Propagation of spherical weak blast waves over rough periodic surfaces

Abstract

Spherical weak blast propagation above a rough periodic surface is investigated by performing numerical simulations of the Euler equations. The study of the reflection pattern shows that waves diffracted by the surface asperities merge to form an effective reflected shock. It is initially detached from the incident shock but gradually catches up with it. If the source energy is sufficient, the reflected shock interacts with the incident one and Mach reflection occurs. Thus, the triple point has a similar trajectory to that over a smooth surface. In addition, the maximal overpressure is shown to be greater for small roughness scales in a layer near the surface. Far from the surface, it is close to that of a smooth surface for small roughness scales and to the free field for the highest ones. The increase in the maximal overpressure is related to oscillations on the waveforms that appear behind the shock. These properties are associated with the existence of a surface wave that propagates along the surface. Comparison of results in the linear regime with an analytic solution confirms this explanation.

Appendices

Appendix 1: Grid convergence

In this appendix, results of a grid convergence study are presented. A rough surface with $h=0.01$ is considered. It corresponds to the smallest roughness scale investigated, with the least number of points per period. It is thus the most restrictive case. Simulations are performed for five grid sizes along the $\xi$-direction, ranging from ${\Delta } \xi = 1\times 10^{-3}$ to $2\times 10^{-3}$, with $1.5\times 10^{-3}$ being the reference grid size used in the paper. The number of points per period of the rough surface $2h/{\Delta } \xi$ is thus between 10 (for ${\Delta } \xi = 2\times 10^{-3}$) and 20 (for ${\Delta } \xi = 1\times 10^{-3}$). For all simulations, the grid size along the $\eta$-direction is $4\times 10^{-3}$ and the CFL number is set to 0.28.

Fig. 22

Waveforms of the overpressure for $h=0.01$ at $r=30$ and $z=0.2$ for several grid sizes

The waveforms of the overpressure at $r=30$ and $z=0.2$ are plotted in Fig. 22 for the five grid sizes. Overall, an excellent agreement is obtained for the different grid sizes. In particular, the peak overpressure only fluctuates by 4% between the finest mesh and the coarsest mesh. The effect of the grid size on the waveforms can be noticed on the oscillations after the first peak, whose period is about 0.18. They appear smoothed for the coarsest mesh, and their peak-to-peak amplitude slightly increases with the reduction in the grid size.

The corresponding energy spectral densities are shown in Fig. 23. Here as well, the curves show only little dependence on the grid size. The main difference is observed on the amplitude of the hump near $f=5.5$, which is related to the oscillations on the waveforms discussed on the previous paragraph.

Fig. 23

Energy spectral density for $h=0.01$ at $r=30$ and $z=0.2$ for several grid sizes

Additional convergence tests have been performed for roughness scales of $h=0.02$ with the same grid sizes than for $h=0.01$ and of $h=0.15$ with grid sizes ${\Delta } \xi$ between $2\times 10^{-3}$ and $3\times 10^{-3}$. Results are not shown for conciseness. The waveforms and the energy spectral densities present even smaller differences with the different grid sizes than for the case exemplified in Figs. 22 and 23.

In conclusion, the grid convergence study has shown that the results in terms of waveforms and spectra only marginally depend on the grid size. This demonstrates that grid convergence is obtained with the mesh used in the study.

Appendix 2: Effective admittance

Fig. 24

Definition of the parameters to calculate I in the boss model

Several methods have been proposed in the literature for deriving effective admittance of rough surfaces. Among them, the boss model [9, 36] is especially well suited for rough periodic surfaces. In this model, the rough surface is composed of identical roughness elements with a characteristic size h placed on an underlying smooth surface and spaced apart by a distance b.

For a 1D rough periodic surface at low frequencies ($\omega h < \omega b \le 1$), the effective admittance can be written at grazing incidence as [9, 36]:

$$\begin{aligned} \beta (\omega ) = -\mathrm{i}\omega \epsilon , \end{aligned}$$

(28)

where $\epsilon$ is related to the geometric properties of the rough surface via:

$$\begin{aligned} \epsilon = V\left( \dfrac{1+K}{\nu }-1\right) , \end{aligned}$$

(29)

with $\nu$ given by:

$$\begin{aligned} \nu = 1 + \dfrac{\pi }{3}V\dfrac{1+K}{b}. \end{aligned}$$

(30)

The parameter V is the cross-sectional area of the roughness elements above the smooth plane per unit length, and K is a hydrodynamic factor that depends on the shape of the roughness element. Following Tolstoy [36] and Lauriks et al. [40], it can be determined by the relation $K=I/(1-I)$ with:

$$\begin{aligned} I = \dfrac{1}{\pi }\int _L \dfrac{{\mathbf {d}}\cdot {\mathbf {t}}}{|{\mathbf {d}}|^2} \mathrm{d} {\mathbf {L}}\cdot {\mathbf {t}}, \end{aligned}$$

(31)

where the integral is performed along the surface of the roughness element. In the above equation, ${\mathbf {t}}$ denotes the vector tangent to the smooth plane and ${\mathbf {d}}$ the vector between the centroid of the roughness element plus its image by the smooth plane and a point on the surface of the roughness element (Fig. 24). Setting ${\mathbf {t}}=(1, 0)$ and using the parametrization ${\mathbf {d}} = (x(t), y(t))$ with $a\le t\le c$, one has $\mathrm{d} {\mathbf {L}} = (-y'(t), x'(t)) \, \mathrm{d} t$, which yields:

$$\begin{aligned} I = -\dfrac{1}{\pi }\int _a^{c} \dfrac{x(t)\,y'(t)}{x^2(t) + y^2(t)}\, \mathrm{d} t. \end{aligned}$$

(32)

The profile in (1) is modeled as a 1D rough surface with roughness elements of sinusoidal shape and of width 2h separated by a distance 2h. This corresponds to $b=2h$ and $V = h/2$. For determining K, the parametrization $x(t) = ht$ and $y(t) = h/2[1+\cos (\pi t)]$ for $-1\le t\le 1$ is chosen. This gives:

$$\begin{aligned} I = \dfrac{1}{2}\int _{-1}^{1} \dfrac{t\sin (\pi t)}{t^2 + \cos ^4(\pi t/2)} \mathrm{d} t\approx 0.55. \end{aligned}$$

(33)

Combining the values of the different parameters to determine $\epsilon$ in (29) finally leads to (20).