Local temperature perturbations in the boundary layer in regime of free viscous-inviscid interaction

Abstract: We analyze the disturbed flow in the supersonic laminar boundary layer when local heated elements are placed on the surface. It is exhibited that these flows are described in terms of free interaction theory for specific sizes of thermal sources. We construct the numerical solution for flat supersonic problem in the viscous asymptotic layer in which the flow is described by nonlinear equations for vorticity, temperature with the interaction condition which provides influence of perturbations to the pressure in the main order.

• The paper's main contribution demonstrates that localized wall heating in supersonic laminar flows triggers nonlinear free viscous-inviscid interactions, leading to significant pressure variations.
• The paper employs a triple-deck asymptotic approach combined with a robust finite-difference numerical scheme to capture the nonlinear dynamics in the viscous sublayer.
• The paper highlights practical implications for thermal control by showing how localized heating can delay flow separation and influence transition on high-speed surfaces.

Local Temperature Perturbations in Supersonic Boundary Layers Under Free Viscous-Inviscid Interaction

Introduction and Theoretical Framework

This work addresses the dynamics of laminar supersonic boundary layers subjected to localized thermal perturbations at the wall, focusing on configurations in which the scale of the heated region triggers a regime of free viscous-inviscid interaction. The setup is a canonical flat-plate with a supersonic, high Reynolds number laminar flow, featuring local heat sources with size scales $a \sim O(\epsilon^{3/4})$, where $\epsilon$ parameterizes the inverse Reynolds number.

The analysis leverages triple-deck theory—a classical multiscale asymptotic approach—decomposing the disturbed region into upper (inviscid), main (Prandtl), and lower (viscous sublayer) decks. The novelty here is the extension from prior linear perturbation analyses to the fully nonlinear regime, with wall temperature perturbations $\Delta T \sim O(1)$ rather than $\Delta T \ll 1$. This generalization necessitates a coupled nonlinear numerical treatment.

In this framework, the interaction between viscous and inviscid domains is encoded by displacement thickness effects: temperature-induced disturbances modulate not only local flow properties but also the external pressure distribution through an interaction condition reflecting the influence of displacement thickness gradients on pressure.

Mathematical Formulation

The governing equations for the lower deck (viscous sublayer) consist of the continuity, momentum, and energy equations, with the following non-dimensionalized forms:

• Continuity: $$\frac{\partial u_b}{\partial x_b} + \frac{\partial v_b}{\partial y_b} = 0$$
• Momentum: $$u_b \frac{\partial u_b}{\partial x_b} + v_b \frac{\partial u_b}{\partial y_b} + T_b \frac{\partial p_b}{\partial x_b} = \frac{\partial^2 u_b}{\partial y_b^2}$$
• Energy: $$u_b \frac{\partial T_b}{\partial x_b} + v_b \frac{\partial T_b}{\partial y_b} = \frac{\partial^2 T_b}{\partial y_b^2}$$

Boundary conditions specify no-slip and prescribed wall temperature at $y_b = 0$, along with matching to the displacement effect and decay of temperature perturbations as $y_b \to \infty$. The pressure is governed nonlocally through the displacement function $A_1(x_b)$ by the interaction law for supersonic flows:

$p(x_b) = -B \frac{\partial A_1(x_b)}{\partial x_b}$

with $B \sim 1$ in the free interaction regime.

The system is further represented in terms of vorticity, facilitating a robust numerical strategy centered on upwind finite-difference discretizations and iterative under-relaxation for convergence.

Numerical Implementation

The vorticity-energy system is solved on a structured $(x, y)$ grid, employing directional finite-difference stencils for convective terms based on velocity sign to maintain numerical stability. The tridiagonal systems arising from spatial marching are resolved efficiently by line-by-line solvers. To enforce the interaction condition, the displacement function $A_1(x_b)$ is updated iteratively, with strict control of the relaxation parameter $r$ ensuring convergence. The analysis notes sharp sensitivity to $r$, with empirical identification of a narrow admissible regime ($r \approx 0.02$ for stability).

Results and Key Findings

The study systematically evaluates distributions of surface pressure and wall shear stress arising from prescribed step-like temperature perturbations localized to a finite span $a$. The core findings include:

• The amplitude and spatial profile of pressure perturbations are strongly dictated by the interaction parameter $B$ and the magnitude of the temperature rise. For $B \ll 1$, even substantial thermal perturbations produce limited pressure effects, while $B \sim 1$ (free interaction) yields significant, non-trivial pressure variations distributed over the interaction region.
• Nonlinear computations reveal substantial quantitative and qualitative deviations from prior linear theory, especially in the downstream relaxation zone and in the vicinity of separations induced by the perturbation. The linear theory underestimates both the strength and spatial extent of pressure and shear modifications for $\Delta T = O(1)$.
• The pressure perturbation profile consists of a positive peak at the heated region’s leading edge, promoting local deceleration and thus enhancing laminar stability by postponing transition or flow separation. Conversely, negative gradients downstream accelerate the flow, increasing shear stress and influencing the reattachment process.
• Two distinct separation points arise at the upstream and downstream boundaries of the heated region, and their mutual interaction must be considered, especially in designs aimed at mitigating boundary layer separation or controlling the transition.

Practical and Theoretical Implications

These results provide a mechanism by which wall thermal management—specifically, the application of localized heating—can modulate laminar boundary layer separation and transition on supersonic vehicles, potentially resulting in enhanced control authority over aerodynamic surfaces. The interaction between displacement thickness perturbations and pressure fields underscores the necessity of capturing nonlinear effects; linear approaches may be grossly inadequate for practical heat input levels.

From a theoretical standpoint, the emergence of elliptic-like behavior in the lower deck’s otherwise parabolic system due to nonlocal interaction underscores the subtleties involved in boundary-layer–inviscid coupling, particularly when dealing with strong nonlinearities and nonuniform wall conditions.

Prospects for Future Work

The current analysis is constrained to a two-dimensional, stationary, flat-plate idealization. A natural progression is extension to three-dimensional configurations, where crossflow effects and lateral diffusion could yield complex interaction patterns with practical ramifications for real-world surface heating strategies. Further, unsteady and time-dependent thermal disturbances remain unexplored in the nonlinear regime, opening a path toward understanding transient control and the initiation/propagation of Görtler or Tollmien-Schlichting instability modes under time-varying wall heating.

Conclusion

This work rigorously establishes the structure of nonlinear free viscous-inviscid interaction in supersonic boundary layers subject to localized wall heating. The findings reveal the necessity of capturing strong nonlinearities to adequately predict the magnitude and distribution of induced pressure and shear alterations. These insights have direct implications for the design of thermal actuation and control methods for high-speed flows and provide a robust computational framework for further generalizations to three-dimensional and unsteady flow regimes (1110.2673).