Intense sound radiation by high-speed flow: Turbulence structure, gas properties, and near-field gas dynamics

Free-shear-flow turbulence with sufficiently fast advection speeds radiates Mach waves, with steepened and skewed pressure profiles. These form within about a mixing layer thickness and dominate the sound field. Their generation and propagation is investigated through comparison of numerical simulations of a temporally developing mixing layer with a series of model-flow simulations designed to isolate physical mechanisms. The first of these are numerical simulations of nonlinearly saturating instability waves, which despite being much simpler than corresponding turbulence reproduce key features of the sound. Motivated in part by this agreement, instability analysis is used to motivate the inclusion of artificial sources in turbulence simulations that are designed to induce specific alterations to the turbulence structures, leaving most of its broadband spectrum unchanged. Comparisons show how insensitive the radiation is to the particular structure. To assess how strongly the near-field sound is coupled to the turbulence, a high dilatational dissipation is imposed to suppress the waves. This significantly reduces radiated pressure intensity, but little changes the Reynolds stresses ($<8\%$), which supports a source-plus-sound perspective. Given this, a low-dimensional nonlinear gas-dynamic mechanism is proposed for the generation and near-field propagation of the waves. The analysis uses a second-order wavy-wall asymptotic solution, and it reproduces the key observations: the sound-field structure, pressure skewness, and even the radiated pressure levels to within a factor of 2.

Figure 1

Iso-surfaces (iso.) of pressure ($p_{\mathrm{iso}.}-p_{\infty }=-0.027{\rho }_{\infty }\mathrm{\Delta }{U}^2$) when ${\delta }_m/{\delta }_m^o=10$: (a) $M=2.5$ and (b) $M=0.9$. [(c), (d)] Corresponding pressure at $y/{\delta }_m=4$ with $p=p_{\infty }\pm 0.4{\rho }_{\infty }\mathrm{\Delta }{U}^2$ for $M=2.5$ and $p_{\infty }\pm 0.1{\rho }_{\infty }\mathrm{\Delta }{U}^2$ for $M=0.9$. Positive perturbations are red. [(e), (f)] Dilatation at $y/{\delta }_m=4$ with $\mathbf{\nabla }·\mathbf{u}=\pm 0.1\mathrm{\Delta }U/{\delta }_m$ for $M=2.5$ and (f) $\pm 0.001\mathrm{\Delta }U/{\delta }_m$ for $M=0.9$. Compressions are black. Only a small part (one-eighth) of the simulation domain in the $y$ direction is shown.

Figure 2

Direct numerical simulation of the unstable mode with $(\alpha ,\beta )=(0.289,0)/{\delta }_m^o$, where ${\delta }_m^o={\delta }_m(t=0)$, for $M=2.5$: [(a), (b)] ${\delta }_m=1.25{\delta }_m^o$ and [(b), (d)] ${\delta }_m=1.5{\delta }_m^o$; panels (a) and (c) show dilatation (grays: $|\mathbf{\nabla }·\mathbf{u}|<0.1\mathrm{\Delta }U/{\delta }_m$) and vorticity (color: $\mathbf{\nabla }·\mathbf{u}<0.5\mathrm{\Delta }U/{\delta }_m$) fields and corresponding streamwise pressure in panels (b) and (d) at the indicated $y$ locations.

Figure 3

[(a), (b)] Growth of $M_t$ (2) at $y=0$ for the discrete modes of Table 3 in Appendix pp1: (a) $M=0.9$ and (b) $M=2.5$. For reference, the horizontal lines indicate the approximately stationary $M_t$ from corresponding turbulence DNS [12]. [(c), (d)] Dependence of pressure skewness on disturbance amplitude $M_t$: (c) $y=0$ and (d) $y=20{\delta }_m$.

Figure 4

Comparison of saturating instabilities and turbulence based on relative free-stream speeds: (a) pressure intensity and (b) pressure skewness at $y/{\delta }_m\left(t\right)=20$. For saturating instabilities, the relative velocity $U=U_c$ is defined using (4). For turbulence, an anticipated range of advection velocities are based on $y$-location mean speed $\overline{u}\left(y\right)$ (see text). The results are shown for $U=\mathrm{\Delta }U/2-\overline{u}\left(y\right)$: $y=0$ (symbol), $y=\pm 1{\delta }_m$ (inner bar), and $y=\pm 2{\delta }_m$ (outer bar). Solid and dashed curves in (a) are $\propto U^8$ and $\propto U^3$, respectively, as labeled.

Figure 5

Effect of source strength $A$ on the $y=0$ pressure spectra when ${\delta }_m\left(t\right)/{\delta }_m^o=10$: (a) streamwise and (b) spanwise directions. The dashed lines indicate the initial and final modulated target wave numbers.

Figure 6

Effect of the source strength $A$ on the Reynolds stresses: (a) $\overline{u^{\prime }{u}^{\prime }}$, (b) $\overline{v^{\prime }{v}^{\prime }}$, (c) $\overline{w^{\prime }{w}^{\prime }}$, and (d) $\overline{u^{\prime }{v}^{\prime }}$.

Figure 7

Effect of the source strength on the sound-field pressure visualized with gray levels spanning $-0.025<p^{\prime }/\left(\mathrm{\Delta }{U}^2{\rho }_{\infty }\right)<0.025$ for $M=2.5$ at $y/{\delta }_m\left(t\right)=10$ when ${\delta }_m\left(t\right)/{\delta }_m^o=10$.

Figure 8

Effect of source strength $A$ on the pressure (a) intensity and (b) skewness for conservative forcing by (8) and nonconservative forcing by (10).

Figure 9

Effect of the advection-based sources on the Reynolds stresses: (a) $\overline{u^{\prime }{u}^{\prime }}$, (b) $\overline{v^{\prime }{v}^{\prime }}$, (c) $\overline{w^{\prime }{w}^{\prime }}$, and (d) $-\overline{u^{\prime }{v}^{\prime }}$.

Figure 10

Effect of the advection-based source modification on the pressure (a) intensity and (b) skewness.

Figure 11

Effect of gas stiffness on the joint pressure and density distribution at $y=0$ when ${\delta }_m/{\delta }_m^o=20$ for (a) $M=0.9$ and (b) $M=2.5$. Isolevels range from $1\%$ to $90\%$ and are filled for the baseline case ($p^s=0$) and shown with just lines otherwise. For reference, isentropic approximations to the gas laws are provided: linearized (dashed) and full (solid).

Figure 12

Effect of gas stiffness on the Reynolds stresses: (a) $\overline{u^{\prime }{u}^{\prime }}$, (b) $\overline{v^{\prime }{v}^{\prime }}$, (c) $\overline{w^{\prime }{w}^{\prime }}$, and (d) $-\overline{u^{\prime }{v}^{\prime }}$.

Figure 13

Effect of gas stiffness on pressure (a) intensity and (b) skewness and the effect of $\gamma$ on the pressure (c) intensity and (d) skewness.

Figure 14

(a) The effect of ${\mu }_b$ on the average dilatation intensity across the mixing layers. [(b)–(d)] Dilatation visualization at $z=L_z/2$ when ${\delta }_m/{\delta }_m^o=10$: (b) ${\mu }_b/\mu =0$, (c) 10, and (d) 100. The gray scale for panels (b) and (c) is $|\mathbf{\nabla }·\mathbf{u}|<0.15\mathrm{\Delta }U/{\delta }_m$ and (d) is $|\mathbf{\nabla }·\mathbf{u}|<0.03\mathrm{\Delta }U/{\delta }_m$, with black indicating compression.

Figure 15

Effect of bulk viscosity on the pressure (a) intensity and (b) skewness.

Figure 16

Uniform flow adjacent to (a) a wavy wall and (b) a corresponding $v$-velocity distribution along $y=0$. (c) The deviation of the boundary pressure distribution from the linear theory ($p_l^{\prime }$) for the two-dimensional wavy wall $M=1.75$ shown in panels (a) and (b). Modifications to the model formulation for entropy change, $\mathrm{\Delta }s=2.5\times {10}^{-2}{c}_p^{★}$, is shown for reference ($c_p^{★}$ is the constant-pressure heat capacity).

Figure 17

(a) Computational domain of supersonic flow adjacent to a plane of cross-stream velocity fluctuations. Spatial distributions of $v_w$ from (b) $M=1.5$, (c) $M=2.5$, (d) $M=3.0$, and (e) $M=3.5$ DNS of mixing layers colored from $-0.3<v_w/\mathrm{\Delta }U<0.3$ velocities.

Figure 18

Dilatation $|\mathbf{\nabla }·\mathbf{u}|<0.1U_{\infty }/{\delta }_m$ at $z=L_z/2$ for the $M_{\infty }=1.75/M=2.5$ case: (a) linear, (b) nonlinear, and (c) turbulence DNS. The dashed line at ${35}^{\circ }$ from the $x$ axis indicates the nominal Mach angle. The color map in panel (c) correspond to $|\mathbf{\nabla }·\mathbf{u}|<0.5\mathrm{\Delta }U/{\delta }_m$. The dilatation in panel (a) has been scaled by ${10}^4$. (d) Pressure traces for panels (a)–(c) at $y=10{\delta }_m$ and data from a Mach 2 jet [57] at $r/D_j\approx 61$ (based on the jet diameter $D_j$ and jet velocity $U_j$).

Figure 19

Streamwise view of dilatation $|\mathbf{\nabla }·\mathbf{u}|<0.1\mathrm{\Delta }U/{\delta }_m$ at $z=L_z/2$ for the steady $M=1.75$ (a) linear and (b) nonlinear model and (c) $y>0$ from the DNS $M=2.5$. The color map in panel (c) corresponds to $|\mathbf{\nabla }·\mathbf{u}|<0.5\mathrm{\Delta }U/{\delta }_m$. The dilatation in panel (a) has been scaled by ${10}^4$.

Figure 20

Domain average skewness dependence on the (a) acoustic intensity at $\left|y\right|=20{\delta }_m$ and (b) the flow velocity $U$. The dotted lines in panel (a), $\propto {\overline{{\left(p^{\prime }\right)}^2}}^{1/2}$, are separated by a factor of two in the pressure intensity. In panel (b), for the wavy-wall model, the $U/c_{\infty }=M_{\infty }$ are listed in Table 2. For turbulence, an anticipated range of advection velocities are based on $y$-location mean speed $\overline{u}\left(y\right)$ (see Sec. 2). The results are shown for $U=\mathrm{\Delta }U/2-\overline{u}\left(y\right)$: $y=0$ (symbol), $y=\pm 1{\delta }_m$ (inner bar), and $y=\pm 2{\delta }_m$ (outer bar).

Figure 21

Two routes to increasing $S_k$ in the model flow: (a) due to mean flow $M_{\infty }$ and (b) due to fluctuation intensity $v_{\mathrm{rms}}/c_{\infty }$. Filled symbols indicate the reference case in Table 2, $M_{\infty }=1.75$ and $v_{\mathrm{rms}}/c_{\infty }=0.255$, based on a corresponding $M=2.5$ turbulence case.

Figure 22

The effect of turbulence modification for $A=2$ on acoustic sources.

Figure 23

(a) Space-time correlations of streamwise velocity perturbations for $M=2.5$ between $0\le y/{\delta }_m\le 3$. Five levels indicate normalized correlations from $0.5\le C_{xt}\le 0.9$ and dashed lines indicate the direction along the maximum integral space-time scale shown in panel (b) with the maximum indicated by triangles. (c) The advection velocity based on the direction of maximum integral space-time correlation compared to the mean streamwise velocity.