[Paper Review] Existence and stability of cylindrical transonic shock solutions under three dimensional perturbations

Shangkun Weng, Zhouping Xin|arXiv (Cornell University)|Apr 5, 2023

Computational Fluid Dynamics and Aerodynamics4 citations

TL;DR

This paper establishes the existence and nonlinear stability of three-dimensional cylindrical transonic shock solutions under general perturbations of the incoming flow and exit pressure, without restrictions on the background shock. Using a novel deformation-curl decomposition of the steady Euler system, it uniquely determines the shock position and strength via algebraic Rankine-Hugoniot conditions and derives a Poisson equation with oblique Neumann boundary conditions, achieving optimal regularity for all physical quantities including velocity, pressure, entropy, and Bernoulli’s function.

ABSTRACT

We establish the existence and stability of cylindrical transonic shock solutions under three dimensional perturbations of the incoming flows and the exit pressure without any further restrictions on the background transonic shock solutions. The strength and position of the perturbed transonic shock are completely free and uniquely determined by the incoming flows and the exit pressure. The optimal regularity is obtained for all physical quantities, and the velocity, the Bernoulli's quantity, the entropy and the pressure share the same regularity. The approach is based on the deformation-curl decomposition to the steady Euler system introduced by the authors to decouple the hyperbolic and elliptic modes effectively. However, one of the key elements in application of the deformation-curl decomposition is to find a decomposition of the Rankine-Hugoniot conditions, which shows the mechanism of determining the shock front uniquely by an algebraic equation and also gives an unusual second order differential boundary conditions on the shock front for the first order deformation-curl system. After homogenizing the curl system and introducing a potential function, this unusual condition on the shock front becomes the Poisson equation with homogeneous Neumann boundary condition on the intersection of the shock front and the cylinder walls from which an oblique boundary condition for the potential function can be uniquely derived.

Motivation & Objective

• To establish the existence and structural stability of three-dimensional cylindrical transonic shock solutions under arbitrary perturbations of the incoming flow and exit pressure.
• To remove prior restrictions on background transonic shock solutions, allowing free and unique determination of shock strength and position.
• To achieve optimal regularity for all physical quantities—velocity, pressure, entropy, and Bernoulli’s function—under the same smoothness conditions.
• To develop a new framework for handling the hyperbolic-elliptic mixed nature of transonic flows via a deformation-curl decomposition of the steady Euler system.
• To derive a novel second-order differential boundary condition on the shock front from the Rankine-Hugoniot conditions, enabling a unique solution via potential function formulation.

Proposed method

• Introduce a deformation-curl decomposition of the steady compressible Euler system to decouple hyperbolic and elliptic modes.
• Derive a new algebraic formulation of the Rankine-Hugoniot conditions that uniquely determines the shock front position through an implicit equation.
• Construct an unusual second-order differential boundary condition on the shock front for the first-order deformation-curl system, which is then homogenized.
• Introduce a potential function to transform the shock-front boundary condition into a Poisson equation with homogeneous Neumann condition on the intersection of the shock and cylinder walls.
• Derive an oblique boundary condition for the potential function from the shock condition, ensuring uniqueness and regularity.
• Apply a fixed-point argument in a suitable function space to prove existence and stability under perturbations of the inflow and exit pressure.

Experimental results

Research questions

• RQ1 Can transonic shock solutions in a three-dimensional cylindrical nozzle be proven to exist and remain stable under arbitrary perturbations of the incoming flow and exit pressure?
• RQ2 How can the shock front position and strength be uniquely determined by the boundary data without imposing constraints on the background solution?
• RQ3 What is the optimal regularity of the velocity, pressure, entropy, and Bernoulli’s function in such transonic shock configurations?
• RQ4 How can the Rankine-Hugoniot conditions be reformulated to yield a second-order differential boundary condition on the shock surface for a first-order system?
• RQ5 Can the deformation-curl decomposition be extended to handle the nonlinear free boundary problem of transonic shocks with full three-dimensional geometry?

Key findings

• The shock front position and strength are uniquely determined by the incoming flow and exit pressure, with no restrictions on the background transonic shock solution.
• All physical quantities—velocity, pressure, entropy, and Bernoulli’s function—achieve the same optimal regularity, which is inherited from the data.
• The Rankine-Hugoniot conditions are reformulated to yield a second-order differential boundary condition on the shock surface, enabling a unique solution to the deformation-curl system.
• The shock-front boundary condition is transformed into a Poisson equation with homogeneous Neumann condition on the intersection of the shock and cylinder walls, from which an oblique boundary condition for the potential function is uniquely derived.
• The system admits a unique solution under small perturbations of the inflow and exit pressure, proving nonlinear stability of the transonic shock configuration.
• The method successfully handles the hyperbolic-elliptic mixed-type nature of the Euler system in three dimensions, overcoming previous limitations in geometric and structural assumptions.