PREDICTION OF FLOW-INDUCED NOISE USING THE EXPANSION
ABOUT INCOMPRESSIBLE FLOW APPROACH
Astrid Schulze and Otto von Estorff Institute of Modelling and Computation
Hamburg University of Technology Denickestrasse 17, 21073 Hamburg, Germany e-mail: a.schulze@tu-harburg.de, estorff@tu-harburg.de
Web page: http://www.mub.tu-harburg.de
Key words: Aero-Acoustics, Incompressible Flow, Nonlinear Acoustics
Abstract. In the present paper an Expansion about Incompressible Flow approach, as
suggested by Shen and Sørensen1, will be used and discussed for the simulation of
flow-induced sound at low Mach numbers. The method consists of two parts: First solving for the incompressible, viscous flow field, and second computing the fluctuating acoustic field. As a result of this splitting approach, one can choose optimized numerical methods and grid sizes for each of the two parts. In the current contribution an explicit finite difference scheme is employed to solve the nonlinear acoustic equations, and the propagation and scattering of the sound are computed simultaneously. The acoustic results are presented for the sound generated by a circular cylinder in a flow at the Reynolds number of 150. In particular, the influence of the discretization parameters and the implementation of numerical filters are discussed and the accuracy of this hybrid method is compared to other approaches.
1 INTRODUCTION
Flow induced noise appears in many technical applications. In the majority of cases, it needs to be reduced or even completely avoided. Therefore, during the early design stage of a product, one of the most essential tasks is the localization and treatment of relevant sound sources. This can be done either by testing expensive prototypes or, nowadays, also by using computational methods which are setup to predict the generation and propagation of flow-induced noise long before a prototype is available. These tools, however, are highly sophisticated and currently under development, since their usage is still limited.
Navier-Stokes equations into mean flow quantities and acoustic perturbations. The acoustic field is assumed to be inviscid and computed from the e.g. linearized Euler equations5 or from acoustic perturbation equations6.
In the present study the generation and propagation of sound at low Mach numbers is calcu-lated by an Expansion about incompressible flow (EIF) approach as suggested by Shen and Sørensen1. The method consists of two parts and the sound sources will be directly computed. First of all, the time dependent, viscous flow field is obtained by solving the incompressible Navier-Stokes-Equation and second, the acoustic quantities are computed as a difference between the compressible and the incompressible equations, neglecting the viscous terms. 2 APPROACH
2.1 Incompressible solution
As a first step, the incompressible Navier-Stokes equations 0 = ∂ ∂ i i x U , (1)
(
)
j i i i j j i i x x U x P x U U t U ∂ ∂ ∂ + ∂ ∂ − = ∂ ⋅ ∂ + ∂ ∂ 2 0 1 ν ρ , (2)with ν defined as kinematic viscosity, need to be solved in order to get the flow field, specified by the incompressible pressure distribution P, the velocity field Ui, and the constant density ρ0 for every moment in time t. As soon as the results at the time t are known, the computation of the acoustic field can be started. Therefore the incompressible solution will be transferred from the hydrodynamic grid to the acoustic grid by an interpolation step.
2.2 Acoustic field
and neglecting the viscous terms of the acoustic perturbations, leads to the nonlinear, first order system of partial differential equations
0 = ∂ ∂ + ∂ ∂ i i c x f t ρ (5)
(
)
0 = ∂ ⋅ + ⋅ ⋅ + ⋅ ∂ + ∂ ∂ j ij c c j i 0 j i i x δ p u U ρ u f t f (6) with c c i i i u U f =ρ⋅ +ρ ⋅ .In general, the well-known energy equation is used to calculate the pressure field, since p is a function of the density ρ and entropy S. Shen and Sørensen pointed out, however, that the effects of viscosity and heat conduction are rather slow on the acoustic time scale, such that only the averaged incompressible pressure leads to losses and the pressure fluctuations can therefore be assumed to be isentropic.
Instead of using the energy equation one gets the relation
t c t p 2 ∂ ∂ ⋅ = ∂ ∂ ρ (7) 1.4 , with 0 2 = + + = γ ρ ρ γ c c p P c
where c represents the local speed of sound and γ is the ratio of the specific heat for an airflow.
Inserting equation (5) into (7) finally yields
t P x f c t p i i c ∂ ∂ − = ∂ ∂ ⋅ + ∂ ∂ 2 (8) The method is based on the assumption, that the flow field actually influences the generation and propagation of sound, but on the other hand there is no feedback from the acoustics to the hydrodynamic field.
3 NUMERICAL METHOD
In the present study an explicit finite difference scheme, namely a MacCormack predictor-corrector scheme of order 2-4, is used to solve the acoustic equations (5), (6), and (8).
In order to eliminate grid-to-grid oscillations and to stabilize the explicit finite-difference scheme, a seven-point spatial filter as recommended by Kim et al.7 is employed. The filter is designed to obtain an amplification factor of zero for a wave length λ equal or smaller than 2 times the grid spacing, and to reach almost 1 for wave lengths λ greater 7 times the grid spacing. In order to minimize the numerical damping, the filter is only applied periodically after a certain number of time steps.
4 EXAMPLE
As an example, the sound generated by the 2-dimensional flow around a circular cylinder at a Reynolds number of Re=150 and a Mach number of M=0.3 is computed. All parameters are made non-dimensional and based on diameter D of the cylinder, the density ρ0, and the
speed of sound c0. Figure 1 shows a sketch of the 2-dimensional geometry.
U0 x φ r y D
Figure 1: 2-Dimensional flow around a circular cylinder
The solution of this example will be compared with the results of a Direct Numerical Simulation performed by Inoue and Hatakeyama8.
The incompressible flow field was obtained by means of a finite element software. After several periods of time, the initial disturbances vanished and a periodic, unsteady solution has been developed. This is the point where the calculation of the acoustic field may be started. Figure 2 shows the incompressible surface pressure P~ as a function of the time for different angles. The Strouhal number St for the Reynolds number 150 is calculated as 0.182.
P~
In Figure 3 the developments of the lift coefficient CL and the drag coefficient CD are shown
as a function of time t*=M·(t-t0). The amplitude of the lift coefficient CL is 0.51 und the mean
value of the drag coefficient CD is 1.31. All values are in very good agreement with the
computational values (St 0.183, CL 0.52, CD 1.32) reported by Inoue and Hatakeyama8.
t*=M (t-t0) CL
CD
Figure 3: Lift coefficient CL and drag coefficient CD
The computation of the acoustic solution can be started when the incompressible solution at the time t is known. In the present study the equations are transformed into cylindrical coordinates and the grid points in radial direction have been clustered by the algebraic function
( )
( )
1 0 with 1 ) 1 ( 1 1 ) 1 ( ) 1 ( ) 1 ( ) r r ( r ) ( r 1 1 min max min ≤η≤ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − β + β + ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − β + β ⋅ − β − + β ⋅ − + = η −η η − , (9)where rmin=0.5 and rmax=140.
Figure 4 shows the grid spacing for different clustering parameters β. The number of grid points in radial direction is 100 and the maximal grid spacing is limited to 1.2.
β =1.15 β =1.10
β =1.15, Δ ≤ 1.2 β =1.10, Δ ≤ 1.2
The results of the incompressible solution are transferred to the acoustic grid and replaced by the undisturbed flow field outside r = 60.
Figure 5 shows the contour plots for the incompressible pressure field and the acoustic pressure field after 80 periods. A filter has been used every 20 time steps for an inner region (20 grid points in r-direction) and every 200 time steps otherwise.
(a) (b)
(c) (d)
Figure 5: Contour plots of the pressure distribution for t=1450 (80 periods) and 100x60 grid points. (a) Incompressible solution P-p0; (b) acoustic field pC; (c) total pressure Δp = P + pC -p0; (d) pressure fluctuations
mean p p p~=Δ −Δ
Δ . The contour levels are [-0.05:0.0025:0.05] for (a) and [-0.1M2.5:0.0025M2.5:0.1M2.5] otherwise. The time history of the pressure fluctuation Δp~=Δp−Δpmean at r=75 and φ=90° is shown in Figure 6. The amplitude of the fluctuating pressure is close to the results presented by Inoue and Hatakeyama8.
intervals between the filtering for three different cases compared to each other. p~ (a) (b) Δ p~ Δ t
Figure 6: Time history of the pressure Δp~=Δp−Δpmean at r=75, φ =90° (a) Result from present calculation (b) Copy of the results from a DNS calculation published by Inoue and Hatakeyama 8
Case 1 Case 2 Case 3
Number of grid points (Nr x Nφ) 100 x 60 100 x 60 100 x 90
Grid clustering parameter β 1.15 1.15 1.10
Inner region 20 grid points 20 grid points 40 grid points
Filter in the inner region Every 20 steps Every 20 steps Every 10 steps Filter in the outer region Every 200 steps Every 20 steps Every 60 steps
Table 1: Number of grid points and filter intervals for different cases
Figure 7 shows the radial pressure distribution and the decay of the pressure peaks pmax=max (Δp~) in the radial direction for φ=90° for the 3 cases.
clearly observe an improvement of the pressure distribution at r greater 60, even if the time intervals between two filter steps have been reduced.
(a) (b) p~ Δ 2.5 M p~ Δ r r
Figure 7: (a) Radial pressure distribution Δp~=Δp−Δpmeanat t=725 (40 periods) and (b) Decay of pressure peaks p max / M2.5 at φ = 90° for different time intervals between filtering the acoustic solution: Case 1 (dashed line); case 2 (dashed dotted line); case 3 (solid line)
5 CONCLUSIONS
A calculation of flow induced sound using an Expansion about Incompressible Flow has been presented. The acoustic solution was obtained by an explicit finite difference scheme and the results of the flow around a circular cylinder at Re=150 and M=0.3 were compared to the computational values of a direct numerical simulation reported in the literature. Compared to the DNS, the acoustic grid (100 x 60 grid points) is quite coarse. Nevertheless, the results for the incompressible solution and the solution of the total pressure field are in good agreement.
It could be shown that the numerical filter influences the results at r>60, where the number of grid points per wave length decreases. As expected, a refinement of the grid in circumferential direction could improve the solution at the expense of applying the filter equations more frequently.
REFERENCES
[1] W.Z. Shen and J.N. Sorensen: “Aeroacoustic modelling of low-speed flows.” The Danish
Center for applied mathematics and mechanics, Report 590, (1998).
[2] N. Curle: “The influence of solid boundaries upon aerodynamic sound”, Proceedings of
the Royal Society of London, A 231 (1955), pp. 505-514.
[4] M.J. Lighthill: “On Sound Generated Aerodynamically: Part I: General Theory”,
Proceedings of the Royal Society of London, A211 (1952), A211, pp. 564-587.
[5] C. Bailly, D. Juve: “Numerical Solution of Acoustic Propagation Problems Using Linearized Euler Equations”. AIAA Journal, Vol. 38, No. 1 (2000), pp. 22-29.
[6] R. Ewert, W. Schröder: “Acoustic Perturbation Equations Based on Flow Decomposition via Source Filtering”, J. Comp. Phys., Vol. 188 (2003), pp. 365–398.
[7] Y. Kim, D.C. Kring, P.D. Sclavounos: “Linear and nonlinear interactions of surface waves with bodies by a three-dimensional Rankine panel method.” Applied Ocean
Research, 19 (1997), pp. 235-249.
