COMPUTATIONAL STUDY OF AEOLIAN TONES by mikeholy

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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.




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                  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 .


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                  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


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                      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.

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                     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.




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                    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.




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