Compressibility effects of supersonic Batchelor vortices

Transition processes in supersonic Batchelor vortices with small disturbances are investigated numerically at Mach numbers 1.5–5.0. The Batchelor vortices for isentropic and uniform stagnation-temperature vortices are considered in terms of thermodynamic properties. In the evolutions, the former vortices are not influenced by compressibility effects; however, the latter have demonstrated slow growth as the Mach number increases. Compressibility effects primarily mean that fluctuation growth related to turbulence is suppressed as the Mach number increases. In spatial fluctuation energy production, isentropic vortices exhibit only kinetic energy fluctuations and uniform stagnation-temperature vortices cause large entropy fluctuation energy with increasing the Mach number. When the Mach number is large, the kinetic fluctuation energy is greatly decreased and the increase in entropy fluctuations is associated with reduced kinetic fluctuation energy. In addition, it is found that the entropy fluctuation property at uniform stagnation temperatures is consistent with the expression derived from the kinetic fluctuation energy based on Morkovin's hypothesis. The present results support the notion that the coefficient of variation of density is appropriate to evaluate compressibility effects. Entropy fluctuation generalizes fluctuation of thermodynamic quantity such as density fluctuation. Therefore, it follows that the property of entropy is strongly related to intrinsic compressibility.

Figure 1

Comparison of Bachelor vortex profiles with experimental measurements in supersonic streamwise vortices for (a) axial velocity and (b) azimuthal velocity. Reprinted with permission from T. Hiejima, Phys. Fluids 25, 114103 (2013). Copyright 2013 American Institute of Physics.

Figure 2

Relations between the growth rate and the axial wave number, $-{\alpha }_i$ vs. $-{\alpha }_r$, obtained from linear stability theory at $M_{\infty }=2.5,q=0.16$ and $\mu =0.5$; (a) isentropic condition, (b) uniform stagnation temperature.

Figure 3

Comparison of results from linear stability theory (LST) and direct numerical simulation (DNS) for eigenfunction profiles of unstable mode $m=-3$ at $M_{\infty }=2.5,q=0.16$, and $\mu =0.5$; (upper) isentropic condition, (lower) uniform stagnation temperature: (a) growth rate, $-{\alpha }_{i}x=ln\left(\right|\varphi |/{\left|\varphi \right|}_{x=0})$; $\varphi =\{R,U,V,W,P$, and ${\mathrm{\Omega }}_x$} are disturbances in the density, radial velocity, azimuthal velocity, axial velocity, pressure, and axial vorticity, respectively. (b) Amplitude and (c) phase at $x=8$.

Figure 4

Isosurfaces of the second invariant of the velocity gradient tensor $\mathcal{Q}$ at $M_{\infty }=2.5,q=0.16$, and $\mu =0.5$: (left) isentropic condition and (right) uniform stagnation temperature; (a), (b) $m=-1$, (c), (d) $m=-2$, (e), (f) $m=-3$, (g), (h) $m=-4$, (i), (j) $m=-5$, (k), (l) $m=-6$.

Figure 5

Contour lines of axial vorticities ${\omega }_x$ in various planes perpendicular to the axial flow: (a) $m=-3$, isentropic condition; (b) $m=-3$, uniform stagnation temperature; (c) $m=-6$, isentropic condition, and (d) $m=-6$, uniform stagnation temperature, at $M_{\infty }=2.5,q=0.16$, and $\mu =0.5$.

Figure 6

Transition processes of Batchelor vortices described by helicity density contours in the cross-sections through the vortex axis: (left) $M_{\infty }=2.5$ and (right) $M_{\infty }=5.0$; (a), (b) $m=-1$, (c), (d) $m=-2$, (e), (f) $m=-3$, (g), (h) $m=-4$, (i), (j) $m=-5$, (k), (l) $m=-6$, for $q=0.16$ and $\mu =0.5$ under isentropic condition. Positive and negative helicities are rendered in black and white, respectively.

Figure 7

Isosurfaces of second invariant of velocity gradient tensor $\mathcal{Q}$: (a) $m=-1$, (b) $m=-2$, (c) $m=-3$, (d) $m=-4$, (e) $m=-5$, (f) $m=-6$, at $M_{\infty }=5.0,q=0.16$ and $\mu =0.5$ under uniform stagnation temperature.

Figure 8

Comparison of velocity fluctuation and average velocity correlation obtained from developments with random disturbances: (a) $\mu =0.5$ and (b) $\mu =0.8$.

Figure 9

Fluctuation energies of kinetic $\widehat{K}$, pressure $\widehat{P}$, entropy $\widehat{S}$, and Eq. (18): (a) $m=-1$, (b) $m=-2$, (c) $m=-3$, (d) $m=-4$, (e) $m=-5$, (f) $m=-6$ at $M_{\infty }=2.5,q=0.16$, and $\mu =0.5$ under uniform stagnation temperature. Note that Eq. (18) is plotted as the closed circles.

Figure 10

Fluctuation energies of kinetic $\widehat{K}$, pressure $\widehat{P}$, entropy $\widehat{S}$, and Eq. (18): (a) $m=-1$, (b) $m=-2$, (c) $m=-3$, (d) $m=-4$, (e) $m=-5$, (f) $m=-6$ at $M_{\infty }=5.0,q=0.16$, and $\mu =0.5$ under uniform stagnation temperature. Note that Eq. (18) is plotted as the closed circles.

Figure 11

Streamwise variations of normalized amplitude of axial velocity disturbance decomposed by modes $m=-1$ to $-12$: (top) isentropic condition and (bottom) uniform stagnation temperature; inflow disturbance mode (a, d) $m=-1$, (b, e) $m=-3$, (c, f) $m=-6$ for $q=0.16$ and $\mu =0.5$ at $M_{\infty }=2.5$.

Figure 12

Evolutions of Batchelor vortices $\left(q=0.16,\mu =0.8\right)$ with random disturbance described by the helicity density contours in the cross-sections through the vortex axis: (a) $M_{\infty }=1.5$, (b) $M_{\infty }=2.5$, (c) $M_{\infty }=3.5$, and (d) $M_{\infty }=5.0$ under isentropic condition.

Figure 13

Evolutions of Batchelor vortices $\left(q=0.16,\mu =0.8\right)$ with random disturbance described by the helicity density contours in the cross-sections through the vortex axis: (a) $M_{\infty }=1.5$, (b) $M_{\infty }=2.5$, (c) $M_{\infty }=3.5$, and (d) $M_{\infty }=5.0$ under uniform stagnation temperature.

Figure 14

Fluctuation energies of kinetic $\widehat{K}$, pressure $\widehat{P}$, entropy $\widehat{S}$, and Eq. (18): (top) axial velocity deficit $\mu =0.5$ and (bottom) $\mu =0.8$; (a), (e) $M_{\infty }=1.5$, (b), (f) $M_{\infty }=2.5$, (c), (g) $M_{\infty }=3.5$, (d), (h) $M_{\infty }=5.0$ under uniform stagnation temperature. Note that Eq. (18) is plotted as the closed circles.

Figure 15

Isentropic conditions: (top) the axial velocity deficit $\mu =0.5$ and (bottom) $\mu =0.8$; (a, d) $\widehat{S}/\widehat{E}$ (ratio of entropy $\widehat{S}$ to total $\widehat{E})$ versus $\widehat{K}/\widehat{E}$ (ratio of kinetic $\widehat{K}$ to total $\widehat{E})$, (b), (e) $M_{\infty }$ versus $\widehat{S}/\widehat{E}$, (c), (f) $M_{\infty }$ versus $\widehat{K}/\widehat{E}$ (the filled circle denotes the average and the error bar indicate the existence span).

Figure 16

Uniform stagnation-temperature conditions: (top) the axial velocity deficit $\mu =0.5$ and (bottom) $\mu =0.8$; (a, d) $\widehat{S}/\widehat{E}$ (ratio of entropy $\widehat{S}$ to total $\widehat{E})$ versus $\widehat{K}/\widehat{E}$ (ratio of kinetic $\widehat{K}$ to total $\widehat{E})$, (b, e) $M_{\infty }$ versus $\widehat{S}/\widehat{E}$, (c, f) $M_{\infty }$ versus $\widehat{K}/\widehat{E}$ (the filled circle denotes the average and the error bar indicate the existence span).

Figure 17

Ratio of $\widehat{K}$ to $\widehat{E}$ and the theoretical curves Eqs. (20) and (21) as a function of $M_0^{*}$; (a) isentropic vortices and (b) uniform stagnation-temperature vortices (the circle denotes the average and the error bar indicates the existence span).

Figure 18

$\widehat{S}/\widehat{E}$ versus $\widehat{K}/\widehat{E}$ plotted using the average value of each case.

Figure 19

Streamwise variations in the growth rate for the unstable mode $m=-6$ at $M_{\infty }=2.5,q=0.16$, and $\mu =0.5$; (a) axial velocity disturbance $W$ and (b) total fluctuation energies $\widehat{E}$.