Kutler and, Hugson-Beran. In order to reach accuracy of second order, Roe and Van Leer scheme apply MUSCL approach and other schemes used Harten s ideas of the construction of a modified flux function to obtain second order accuracy and TVD characteristics. This paper also offered solution in both of formulation: implicit and explicit. Implicit solved through ADI Alternating Direction Implicit approximate factorization while the explicit one s used a time splitting method. Lastly, study concluded that Roe s scheme exhibit the best agreement to the experimental results both in the implicit and explicit formulation due to the best estimative of the shock angle. III. THE GOVERNING EQUATION A. Description of Euler Solution The governing equation of inviscid flow domain for the case of compressible, non-viscous and two dimensional unsteady flows in conservative form is 23: Where: II. PROCEDURE FOR COMPUTATION Present airfoil analysis is employing with Euler equation to deal with two-dimension inviscid flow over airfoil NACA Euler equation will be treated in explicit formulation. Roe s TVD scheme is utilized to resolve this explicit Euler equation with MUSCL s scheme is exploited to increase accuracy of second order formulation. In order to apply these methods to complex geometric configurations, the finite volume formulation has been used to develop the space discretization, and allows the implementation of an arbitrary grid. Structured numerical grid generation is used since the problem of single airfoil NACA 0012 is considered as relatively straightforward configuration. To accomplish the goal above computer codes for TVD scheme and grid generation were utilized, which had taken from Blazek 20. The criterion must be satisfied by grid generation process were 1 they domain is completely covered by the grid, 2 there is no free space left between the grid cells, and 3 the grid cells do not overlap each other. The detail about its governing equation would be described in the next section. The results of the study are mainly focused on pressure distribution, lift coefficient and moment coefficient. Due to Euler solution for inviscid flow domain, aerodynamics characteristics mentioned is sufficient enough without taking account of drag coefficient since the viscosity effect that affected the airfoil surface characteristics are neglected. Computer code that introduced in 20 is utilized namely AIRFOIL_ROE_SCHEME in solving the objective of study. There are plenty of experimental data can be used as a weighing scale for analysis, however it must be chosen depends on fundamental of experiment. The work of Gregory and Wilby 10 and, Gregory and O Riley 11 were reliable and matched to be an assessment data set since experiment was conducted at subsonic flow and Reynolds number about 2.88 X 106. Moreover, it will facilitate comparison process where the results of experiment represent pressure coefficient distribution CP 11, lift coefficient CL, and moment coefficient CM 10. For the high Mach number, experimental With: Above equation is known as Compressible Euler Equation and represents a highly nonlinear partial differential equation and there is no analytical solution. denotes as the ratio of specific heat capacities of the gas. In a two dimensional, Euler equation is wrote in hyperbolic equation form. Where A and B is Jacobian matrix system For more convenience, it is wise if Euler equation is derived in one dimensional then for future use, one can simply extend to multi-dimensional. One dimensional explicit time stepping formulation read as: Following the step of Roe s scheme, each term in 6 are derived as follow 24: 52

3 International Conference Recent treads in Engineering & Technology ICRET 2014 Feb 13-14, 2014 Batam Indonesia First terms of right hand side equation represents convective flux while the second terms are dissipative flux. Convective flux is treated by upwind scheme, and dissipative flux will follow Roe high resolution scheme. is eigenvector matrix correspondents to matrix eigenvalues respects to similarity such that 25: D and X -1 are matrix diagonal and inverse eigenvector matrix respectively and speed of sound,. All quantities with the hat that appears in 9 are evaluated by Roe average: As an Riemann approximation solver, Roe s scheme reads 26. With MUSCL s interpolation, velocity terms in are formulated as follow 28. Where: is a free parameter lies in interval -1,1, where for, is a central difference approximation, multiplied by, to the first spatial derivative of the numerical of the numerical solution at the time level n. MUSCL s interpolation can be more accurate with quadratic reconstruction, that are 20: With and, and following definition: Thus: According to 24 and 27 Roe s vector terms in is formulated as: It can be written as: Hence, turns to the following forms. With: 53

4 International Conference Recent treads in Engineering & Technology ICRET 2014 Feb 13-14, 2014 Batam Indonesia The 15 can be simplified if we consider slope limiters with the symmetry property as: Thus, 15 becomes: With limiter function is defined as: MUSCL scheme is divided into two category where it is determined by value of. MUSCL0 represents and MUSCL3 for For the MUSCL3 for, Van Albada flux limiter, and limiter function, followed below expression. For simplification purpose, Eq 26 is written in this form: Generating grid for computational space can be undergone in various techniques. Present study uses structured grid C- type as obtained by Blazek 20 namely C_GRID_GENERATOR. This method is dealt with elliptic partial differential equation or specifically Poisson equation. C type is one of the grid topology which is enclosed by one family of grid lines and also forms the wake region. The situation is shown is Fig 3.1 where lines start at the farfield, follow the wake, pass the trailing edge node b, surround the body in clockwise, then reach farfield again at. For the other grid lines exudes in normal direction from the body and wake. The coordinate cut that is represented by segment of a-b of grid lines at physically map onto two segments in the computational space namely and for lower space and upper space respectively. Where: The additional parameter in 29 prevents the activation of the limiter in smooth flow regions due to small-scale oscillations 20. This is sometimes necessary in order to achieve a fully converged steady-state solution. For this purpose, is set at while other alphabets are defined as follow. Fig. 3.1: C-grid topology in two-dimension Elliptic equations for the two-dimension grid generation are: In order to increase the accuracy and to extend the stability region 17, solution is enhanced by Runge-Kutta multistep method. It first has been developed by Jameson 18 with applying a five-stage Runge-Kutta to advance the solution in time. Updating solution due to Runge-Kutta methods, it follows steps below. Where metrics coefficient in equation are: 54

5 International Conference Recent treads in Engineering & Technology ICRET 2014 Feb 13-14, 2014 Batam Indonesia IV. RESULT AND ANALYSIS Numerical high resolution of Euler s solver scheme namely Roe s scheme is represented in this section. An illustration about computational space is portrayed in Fig 4.1, where it is a result from C_GRID_GENERATOR code. Pressure coefficient of airfoil NACA 0012 is observed as shown in Fig 4.2, an agreement between experiment and computation is achieved in excellent manner for low angle of attack and low Mach number M = 0.3 cases. At high angle of attack within low Mach number, Roe s solver still provides fairly prediction as XFOIL did. At Mach number 0.8, a wide deviation is occurred due to shock wave presence. As shown in Fig 4.2 a good agreement is achieved ahead shock wave come off, and Roe s scheme exposes poor capability to capture such phenomenon. Nonetheless, compare to software XFOIL, Roe s scheme remains exceptional since XFOIL software was limited to flow at considerably low speed. From Figs it can be observed that two parameters impede computational are high angle of attack and large Mach number. Higher angle of attack affected calculation with overprediction occurred in tracking maximum pressure coefficient, CP. In similar manner as present method, software XFOIL also poses alike fashion even in determining the point of maximum CP, XFOIL remains greater than prediction of present method. From point of view, it simply can be realized that the present method with no viscosity effect exhibit a good quality in emulating experimental data. Another parameter mentioned is Mach number. As depicted in Fig 4.4 and 4.5 for Mach number 0.7 and respectively, error deviates proportionally to Mach number, where at Mach number 0.799, with existence of shock wave, error radically exceeded 20% chord. It implies for relatively larger Mach number as transonic, present method remains unrealistic to be applied. On the other hand, an excellent work done can be seen in the Fig 4.5 where assessment of MUSCL s interpolation scheme plays role in diminishing spurious oscillation of shock wave. Fig. 4.2: Pressure coefficient distribution along NACA 0012 surface at angle of attack 6 and Mach number 0.3. Fig 4.3: Pressure coefficient distribution along NACA 0012 airfoil surface at angle of attack 16.5 and Mach number 0.3. Fig 4.4: Pressure coefficient distribution along NACA 0012 airfoil surface at angle of attack 1.49 and Mach number 0.7. Fig 4.1: View of structured grid about NACA

6 International Conference Recent treads in Engineering & Technology ICRET 2014 Feb 13-14, 2014 Batam Indonesia Fig 4.5: Distribution of pressure coefficient along NACA 0012 airfoil surface at angle of attack 2.26 and Mach number Fig 4.7: Distribution of lift coefficient along NACA 0012 airfoil surface at various angles of attack and various Mach number. For more general, it is convenient if we look at flow behavior by viewing lift coefficient characteristic at various angles of attack. Fig 4.6 illustrates lift data in range of angle of attack between 0 to12 for three difference approach. Roe s scheme shows relatively good estimation to XFOIL for Mach number 0.3. As angle of attack arose, difference between present method and experiment becoming more evident and gains its peak at maximum lift point. Nevertheless, this fine performance of present method descending as Mach number is increased towards to transonic flow. Apparently, it can be seen in the Fig 4.7 how the Roe s scheme performed a better prediction than XFOIL software in presuming lift coefficient, CL. Fig 4.8: Distribution of lift coefficient along NACA 0012 airfoil surface at various angles of attack and Mach number 0.3. The same tendency of CL can be found for the moment coefficient CM tracing, where inadequate of prediction as shown in Fig 4.8 happened at relatively high angle of attack and at large Mach number. Typically for this study, neglecting of viscous effect is identified as a major factor for this lacking since viscosity plays a big part for compressible flow especially at high Mach number. Fig 4.8 also shows XFOIL made a better imitation than present method while at Mach number equal to 8, as depicted in Fig 4.9 XFOIL remains unreliable method to be used in computing such flow behavior. Fig 4.6: Distribution of lift coefficient along NACA 0012 airfoil surface at various angles of attack and Mach number

7 International Conference Recent treads in Engineering & Technology ICRET 2014 Feb 13-14, 2014 Batam Indonesia Fig 4.9: Distribution of lift coefficient along NACA 0012 airfoil surface at various angles of attack and Mach number 0.8. V. CONCLUSION Present study has proposed Roe s scheme as a computational method to deal with flow around 2D airfoil NACA 0012 at Reynolds number 3 million and Mach number from 0.3 to 0.8. As discussed in previous section, present method reveals a good ability in emulating experimental results as provided by Gregory and Wilby 10, and Harris 12. Viscosity effect is detected as causal factor for inaccurate prediction particularly for the high Mach number occasion. Generally, computational results were outstanding instead of XFOIL software. In linearizing the second order of Roe s scheme, MUSCL with Van Albada limiter exposed an excellent performance due to diminishing spurious oscillation. Artificial viscosity is suggested to be included in governing equation to pursue the accuracy or another technique can be used is interaction boundary layer approach as alternative. ACKNOWLEDGMENT This research is sponsored by University Tun Hussein Onn Malaysia under postgraduate faculty. Present work is provided according to requirement of International Institute of Engineering IIENG and to be presented at Batam, Indonesia. REFERENCES 1 Jacobs, N.E., and Abbott, I.H. The NACA Variable-Density Wind Tunnel, NACA Tech. Report, No. 416, Langley Field, Vancouver, Jacobs, N.E., and Sherman, A. Airfoil Section Characteristics as Affected by Variation of the Reynolds Number, NACA Tech. Report, No. 586, Langley Field, Vancouver, Wenzinger, C.J. Pressure Distribution Over NACA Airfoil With an NACA External-Airfoil Flap, NACA Tech. Bartnoff, S, and Gelbart, A. Application of Methods to Studies of Flow with Circulation about a Circular Cylinder. NACA TN No. 1171, Jacobs, N.E., and Clay, W.C. Characteristics of the NACA Airfoils from Tests in the Full-Scale and Variable-Density Tunnel NACA Tech. Report, No. 530, Langley Field, Vancouver, Abbott, I.H., Doenhoff, A.E.V., and Stivers, L.S.J. Summary of Airfoil Data, NACA Tech. Bartnoff, S, and Gelbart, A. Application of Methods to Studies of Flow with Circulation about a Circular Cylinder. NACA TN No. 1171, Moyers, F.N. A Comparison of Theoretical and Experimental Pressure Distribution at High Speed About the NACA 4412, Thesis of Master Degree, Pasadena, California, Ferri, A. Experimental Results with Airfoil tested in the High Speed Tunnel at Guidonia, NACA Tech. Report, No. 946, Washington, Korn, D.G. Computation of Shock Free Transonic Flow for Airfoil Design, AEC Research and Development Report, NYO , Courant Inst. of Mathematics and Science, New York, Taylor, T.D. Numerical Method for Predicting Subsonic, Transonic, and Supersonic Flow, AGARD-AG-187, France, Gregory, N., and Wilby, P.G. NPL 9615 and NACA 0012, A Comparison of Aerodynamic Data, Aeronautical Research Council papers, no. 1261, Bedford, Gregory, N., and O Reilly, C.L. Low-Speed Aerodynamic Characteristics of NACA 0012 Aerofoil Section, including the Effects of Upper-Surface Roughness Simulating Hoar Frost, Aeronautical Reports and Memoranda, no. 3729, Harris, C.D. Two-Dimensional Aerodynamic Characteristics of the NACA 0012 Airfoil in the Langley 8-Foot Transonic Pressure Tunnel, NACA TN, pp , Langley, Langley, M.J. Numerical Method for Two Dimensional Axisymmetric Transonic Flows, Aeronautical Research Council papers, no. 1376, Stockhom, Nash, J.F, uincey, V.G, and Callinan, J. Experiments on Two- Dimensional Base Flow at Subsonic, and Transonic Speeds, Aeronautical Reports and Memoranda, no. 3427, Moss, G.F., and Murdin, P.M. Two-Dimensional Low-Speed Tunnel Tests on the NACA 00 I2 Section Including Measurements Made During Pitching Oscillations at the Stall, Aeronautical Research Council papers, pp. 1145, Fanborough, Jameson, A. Iterative Solution of Transonic Flows over Airfoils and Wings, Including Flows at Mach 1, Courant Inst. of Mathematics and Science, New York, Jameson, A., Schimidt, W., and Turkel, E. Numerical solution of the Euler Equations by finite volume methods using runge-kutta timestepping schemes, AIAA Journal, pp , Arias, A., Falcinelli, O., Fico, N.J, and Elaskar, S. Finite Volume Simulation of a Flow Over a Naca 0012 Using Jameson, Maccormack, Shu and Tvd Esquemes, Mechanical Computational, vol. 16, pp , Córdoba, Maciel, E.S.D.G. Explicit and Implicit TVD High Resolution Schemes in 2D, Journal of WSEAS Transactions on Applied and Theoretical Mechanics, E-ISSN: , Brazil, Blazek, J. Computational Fluid Dynamics: Principles and Applications, Elsevier Science Ltd, Oxford, United Kingdom, Maksymiuk, C.M., and Pulliam, T.H. Viscous Transonic Airfoil Workshop Results Using Arc2d, AIAA Journal, pp , McCroskey, W.J. A Critical Assessment of Wind Tunnel Results for the NACA 0012 Airfoil, NASA Tech. Memorandum, pp , USAAVSCOM Tech. Rep. 87-A-5, Moffett Field, Toro, E. F. Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin, Nishikawa, H. A Comparison of Numerical Flux Formulas for the Euler Equation, Final Assigment, Hoffmann, K.A., & Chiang, S.T, Computational Fluid Dynamics, vol. 2, Wichita, Kansas, Nishikawa, H. A Comparison of Numerical Flux Formulas for the Euler Equation, Final Assigment, Kroll, C., Aftosmis, M., and Gaitonde, D. An Examination of Several High Resolution Schemes Applied to Complex Problems in High Speed Flows, Final Report Wright Laboratory, AD-A , Ohio, Maciel, E.S.D.G. Explicit and Implicit TVD High Resolution Schemes in 2D, Journal of WSEAS Transactions on Applied and Theoretical Mechanics, E-ISSN: , Brazil,