3. Validation of computational methods

### 3.1. Validation of direct simulations

Figure 5 shows the Strouhal number of vortex shedding predicted by the present direct simulations. The Strouhal number St is the frequency nondimensionalized by the freestream velocity U<sup>0</sup> and the side length of the cylinder d. The present results are compared with the results of the past incompressible simulation (St = 0.155) [13] and those of the past experiment (St = 0.162) [14] for the same Reynolds number. The flow condition of the experiment is approximately incompressible. The present Strouhal numbers for all the Mach numbers are slightly lower than those in past results. The present computational results show that the Strouhal number becomes lower as the freestream Mach number becomes higher. This is

Figure 5. Effects of Mach number on frequency of vortex shedding.

related to the variation of vortices with the variation of the Mach number as discussed in detail in Section 4.1. Also, based on the present results, the extrapolated Strouhal number at M = 0.0 is 0.155. This value agrees well with the past computational data [13], although it is not clear why the past experimental value [14] is slightly higher. Consequently, the present direct simulations are confirmed to be validated.

### 3.2. Validation of hybrid simulation

Figure 6 shows the polar plots of the sound pressure levels at r/d = 30.0 predicted by direct and hybrid simulations for M = 0.4. The acoustic field by the hybrid simulation is approximately in good agreement with that by the direct simulation. It has been confirmed that the two fields also agree for other Mach numbers such as M = 0.2 and 0.6. The above-mentioned methods of hybrid simulation are clarified to be validated.

is 0.151 and that for M = 0.6 is 0.144. Here, to clarify the reason, the Strouhal number becomes

Direct and Hybrid Aeroacoustic Simulations Around a Rectangular Cylinder

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Figure 8(a) shows the mean streamwise velocity at x/D = 1.0 for M = 0.2, 0.4, and 0.6. Figure 8(b) shows the half-value width of that profile, dh, for M = 0.2–0.6. The half-value width is shown to increase as the freestream Mach number becomes higher. Also, Figure 9 shows the mean streamwise Reynolds stress u1rms/U0. This figure shows that the Reynolds stress becomes larger as the freestream Mach number becomes higher. This means that the velocity fluctuations of the vortices intensify. Due to this intensification, the recovery of the mean streamwise in the wake becomes more rapid and the wake becomes wider as mentioned above. This change is different

Figure 8. (a) Mean streamwise velocity at x/D = 1.0 (M = 0.2, 0.4, and 0.6) and (b) half-value width of mean streamwise

lower, and the flow fields are discussed.

velocity at x/D = 1.0 for M = 0.2–0.6.

Figure 7. Contours of vorticity ωz/(U0/d). (a) M = 0.2, (b) M = 0.4, and (c) M = 0.6.

Figure 6. Polar plots of sound pressure levels by direct and hybrid simulations at r/d = 30.0 for M = 0.4.
