Statistical behavior of post-shock overpressure past grid turbulence

Abstract

When a shock wave ejected from the exit of a 5.4-mm inner diameter, stainless steel tube propagated through grid turbulence across a distance of 215 mm, which is 5–15 times larger than its integral length scale $L_{u}$, and was normally incident onto a flat surface; the peak value of post-shock overpressure, $\Delta P_{\mathrm{peak}}$, at a shock Mach number of 1.0009 on the flat surface experienced a standard deviation of up to about 9 % of its ensemble average. This value was more than 40 times larger than the dynamic pressure fluctuation corresponding to the maximum value of the root-mean-square velocity fluctuation, $u^{\prime }= 1.2~\hbox {m}/\hbox {s}$. By varying $u^{\prime }$ and $L_{u}$, the statistical behavior of $\Delta P_{\mathrm{peak}}$ was obtained after at least 500 runs were performed for each condition. The standard deviation of $\Delta P_{\mathrm{peak}}$ due to the turbulence was almost proportional to $u^{{\prime }}$. Although the overpressure modulations at two points 200 mm apart were independent of each other, we observed a weak positive correlation between the peak overpressure difference and the relative arrival time difference.

Appendix: Recovering the pressure signal through deconvolution using numerical Laplace transform

The basic scheme of the following method of signal recovery is referenced from Refs. [37] and [38], while the details in the formulation are the original contribution by this paper. The Laplace transform of a device transfer function, $H\left( s \right)$, was obtained from shock tube experiments:

$$\begin{aligned}&g_0 (t)=\int _{-\infty }^\infty {h(\tau )} f_0 (t-\tau )\mathrm{d}\tau ,\end{aligned}$$

(12)

$$\begin{aligned}&G_0 (s)=H(s)F_0 (s), \end{aligned}$$

(13)

where $g (t)$ denotes a signal as a function of recorded time, $f(t)$ is the signal to be recovered, and $h(\tau )$ is the transfer function. The functions $F$(s), $G$(s), and $H$(s) are the Laplace transforms of $f$(t), $g$(t), and $h$(t), respectively. The subscript 0 represents the data obtained in the calibration experiment. In this study, the calibration experiment was conducted using a shock tube. The combination of a pressure transducer and mount holder that was identical to that used in the wind tunnel was flush-mounted on the end wall of the shock tube. Assuming that $f_0 (t)$ is expressed by a Heaviside function up to a finite time of $t=T$, we obtain:

$$\begin{aligned} f_0 (t)=\Delta P_0 \chi (t)=\left\{ {\begin{array}{ll} 0&{} \quad (t\le 0) \\ \Delta P_0 &{} \quad (0\le t\le T) \\ \end{array}} \right. , \end{aligned}$$

(14)

where $\Delta P_0$ is obtained from Rankine–Hugoniot relation applied to the reflected shock wave, having a constant value.

$H(s)$ is numerically obtained from Eq. (13) as follows:

$$\begin{aligned} \frac{1}{H(s)}&= \frac{F_0 (s)}{G_0 (s)},\end{aligned}$$

(15)

$$\begin{aligned} F_0 (s)&= \frac{\Delta P_0 }{s}. \end{aligned}$$

(16)

For numerical transform, $t$ and $s$are discretized as follows:

$$\begin{aligned} t_m&= \left( {2m-1} \right) \Delta t=\frac{\left( {2m-1} \right) T}{4n},\end{aligned}$$

(17)

$$\begin{aligned} \Delta t&= \frac{T}{2\left( {2n} \right) },\end{aligned}$$

(18)

$$\begin{aligned} s_k&= \left\{ {\begin{array}{ll} \alpha +j\left( {2k-1} \right) \Delta \omega ,&{} \quad k>0 \\ \alpha +j\left( {2k+1} \right) \Delta \omega =s_{\left| k \right| } *&{} \quad k<0 \\ \end{array}} \right. . \end{aligned}$$

(19)

The reciprocity conditions are set to:

$$\begin{aligned}&N=n,\end{aligned}$$

(20)

$$\begin{aligned}&\Delta \omega =\frac{\pi }{T}. \end{aligned}$$

(21)

Following Ref. 37, $\alpha$ is set to:

$$\begin{aligned} \alpha =\frac{2\pi }{T}=2\Delta \omega . \end{aligned}$$

(22)

From Eqs. (16) and (19):

$$\begin{aligned} F_0 (k)=\frac{\Delta P_0 }{\alpha +j\left( {2k-1} \right) \Delta \omega }. \end{aligned}$$

(23)

By definition, $G_{0}$ is:

$$\begin{aligned}&G_0 (s)=\sum _{m=1}^{2n} {g_0 \left( {t_m } \right) \mathrm{e}^{-s_k t_m }\left( {2\Delta t} \right) } \nonumber \\&\qquad \;\,\quad =\frac{T}{2n}\sum _{m=1}^{2n} {g_0 \left( {t_m } \right) \mathrm{e}^{-s_k t_m }} , \nonumber \\&G_0 (k)=\frac{T}{2n}\sum _{m=1}^{2n} {\mathrm{e}^{-\alpha t_m }g_0 \left( {t_m } \right) \mathrm{e}^{-j\frac{\left( {2k-1} \right) \left( {2m-1} \right) }{4n}\pi }} ,\end{aligned}$$

(24)

$$\begin{aligned}&\hbox {Re}\left[ {G_0 (k)} \right] =\frac{T}{2n}\sum _{m=1}^{2n} {\mathrm{e}^{-\alpha t_m }g_0 \left( {t_m } \right) }\nonumber \\&\qquad \qquad \qquad \quad \;\times \,{\cos \left\{ {\frac{\left( {2k-1} \right) \left( {2m-1} \right) }{4n}\pi } \right\} } ,\end{aligned}$$

(25)

$$\begin{aligned}&\hbox {Im}\left[ {G_0 (k)} \right] =-\frac{T}{2n}\sum _{m=1}^{2n} {\mathrm{e}^{-\alpha t_m }g_0 \left( {t_m } \right) }\nonumber \\&\qquad \qquad \qquad \quad \;\times \,{\sin \left\{ {\frac{\left( {2k-1} \right) \left( {2m-1} \right) }{4n}\pi } \right\} } . \end{aligned}$$

(26)

From Eqs. (15), (23), (25), and (26), we obtain the components in the Laplace transform of the device transfer function:

$$\begin{aligned}&\hbox {Re}\left[ {\frac{1}{H(k)}} \right] ==\frac{\Delta P_0 \left\{ {\alpha \hbox {Re}\left[ {G_0 (k)} \right] -\left( {2k-1} \right) \Delta \omega \hbox {Im}\left[ {G_0 (k)} \right] } \right\} }{\left\{ {\alpha \hbox { Re}\left[ {G_0 (k)} \right] -\left( {2k-1} \right) \Delta \omega \hbox { Im}\left[ {G_0 (k)} \right] } \right\} ^{2}+\left\{ {\alpha \hbox { Im}\left[ {G_0 (k)} \right] +\left( {2k-1} \right) \Delta \omega \hbox { Re}\left[ {G_0 (k)} \right] } \right\} ^{2}},\end{aligned}$$

(27)

$$\begin{aligned}&\hbox {Im}\left[ {\frac{1}{H(k)}} \right] =\frac{-\Delta P_0 \left\{ {\alpha \hbox {Im}\left[ {G_0 (k)} \right] +\left( {2k-1} \right) \Delta \omega \hbox {Re}\left[ {G_0 (k)} \right] } \right\} }{\left\{ {\alpha \hbox { Re}\left[ {G_0 (k)} \right] -\left( {2k-1} \right) \Delta \omega \hbox { Im}\left[ {G_0 (k)} \right] } \right\} ^{2}+\left\{ {\alpha \hbox { Im}\left[ {G_0 (k)} \right] +\left( {2k-1} \right) \Delta \omega \hbox { Re}\left[ {G_0 (k)} \right] } \right\} ^{2}}. \end{aligned}$$

(28)

Next, in the wind tunnel experiment, the Laplace transform of a recovered overpressure, $F(s)$, is obtained from a recorded signal, $g(t)$, by:

$$\begin{aligned} F(s)&= \frac{G(s)}{H(s)},\end{aligned}$$

(29)

$$\begin{aligned} G(s)&= \int _0^\infty {g(t)} \mathrm{e}^{-st}\mathrm{d}t,\end{aligned}$$

(30)

$$\begin{aligned} G(k)&= \frac{T}{2n}\sum _{m=1}^{2n} {\mathrm{e}^{-\alpha t_m }g\left( {t_m } \right) \mathrm{e}^{-j\frac{\left( {2k-1} \right) \left( {2m-1} \right) }{4n}\pi }} ,\end{aligned}$$

(31)

$$\begin{aligned} F(k)&= \hbox {Re}\left[ {F(k)} \right] +j\hbox { Im}\left[ {F(k)} \right] ,\\&\left\{ {\begin{array}{l} \hbox {Re}[F(k)]=\hbox { Re}\left[ {G(k)} \right] \hbox { Re}\left[ {\frac{1}{H(k)}} \right] \\ \qquad \qquad \qquad \,\, -\hbox {Im}\left[ {G(k)} \right] \hbox { Im}\left[ {\frac{1}{H(k)}} \right] \\ \hbox {Im}[F(k)]=\hbox {Re}\left[ {G(k)} \right] \hbox { Im}\left[ {\frac{1}{H(k)}} \right] \\ \qquad \qquad \qquad \,\, +\hbox {Im}\left[ {G(k)} \right] \hbox { Re}\left[ {\frac{1}{H(k)}} \right] \\ \end{array}} \right. .\nonumber \end{aligned}$$

(32)

Finally, the overpressure $f(t)$ is recovered through inverse Laplace transform:

$$\begin{aligned} f(t_m )&= \frac{2\mathrm{e}^{\alpha t_m }}{T}\hbox { Re}\left[ {\sum _{k=1}^N {F\left( k \right) \mathrm{e}^{j\frac{\left( {2k-1} \right) \left( {2m-1} \right) }{4n}\pi }} } \right] ,\nonumber \\&= \frac{2\mathrm{e}^{\alpha t_m }}{T}\sum _{k=1}^N \left\{ \hbox {Re}\left[ {F\left( k \right) } \right] \cos \frac{\left( {2k-1} \right) \left( {2m-1} \right) }{4n}\pi \right. \nonumber \\&\left. -\hbox {Im}\left[ {F\left( k \right) } \right] \sin \frac{\left( {2k-1} \right) \left( {2m-1} \right) }{4n}\pi \right\} . \end{aligned}$$

(33)