Self-consistent feedback mechanism for the sudden viscous dissipation of finite-Mach-number compressing turbulence

Previous work [Davidovits and Fisch, Phys. Rev. Lett. 116, 105004 (2016)] demonstrated that the compression of a turbulent field can lead to a sudden viscous dissipation of turbulent kinetic energy (TKE), and that paper suggested this mechanism could potentially be used to design new fast-ignition schemes for inertial confinement fusion (ICF). We expand on previous work by accounting for finite Mach numbers, rather than relying on a zero-Mach-limit assumption as previously done. The finite-Mach-number formulation is necessary to capture a self-consistent feedback mechanism in which dissipated TKE increases the temperature of the system, which in turn modifies the viscosity and thus the TKE dissipation itself. Direct numerical simulations with a tenth-order accurate Padé scheme were carried out to analyze this self-consistent feedback loop for compressing turbulence. Results show that, for finite Mach numbers, the sudden viscous dissipation of TKE still occurs, for both the solenoidal and dilatational turbulent fields. As the domain is compressed, oscillations in dilatational TKE are encountered due to the highly oscillatory nature of the pressure dilatation. An analysis of the source terms for the internal energy shows that the mechanical-work term dominates the viscous turbulent dissipation. As a result, the effect of the suddenly dissipated TKE on temperature is minimal for the Mach numbers tested. Moreover, an analytical expression is derived that confirms the dissipated TKE does not significantly alter the temperature evolution for low Mach numbers, regardless of compression speed. The self-consistent feedback mechanism is thus quite weak for subsonic turbulence, which could limit its applicability for ICF.

Figure 1

Schematic of energy transfer between the four energy components: mean kinetic energy, mean internal energy, solenoidal TKE, and dilatational TKE.

Figure 2

The dissipation spectrum, normalized by the Kolmogorov velocity $u_{\eta }={\left(\epsilon \nu \right)}^{1/4}$, at the final time of the linearly forced simulation. Plots using (a) a linear scale and (b) a log-log scale are included.

Figure 3

Evolution of (a) solenoidal and (b) dilatational TKE as a function of the size of the domain $L$. $S_0^{*}$ is the initial value of the shear parameter $S^{*}=Sk/\epsilon$, where $S=\dot{L}/L$. $k_0^{\left(s\right)}$ and $k_0^{\left(d\right)}$ are the initial values of $k^{\left(s\right)}$ and $k^{\left(d\right)}$, respectively. The initial length of the domain is $L=1$, which decreases as time progresses.

Figure 4

(a) Solenoidal TKE against the solenoidal shear parameter $S_s^{*}=S{k}^{\left(s\right)}/{\epsilon }^{\left(s\right)}$ and (b) dilatational TKE against dilatational shear parameter $S_d^{*}=S{k}^{\left(d\right)}/{\epsilon }^{\left(d\right)}$. $S_0^{*}$ is the initial value of the shear parameter $S^{*}=Sk/\epsilon$, where $S=\dot{L}/L$. $k_0^{\left(s\right)}$ and $k_0^{\left(d\right)}$ are the initial values of $k^{\left(s\right)}$ and $k^{\left(d\right)}$, respectively. The vertical dashed line in (a) corresponds to the time at which $P^{\left(s\right)}=〈\rho 〉{\epsilon }^{\left(s\right)}$, and the vertical dashed line in (b) corresponds to the time at which $P^{\left(d\right)}=〈\rho 〉{\epsilon }^{\left(d\right)}$. The dashed diagonal lines correspond to Eq. (25).

Figure 5

TKE budget for the (a) solenoidal and (b) dilatational flow fields, given $S_0^{*}=0.5$. The initial length of the domain is $L=1$, which decreases as time progresses. All terms have been normalized by ${\rho }_0{U}_0^3/L_0$.

Figure 6

TKE budget for the (a) solenoidal and (b) dilatational flow fields, given $S_0^{*}=5.0$. The initial length of the domain is $L=1$, which decreases as time progresses. All terms have been normalized by ${\rho }_0{U}_0^3/L_0$. The same legend as that of Fig. 5 applies to the plots above. The vertical dashed line corresponds to the domain size at which peak solenoidal TKE (a) and peak dilatational TKE (b) are achieved.

Figure 7

TKE budget for the (a) solenoidal and (b) dilatational flow fields, given $S_0^{*}=50$. The initial length of the domain is $L=1$, which decreases as time progresses. All terms have been normalized by ${\rho }_0{U}_0^3/L_0$. The same legend as that of Fig. 5 applies to the plots above. The vertical dashed line corresponds to the domain size at which peak solenoidal TKE (a) and peak dilatational TKE (b) are achieved.

Figure 8

TKE budget for the (a) solenoidal and (b) dilatational flow fields, given $S_0^{*}=500$. The initial length of the domain is $L=1$, which decreases as time progresses. All terms have been normalized by ${\rho }_0{U}_0^3/L_0$. The same legend as that of Fig. 5 applies to the plots above. The vertical dashed line corresponds to the domain size at which peak solenoidal TKE (a) and peak dilatational TKE (b) are achieved.

Figure 9

Energy spectra for the (a) solenoidal and (b) dilatational TKE, at different times (or domain lengths) throughout the compression. The spectra correspond to the $S_0^{*}=5.0$ case.

Figure 10

Evolution of temperature as a function of the size of the domain. The initial temperature is denoted by ${\tilde{T}}_0$.

Figure 11

Mean internal energy budget for the four compression speeds (a) $S^{*}=0.5$, (b) $S^{*}=5.0$, (c) $S^{*}=50$, and (d) $S^{*}=500$. The initial length of the domain is $L=1$, which decreases as time progresses. All terms have been normalized by ${\rho }_0{U}_0^3/L_0$.