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European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2004 a P. Neittaanm¨ki, T. Rossi, K. Majava, and O. Pironneau (eds.) e rıˇ J. P´riaux and M. Kˇ´zek (assoc. eds.) a a Jyv¨skyl¨, 24–28 July 2004 COMPUTATIONAL STUDY OF AEOLIAN TONES Osamu Inoue , Nozomu Hatakeyama , Ayaka Imamura , Tomohiro Irie , and Sakari Onuma Institute of Fluid Science, Tohoku University, 2-1-1 Katahira, Aoba-ku, Sendai 980-8577, Japan e-mail: inoue@ifs.tohoku.ac.jp, hatakeyama@ifs.tohoku.ac.jp, imamura@miro.ifs.tohoku.ac.jp, irie@miro.ifs.tohoku.ac.jp, ohnuma@ifs.tohoku.ac.jp Key words: Computational Aeroacoustics, DNS, Aeolian Tones. Abstract. Direct Numerical Simulation results of aeolian tones generated by a two- dimensional obstacle (square cylinder and NACA0012 airfoil) in a uniform ﬂow are pre- sented and the generation and propagation mechanisms of the sound are discussed. The two-dimensional unsteady compressible Navier-Stokes equations are solved by a highly- accurate ﬁnite diﬀerence scheme over the entire region from near to far ﬁelds. Results show that sound pressure waves are generated in response to vortex shedding. In both cases of square cylinder and NACA0012 airfoil, the generation and propagation mechanisms are aﬀected by the angle of attack. 1 O. Inoue, N. Hatakeyama, A. Imamura, T. Irie and S. Onuma 1 INTRODUCTION Recent development of a high-performance supercomputer and highly-accurate numeri- cal schemes makes it possible to simulate a sound ﬁeld by directly solving the compressible Navier-Stokes equations over the entire region from near to far ﬁelds [1] [2] [3]. Unlike the hybrid method which uses the acoustic analogy, direct numerical simulation (DNS) does not suﬀer from restrictions such as low Mach number and compactness of the source region, but requires a large amount of computer resources; the studies using DNS are few. Aeolian tone is the sound generated by a ﬂow around an obstacle and one of the most important problems in aeroacoustics. Recently, Inoue and Hatakeyama [3] studied by DNS the aeolian tone generated by a two-dimensional (2D) circular cylinder in a uniform ﬂow and clariﬁed the generation and propagation mechanisms in some detail. In this paper, we apply the method of Inoue and Hatakeyama to the aeolian tone problems generated by a 2D square cylinder and a 2D NACA0012 airfoil in a uniform ﬂow, and try to clarify the generation and propagation mechanisms of these sounds. 2 NUMERICAL METHOD Mathematical formulation and numerical procedure are quite similar, in many respects, to those in the circular cylinder case [3]. The 2D unsteady compressible Navier-Stokes equations are solved by a highly-accurate ﬁnite diﬀerence scheme over the entire region e from near to far ﬁelds. A sixth-order-accurate compact Pad´ scheme [4] was used for spatial derivatives. The fourth-order-accurate Runge–Kutta scheme was applied for time- marching. Non-reﬂecting boundary conditions [5] were applied for outer boundaries, and non-slip and adiabatic conditions were used for obstacle surfaces. The molecular viscosity and the thermal conductivity are assumed to be constant. The Prandtl bumber is assumed to be 0.75, and the ratio of speciﬁc heats is 1.4. For the case of a square cylinder, the Mach number M of the uniform ﬂow is prescribed to be either 0.1 or 0.2. The Reynolds number based on the diameter of the cylinder is ﬁxed to be Re = 150. The angle of attack α of the square cylinder is prescribed to be less than α ≤ 45◦ , taking into account the symmetry of the body shape. We adopt a rectangular grid (L-grid) system with non-uniform meshes. The total number of grid points is about 3 × 106 . For the case of an NACA0012 airfoil, the Mach number of the uniform ﬂow is prescribed to be M = 0.1 and 0.2, again. The Reynolds number based on the chord length is prescribed to be Re = 300 and 5000. The angle of attack α is prescribed to be less than 20◦ . In this case, we used a generalized coordinate system (ξ, η), where ξ is the coordinate tangential to the airfoil surface measured from the origin (the leading edge of the airfoil) and η is the coordinate normal to the airfoil surface. The grid system consists of two parts; a C-grid part around the airfoil and a rectangular-grid (H-grid) part in the downstream region. The two grid systems are connected smoothly at the plane of the trailing edge. The total number of grid points is, typically, 1.3 × 106 . 2 O. Inoue, N. Hatakeyama, A. Imamura, T. Irie and S. Onuma 3 RESULTS AND DISCUSSION 3.1 Square cylinder Typical examples of computational results for the case of a square cylinder are pre- sented in Figures 1 and 2; an instantaneous vorticity ﬁeld is shown in Figure 1 and an instantaneous ﬂuctuation pressure ﬁeld is in Figure 2, respectively. The Mach number is M = 0.2, the Reynolds number is Re =150, and the angle of attack is α = 20◦ . The p p ﬂuctuation pressure ∆˜ is deﬁned by ∆˜ = ∆p − ∆pmean , where ∆p(= p − p∞ ) is the pressure, p∞ is the ambient pressure and ∆pmean is the time-averaged pressure. In the ﬁgures, clockwise vortices and positive ﬂuctuation pressures are shown by dashed lines; anticlockwise vortices and negative ﬂuctuation pressures are shown by solid lines. As shown by Imamura, Hatakeyama and Inoue [6], when α = 0◦ , clockwise and an- ticlockwise vortices are shed alternately from the upper and lower sides of the square cylinder, and pressure waves are generated periodically in response to vortex shedding, as in the case of a circular cylinder. When a vortex is shed from the lower side of the cylinder, a negative pressure pulse is generated on the lower side whereas a positive pres- sure pulse is generated on the upper side; CL becomes negative in this half-cylce of vortex shedding. On the other hand, when a vortex is shed from the upper side of the cylinder, a negative pressure pulse is generated on the upper side whereas a positive pressure pulse is generated on the lower side; CL becomes positive in this half-cylce. Therefore, the pres- sure ﬁeld is dominated by dipoles, especially by lift dipole, and the generation frequency of the pressure pulses is equal to the vortex shedding frequency. When M = 0.2 and Re =150, the Strouhal number St is 0.151 which is smaller than St = 0.183 in the circular cylinder case. With an increasing angle of attack α, the boundary layer on the upper surface of the cylinder separates from the leading corner of the cylinder whereas that on the lower surface separates from the trailing corner, as shown in Figure 1. Figure 2 shows that the generated pressure pulses propagate asymmetrically with respect to the y=0-plane; the pressure pulses propagate more upstream on the upper plane whereas less upstream on the lower plane than in the α = 0◦ case. With increasing α, the amplitude of CL increases p and the amplitude of the generated ﬂuctuation pressure ∆˜ also increases. 3.2 NACA0012 airfoil A typical example of an instantaneous ﬂow ﬁeld for the case of an NACA0012 airfoil is shown in terms of the vorticity ω in Figure 3 and in terms of the ﬂuctuation pressure ∆˜ in Figure 4, respectively, for M = 0.2, Re=5000, and α = 5◦ . In this case as well p as in the square cylinder case, sound pressure waves are generated in response to vortex shedding. As we can see from Figure 3, the upper boundary layer (on the suction side) separates from the airfoil surface and rolls up to form a clockwise vortex. On the other 3 O. Inoue, N. Hatakeyama, A. Imamura, T. Irie and S. Onuma 2 0 -2 0 5 10 15 Figure 1: vorticity ω ﬁeld for the case of a square cylinder. M = 0.2, Re=150, α = 20◦ . t = 1372. hand, the lower boundary layer (on the pressure side) separates from the trailing edge and rolls up to form an anticlockwise vortex. When a vortex rolls up on the suction side, a negative pressure pulse is generated on the suction side and propagates upward whereas a positive pressure pulse is generated near the trailing edge and propagates downward on the pressure side. On the other hand, when the lower boundary layer rolls up to form an anticlockwise vortex, a negative pressure pulse is generated near the trailing edge and propagates downward on the pressure side whereas a positive pressure pulse is generated on the suction side and propagates upward; the generated pressure pulses have a dipolar nature. With an increasing α, the separation and roll-up points of the upper boundary layer on the suction side proceeds upstream and the generation point of the pressure waves also proceeds upstream. On the other hand, the roll-up point of the lower boundary layer on the pressure side is ﬁxed at the trailing edge, independent of α; negative pressure waves are also generated near the trailing edge, irrespective of α. 4 CONCLUSIONS Aeolian tones generated by a two-dimensional square cylinder and NACA0012 airfoil in a uniform ﬂow are simulated by DNS. Results have shown that the generation and propagation mechanisms are aﬀected by the angle of attack in both cases of square cylinder and NACA0012 airfoil. 4 O. Inoue, N. Hatakeyama, A. Imamura, T. Irie and S. Onuma 20 10 0 -10 -20 -20 -10 0 10 20 Figure 2: Fluctuation pressure ∆˜ ﬁeld for the case of a square cylinder. M = 0.2, Re=150, α = 20◦ . t p = 1372. REFERENCES [1] T. Colonius, S. K. Lele and P. Moin. The Scattering of sound waves by a vortex: numerical simulation and analytical solutions. J. Fluid Mech., 260, 271–298, 1994. [2] B. E. Mitchell, S. K. Lele and P. Moin. Direct computation of sound generated by vortex pairing in an axisymmetric jet. J. Fluid Mech., 383, 113–142, 1999. [3] O. Inoue and N. Hatakeyama. Sound generation by a two-dimensional circular cylin- der in a uniform ﬂow. J. Fluid Mech., 471, 285–314, 2002. [4] S. K. Lele. Compact ﬁnite diﬀerence schemes with spectral-like resolution. J. Comput. Phys., 103, 16–42, 1992. [5] T. Poinsot and S. K. Lele. Boundary conditions for direct simulation of compressible viscous ﬂows. J. Comput. Phys., 101, 104–129, 1992. [6] A. Imamura, N. Hatakeyama and O. Inoue. Numerical analysis of sound generation by a two-dimensional cylinder in a uniform ﬂow. Fourth ASME/JSME Joint Fluids Engineering Conference, Paper No. FEDSM2003-45754, 2003. 5 O. Inoue, N. Hatakeyama, A. Imamura, T. Irie and S. Onuma 0.3 0 -0.3 0 0.5 1 1.5 2 Figure 3: Instantaneous vorticity ﬁeld for the case of NACA0012 airfoil. M = 0.2, Re=5000, α = 5◦ . t = 1001. 3 2 1 0 -1 -2 -3 -2 -1 0 1 2 3 Figure 4: Fluctuation pressure ∆˜ ﬁeld for the case of NACA0012 airfoil. M = 0.2, Re=5000, α = 5◦ . t p = 1001. 6