*3.3. Scattering at a Cylinder*

While the one-dimensional setups served well to demonstrate the basic numerical properties of the penalization scheme, they did not show the benefit of this approach. Only with multiple dimensions, the mesh generation was problematic for high-order schemes. Thus, we now turn to the scattering of a two-dimensional acoustic wave at a cylinder. The result was compared against the analytical solution of the linearized equations presented in [20]. In this case, the surface of the object was curved, and the wave did not only impinge in the normal direction of the obstacle. The expected symmetric scattering pattern of the reflected pulse eased the identification of numerical issues introduced by the modeling of the cylinder wall. Thus, this setting illustrated the treatment of curved boundaries in the high-order approximation scheme by penalization within simple square elements. The problem setup is depicted in Figure 10 and consisted of a cylinder of diameter *d* = 1.0 with its center lying at the point *P* with the coordinates (10, 10).

**Figure 10.** Test case setup for the wave scattering; only the section containing the cylindrical obstacle, the probing points, and the initial pulse is shown, and the actual computational domain is larger. The cylindrical obstacle is represented by the black circle located at *P*(10, 10). Five observation points (*A*, ... , *E*) around the obstacle are shown as circles. The initial pulse in pressure is indicated by the black dot with a turquoise circle around it located at *S*(14, 10).

The initial condition prescribed a circular Gaussian pulse in pressure with its center at the point *S* = (14, 10) and a half-width of 0.2. Thus, the initial condition for the perturbation of pressure is given by:

$$p' = \varepsilon \exp\left[-\ln(2)\frac{(x-14)^2 + (y-10)^2}{0.04}\right].\tag{41}$$

The amplitude = 10−<sup>3</sup> of the pulse was chosen to be sufficiently small to nearly match the full compressible Navier–Stokes simulation with the linear reference solution.

The initial condition in terms of the conservative variables was given as:

$$
\rho = \rho\_0 + p',\\
m\_1 = m\_2 = 0,\\
e = c\_p T - \frac{p}{\rho}.\tag{42}
$$

Here, *ρ*<sup>0</sup> is the background density chosen as *ρ*<sup>0</sup> = 1.0. *m*1, *m*<sup>2</sup> are the momentum in the *x* and *y* direction, respectively, and *e* is the total energy. *T* and *cp* are the temperature and specific heat at constant pressure, respectively. The ratio of specific heats was chosen to be *γ* = 1.4. The Reynolds number used was *Re* = <sup>5</sup> × <sup>10</sup><sup>5</sup> , calculated using the diameter of the cylinder *<sup>d</sup>* = 1.0 as the characteristic length. Figure 10 shows the test case setup magnified around the area of interest.

The overall simulation domain was Ω = [0, 24] × [0, 20] to ensure that the boundaries were sufficiently far away to avoid interferences from reflections during the simulated time interval. To test the accuracy of the simulation, five probing points were chosen around the obstacle. The points were located in different directions with respect to the obstacle *P* and the source *S*. The incident and the reflected acoustic wave passed through these probes at different points in time. This intends to address both phase and amplitude errors that arise from the Brinkman penalization. The porosity was set to *φ* = 1.0. The viscous and thermal permeability *η* and *η<sup>T</sup>* were defined respectively with the help of the scaling parameter *β* = 10−<sup>6</sup> according to Equations (33) and (34). Results were obtained solving the compressible Navier–Stokes equations in two dimensions with a spatial scheme order of *O* = 8, i.e., 64 degrees of freedom per element. Cubical elements with an edge length of *dx* = 1/64 were used to discretize the complete domain. The simulation was carried out for a total time of *tmax* = 10 s.

The pressure perturbation in the initial condition resulted in the formation of an acoustic wave that propagated cylindrically outwards as depicted in Figure 11a. Eventually, the wave impinged on the obstacle, where it was reflected as shown in Figure 11b with the pressure perturbation at *t* = 4. The quality of this reflected wave was completely dependent on the quality of the obstacle representation.

(**c**) Pressure perturbation at *t* = 6 (**d**) Pressure perturbation at *t* = 8

**Figure 11.** Simulation snapshots of pressure perturbations captured at successive points in time. The cylindrical obstacle is visible as a black disk and the probe points surrounding it as white dots. The scale of the pressure perturbation is kept constant for all snapshots.

A third wave was generated when the initial wave, disrupted by the obstacle, traveled further to the left and joined again. This is visible in Figure 11c, and its further evolution is visible in Figure 11d, which shows the pressure perturbation at *t* = 8. These three circular acoustic waves had different centers (shifted along the x-axis), but coincided left of the obstacle. As can be seen from these illustrations, the expected reflection pattern was nicely generated by the obstacle representation via the penalization. For a more quantitative assessment of the resulting simulation, we looked at the time evolution of the pressure perturbations at the chosen probing points.

Figure 12 shows the time evolution of the pressure fluctuations monitored at each of the five observation points around the cylinder. The numerical results were compared with the analytical solution for linear equations at these points. Here, we can observe the principal wave and the reflected wave arriving at different probing points at different times. We also observed that the computational results obtained showed an excellent agreement with the analytical solution for all the probes. It nearly perfectly predicted all the amplitudes and pressure behavior without showing phase shifts.

(**e**) Time evolution at Point *E*(8, 10)

**Figure 12.** Time evolution of pressure perturbations at all five observation points surrounding the cylinder up to *t* = 10. Be aware that the perturbation pressure plotted along the *y* axis is scaled differently from probe to probe to illustrate the pressure profile better.
