Investigation of supersonic heat-conductivity hyperbolic waves in radiative ablation flows

We carry out a numerical investigation of three-dimensional linear perturbations in a self-similar ablation wave in slab symmetry, representative of the shock transit phase of a pellet implosion in inertial confinement fusion (ICF). The physics of ablation is modeled by the equation of gas dynamics with a nonlinear heat conduction as an approximation for radiation transport. Linear perturbation responses of the flow, its external surface, and shock-wave front, when excited by external pressure or heat-flux perturbation pulses, are computed by fully taking into account the flow compressibility, nonuniformity, and unsteadiness. These responses show the effective propagation, at supersonic speeds, of perturbations from the flow external surface through the whole conduction region of the ablation wave, beyond its Chapman-Jouguet point, and up to the ablation front, after the birth of the ablation wave. This supersonic forward propagation of perturbations is evidenced by means of a set of appropriate pseudocharacteristic variables and is analyzed to be associated to the “heat-conductivity” waves previously identified by Clarisse et al. [J. Fluid Mech. 848, 219 (2018)]. Such heat-conductivity linear waves are found to prevail over heat diffusion as a feedthrough mechanism [Aglitskiy et al., Philos. Trans. R. Soc. A 368, 1739 (2010)] for perturbations of longitudinal characteristic lengths of the order of—or larger than—the conduction region size, and long transverse wavelengths with respect to this region size, and over time scales shorter to much shorter than the shock transit phase duration. This mechanism, which results from the dependency of the heat conductivity on temperature and density in conjunction with a flow temperature stratification, is expected to occur for other types of nonlinear heat conductions—e.g., electron heat conduction—as well as to be efficient at transmitting large scale perturbations from the surrounding of an ICF pellet to its inner compressed core at later times of its implosion. Besides, the proposed set of pseudocharacteristic variables are recommended for analyzing perturbation dynamics in an ablation flow as it furnishes additional propagation information over the fundamental linear modes of fluid dynamics [Kovásznay, J. Aeronautic. Sci. 20, 657 (1953)], especially in the flow conduction region.

Figure 1

(a) Base flow profiles in the Lagrangian coordinate $m$ scaled by the function defined by ${\mathrm{slog}}_1(·)\equiv \text{sgn}(·)log(1+|·\left|\right)$. Density ($\rho$), longitudinal velocity ($v$), and heat flux ($\phi$). (b) Characteristic wave speeds [Eq. (9)] (${\lambda }_{1,2,3,4}$) and fluid momentum relative to the ablation front ($-\rho u^{\prime }$).

Figure 2

Projection of the solution to Eq. (4) for a heat-flux perturbation $(n,k_{\perp })=(4,1)$ on pseudocharacteristic ${\widehat{\mathcal{W}}}_1$ in the conduction region in logscale. The dashed line represents a ${\mathcal{C}}_1$ characteristic originating from the pulse half height [Eq. (11)]. Trajectory of the CJ point ($*$) is also indicated.

Figure 3

Perturbation in (a) density in logscale, (b) longitudinal velocity in logscale, and (c) temperature in ${\text{slog}}_1$ scale, for a heat-flux perturbation with $(n,k_{\perp })=(4,0)$. The dashed line represents a ${\mathcal{C}}_1$ characteristic originating from the pulse half height [Eq. (11)].

Figure 4

Response of Eq. (4) to a heat-flux perturbation [Eq. (11b)] with $(n,k_{\perp })=(4,1)$ visualized in $(m,t)$ coordinates. Pseudocharacteristic variable: (a) $log|{\widehat{\mathcal{W}}}_1|$, (b) ${\text{slog}}_{{10}^3}\left({\widehat{\mathcal{W}}}_2\right)$, (c) ${\text{slog}}_{10}\left({\widehat{\mathcal{W}}}_4\right)$, (d) ${\text{slog}}_{{10}^3}\left({\widehat{\mathcal{W}}}_3\right)$; Kovásznay modes: (e) pressure ${\text{slog}}_{{10}^3}\left(\widehat{p}\right)$, (f) entropy ${\text{slog}}_{{10}^3}\left(\widehat{s}\right)$, and (g) potential vorticity ${\text{slog}}_{{10}^2}({\widehat{\omega }}_{\perp }/\rho )$ [28]. Trajectories of the mean position of the ablation front ($\times$), CJ point ($*$), and shock front ($\square$). Dash lines represent characteristic trajectories: ${\mathcal{C}}_1$, ${\mathcal{C}}_2$, ${\mathcal{C}}_4$, and ${\mathcal{C}}_3$ characteristics are constant $m$ lines. The first ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ characteristics originate from the pulse half-height [Eq. (11a)], while the subsequent characteristics originate from the interactions of significant perturbation signals with the shock front, the external surface, and the ablation front.

Figure 5

Response of Eq. (4) to a pressure perturbation [Eq. (11a)] with $(n,k_{\perp })=(4,0)$ projected on (a) $log|{\widehat{\mathcal{W}}}_1|$ and (b) ${\text{slog}}_1\left({\widehat{\mathcal{W}}}_2\right).$ Conventions similar to Fig. 4. The dash lines represent ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ characteristics originating from the pulse half maximum [Eq. (11b)].

Figure 6

Schematic representation of perturbation propagation in the $(m,t)$ plane originating from a heat-flux perturbation at the external surface. Perturbation trajectories are sketched as colored lines: ${\mathcal{C}}_1$ (green, densely dotted), ${\mathcal{C}}_2$ (cyan, loosely dashed), and ${\mathcal{C}}_4$ (red, solid). Arrows indicate the propagation direction. The thickness and number of arrows render the intensity of the corresponding signal. Trajectories of CJ point, ablation front (“$\mathrm{af}$”), and shock front (“$\mathrm{sf}$”) also indicated.

Figure 7

Baroclinic term [Eq. (12)] in logscale for an external heat-flux perturbation $(n,k_{\perp })=(4,1)$. Dash lines represent characteristic trajectories as in Fig. 4.

Figure 8

Deformations of the shock front ${\widehat{X}}_{\mathrm{sf}}$ (red, solid), the ablation front ${\widehat{X}}_{\mathrm{af}}$ (green, dashed), and the external surface ${\widehat{X}}_{\mathrm{es}}$ (blue, dotted) for (a) a longitudinal heat-flux and (b) pressure perturbation $(n,k_{\perp })=(4,0)$, normalized to unity [Eq. (13)]. These deformations are first order perturbations of mean positions appearing on Fig. 4. The insets illustrate long time behaviors.

Figure 9

Deformation of the ablation front ${\widehat{X}}_{\mathrm{af}}$ at early times for different longitudinal characteristic ${\lambda }_x/l_{\mathrm{cond}}=\left\{1/2,4\right\}$ and transverse wave numbers $k_{\perp }=\left\{0,1,10\right\}.$

Figure 10

Contribution from (a) advection, (b) diffusion, and (c) remaining terms from Eq. (10a) to the variation rate of ${\widehat{\mathcal{W}}}_1(n,k_{\perp })=(4,1)$. The dash line represents a ${\mathcal{C}}_1$ characteristic trajectory originating from the top of the heat-flux pulse [Eq. (5)].

Figure 11

Contributions of primitive variables $(\widehat{\rho },{\widehat{v}}_x,\widehat{T})$ in a unit heat-conductivity wave quantity $\left({\widehat{\mathcal{W}}}_1\right)$ conserved along ${\mathcal{C}}_1$ in Eq. (9), with no contribution from transverse velocity. Zoom in the conduction region.

Figure 12

Dominating terms in Eq. (10a) for (a) ${\widehat{\mathcal{W}}}_i\to {\widehat{\mathcal{W}}}_1$, (b) ${\widehat{\mathcal{W}}}_i\to {\widehat{\mathcal{W}}}_2$, and (c) ${\widehat{\mathcal{W}}}_i\to {\widehat{\mathcal{W}}}_4$, for $1/n$ [Eq. (11)] ranging from ${10}^{-3}$ to 10 and $m=0.25$. Contribution from second order derivatives ${\mathcal{A}}_{ij}{\partial }_m^2{\widehat{\mathcal{W}}}_j$ (solid), first order derivatives ${\mathrm{\Lambda }}_{ii}{\partial }_m{\widehat{\mathcal{W}}}_i$ (dashed), and amplification terms $({\mathcal{C}}_{ij}+{\mathrm{\Delta }}_{ij}){\widehat{\mathcal{W}}}_j$ (dotted).

Figure 13

Magnitude of dominating amplification terms in the ablation layer, (a) ${\widehat{\mathcal{W}}}_i\to {\widehat{\mathcal{W}}}_1$, (b) ${\widehat{\mathcal{W}}}_i\to {\widehat{\mathcal{W}}}_2$, and (c) ${\widehat{\mathcal{W}}}_i\to {\widehat{\mathcal{W}}}_4$.