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2 공학석사학위논문 임계받음각에서 NACA4412 에어포일주위유동의큰에디모사 Large eddy simulation of flow over NACA4412 airfoil at the critical angle of attack 2014 년 2 월 서울대학교대학원 기계항공공학부 안은혜

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4 Large eddy simulation of flow over NACA4412 airfoil at the critical angle of attack Eunhye Ahn Department of Mechanical & Aerospace Engineering Seoul National University Abstract The flow characteristics over NACA4412 airfoil near stall are investigated using large eddy simulation (LES) at the Reynolds number of based on the chord length (c) and free stream velocity (u ). This flow is challenging to simulate because it contains thin turbulent boundary layer, separation near the trailing edge, and wake. Various flow characteristics are presented at the critical angle of attack of 12, and results are compared with those from Wadcock s experiment. Moreover, flow phenomena at few more angles of attack before and after stall are presented. After the stall occurred, we confirmed that the drag and lift coefficient fluctuated hardly, so the angle of attack of 12 is the critical angle of attack at Re c = Keywords: airfoil, large eddy simulation (LES), NACA4412, separation bubble, turbulent boundary layer, trailing edge vortex, stall Student number: i

5 Contents Abstract i Contents ii List of Figures iii List of Tables v Nomenclature vi Chapter 1 Introduction 1 2 Numerical Details Governing equations Computational domain and boundary conditions NACA4412 airfoil Numerical Results Mean flow statistics Flow characteristics at a pre-stall and post-stall angles of attack Stall phenomenon Conclusions 23 Bibliography ii

6 List of Figures Figure 1.1 NACA4412 airfoil characteristics (Wadcock, 1987): (a) lift coefficient (b) drag coefficient. This figure indicates the critical angle of attack Schematic diagram of the computational domain and boundary conditions with the trip Geometry of the NACA4412 airfoil. This airfoil has a maximum thickness of 12% and a maximum camber of 4% of the chord length Mean pressure coefficients along the airfoil surface at α = 12. Also, This illustrates a laminar separation bubble and separation point. Red line is a present result of LES, and square and circle symbols mean Hastings and Wadcock s experiment results, respectively Mean streamwise velocity profile at α = 12 (a) in various chord lengths: x/c = 0.529, 0.74, 0.815, 0.85, (b) in x/c = more specifically. Velocity profiles are obtained along normals to the airfoil surface Mean pressure coefficients along the airfoil surface at (a) the prestall angle of attack, α = 10 (b) the post-stall angle of attack, α = Time traces at the pre-stall, critical and post-stall angles of attack: (a) the drag coefficient; (b) the lift coefficient iii

7 3.5 Instantaneous vorticity contour at (a) the pre-stall angle of attack, α = 10 (b) the critical angle of attack, α = 12 (c) the post-stall angle of attack, α = Instantaneous streamlines at (a) the pre-stall angle of attack, α = 10 (b) the critical angle of attack, α = 12 (c) the poststall angle of attack, α = Mean streamlines and separation bubble at (a) the pre-stall angle of attack, α = 10 (b) the critical angle of attack, α = 12 (c) the post-stall angle of attack, α = Instantaneous pressure contour and streamlines at (a) the state of (1), minimum lift coefficient; (b) the state of (2), maximum lift coefficient. Here, figure at the top side shows the time trace of the lift coefficient. The lift coefficient fluctuates with a large amplitude, and states of (1) and (2) indicates the minimum and maximum lift coefficients, respectively Instantaneous pressure and vorticity contours at (a) the minimum lift coefficient (b) the maximum lift coefficient iv

8 List of Tables Table 3.1 Flow statistics at α = 12. Here, CD, CL, and x sp are timeaveraged values The reverse flow at α = 12 and x/c = Here, U e indicates the velocity at the edge of profile and δ e is the boundary layer thickness at the position of 0.995U e v

9 Nomenclature Roman Symbols c C C D C Du C L C Lu C p airfoil chord length wind tunnel section area drag coefficient uncorrected drag coefficient lift coefficient uncorrected lift coefficient pressure coefficient, C p = (p p )/ 1 2 ρu2 C f f i h n N x, N y, N z p q Re t u u e skin friction coefficient, C f = (τ ω )/ 1 2 ρu2 momentum forcing wind tunnel height normal direction to the airfoil surface number of grid points pressure mass source/sink Reynolds number time free-stream velocity edge velocity u, v, w streamwise, wall-normal and spanwise velocity components, respectively V airfoil volume x, y, z Cartesian coordinate (streamwise, wall-normal, spanwise direcvi

10 x sp tion, respectively) separation point Greek Symbols α δ δ e ν ρ τ w ε angle of attack boundary layer thickness boundary layer thickness of the edge velocity kinematic viscosity density wall shear stress blockage correction Superscripts ( ) root-mean-square ( ) mean quantity Subscripts ( ) free-stream ( ) lower lower surface of airfoil ( ) max maximum ( ) min minimum ( ) u uncorrected ( ) upper upper surface of airfoil vii

11 Abbreviations CD CN2 LES QUICK rms second-order central difference scheme second-order Crank-Nicolson method large eddy simulation quadratic upwind interpolation root-mean-square viii

12 Chapter 1 Introduction Airfoils have been used widely as a device to generate hydro- or aero- dynamic performances such as a wing section, helicopter rotor, wind tunnel, and tunnel blade etc. The characteristics indicating the aerodynamic performances of an airfoil are the drag and lift coefficients, and the lift to drag ratio. It is important to understand flow around an airfoil for enhancing these performances. Figure 1.1 shows NACA4412 airfoil characteristics (Wadcock, 1987). The flow around an airfoil shows different characteristics depending on the angle of attack. When the angle of attack increases, both the drag and lift coefficients increase before a stall occurs. However, after the stall, the lift coefficient decreases and the drag coefficient increases rapidly with further increase of the angle of attack. The angle of attack at the maximum lift coefficient is called the critical angle of attack. On the other hand, there are different applications in the wide range of Reynolds numbers and many studies have been made using an airfoil. Especially in Re O(10 6 ), major improvements have been made in an airfoil design, and we find interesting characteristics that captured in flight by both man and nature (Carmichael, 1981; Lissaman, 1983). But previous researches for an airfoil are focused mostly on low Reynolds numbers (Ahmed et al., 2007; Boutilier et al., 2013; Rodriguez et al., 2013; Joshua et al., 2013). This is because performing experiments or numerical simulations accurately at high angles 1

13 of attack for Re O(10 6 ) is difficult. There exist various flow characteristics including small-scale turbulence and flow separation, for example thin turbulent boundary layer, trailing edge separation, separation bubble and wake. For the accurate solution of complex flows, large eddy simulations (LES) of flow over an airfoil at high angles of attack have been performed since 1990 s. The first LES of an airfoil at the high angle of attack was attempted by Kaltenbach and Choi (1995) using NACA4412 airfoil. they demonstrated the possibility of the LES, but the resolution was too coarse to simulate the separation accurately. And Mary and Sagaut (2002) investigated the effects of LES modeling using A-airfoil at α = But after these researches, there is no more study using LES around an airfoil at high angles of attack. There is an issue of the critical angle of attack. Most CFD researches about the NACA4412 airfoil had been conducted at α = and Re c = based on the experiment by Coles and Wadcock (1979), but they performed the experiment in the cylindrical test section of a wind tunnel and inserted flat plates horizontally. It was inappropriate to the 2-dimensional model and suffered from severe blockage interferences. New experiments on the NACA4412 airfoil were performed at α = 12 in the 7ft 10ft and 13ft 9ft wind tunnels, respectively, by Wadcock (1987) and Hastings & Williams (1987). So, we choose Wadcock s experiment conditions, that is the Reynolds number of and the angle of attack of 12. In the present study, We conduct numerical simulation of flow around the NACA4412 airfoil at the critical angle of attack using large eddy simulation and compare the flow characteristics with those from previous studies (Wadcock, 1987; Hastings and Williams, 1987). The characteristics of the NACA4412 airfoil were measured with LV and pitot-rake measurements in experiment s results. Also we investigate the flow characteristics at three different angle of 2

14 attack; α = 10, 12, and Finally, we confirm the critical angle of attack at Re c = and consider a stall phenomenon. 3

15 critical angle of attack, (a), (b) Figure 1.1. NACA4412 airfoil characteristics (Wadcock, 1987): (a) lift coefficient (b) drag coefficient. 4

16 Chapter 2 Numerical Details 2.1 Governing equations The filtered governing equations of unsteady incompressible viscous flow for large eddy simulation on the Cartesian coordinate system are ū i t + ū iū j x j = p ū i + τ ij + x i Re x j x j x f i, (2.1) j where ū i x i q = 0, (2.2) ( ) indicates the filtering operation, x i s are the Cartesian coordinates, u i s are the corresponding velocity components, p is the pressure, f i and q are the momentum forcing and mass source/sink defined on the immersed boundary or inside the body (Kim et al., 2001). Dynamic global model proposed by Park et al. (2006) and Lee et al. (2010) are used for obtaining the sub-grid scale stress tensor τ ij = u i u j ū i ū j. All the variables are nondimensionalized by the chord length c, and the free stream velocity u. The Reynolds number is defined as Re c = u c/ν, where ν is the kinematic viscosity. A staggered grid system is employed in this study, and thus u i and f i are defined at the cell face, whereas p and q are defined at the cell center. The time integration method used to solve (2.1) and (2.2) is based on a fractional- 5

17 step method. Hence, a pseudo-pressure is used to correct the velocity field and then the continuity equation is satisfied at each computational time step. In this study, we use a second-order fully implicit time advancement scheme as in Kim et al. (2002) to relieve the Courant-Friedrichs-Lewy (CFL) restriction. Consequently, the second-order Crank-Nicolson method is used for both the diffusion and convection terms. Also, spatial derivative terms are handled with hybrid scheme: third-order QUICK scheme used at laminar accelerating region (x/c 0.02) and second-order central difference elsewhere (Yun et al. (2006)). 2.2 Computational domain and boundary conditions Figure 2.1 shows the schematic diagram of the computational domain and boundary conditions. As explained before, we use the Cartesian coordinate system, where x, y, and z, respectively, denote the streamwise, wall-normal, and spanwise directions. Reynolds number based on the chord length and freestream velocity is , and we changed the angle of attack from 10 to The computational domain size in free-stream condition is 7.5 x/c 3.0, 3.0 y/c 4.0, and 0.0 z/c 0.05, where x/c = 0 and y/c = 0 corresponds to the leading edge of the airfoil. The domain size of the top boundary is set up to 4c which has same values of slip velocity. After the considerable grid convergence test, the number of grid points are 640(x) 864(y) 64(z) for α = 12. Non-uniform meshes are used with dense resolution in the vicinity of the airfoil to accurately capture the transition, thin turbulent boundary layer, separation and wake, as well as the average pressure and skinfriction coefficient profiles along the airfoil surface. And we use a trip based on Wadcock s experiment at the leading edge for ensuring uniform transition across the span. The angle of attack of the airfoil applies to the velocity at the 6

18 inlet boundary, and the grid system is fixed. This is to minimize the change of grid at the Cartesian coordinate, and to describe efficiently the turbulent flow near the trailing edge. The computational domain has periodic boundary condition along the spanwise direction in order to simulate an infinite span wing. A Dirichlet boundary condition considering the angle of attack (u = u cos α, v = u sin α, w = 0) is used at the inflow. For the outflow boundary, a convective boundary condition ( u i / t+ū u i / x = 0) is used, where ū is the space-averaged streamwise velocity at the exit. And for the bottom boundary, a Dirichlet boundary condition is imposed. At the far boundary side, we used a Dirichlet boundary condition for streamwise velocity and a Neumann boundary condition for normal and spanwise velocities. 2.3 NACA4412 airfoil In the present work, we use NACA4412 airfoil which has a maximum thickness of 12% and a maximum camber of 4% of the chord length. Figure 2.2 presents the geometry of the NACA4412 airfoil. Especially, this airfoil has been used in several experiments. This is appropriate for researches of flow around the airfoil because it shows a gradually wider region of the separation with increase of the angle of attack and forms a separation bubble at the trailing edge rather than an abrupt leading edge separation. 7

19 =, = = 0 Trip ~ c + = 0 = 12 = = = 0 (x, y) = (0, 0) (1, 0) = = = 0 Figure 2.1. Schematic diagram of the computational domain and boundary conditions with the trip. y / c x / c Figure 2.2. Geometry of the NACA4412 airfoil. 8

20 Chapter 3 Numerical Results 3.1 Mean flow statistics In this study, we define the drag and lift coefficients as follows: C D = F y sin α + F x cos α 1 2 ρu 2 A (3.1) C L = F y cos α F x sin α 1 2 ρu 2 A (3.2) where F x and F y are total forces exerted on the airfoil by the fluid in the stream-wise and wall-normal directions, respectively and A is a planform area of the airfoil. Forces are obtained by integrating the momentum forcing which directions are stream-wise and wall-normal. Table 3.1 shows the drag and lift coefficients, and separation point which defined that C f = 0 from the numerical simulation at Re c = and previous experiments (Wadcock, 1987; Hastings and Williams, 1987). Previous results are modified in the way to consider the blockage effect that Pope and Harper suggested. The drag and lift coefficients, and the separation point agree well with those of previous experiments. At this point, the shape factor is H = 4.2. Figure 3.1 shows mean pressure coefficients, C p α = 12. = (p p )/( 1 2 ρu 2 ), at On the suction side of the airfoil, the laminar separation bubble 9

21 Table 3.1. Flow statistics at α = 12. C D CL x sp /c Present (LES) Wadcock (1987) Hastings & Williams (1989) occurs at 0.02 x/c 0.09 with 7% of chord length due to the trip at the leading edge. This phenomenon generates the pressure loss a little near the leading edge. And we can see the flat region of the mean pressure coefficient after the separation occurs. The present result of LES shows good agreement near the trailing edge. Figure 3.2(a) shows mean streamwise velocity profile at α = 12 in various chord lengths. Velocities are non-dimensionalized by the velocity at the edge of profile, and the boundary layer thickness indicating 99.5% of the edge velocity is defined to δ edge. This is the way Hastings and Williams (1987) suggested. Mean streamwise velocity profiles agree well with those of previous experiments. And these show similar tendency that velocity changes when it comes vertically to U e at the surface of the airfoil. Then, let me show the mean streamwise velocity of the separated region more specifically. Figure 3.2(b) illustrates mean streamwise velocity profile at α = 12 and x/c = Also, Table 3.2 shows the thickness and magnitude of the reverse flow. According to Wadcock s experiment, the reverse flow which was separated at x/c = 0.85 took around 25% of the boundary layer thickness δ e, and it had the minimum velocity of 0.16U edge. When it comes to Hastings and Williams (1987), the minimum velocity was 0.12U e and it took 23% of the boundary layer thick- 10

22 Table 3.2. The reverse flow at α = 12 and x/c = Thickness of Magnitude of the reverse flow the reverse flow Present (LES) 0.21δ e -0.12U e Wadcock (1987) 0.25δ e -0.16U e Hastings & Williams 0.29δ e -0.13U e (1989) ness. As the result of the numerical simulation, the reverse flow took 21% of δ e and the minimum velocity was 0.12 of U e. This result is reasonable within a numerical error, so the separation near the trailing edge is predicted well. 3.2 Flow characteristics at the pre-stall and post-stall angles of attack We simulate at the pre-stall and post-stall angles of attack to investigate the stall phenomenon. Figure 3.3(a) & (b) show mean pressure coefficient at α = 10 and α = Similar to the case of α = 12, results of LES at α = 10 and α = agree well near the trailing edge. Figure 3.4(a) & (b) show time traces of the drag and lift coefficients at three different angles of attack. In the case of increasing from α = 10 to α = 12, both the mean drag and lift coefficients increase. However, when the angle of attack increases from α = 12 to α = 13.87, the mean drag coefficient increases rapidly, while the mean lift coefficient decreases. At this point, we confirm that the angle of attack of 12 is the critical angle of attack at Re = Before the stall occurs, the drag and lift coefficients fluctuate at high frequency 11

23 with a small amplitude, but after the stall occurs, these fluctuate with a large amplitude. This phenomenon shows that the performance of the airfoil drops suddenly and the flow control is more difficult after the stall occurs. Figure 3.5 and 3.6 illustrate instantaneous vorticity contour and streamlines at the pre-stall, critical and post-stall angles of attack. Before the stall occurs, the flow separates around the trailing edge and attaches by the flow of the pressure side. So, the separation bubble is formed and figure 3.7 shows mean streamlines and separation bubble at three different angles of attack. In the shear layer above the separation bubble at α = 10 and α = 12, small-scale turbulence structures evolve. This phenomenon appears more severe just before the stall occurs at α = 12. This is because the flow runs along the shear layer after the turbulent boundary layer separation occurs. After the stall, the massive separation occurs and large-scale vortical structures evolve in the wake. This is a different characteristics between the pre-stall and post-stall. 3.3 Stall phenomenon Here, let s see why the large fluctuation of the drag and lift coefficients occurs. Figure 3.8 shows instantaneous pressure fields and streamlines. Two contours indicate different states. At this point, we can categorize the characteristics of stall with the strong vortex at the trailing edge. Figure 3.8(a) is the state of that the lift coefficient is minimum. At this figure, the trailing edge vortex has the maximum size, and the pressure of the lower surface decreases because this vortex has lower pressure and high momentum at the center. On the other hand, figure 3.8(b) is the state of that the lift coefficient is maximum. At this figure, the trailing edge vortex indicating low pressure is shedding off and does not affect the pressure of the lower surface. Then, the difference of the 12

24 pressure between the pressure side and suction side becomes maximum before the trailing edge vortex generates again, and the lift coefficient also increases. Figure 3.9 shows the instantaneous pressure and vorticity contours with a different time. Same as mentioned before, the flow repeats that the strong vortex is shedding off and forming again at the trailing edge and the pressure changes with the trailing edge vortex. Strong fluctuations of the aerodynamic force occur because of this interaction with trailing edge vortices that shedding and forming periodically. This is similar to Karman vortex shedding but in the case of airfoil, the vorticity created at the pressure side is stronger than that at the suction side. 13

25 Laminar separation bubble Separation point (LES, Wadcock (1987)) Present (LES) Hastings Wadcock Suction side / Pressure side Figure 3.1. Mean pressure coefficients along the airfoil surface at α =

26 x/c = x/c = 0.74 x/c = x/c = 0.85 x/c = Present (LES) Hastings Wadcock / / (a) / / (b) Figure 3.2. Mean streamwise velocity profiles at α = 12 (a) in various chord lengths: x/c = 0.529, 0.74, 0.815, 0.85, (b) in x/c = more specifically. 15

27 Present (LES) Wadcock / (a) Present (LES) Wadcock / (b) Figure 3.3. Mean pressure coefficients along the airfoil surface at (a) the prestall angle of attack, α = 10 (b) the post-stall angle of attack, α =

28 =13.87 =12 =10 / (a) =12 =10 =13.87 / (b) Figure 3.4. Time traces at the pre-stall, critical and post-stall angles of attack: (a) the drag coefficient (b) the lift coefficient. 17

29 (a) = (b) = (c) =. Figure 3.5. Instantaneous vorticity contours at (a) the pre-stall angle of attack, α = 10 (b) the critical angle of attack, α = 12 (c) the post-stall angle of attack, α =

30 (b) = (b) = (c) =. Figure 3.6. Instantaneous streamlines at (a) the pre-stall angle of attack, α = 10 (b) the critical angle of attack, α = 12 (c) the post-stall angle of attack, α =

31 a b c Figure 3.7. Mean streamlines and separation bubble at (a) the pre-stall angle of attack, α = 10 (b) the critical angle of attack, α = 12 (c) the post-stall angle of attack, α =

32 Low pressure (a) (b) Figure 3.8. Instantaneous pressure contours and streamlines at (a) the state of (1), minimum lift coefficient (b) the state of (2), maximum lift coefficient. 21

33 (a) (b) Figure 3.9. Instantaneous pressure and vorticity contours of (a) the minimum lift coefficient (b) the maximum lift coefficient. 22

34 Chapter 4 Conclusions In the present study, flow around the NACA4412 airfoil was investigated using large eddy simulation at Re c = and three different angles of attack including the critical angle of attack. Mean flow statistics at the critical angle of attack, such as the drag and lift coefficients, the separation point, the pressure distribution, and the velocity profile, were compared with those of Wadcock s and Hastings experiments, showing good agreements. Moreover, flow characteristics at the pre-stall and post-stall angles of attack are presented. At this point, we confirmed that the angle of attack of 12 is the critical angle of attack at Re c = After the stall occurred, the drag and lift coefficients fluctuated hardly, and this phenomenon was due to the interaction with trailing edge vortices that was shed off periodically. 23

35 Bibliography Barlow, J. B., Rae, W. H. & Pope, A Low speed wind tunnel testing: third Edition. John Wiley & Sons. Carmichael, B. H Low Reynolds number airfoil survey. NACA CR 1, Coles, D. & Wadcock, A. J Flying-Hot-Wire study of flow past an NACA4412 airfoil at maximum lift. AIAA J. 17, Germano, M., Piomelli, U., Moin, P. & Cabot, W. H A dynamic subgridscale eddy viscosity model. Phys. Fluids A 3, Hastings, R. C. & Williams, B. R Studies of the flow field near a NACA4412 aerofoil at nearly maximum lift. Aeronautical J. 91, Kaltenbach, H. K. & Choi, H Large eddy simulation of flow around an airfoil on a structured mesh. CTR Ann. Research Briefs, Kim, K., Baek, S.-J. & Sung, H. J An implicit velocity decoupling procedure for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Eng. 28, 2, Kim, J., Kim, D. & Choi, H An immersed-boundary finite-volume method for simulations of flow in complex geometries. J. Comput. Phys. 171, 1, Lee, J., Choi, H. & Park, N Dynamic global model for large eddy simulation of transient flow. Phys. Fluids 22, Lissaman, P. B. S Low-Reynolds-number airfoils. Ann. Rev. Fluid Mech. 15,

36 Mary, I. & Sagaut, P Large eddy simulation of flow around an airfoil near stall. AIAA J. 40, Park, N., Lee, S., Lee, J. & Choi, H A dynamic subgrid-scale eddy viscosity model with a global model coefficient. Phys. Fluids 18, Rhie, C. M. & Chow, W. L Numerical study of the turbulent flow past an airfoil with trailing edge separation. AIAA J. 21, Wadcock, A. J Investigation of low-speed turbulent separated flow around airfoils. NASA report. Yun, G., Kim, D. & Choi, H Vortical structures behind a sphere at subcritical Reynolds numbers. Phys. Fluids 18,

37 임계받음각에서 NACA4412 에어포일주위 유동의큰에디모사 서울대학교대학원 기계항공공학부 안은혜 요약 본연구에서는실속근처에서 NACA4412 에어포일주위의유동 특성을큰에디모사방법을사용하여살펴보았다. 레이놀즈수는시위 길이와자유유동속도를기준으로하여 Re = 1,640,000 이다. 이 유동은얇은난류경계층, 뒷전에서의유동박리, 후류등의복잡한 특성을갖고있기때문에모사하는것이도전적인문제이다. 임계 받음각인 α = 12⁰ 에서다양한유동특성들을관찰하였고, 그결과를 Wadcock 의실험결과와비교하였다. 그리고실속이전과이후의 각도에서유동현상들을관찰하였다. 실속이발생한후에는항력계수와 양력계수가심하게변화하는것을확인하였고, 따라서레이놀즈수 Re = 1,640,000 에서받음각 α = 12⁰ 가임계받음각이다. 주요어 : 에어포일, 큰에디모사, NACA4412, 유동박리기포, 난류경계층, 뒷전와류, 실속 학번 :