Transonic shock solutions for steady 3-D axisymmetric full Euler flows with large swirl velocity in a finite cylindrical nozzle

Technical Summary: Transonic Shock Solutions for Steady 3-D Axisymmetric Full Euler Flows with Large Swirl Velocity

Problem Statement
This paper addresses the mathematical existence and location of three-dimensional (3-D) axisymmetric transonic shock solutions for steady compressible full Euler flows within a finite cylindrical nozzle. The study focuses specifically on flows characterized by large swirl velocity (and consequently large vorticity) under prescribed boundary conditions: a given supersonic state at the nozzle entrance, a specific receiver pressure at the exit, and a slip condition on the nozzle wall.

The core challenge lies in determining the position of the shock front (a free boundary) for flows where the swirl velocity is not negligible. Unlike flows with zero or small swirl, where background shock solutions can be trivially constructed as piecewise constant states, flows with large swirl do not admit such simple background solutions. Furthermore, the presence of large swirl introduces strong coupling between the elliptic and hyperbolic parts of the Euler system in the subsonic region behind the shock, specifically through lower-order terms, preventing standard decoupling techniques used in previous literature.

Methodology
The authors employ a two-step strategy to establish the existence of shock solutions with a determined location:

1. Construction of Special Background Solutions (Step 1):

   • The authors first establish the existence and uniqueness of "special shock solutions" where the flow states depend only on the radial distance from the symmetry axis.
   • They derive necessary and sufficient conditions for these solutions to exist, showing that the Mach number function of the incoming flow must satisfy specific ordinary differential equations.
   • Crucially, they demonstrate that for these special solutions, the shock front location can be arbitrarily shifted within the nozzle, meaning they cannot serve directly as a fixed background for perturbation analysis.

2. Perturbation Analysis and Free Boundary Determination (Step 2):

   • The main problem is treated as a perturbation of the special shock solutions established in Step 1. The boundary data (entrance state and exit pressure) are perturbed by a small parameter $\sigma$.
   • Decomposition Technique: To handle the strong coupling in the elliptic-hyperbolic composite system behind the shock, the authors develop new decomposition techniques. They introduce two auxiliary problems:
     • A boundary value problem for an elliptic system derived by removing coupling zero-order terms.
     • A remaining problem handling the strongly coupled non-local terms.
   • Singularity Handling: To address singularities along the symmetry axis ($r=0$), the authors lift the problem to higher dimensions (5D for the hyperbolic part ahead of the shock and 5D for the elliptic part behind the shock) to utilize standard regularity theories and energy estimates.
   • Shock Location Mechanism: The location of the shock front is determined via a solvability condition derived from the elliptic auxiliary problem. This condition relies fundamentally on the non-zero swirl velocity of the unperturbed solution. The authors show that the swirl velocity contributes essential terms that allow the shock position to be uniquely fixed by the boundary data, contrasting with mechanisms in flat nozzles without swirl where geometry often dictates the location.

Key Results

• Existence of Special Solutions: The paper proves the existence of unique special steady normal shock solutions with large non-zero swirl, dependent only on the radial coordinate, under specific regularity and compatibility conditions on the Mach number and flow parameters.
• Existence of Perturbed Solutions: Under the assumption that the boundary data are small perturbations of the special solutions, the authors prove the existence of a unique transonic shock solution to the full nonlinear free boundary problem.
• Determination of Shock Location: The paper establishes that the shock front position is uniquely determined by the perturbed boundary data. The non-zero swirl velocity plays an essential and fundamental role in this determination, providing a mechanism distinct from those observed in zero-swirl or small-swirl scenarios.
• Regularity: The solutions are shown to possess high regularity ($C^{2,\alpha}$ for the subsonic region and $C^3$ for the supersonic region), with careful estimates provided to handle the singularities at the symmetry axis and the nozzle wall.

Significance and Claims
The authors claim this work represents the first rigorous mathematical result concerning 3-D transonic shocks with either large vorticity or large swirl velocity.

• Novelty in Coupling: The paper highlights that the strong coupling of elliptic and hyperbolic parts in the lower-order terms due to large swirl prevents the application of classical theories used for zero-swirl flows. The developed decomposition techniques are presented as a necessary innovation to overcome this.
• Mechanism for Shock Location: A significant contribution is the identification of a new mechanism for determining the shock location. The authors assert that, unlike in flat nozzles where the shock location might be non-unique or determined by boundary geometry, the non-zero swirl velocity here provides the essential terms in the solvability condition that uniquely fix the shock position.
• Overcoming Lack of Background Solutions: The work addresses the difficulty that no trivial piecewise constant background solutions exist for large swirl flows, requiring the construction of non-trivial special solutions as a prerequisite for the perturbation analysis.

The paper concludes that the non-zero swirl velocity is not merely a perturbation but a fundamental factor that enables the solvability and uniqueness of the shock location in this specific class of 3-D axisymmetric flows.