Spectral energy cascade and decay in nonlinear acoustic waves

We present a numerical and theoretical investigation of the nonlinear spectral energy cascade of decaying finite-amplitude planar acoustic waves in a single-component ideal gas at standard temperature and pressure. We analyze various one-dimensional canonical flow configurations: a propagating traveling wave (TW), a standing wave (SW), and randomly initialized acoustic wave turbulence (AWT). Due to nonlinear wave propagation, energy at the large scales cascades down to smaller scales dominated by viscous dissipation, analogous to hydrodynamic turbulence. We use shock-resolved mesh-adaptive direct numerical simulation (DNS) of the fully compressible one-dimensional Navier-Stokes equations to simulate the spectral energy cascade in nonlinear acoustic waves. The simulation parameter space for the TW, SW, and AWT cases spans three orders of magnitude in initial wave pressure amplitude and dynamic viscosity, thus covering a wide range of both the spectral energy cascade and the viscous dissipation rates. The shock waves formed as a result of the energy cascade are weak $\left(M<1.4\right)$, and hence we neglect thermodynamic nonequilibrium effects such as molecular vibrational relaxation in the current study. We also derive a set of nonlinear acoustics equations truncated to second order and the corresponding perturbation energy corollary yielding the expression for a perturbation energy norm $E^{\left(2\right)}$. Its spatial average, $\langle E^{\left(2\right)}\rangle$, satisfies the definition of a Lyapunov function, correctly capturing the inviscid (or lossless) broadening of spectral energy in the initial stages of evolution—analogous to the evolution of kinetic energy during the hydrodynamic breakdown of three-dimensional coherent vorticity—resulting in the formation of smaller scales. Upon saturation of the spectral energy cascade, i.e., a fully broadened energy spectrum, the onset of viscous losses causes a monotonic decay of $\langle E^{\left(2\right)}\rangle$ in time. In this regime, the DNS results yield $\langle E^{\left(2\right)}\rangle \sim t^{-2}$ for TWs and SWs, and $\langle E^{\left(2\right)}\rangle \sim t^{-2/3}$ for AWT initialized with white noise. Using the perturbation energy corollary, we derive analytical expressions for the energy, energy flux, and dissipation rate in the wave number space. These yield the definitions of characteristic length scales such as the integral length scale $\ell$ (characteristic initial energy containing scale) and the Kolmogorov length scale $\eta$ (shock thickness scale), analogous to the K41 theory of hydrodynamic turbulence [A. N. Kolmogorov, Dokl. Akad. Nauk SSSR 30, 9 (1941)]. Finally, we show that the fully developed energy spectrum of the nonlinear acoustic waves scales as ${\widehat{E}}_k{k}^2{\epsilon }^{-2/3}{\ell }^{1/3}\sim Cf\left(k\eta \right)$, with $C\approx 0.075$ constant for TWs and SWs but decaying in time for AWT.

Figure 1

$Q$-criterion isosurfaces colored with the local velocity magnitude obtained from a direct numerical simulation of a Taylor-Green vortex in a triply periodic domain ${\left[-\pi ,\pi \right]}^3$ [14], exhibiting breakdown into hydrodynamic turbulence (a), velocity perturbation field in a high-amplitude nonlinear traveling acoustic wave (b), evolution of normalized spatial average of $u^2$ (c), (e), and velocity spectra $|{\widehat{u}}_k{|}^2$ (d), (f) at times $t_0,t_1,t_2,t_3$. The spectral broadening occurs due to the nonlinear terms in the governing equations generating smaller length scales resulting in energy cascade from larger to smaller length scales.

Figure 2

Weak shock wave structure (a) pressure $p^{*}$ and (b) entropy $s^{*}$ propagating with a speed $a_s^{*}>a_0^{*}$ obtained from DNS data (see Sec. 4). $\mathrm{\Delta }{s}^{*}/R^{*}$ and $s_{\max }^{*}/R^{*}$ are the entropy jump and maximum entropy, respectively. With increasing viscosity, the peak in entropy remains constant. The DNS data have been obtained for base state viscosity values given in Table 1.

Figure 3

Entropy jump $\mathrm{\Delta }{s}^{*}=s_2^{*}-s_1^{*}$ and maximum entropy generated $s_{\max }^{*}$ versus pressure jump $\mathrm{\Delta }{p}^{*}$ across a planar shock wave. In the labeled region $(\mathrm{\Delta }{p}^{*}/\gamma p_1^{*}<1$, referred to as “weak shocks” hereafter), the entropy jump $\mathrm{\Delta }{s}^{*}$ scales as $O\left(\mathrm{\Delta }{p}^{*3}\right)$, whereas the maximum entropy generated $s_{\max }^{*}$ scales as $O\left(\mathrm{\Delta }{p}^{*2}\right)$, approximately. Markers denote DNS data (see Sec. 4): $(•,\blacksquare ){\mu }^{*}=7.5\times {10}^{-3}\text{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$; $(•,\blacksquare )$, ${\mu }^{*}=7.5\times {10}^{-4}\text{kg}{\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$; $(○,\square )$, ${\mu }^{*}=7.5\times {10}^{-5}$ kg ${\mathrm{m}}^{-1}{\mathrm{s}}^{-1}$ for varying values of $\mathrm{\Delta }{p}^{*}$. Solid lines correspond to Eqs. (7) and (9).

Figure 4

Comparison of perturbation energy function for nonlinear acoustic waves $E^{\left(2\right)}$ with the linear acoustic energy $E^{\left(1\right)}$ in the case of $p^{\prime }=u^{\prime }$ (assumed for illustrative purposes). The correction $f\left(p^{\prime }\right)$ is independent of $u^{\prime }$.

Figure 5

Spatial profile of finite-amplitude waves (top) for TWs (a), SWs (b), and AWT (c). Evolution of the average perturbation energy $[\langle E^{\left(2\right)}\rangle$ (solid lines); $\langle E^{\left(1\right)}\rangle$ (dashed lines)] evaluated from the DNS data (bottom) scaled by the initial value against scaled time $t/\tau$ [cf. Eq. (55)] for increasing values of perturbation amplitude $A_{\mathrm{rms}}$ defined in Eq. (38) at ${\nu }_0=1.836\times {10}^{-7}$ (see Table 1). The curves are shifted vertically by 0.25 for illustrative purposes only. With increasing perturbation amplitude $A_{\mathrm{rms}}$, the variation of linear acoustic energy norm $\langle E^{\left(1\right)}\rangle$ becomes increasingly nonmonotonic. The vertical dashed line (bottom) highlights the end of the approximately inviscid spectral energy cascade regime. In this regime, the energy is primarily redistributed in the spectral space due to the nonlinear propagation $\left(\epsilon \simeq 0\right)$.

Figure 6

Illustration of the binary tree implementation of the adaptive mesh refinement technique (top left). The mesh is refined based on the resolution error in pressure field in each cell acting as a node of a binary tree. The pressure field shown (middle) corresponds to the randomly initialized AWT case with $A_{\mathrm{rms}}={10}^{-1},{\nu }_0=1.836\times {10}^{-6}$ (Table 1) at $t/\tau =0.04$. The inset shows the resolved shock wave with $(+)$ denoting the cell interfaces. The mesh refinement levels (bottom) show the depth $d$ of the binary tree.

Figure 7

Schematic illustrating the global picture of various length scales associated with spectral energy cascade in nonlinear acoustics in both spatial (a) and spectral (b)–(d) space. (a) shows the perturbation velocity $u^{\prime }$ (dashed lines) and pressure $p^{\prime }$ (solid lines) fields in AWT obtained from the DNS data for ${\nu }_0=1.836\times {10}^{-7}$ and $A_{\mathrm{rms}}={10}^{-1}$. (b) shows the corresponding spectral energy ${\widehat{E}}_k$ in log-log space. The spectral flux ${\widehat{\mathrm{\Pi }}}_k$ and dissipation ${\widehat{\mathcal{D}}}_k$ are shown in (c) and (d), respectively. The integral length scale $\ell$ corresponds to the characteristic distance between the shock waves traveling in the same direction. The Kolmogorov length scale $\eta$ corresponds to the shock wave thickness. The Taylor microscale $\lambda$ is the diffusive length scale and satisfies $\ell \gg \lambda \gg \eta$. $L$ corresponds to the length of the domain.

Figure 8

Spectrotemporal evolution of ${\widehat{E}}_k$ (top) and spectral flux ${\widehat{\mathrm{\Pi }}}_k$ (bottom) for a TW (a), SW (b), and AWT (c). Spectral flux ${\widehat{\mathrm{\Pi }}}_k$ for a traveling wave simply increases towards high wave numbers. For a standing wave, ${\widehat{\mathrm{\Pi }}}_k$ oscillates at low wave numbers cyclically due to collisions of oppositely traveling shock waves while high-wave-number behavior resembles that of a traveling shock. For AWT, the spectral broadening occurs for $k>k_E$ with small fluctuations in time for $k<k_E$.

Figure 9

Temporal evolution of scaled total energy $\langle E^{\left(2\right)}\rangle A_{\mathrm{rms},0}^{-2}$ (solid lines) (top), dissipation rate $\epsilon A_{\mathrm{rms},0}^{-3}$ (dashed lines) (middle), and normalized Taylor microscale $\lambda /\sqrt{\delta \tau }$ (bottom) for TWs (a), SWs (b), and AWT (c) against the scaled time $t/\tau$ for varying perturbation Reynolds number ${\mathrm{Re}}_L$. The time $t_0$ signifies fully broadened spectrum of the perturbation field.

Figure 10

Evolution of the integral length scale $\ell$ (a) and the Reynolds number ${\mathrm{Re}}_{\ell }$ (b) defined in Eqs. (60) and (69), respectively, for all the cases of AWT considered. For small thermoviscous diffusivity, $\ell$ increases approximately as $t^{1/3}$ before saturating to the dimensionless domain length $L=1$ and ${\mathrm{Re}}_{\ell }$ remains approximately constant.

Figure 11

Fully developed spectra of compensated energy (a), spectral energy flux (b), and cumulative dissipation (c) for TWs at time instant $t_0\approx 0.03$. Harmonics with wave numbers such that $k\eta <1$ contain all the energy. The spectral energy flux vanishes at $k\eta \approx 1$ thus indicating numerical resolution of all the energy-containing harmonics. The marked regime $0.1<k\eta <1$ signifies the dissipation range. The constant $C\approx 0.075$. Solid lines: $A_{\mathrm{rms},0}={10}^{-1}$; dashed lines: $A_{\mathrm{rms},0}={10}^{-2}$; dotted lines: $A_{\mathrm{rms},0}={10}^{-3}$.

Figure 12

Fully developed spectra of compensated energy (a), spectral energy flux (b), and cumulative dissipation (c) for SWs averaged over one time cycle after $t_0\approx 0.04$. Harmonics with wave numbers $k\eta <1$ contain all the energy. The spectral energy flux vanishes at $k\eta \approx 1$ thus indicating numerical resolution of all the energy-containing harmonics. The marked regime $0.1<k\eta <1$ signifies the dissipation range. The constant $C\approx 0.075$. Solid lines: $A_{\mathrm{rms},0}={10}^{-1}$; dashed lines: $A_{\mathrm{rms},0}={10}^{-2}$; dotted lines: $A_{\mathrm{rms},0}={10}^{-3}$.

Figure 13

Fully developed spectra of compensated energy (a), spectral energy flux (b), and cumulative dissipation (c) against scaled wave number $k\eta$ for the randomly initialized broadband noise (AWT) cases with $A_{\mathrm{rms},0}$ and ${\nu }_0$ listed in Table 1 at dimensionless time $t/\tau ={\tau }_0\approx 6\times {10}^{-4}$. The marked regime $0.1<k\eta <1$ signifies the dissipation range. Solid lines: $A_{\mathrm{rms},0}={10}^{-1}$; dashed lines: $A_{\mathrm{rms},0}={10}^{-2}$; dotted lines: $A_{\mathrm{rms},0}={10}^{-3}$.