2.2. Direct simulation

### 2.2.1. Governing equations and finite difference formulation

Both flow and acoustic fields are solved by the two-dimensional compressible Navier-Stokes equations in a conservative form, which is written as:

$$\left(\mathbf{Q}\_t + (\mathbf{E} - \mathbf{E}\_\mathbf{v})\_x + (\mathbf{F} - \mathbf{F}\_\mathbf{v})\_y = \mathbf{0},\tag{3}$$

where Q is the vector of the conservative variables, E and F are the inviscid fluxes, and E<sup>v</sup> and F<sup>v</sup> are the viscous fluxes. The spatial derivatives and time integration were evaluated by the sixth-order accurate compact finite difference scheme [15] and a third-order accurate Runge-Kutta method. To suppress the numerical instabilities associated with the central differencing in the compact scheme, we use a tenth-order accurate spatial filter shown below:

Figure 1. Configurations for flow around a two-dimensional square cylinder.

$$a\_f \hat{\boldsymbol{\varphi}}\_{i-1} + \hat{\boldsymbol{\varphi}}\_i + a\_f \hat{\boldsymbol{\varphi}}\_{i+1} = \sum\_{n=0}^5 \frac{a\_n}{2} (\boldsymbol{\varphi}\_{i+n} + \boldsymbol{\varphi}\_{i-n}) \tag{4}$$

where <sup>φ</sup> is a conservative quantity and <sup>φ</sup><sup>b</sup> is the filtered quantity. The coefficients an are the same as the values used by Gaitonde and Visbal [16], and the parameter α<sup>f</sup> is 0.47.

### 2.2.2. Computational grids and boundary conditions

Figure 2 shows the computational domain and boundary conditions. The coordinates originate from the center of the cylinder. Generally, the nonreflecting boundary conditions based on the characteristic wave relations [17–19] are used at the inflow, upper, and outflow boundaries along with a buffer region. The role of the buffer region is similar to that of the "sponge region" of Colonius et al. [20]. At the wall, the nonslip and adiabatic boundary conditions are used.

For all the cases of M = 0.2–0.6, the same grids are used. The computational domain is divided into three regions of different grid spacings as shown in Figure 2: a vortex region [�4.0 ≤ x/d ≤ 30.0, �4.0 ≤ y/d ≤ 4.0], a sound region [�70.0 ≤ x/d < �4.0, 30.0 < x/d ≤ 70.0, �70.0 ≤ y/d < �4.0, 4.0 < y/d ≤ 70.0], and a buffer region [�500.0 ≤ x/d < �70.0, 70.0 < x/ d ≤ 500.0, �500.0 ≤ y/d < �70.0, 70.0 < y/d ≤ 500.0].

The spacing in the vortex region is prescribed to be fine enough to analyze the separated shear layer and the vortical structures in the wake of the cylinder. Figure 3 shows the computational grid near the cylinder. The spacing adjacent to the cylinder surface is Δxmin/d and Δymin/ d = 0.0025. With this grid distribution, the number of grid points within the separated shear layer for Re = 150 is 22 in the x, y direction (the thickness of the separated shear layer was estimated by δ/d ~ 1/Re0.5 and 0.08 for Re = 150 like the circular cylinder [6]), and the separated shear layer can be sufficiently captured. In the whole vortex region, Δx/d and Δy/d are less than 0.2, where the

number of grid points within a shed vortex is about 15 in the x, y direction (the size of the vortex is estimated to be about 3d by the spacing of the local maxima of vorticity in the computational

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Figure 3. Computational grid near cylinder for direct simulations. Every 10th grid line is shown for clarity.

In the sound region, the spacing is prescribed to be larger than that in the vortex region but still fine enough to capture the acoustic waves. The spacings are Δx/d, Δy/d ≤ 0.23 except for the downstream region of the cylinder. In the downstream region, the largest spacings are Δx/d and Δy/d = 0.46. This is because the acoustic wavelength becomes longer than that in the upstream region due to the Doppler effects. In the whole sound region, more than 20 grid points are used per one acoustic wavelength of the tonal sound at the frequency of the vortex

After many preliminary tests, grid- and domain-size independence has been established for

The two-dimensional Lighthill's equation [Eqs. (1) and (2)] is solved based on the wave equation. Here, the open-source software, FrontFlow/blue-ACOUSTICS, was used. Here, the acoustic simulations are performed in a frequency domain using finite-element methods. A

results) and the vortices were sufficiently analyzed.

shedding, and the acoustic waves are sufficiently captured.

2.3.1. Governing equations and discretization formulation

component perturbed at the frequency f of quantify gf can be written as:

the solutions presented in this chapter.

2.3. Hybrid simulation

Figure 2. Computational domain and boundary conditions for direct simulations.

Figure 3. Computational grid near cylinder for direct simulations. Every 10th grid line is shown for clarity.

number of grid points within a shed vortex is about 15 in the x, y direction (the size of the vortex is estimated to be about 3d by the spacing of the local maxima of vorticity in the computational results) and the vortices were sufficiently analyzed.

In the sound region, the spacing is prescribed to be larger than that in the vortex region but still fine enough to capture the acoustic waves. The spacings are Δx/d, Δy/d ≤ 0.23 except for the downstream region of the cylinder. In the downstream region, the largest spacings are Δx/d and Δy/d = 0.46. This is because the acoustic wavelength becomes longer than that in the upstream region due to the Doppler effects. In the whole sound region, more than 20 grid points are used per one acoustic wavelength of the tonal sound at the frequency of the vortex shedding, and the acoustic waves are sufficiently captured.

After many preliminary tests, grid- and domain-size independence has been established for the solutions presented in this chapter.

### 2.3. Hybrid simulation

<sup>α</sup>fφb<sup>i</sup>�<sup>1</sup> <sup>þ</sup> <sup>φ</sup>b<sup>i</sup> <sup>þ</sup> <sup>α</sup>fφb<sup>i</sup>þ<sup>1</sup> <sup>¼</sup> <sup>X</sup>

2.2.2. Computational grids and boundary conditions

330 Numerical Simulations in Engineering and Science

d ≤ 500.0, �500.0 ≤ y/d < �70.0, 70.0 < y/d ≤ 500.0].


Figure 2. Computational domain and boundary conditions for direct simulations.

Buffer region

Non-reflecting


same as the values used by Gaitonde and Visbal [16], and the parameter α<sup>f</sup> is 0.47.

5

an

<sup>2</sup> <sup>φ</sup><sup>i</sup>þ<sup>n</sup> <sup>þ</sup> <sup>φ</sup><sup>i</sup>�<sup>n</sup>

� �, (4)

n¼0

where <sup>φ</sup> is a conservative quantity and <sup>φ</sup><sup>b</sup> is the filtered quantity. The coefficients an are the

Figure 2 shows the computational domain and boundary conditions. The coordinates originate from the center of the cylinder. Generally, the nonreflecting boundary conditions based on the characteristic wave relations [17–19] are used at the inflow, upper, and outflow boundaries along with a buffer region. The role of the buffer region is similar to that of the "sponge region" of Colonius et al. [20]. At the wall, the nonslip and adiabatic boundary conditions are used.

For all the cases of M = 0.2–0.6, the same grids are used. The computational domain is divided into three regions of different grid spacings as shown in Figure 2: a vortex region [�4.0 ≤ x/d ≤ 30.0, �4.0 ≤ y/d ≤ 4.0], a sound region [�70.0 ≤ x/d < �4.0, 30.0 < x/d ≤ 70.0, �70.0 ≤ y/d < �4.0, 4.0 < y/d ≤ 70.0], and a buffer region [�500.0 ≤ x/d < �70.0, 70.0 < x/

The spacing in the vortex region is prescribed to be fine enough to analyze the separated shear layer and the vortical structures in the wake of the cylinder. Figure 3 shows the computational grid near the cylinder. The spacing adjacent to the cylinder surface is Δxmin/d and Δymin/ d = 0.0025. With this grid distribution, the number of grid points within the separated shear layer for Re = 150 is 22 in the x, y direction (the thickness of the separated shear layer was estimated by δ/d ~ 1/Re0.5 and 0.08 for Re = 150 like the circular cylinder [6]), and the separated shear layer can be sufficiently captured. In the whole vortex region, Δx/d and Δy/d are less than 0.2, where the

500

70

4



*y*/*D*



500

70

30

Non-slip, adiabatic wall

*x*/*D*

Vortex region

Sound region

### 2.3.1. Governing equations and discretization formulation

The two-dimensional Lighthill's equation [Eqs. (1) and (2)] is solved based on the wave equation. Here, the open-source software, FrontFlow/blue-ACOUSTICS, was used. Here, the acoustic simulations are performed in a frequency domain using finite-element methods. A component perturbed at the frequency f of quantify gf can be written as:

$$\mathbf{g}\_f = \tilde{\mathbf{g}}\_f(\mathbf{x}) \mathbf{e}^{i2\pi ft}.\tag{5}$$

r0=d ¼ 80:0, L=d ¼ 20, (10)

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Figure 4 shows the computational grid for the acoustic simulations. The spacing adjacent to the cylinder surface is Δxmin/d and Δymin/d = 0.1. In the whole domain, the grid spacing is less than 0.52, and more than 20 grid points are used per acoustic wavelength. The preliminary computations confirmed that the acoustic waves are sufficiently analyzed with these grid resolutions. The reflecting conditions are adopted on the cylinder wall. On the other boundaries, the

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

where r is the distance from the center of the cylinder.

2.3.2. Computational grids and boundary conditions

nonreflecting boundary conditions are adopted.

3. Validation of computational methods

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

3.1. Validation of direct simulations

Using Eq. (5), Lighthill's equation can be written as:

$$\frac{\partial^2 \tilde{\rho}\_f}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} + k^2 \tilde{\rho}\_f = -\frac{1}{a\_0^2} \frac{\partial^2 \tilde{T}\_{i\sharp f}}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} \,. \tag{6}$$

where k = 2πf/a<sup>0</sup> is the wavenumber. The right-hand side of Eq. (6) is computed by the results of the direct simulation in the present chapter. Also, acoustic waves at the frequency of the vortex shedding are focused on. To minimize the spurious errors, the computed acoustic sources are reduced smoothly to zero near the outflow boundary of the acoustic simulation by using the filter. Figure 4 shows the computational domain for the acoustic simulations and the outer shape of computational domain is circular with the cylinder at its center, and the radius is 100D. The above-mentioned filter is defined as:

$$
\widehat{A} = A \times \mathbb{G}(r - r\_0),
\tag{7}
$$

$$A = \mathfrak{d}^2 \frac{\tilde{T} \ddot{\eta}\_{\prime} f}{\partial \text{xi} \ddot{\partial} \text{x} \dot{\jmath}^{\prime}} \, \tag{8}$$

$$G(r\_{\rm d}) = \begin{cases} \frac{1}{2} \left( 1 + \cos \frac{r\_{\rm d}}{L} \pi \right) (80.0 < r \le 100.0) \\\\ 1.0 & (r \le 80.0) \end{cases} \tag{9}$$

Figure 4. Computational grids and boundary conditions for acoustic simulations. (a) Overall grids (every fifth grid line is shown for clarity) and (b) grids near cylinder (every grid line is shown).

$$r\text{0/d} = 80.0, \text{L/d} = 20,\tag{10}$$

where r is the distance from the center of the cylinder.

### 2.3.2. Computational grids and boundary conditions

gf <sup>¼</sup> <sup>g</sup>~fð Þ<sup>x</sup> <sup>e</sup><sup>i</sup>2<sup>π</sup> ft: (5)

Ab ¼ A � G rð Þ � r<sup>0</sup> , (7)

ð Þ 80:0 < r ≤ 100:0

<sup>∂</sup>xi∂xj , (8)

,

(9)

, (6)

Using Eq. (5), Lighthill's equation can be written as:

332 Numerical Simulations in Engineering and Science

100D. The above-mentioned filter is defined as:

G rð Þ¼ <sup>d</sup>

shown for clarity) and (b) grids near cylinder (every grid line is shown).

1 2

8 >><

>>:

∂<sup>2</sup>ρ~<sup>f</sup> ∂xi∂xj

þ k 2 <sup>ρ</sup>~<sup>f</sup> ¼ � <sup>1</sup> a2 0

where k = 2πf/a<sup>0</sup> is the wavenumber. The right-hand side of Eq. (6) is computed by the results of the direct simulation in the present chapter. Also, acoustic waves at the frequency of the vortex shedding are focused on. To minimize the spurious errors, the computed acoustic sources are reduced smoothly to zero near the outflow boundary of the acoustic simulation by using the filter. Figure 4 shows the computational domain for the acoustic simulations and the outer shape of computational domain is circular with the cylinder at its center, and the radius is

> <sup>A</sup> <sup>¼</sup> <sup>∂</sup><sup>2</sup> Tij, f ~

rd L π � �

Figure 4. Computational grids and boundary conditions for acoustic simulations. (a) Overall grids (every fifth grid line is

1:0 ð Þ r ≤ 80:0

1 þ cos

∂<sup>2</sup>T~ij,f ∂xi∂xj

Figure 4 shows the computational grid for the acoustic simulations. The spacing adjacent to the cylinder surface is Δxmin/d and Δymin/d = 0.1. In the whole domain, the grid spacing is less than 0.52, and more than 20 grid points are used per acoustic wavelength. The preliminary computations confirmed that the acoustic waves are sufficiently analyzed with these grid resolutions.

The reflecting conditions are adopted on the cylinder wall. On the other boundaries, the nonreflecting boundary conditions are adopted.
