Experimental study of shock wave modulation caused by velocity and temperature fluctuations in cylinder wakes

Experiments on a spherical shock wave propagating across an unheated- or a heated-cylinder wake are performed in a wind tunnel to investigate the effects of velocity and temperature fluctuations of turbulence on the shock wave. The temperature of the heated cylinder is low enough for the buoyancy effect to be negligible in the wake development, and comparisons between the heated- and unheated-cylinder experiments highlight the effects of temperature fluctuations on the shock wave. Peak overpressure of the spherical shock wave is measured on a wall after the shock wave has passed the wake. Along with the overpressure measurement, temperature and velocity are measured in the heated and unheated wakes, respectively. Larger peak-overpressure fluctuations are obtained when the shock wave interacts with the heated-cylinder wake than with the unheated-cylinder wake. Correlation coefficients are calculated between the velocity/temperature fluctuations of the unheated/heated-cylinder wakes and peak-overpressure fluctuations. The temperature fluctuations and overpressure fluctuations are found to be negatively correlated, which is explained by the shock deformation caused by speed-of-sound fluctuations in front of the shock wave. By comparing the correlation coefficients between velocity and overpressure fluctuations with those between temperature and overpressure fluctuations, it is also discovered that the temperature fluctuations of the heated-cylinder wake have a stronger correlation with the overpressure fluctuations than the velocity fluctuations of the unheated-cylinder wake.

Figure 1

(a) Schematic of the experimental setup. (b) Side view of the test section. Orange and blue lines show the centerline of the cylinder wake and shock ray from the shock tube end to the pressure transducer, respectively. The red cross shows the intersection of the wake centerline and the shock ray. The distance between the cylinder center and the red cross is 561 mm. (c) Schematic of the heater-wrapped cylinder. Lengths are shown in mm.

Figure 2

Schematic of shock wave deformation due to local velocity fluctuation $u$ and temperature fluctuation $T^{\prime }$. The orange area shows $0\le r\le r_0\mathrm{and}0\le x\le x_l$, where $u_{\mathrm{M}}=u$ and $T_{\mathrm{M}}=T_0+T^{\prime }$ at $t=0$.

Figure 3

(a) Time-averaged velocity $U_{\mathrm{ave}}/U_0$ and (b) rms velocity fluctuation $u_{\mathrm{rms}}/U_0$ of the unheated-cylinder wake. (c) Time-averaged temperature $T_{\mathrm{ave}}/T_0$ and (d) rms temperature fluctuation $T_{\mathrm{rms}}/T_0$ of the heated-cylinder wake. See Table 1 for symbols.

Figure 4

Examples of pressure waveforms in cases (a) 1-OFF, (b) 1-ON, (c) 3-OFF, and (d) 3-ON.

Figure 5

$\langle \mathrm{\Delta }p\rangle$ plotted against $\mathrm{\Delta }U$. The broken line shows $\langle \mathrm{\Delta }p\rangle$ obtained in the flow at $U_0=10$ m/s without the cylinder. See Table 1 for symbols.

Figure 6

PDFs of cases (a) 1-OFF, (b) 1-ON, (c) 2-OFF, (d) 2-ON, (d) 3-OFF, and (f) 3-ON. Solid lines show the Gaussian profile. See Table 1 for symbols.

Figure 7

${\sigma }_{\mathrm{\Delta }p}/\langle \mathrm{\Delta }p\rangle$ plotted against (a) $f(u_{\mathrm{rms}}/a_0,0,0)/\sqrt{M_{\mathrm{S}0}^2-1}$, (b) $f(u_{\mathrm{rms}}/a_0,T_{\mathrm{rms}}/2T_0,0)/\sqrt{M_{\mathrm{S}0}^2-1}$, (c) $f(u_{\mathrm{rms}}/a_0,T_{\mathrm{rms}}/2T_0,0.5)/\sqrt{M_{\mathrm{S}0}^2-1}$, and (d) $f(u_{\mathrm{rms}}/a_0,T_{\mathrm{rms}}/2T_0,1)/\sqrt{M_{\mathrm{S}0}^2-1}$. Solid lines in each figure are calculated with Eq. (11). The coefficients of determination $R^2$ calculated between ${\sigma }_{\mathrm{\Delta }p}/\langle \mathrm{\Delta }p\rangle$ and $f/\sqrt{M_{\mathrm{S}0}^2-1}$ of each figure are also shown. See Table 1 for symbols.

Figure 8

Correlation coefficients $R_u(d,\mathrm{\Delta }d)$ and $R_T(d,\mathrm{\Delta }d)$ of cases (a) 1-OFF, (b) 1-ON, (c) 2-OFF, (d) 2-ON, (d) 3-OFF, and (e) 3-ON. Note that $R_u$ is shown in (a), (c), and (e) while $R_T$ is shown in (b), (d), and (f). The crosses show $(d_{\max },\mathrm{\Delta }{d}_{\max })$.

Figure 9

(a) Schematic of the shock wave surface and the temperature fluctuating field in front of the shock wave before the interaction. (b) Shock wave deformation caused by the interaction.

Figure 10

Relation between $d_{\max }$ and the shock ray from the open end of the shock tube to the pressure transducer location. See Table 1 for symbols.

Figure 11

Relation between ${\left(R_u\right)}_{\max }$ and $u_{\mathrm{rms}}/a_0$ (blue) and relation between ${\left(R_T\right)}_{\max }$ and $T_{\mathrm{rms}}/2T_0$ (red) in each case. See Table 1 for symbols.