1 Introduction

The interaction between shock wave (SW) and turbulence is a ubiquitous problem in science (e.g., star formations in galaxies [1] and inertial confinement fusion [2]) and in engineering applications (e.g., sonic boom generated by supersonic transports [3]). The post-SW overpressure modulation observed as a peak-overpressure fluctuation induced by the effects of turbulence is known to be one of the most important phenomena of this problem [4,5,6,7].

Despite the importance of this problem in such various fields, the detailed mechanisms of the modulation are not clear yet because turbulent velocity and the SW overpressure were not measured simultaneously in most previous studies. For further investigations, it is necessary to reveal the relation between the instantaneous spatial distributions of turbulent velocity and the SW overpressure modulation.

In this study, we perform simultaneous measurements of the velocity fluctuations of grid turbulence and the spherical SW overpressure in wind tunnel experiments. Taylor hypothesis is used to estimate the spatial distributions of velocity fluctuations. The modulation of the SW overpressure is investigated in relation to the estimated spatial profile of velocity.

2 Methods

The experimental setup is shown in Fig. 1a. A square grid with a mesh size M (solidity, 0.36) is installed at the entrance to the test section of the wind tunnel (the detail of wind tunnel can be seen in [6, 8]) to generate turbulent velocity field. We use two square grids with M = 50 mm and 100 mm. On the other hand, a spherical SW is ejected vertically downward from an open end of a tube using a quick piston valve. The total length and the inner diameter of the tube are 3.4 m and 23.3 mm, respectively. The spherical SW generator was also used in our previous studies [6]. Between the grid and the open end of the shock tube, we install a measurement plate with the thickness of 5 mm, on which a piezoelectric pressure transducer (PCB Piezotronics Inc. 113B27) and an I-type hot-wire probe (Dantec Dynamics 55P11) are mounted (see Fig. 1a) for the measurements of overpressure P(t) and velocity U(t). The measurement plate is fixed 75 mm above the wall of the wind tunnel to prevent the boundary layer from affecting the measurement. The shock Mach number is M S0 = 1.004 on the measurement plate. Table 1 summarizes the experimental conditions including turbulence characteristics at the measurement location. Further detail of grid turbulence studied here can be found in [8]. The signals of P(t) and U(t) are sampled with the oscilloscope (Yokogawa DL750) at the sampling rate of 1 MHz. The SW generator and the measurement system are controlled by a computer: the sampling is started 50 ms before the SW is ejected from the open end. The velocity measurement height h defined in Fig. 1a is ranged from 15 to 125 mm. In each measurement condition, 500 runs of the SW ejection into the grid turbulence were conducted for the statistical analysis. We also confirmed that 300 runs were enough to obtain the results discussed below.

Fig. 1
figure 1

(a) Experimental setup (lengths are shown in mm). (b) Examples of time histories of overpressure P for M100-U10 (red/blue line shows strong/weak case) and the ensemble average of peak overpressure ⟨Δp⟩ and peak-overpressure fluctuation Δp ′′ = Δp − ⟨Δp⟩ are also shown

Full size image
Table 1 Experimental conditions of the spherical SW and grid turbulence interaction experiments, where Re M = UM/ν is the mesh Reynolds number (ν, the kinematic viscosity; M, the mesh size; U 0, the streamwise mean velocity); the turbulence statistic at the measurement location is shown for the rms streamwise velocity fluctuation 𝑢rms, Kolmogorov microscale η, Taylor microscale λ, and longitudinal integral length scale L u calculated from the longitudinal autocorrelation function (the definitions of scales can be seen in [8]); and the table also includes the rms peak-overpressure fluctuation p rms obtained in the experiments
Full size table

Figure 1b shows examples of time histories of P. The peak overpressure Δp is the increase in P due to the SW defined with the peak value of P. Δp′′ (=Δp−⟨Δp⟩) is the fluctuation of Δp from ⟨Δp⟩. Here, ⟨ ⟩ denotes the ensemble average of all runs of the SW. We analyze Δp in relation to U(t) measured before the SW reaches the measurement plate. Table 1 also shows the rms values of the velocity fluctuation u(t)=U(t)−U 0 and of the peak-overpressure fluctuation Δp′′, denoted as 𝑢rms and p rms, respectively. We can find that p rms increases with 𝑢rms which is consistent with previous studies [4, 6, 7].

The turbulence past the probe is advected downstream by the mean flow. The spatial distribution of the velocity U(𝑑) at the probe height ℎ can be estimated from 𝑈(t) with the Taylor hypothesis, where 𝑑 is the streamwise distance from the pressure transducer (see Fig. 1a). Since the fluid motion is much slower than the SW, the turbulence can be assumed to be frozen and hardly changes while the SW is propagating. This enables us to estimate the velocity distribution at the moment when the SW is ejected. The decay of grid turbulence is not important in the range of 𝑑 considered here because the measurement location is far from the grid, while turbulence intensity changes with a power law of the streamwise distance from the grid [8].

The SW propagates upstream through the turbulence whose velocity profile at h is represented by U(𝑑) and reaches the pressure transducer location. We apply the low-pass filter with a cutoff length Δ𝑑 to the velocity as \( \overline{U}\left(d,\Delta d\right)=\left(1/\Delta d\right){\int}_{-\Delta d/2}^{\Delta d/2}U\left(d+\delta \right) \mathrm{d}\delta \), which is the velocity of motions in scales larger than Δ𝑑 at the location 𝑑. This low-pass filtering is used for investigating the scale dependence of the pressure modulation by the turbulence.

The correlation coefficient defined with the ensemble average is calculated for Δp′′ and \( \overline{u}\left(d,\Delta d\right)=\overline{U}\left(d,\Delta d\right)-{U}_0 \) as \( R\left(d,\Delta d\right)=\left\langle \overline{u}\Delta {p}^{\prime \prime}\right\rangle /\sqrt{\left\langle {\left(\overline{u}\right)}^2\right\rangle \left\langle {\left(\Delta {p}^{\prime \prime}\right)}^2\right\rangle } \) to evaluate the relation between the turbulence velocity fluctuation and the peak-overpressure fluctuation. This correlation coefficient is computed at all of the velocity measurement height ℎ.

3 Results

Probability density functions (PDFs) of the velocity fluctuation and the peak-overpressure fluctuation, normalized by their rms values, are shown in Fig. 2a, b, respectively. We can find that both follow the Gaussian profile.

Fig. 2
figure 2

PDFs of (a) velocity fluctuation 𝑢/𝑢rms and (b) peak-overpressure fluctuation Δp′′/p rms; red line shows the Gaussian profile; see Table 1 for symbols

Full size image

Figure 3 shows an example of the scatterplot of the velocity fluctuation \( \overline{u}\left(d,\Delta d\right)/{U}_0 \) and the peak-overpressure fluctuation ∆p ′′/〈∆p〉 at the location where the highest value is obtained for the correlation coefficient R(d, ∆d) in the case M100-U10. The scatterplot confirms that there exists positive correlation between the velocity and the peak-overpressure fluctuation.

Fig. 3
figure 3

Example of scatterplot of velocity fluctuation \( \overline{u}\left(d,\Delta d\right)/{U}_0 \) and peak-overpressure fluctuation Δp′′/⟨Δp⟩ of M100-U10, ℎ = 50 mm, (d, ∆d) = (173 mm, 162 mm); R(d, ∆d) = 0.330

Full size image

We consider the set of (d, ∆d) for which the highest correlation coefficient R is obtained in all cases in Table 1. Hereafter, ∆d P denotes the value of ∆d with the highest correlation. ∆d P is related to the length scale of turbulent motions which is the most effective to the modulation of the peak overpressure. In Fig. 4, ∆d P/L u is plotted against M T/M S0 (M T = u rms/a, the turbulent Mach number; 𝑎, the sound speed in front of the SW). We can find that ∆d P are of the order of the integral length scale, suggesting that the large scale of turbulent motions is related to the peak-overpressure modulation. It is also found that ∆d P/L u decreases as M T/M S0 increases.

Fig. 4
figure 4

Relation between ∆d P/L u and M T/M S0; see Table 1 for symbols

Full size image

4 Conclusions

In this study, we simultaneously measured the flow velocity and the overpressure of the SW in grid turbulence where the spherical SW propagates. PDFs of the velocity fluctuation and the peak-overpressure fluctuation follow the Gaussian profile. The correlation coefficients calculated for the velocity and the peak-overpressure fluctuation confirm that the turbulent velocity is positively correlated to the peak-overpressure fluctuation and that the peak-overpressure modulation is induced by the turbulent motions at large scale.