Surface instabilities in laminar compressible boundary layers with sublimation

TL;DR

This work addresses surface pattern formation on ablating materials by performing a local linear stability analysis of a compressible laminar boundary layer over a sublimating camphor surface, incorporating sublimation and conjugate heat transfer. A coupled fluid–solid–interface model is developed, utilizing a uniformly receding baseflow and a modal decomposition in wavenumbers $\alpha$ and $\beta$ to solve the resulting eigenvalue problem. The key findings show a single surface mode that becomes unstable only when the wall temperature exceeds the adiabatic level $T_{ad}$, with the dominant orientation transitioning from $\psi=0$ to near the sonic angle $\psi_s$ and then to $\psi=90^{\circ}$ as the wall-temperature ratio $T_r$ increases; a finite critical wavenumber is also identified, consistent with observations in turbulent flows. These results support differential ablation as a plausible laminar mechanism for surface patterns on hot-walled sublimators and provide a framework for extending to turbulent regimes and other surface-process hypotheses.

Abstract

Surface patterns on ablating materials are observed in high-speed ground and flight tests, but the mechanisms behind their formation are not known. In this paper, the origin of surface patterns is investigated via a local linear stability analysis of compressible laminar boundary layers over a flat camphor plate. The effects of sublimation and conjugate heat transfer are included both on the baseflow and the linear fluctuations. This framework identifies one mode that fully characterizes the stability of the surface, which becomes unstable when the wall temperature exceeds that of an adiabatic wall, $T_{ad}$. These findings are consistent with experimental observations, where laminar flow conditions at adiabatic wall temperatures are found to be stable. The analysis also reveals that the nature of this surface mode varies as a function of the oblique angle $ψ= \tan^{-1}{β/α}$, where $α$ and $β$ are the streamwise and spanwise wavenumbers. Specifically, for baseflow temperatures below $\approx 1.15~T_{ad}$, the surface mode is most unstable at $ψ= 0$. Conversely, above $\approx 1.15~T_{ad}$ the surface is most unstable near the sonic angle $ψ_s = \cos^{-1}(1/M_e)$, which is the angle at which the normal Mach number equals one. Finally, a critical wavenumber is identified (i.e., one at which the temporal growth rate reaches a maximum) that is in good agreement with available experimental observations of turbulent flows.

Figures (9)

• Figure 1: Ablation patterns showing streamwise grooves (left), turbulent wedges (middle), and crosshatching (right). Images taken from Figure 1 of stock1973cross.
• Figure 2: Schematic of local linear stability of an ablating surface
• Figure 3: Self-similar baseflows with binary sublimation of camphor at $M=3$, $Re=785$, $\tilde{T}_0=0.65$, and $\tilde{p}_0=2$.
• Figure 4: Eigenvalues for a sublimating (black dots) and non-sublimating (red triangles) interface (left) and pressure component of eigenmodes (right). The baseflow conditions are $M=3$, $Re=5000$, $\tilde{T}_0=0.65$, $\tilde{p}_0=2$, and $T_r=1.0$.
• Figure 5: Growth rates for 2D surface modes for variations in the baseflow conditions with respect to a common condition of $M=3$, $Re=785$, $\tilde{T}_0=0.65$, $\tilde{p}_0=2$, and $T_r=1.0$. The single condition varied is indicated in the legend of each panel. The black lines mark neutral stability.