Computational Aeroacoustics (CAA) : Fundamental and its applications

Computational Aeroacoustic (CAA) deals about capturing radiated acoustic quantities generated from flow fluctuations numerically. In general, the amplitude of acoustics is less than 4 th order of the flow. Therefore, a higher order scheme, such as compact scheme, is employed to capture the acoustic and flow at the same time. To minimize the numerical phase error in the radiated acoustics, the coefficients of the high order scheme are optimized to have minimum dispersion error. Several high order high resolution schemes are compared in the point of numerical wave resolutions. The high order optimized compact schemes are applied in the acoustic from the cylinder, cavity tone and screech tone in the supersonic jet. The result of cylinder shows that the optimized compact scheme yields precise results and results of cavity tone and screech tone shows feedback phenomena. Additionally, other effective methods for numerical analysis of incompressible flow noise are addressed and discussed. The need for a finite difference scheme which has a spatial and temporal high-order of truncation has been increased for computational aeroacoustics (CAA). In addition to the high-order of truncation, the highresolution of the scheme has been emphasized because it will determine the number of grid points per wave length required to resolve the shortest wave component in the actual computation. Unsteady compressible Navier-Stokes or Euler equations are used to solve the flow and acoustics. Acoustic radiations from a circular cylinder at low Reynolds number are calculated using the method. Resonant phenomena owing to the feedback from the radiated acoustics propagating to the flow are also calculated. Special treatments at low speed incompressible flow region are discussed including the flow-acoustic resonant problem. Optimized Compact Scheme Generally, the amplitude of acoustics is less than 4 th order of the flow. And there exist many methods (Tam & Webb, 1993, and Lele, 1992) to analyze noise of small amplitude precisely. In this paper one of schemes, optimized high order compact (OHOC) scheme (Kim & Lee, 1995), is introduced. These methods are validated in the benchmark problem workshop of CAA. High-order and high-resolution numerical schemes are applied in a generalized structured grid system to analyze the flow and the acoustic waves. OFOP (optimized fourth-order penta-diagonal) compact scheme is used for evaluating the flux derivatives. The optimized coefficients of OHOC scheme provide the high-order accuracy and the maximum resolution for the central compact schemes. The maximum resolution characteristics of the OHOC schemes are compared with those of other standard central schemes in Figure 1. Combined with high-order finite difference schemes in space, the LDDRK (low dissipation and dispersion Runge-Kutta) scheme is used for integrating the governing equations in time. The adaptive nonlinear artificial dissipation model (Kim & Lee, 1997) and generalized characteristic boundary conditions (Kim & Lee, 2000) are used to prevent unwanted non-physical results. a : second-order central differences b : fourth-order central differences c : sixth-order central differences d : Tam's DRP scheme in space e : exact differentiation 1 : standard Pade' scheme 2 : sixth-order tridiagonal scheme ( c = 0 ) 3 : OSOT (optimized sixth-order tridiagonal) scheme 4 : eighth-order tridiagonal scheme 5 : Lele's fourth-order spectral-like pentadiagonal scheme 6 : OFOP (optimized fourth-order pentadiagonal) scheme 7 : tenth-order pentadiagonal scheme Figure 1. Maximum resolution characteristics of the OHOC schemes in comparison with the other schemes Figure 2 shows an example of application of OHOC scheme to cylinder problem. The OHOC scheme yields good reproduction of sound around a cylinder. (a) Figure 2. Pressure contour around a cylinder Feedback Mechanism Screech tone and cavity tone are generated due to the feedback between flow and acoustic wave. It is recognized that the period is determined by the time required for the flow convection in one direction, the time required for the acoustic propagation in the other direction and the time for phase shift depending on the flows and mode. Using the numerical algorithm described earlier, we have simulated the screech tone of a supersonic jet (Lee et al., 2004) numerically. Figure 3 shows the instantaneous pressure contour of a Mach number 1.15 cold jet. From the contour, we observe not only the Mach wave that propagates downstream but also screech tone, which propagates upstream. Figure 3. Density contour of supersonic jet (Mj=1.15) Figure 4. Acoustic field of cavity tones (L/D=2, M∞=0.5, θ/D=0.04 and Reθ=U∞θ/ν=200) Figure 4 is the acoustic field of cavity flow. (Heo et al., 2004) Vortices generated from the leading edge convect downstream in the shear layer and impinge on the downstream edge. This impingement induces a large pressure fluctuation, which becomes the major acoustic source. The acoustic waves are generated near the downstream edge in the cavity and the most intense waves radiate in a forward-upper direction. Incompressible Flow Treatments There are many treatments to handle the low speed flow noise problems. Lee & Koo (1995) used splitting method for noise prediction of a spinning vortex. It splits the flow and acoustic variables to resolve difficulties in scale disparity, especially for low Mach number flows. Hydrodynamic density concepts (Hardin & Pope, 1990) are used for the quadrupole noise. Moon & Seo (2003) used incompressible CFD (computational fluid dynamics) solver for the flow solver, and perturbed acoustic equation for the noise radiation. It can be applied to complicated noise source problems. A new method for simulating phenomena of flow locking is developed by the authors. It uses both acoustic mode analysis and CFD solver. Figure 5 shows the results of calculation for the flow locking in a whistle geometry. Using this method, we can simulate acoustic flow locking, which is dominant in specific resonance frequency region. 100