Aerodynamics of Rotating Discs

Proceedings of ICFD 10: Tenth International Congress of FluidofDynamics Proceedings ICFD 10: December 16-19, 2010, Stella Di MareTenth Sea Club Hotel, Ain Soukhna, Egypt International Congress of Red FluidSea, Dynamics December 16-19, 2010, Ain Soukhna, Red Sea, Egypt ICFD10-EG-3901 Aerodynamics of Rotating Discs Shereef A. Sadek1 Saad A. Ragab2 Post-Doctoral Associate Professor Department of Engineering Science and Mechanics Virginia Tech, Blacksburg, VA, USA ABSTRACT Nomenclature In this paper we present a numerical study of three-dimensional flow over a rotating circular disc in a uniform flow at different Mach numm ber ratios λ,(λ = MT ip ). The Reynolds Averaged Navier-Stokes equations and a one-equation turbulence closure model for the Reynolds stresses are solved using OVERFLOW software. The flowfield around the disc is analyzed at two agles of attack α = 0, 10 for a range of λ. Comparison of surface pressure coefficient, friction velocity and wake structure are made between the rotating and non-rotating disc. For the non-rotating disc the flow field resembles that of a small aspect ratio wing with symmetric wake and trailing vortices. For the rotating disc the surface pressure coefficient, friction velocity and wake develop asymmetry due to disc rotation. Depending on the disc rotation the friction velocity develops a minima on both disc surfaces, the location of which depends on the Mach number ratio λ. Variation of drag, side force and yawing moment coefficients with tip speed ratio are presented. This flow problem proves to be a very chalanging one in terms of turbulence modelling and grid resolution. α Disc angle of attack λ Ratio of tip Mach number to free stream Mach number Ωx Streamwise vorticity ρ Flow density τ Wall shear stress a Free stream speed of sound CD Drag coefficient CDp Drag coefficient due to pressure CDv Drag coefficient due to shear stress CL Lift coefficient D Disc diameter M Free stream Mach number Mtip Rotating disc tip Mach number n+ ReD Reynolds number based on disc diameter uτ 1 Post-Doctoral 2 Professor, Distance normal to the wall normalized by wall scales Associate, email: ssadek@vt.edu email: ragab@vt.edu Friction velocity x, y and z Cartesian coordinates 1 Copyright 2010 by ICFD 10

1 INTRODUCTION Flow over rotating discs has been studied by many researchers in the last few decades mainly in effort to study transition and turbulent flows. Flow over rotating discs appears in many engineering and recreational applications. Zdravkovich et al. [1], studied the aerodynamics of coin-like cylinders of varying aspect ratios in a uniform flow but with no rotation. Forces and moments were measured for different shapes of cylinder edge and the flow topolgy was examined. Potts and Cowther [2], studied the aerodynamics of Frisbee like discs that had an approximately elliptic cross-section and hollowed out underside cavity. The lift and drag coefficients have been found to be independent of Reynolds number for the range of tunnel speeds tested. The upper surface flow is characterised by separation at a line (arc) of constant radius on the leading edge rim, followed by reattachment at a line of similar geometry. Trailing vortices detach from the trailing edge rim. The cavity flow is characterised by separation at the leading edge lip, followed by straight-line reattachment. Badalamenti and Prince [3],studied the effects of endplates on a rotating cylinder in crossflow for a range of Reynolds number. They found that endplates can enhance the lift/drag ratio and increase lift up to a limiting value. In this study the effects of rotation on the flow over a finite disc will be examined in some depth. The ratio of tip Mach number to freestream Mach number, λ, will be varied as well as the angle of attack. Changes in the surface pressure distribution and wake development will also be examined. Force and moment coefficients will be presented as a function of λ. 2 MATHEMATICAL FORMULA- TION The flow solver used in this study, OVERFLOW, is a three-dimensional flow solver that uses structured overset grids. It solves the Navier-Stokes equations in generalized coordinates,1. It is capable of obtaining time-accurate as well as steady state solutions. where Q t + E ξ + F η + G ζ = 0 (1) Q is the vector of conserved variables in genralized coordinates ξ, η and ζ are the generalized coordinates G, F and G are the total flux (convective and viscous) in the direction of generalized coordinates ξ, η and ζ, respectively. 2.1 Computational Domain The computational domain consists of three grid levels. The near body grid, an intermediate grid and farfield grid. The near body grid, shown in Figure 1, is an o-grid that extends 0.4D normal to the surface. The disc is centered at the origin and the disc symmetry axis (axis of rotation) is aligned with the z-axis. The number of grid points in radial, azimuthal and normal direction is 257, 361 and 155, respectively. Constant height layers were used for the first 35 layers, above that the grid was stretched with a maximum stretching ratio of 1.1. The first grid point off the surface was estimated to be at n + = 0.5. The other two grids are Cartesian grids. The intermediate grid covered a cube of side length equal to 3D, to capture the wake details and the farfield extended about 12Din all direction from the disc center. The total number of grid points was about 14.4 million. 2.2 Flow Boundary Condition The free stream Mach number was set to M = 0.2 and the pressure and temperature were set to standard sea level values. The Reynolds number 2

based on free stream conditions and disc diameter, Re D = 18.5x10 6 The disc rotational tip Mach number was varied as a factor of the free stream Mach number, M tip = λm. The ratio λ took on the values, 0.0, 0.38, 0.76, 1.3, 1.51 and 1.89. Freestream values were implemented using a freestream characteristic boundary condition applied to the farfield grid at the upstream and spanwise faces while a characteristic boundary condition based on Riemann invariants is applied to the downstream face. For the intermediate grid, no boundary conditions were applied since the flow variables are interpolated from neighboring grids. For the near body grid, a no slip boundary condition with specified rotational velocity around the vertical axis is specified at the disc wall. Along the vertical grid lines above and below the disc an axis of symmetry boundary conditions is also specified. 3 RESULTS AND DISCUSSION In this section the main flow features are presented. Of interest, are the fricition velocity, u τ, surface pressure coefficient, C P, and the force and moment coefficients acting on the disc. Friction Velocity Figures 2-5 show contours of friction velocity, defined in equation 2, τ u τ = (2) ρ on the disc surface at α = 0.0 for a range of λ = 0.0 1.89. For the non-rotating disc the minimum friction velocity occurs at the leading and trailing edges of the disc. At the leading edge the flow is stagnant and the boundary layer is very thin. At the trailing edge there is a small separation region where the flow is recirculating at a very low speed. As the rotation speed is increased in Figure 3, λ = 0.78, the contour lines become asymmetric and the location of the minimun value moves counterclockwise along the circumference in the negative x-y plane. The maxima of u τ increases with the increase in λ but remains in the positive x-y plane along the circumference near the leading edge. This is due to the fact that the net relative flow velocity to the disc surface increases in the positive x-y plane while it is reduced in the negative x-y plane. As the rotation speed is increased above the free stream speed in Figure 4, λ = 1.51, a double minima is developed on the upper and lower disc surfaces and the location of the minima moves inward towards the center at 90 with the positive x-axis direction. This is due to the fact that the surface speed matches the flow speed at some radial distance less than the disc radius. Finally in Figure 5, λ = 1.89, a well defined minima is present on the upper an lower surfaces approximately half a radius away from the center. This is a unique feature of this flowfield. On the other hand the maximum u τ is always located on the disc s circumferance since the relative flow velocity is always positive in the positive x-y plane. Surface Pressure Figures 6 and 8 show the lower and upper surface pressure coefficient for the non-rotating disc at α = 10.0, while Figures 7 and 9 show the surface pressure coefficient for the rotating disc at α = 10.0 and λ = 1.89. For the non-rotating case the disc behaves like a finite wing with low aspect ratio. Upstream of the disc center on the upper side we can see the typical leading edge suction and on the lower side we can see high pressure region. A common feature of finite wings are the tip vortices, which are present here as well. Low pressure coefficient can be seen at the disc tips at an approximate angle of 75 from the positive x-direction. These suction peaks are due to the tip vortices formation that is typical of such flowfields. For the rotating-disc, the first noticable effect due to rotation is the breaking of the flow symmetry. Upstream of the disc center, the pressure coefficient looks similar to the non-rotating case, this might be due to the fact that the boundary 3

layer is very thin and still developing. On the other hand, downstream of the disc center; the rotation effects are more significant. This is evident by the degree of asymmetry in the pressure coefficient contours downstream of the disc center on the upper and lower surfaces. This asymmetry is because the disc rotation affects the surface shear stress distribution along the disc surface as shown above, the surface shear stress being higher in the positive x-y half-plane than in the negative x-y half-plane. This in turn changes the boundary thickness distribution along the disc surface in an asymmetric manner. The asymmetric surface shear stress creates more adverse pressure gradient in the positive x-y half-plane and more favorable pressure gradient in the negative x-y half-plane. Hence, to the incoming inviscid flow the effective disc shape is asymmetric. Tip Vortices and Wake As mentioned above, the asymmetric surface shear stress affects the surface pressure distribution, it also affects the shape of the tip vortices and the shed wake. Figures 10a and 10b show crossflow streamlines in a plane at 70 with the x-z plane along with pressure coefficient contours for the advancing disc tip. It is clear that the vortex size is greater for the rotating disc, which also indicates that the separation at the tip occurs earlier than in the non-rotating case. Figures 11a and 11b show crossflow streamlines in a plane at 70 with the x-z plane along with pressure coefficient contours for the retreating disc tip. For the non-rotating disc the picture is similar to Figure 10a, however for the rotating disc, the vortex size is reduced and its center is further off the disc surface. Figures 12a and 12b show the disc wake development for the non-rotating and rotating disc, respectively. The figures show surface pressure coefficient contours as well as contours of the normalized streamwise vorticity, Ω x D/a at three different values of x/d. For the rotating disc the wake diffuses much faster than the non-rotating case. This suggests that the rotation has the effect of weakening the tip vortices which means that the disc loses lift. Aerodynamic Coefficients Figure 13 shows the disc drag, side force and yawing moment coefficients at α = 0.0. In general, the rotation increases the disc s drag. The side force is directed in the negative y-direction and the moment direction is opposite to the direction of rotation as expected. The increase in drag coefficient is mainly due to increase in the shear stress; this is shown in Table 1. For α = 0.0, the viscous drag increases by about 40% and by 50% for α = 10.0. On the other hand, disc rotation increases the pressure drag at α = 0.0 by about 380% for the same rotation speed; but it has insignificant influence at α = 10.0. In summary, the drag coefficient increases with disc rotation; however, disc rotation reduces the increase in drag with α. More data points are needed to confirm this. Finally, also shown in Table 1 is the lift coefficient. It is shown that disc rotation reduces the lift coefficient by about 4% of the non-rotating disc value. This is in agreement with the previous observation that for the rotating-disc the tip vortices are weaker than in the case of nonrotating disc. CONCLUSIONS In this paper, the flow over a circular rotating disc was analyzed by solving compressible Navier- Stokes equations using OVERFLOW. One equation Spalart-Allmaras turbulence model was used to close the system of equations. The flow was analyzed at different disc rotational speed and at two angles of attack. It was shown that, Disc rotation ulters the surface shear stress distribution which in turn changes the boundary layer thickness and structure The asymmetric surface shear stress creates asymmetric boundary layer thickness, and 4

hence viscous-invicid interaction leads to asymmetric surface pressure distribution Disc rotation also affects the tip vortices which leads to weakining of the shed vorticity, this in turn reduces the lift coefficient Drag coefficient increase significantly due to rotation at zero angle of attack but is reduced significantly at 10.0 angle of attack More simulations at different angles of attack and at higher values of λ are needed to have a full picture of the flowfield Figure 1: Near disc computational grid Figure 2: Friction velocity uτ, λ = 0.00 5

Figure 3: Friction velocity uτ, λ = 0.78 Table 1: Drag, side force, yawing moment and lift coefficients α λ CDp CDv CD CL 0.0 0.0 0.00113 0.00630 0.00743 0.00000 1.89 0.00426 0.00887 0.01312 0.00000 Figure 4: Friction velocity uτ, λ = 1.51 10.0 0.0 0.02013 0.00589 0.02602 0.21790 1.89 0.02092 0.00894 0.02985 0.2098 REFERENCES [1] M. M. Zdravkovich, A. J. Flaherty, M. G. Pahle and I. A. Skellhorne, Some aerodynamic aspects of coin-like cylinders, Journal of Fluid Mechanics, 1998, vol 360, pp 73-84. [2] J. R. Potts and W. J. Crowther, The flow over a rotating disc-wing, RAeS Aerodynamics Research Conference, London, UK, 17-18 Apr. 2000. [3] J. R. Potts and W. J. Crowther, Frisbee Aerodynamics, 20th AIAA Applied Aerodynamics Conference, 24-26 June 2002, St. Louis, Missouri. Figure 5: Friction velocity uτ, λ = 1.89 6

Figure 6: Lower surface pressure coefficient, λ = 0.0, α = 10 Figure 8: Upper surface pressure coefficient, λ = 0.0, α = 10 Figure 7: Lower surface pressure coefficient, λ = 1.89, α = 10 Figure 9: Upper surface pressure coefficient, λ = 1.89, α = 10 7

(a) λ = 0.00, α = 10 (a) λ = 0.00, α = 10 (b) λ = 1.89, α = 10 (b) λ = 1.89, α = 10 Figure 10: Advancing tip vortex Figure 11: Retreating tip vortex 8

(a) λ = 0.0 Figure 13: Drag, side force and yawing moment coefficients at α = 0.0 (b) λ = 1.89 Figure 12: Surface pressure coefficient and wake streamwise normalizied vorticity, α = 10 9