[Paper Review] Subsonic flows passing a duct for three-dimensional steady compressible Euler system with friction

Hairong Yuan, Qin Zhao|arXiv (Cornell University)|Nov 28, 2017

Navier-Stokes equation solutions5 references3 citations

TL;DR

This paper establishes the existence and stability of symmetric subsonic, supersonic, and transonic-shock solutions in a 3D duct for the steady compressible Euler system with friction (Fanno flow). Using a novel iteration scheme involving a nonlocal elliptic equation and leveraging a framework for elliptic-hyperbolic composite-type systems, it proves the existence of unique solutions under small multidimensional perturbations of boundary data.

ABSTRACT

For the three-dimensional steady non-isentropic compressible Euler system with friction, we show existence of a class of symmetric subsonic, supersonic and transonic-shock solutions in a straight duct with constant square-section. Such flows are called Fanno flow in engineering. We formulate a boundary value problem for subsonic flows, and study their stability under multidimensional small perturbations of boundary conditions. Since the subsonic Euler system is of elliptic-hyperbolic composite-mixed type, this is achieved by using the framework established in [L. Liu; G. Xu; H. Yuan: Stability of spherically symmetric subsonic flows and transonic shocks under multidimensional perturbations. Adv. Math. 291 (2016), 696--757], and establishing an iteration scheme, which involves solving a second order nonlocal elliptic equation.

Motivation & Objective

• To rigorously analyze the effects of friction in three-dimensional steady compressible Euler flows within a duct.
• To formulate and solve a boundary value problem for subsonic flows in a duct with constant square cross-section.
• To establish the stability of subsonic flows under multidimensional small perturbations of boundary conditions.
• To extend the framework for elliptic-hyperbolic composite-type systems to include friction-induced nonlocality.
• To provide a mathematical foundation for understanding transonic shocks and Fanno flow behavior in 3D geometry.

Proposed method

• Formulates a boundary value problem for the 3D steady non-isentropic compressible Euler system with a friction term proportional to $(u^0)^2$.
• Employs a decomposition framework from prior work to linearize the system around symmetric solutions.
• Introduces an iterative scheme involving the solution of a second-order nonlocal elliptic equation to handle the friction-induced nonlocality.
• Uses weighted Hölder norms and a fixed-point argument in a complete metric space to prove existence and uniqueness.
• Applies the Banach fixed-point theorem to the iteration map, showing contraction under small perturbations.
• Establishes a priori estimates via Lemmas 5.1–5.4 to control differences in velocity, pressure, entropy, and energy across iterations.

Experimental results

Research questions

• RQ1 Can symmetric subsonic, supersonic, and transonic-shock solutions exist in a 3D duct with friction for the compressible Euler system?
• RQ2 How does friction influence the structure and stability of subsonic and transonic flows in three dimensions?
• RQ3 Can a well-posed boundary value problem be formulated for 3D subsonic Euler flows with friction?
• RQ4 What is the role of nonlocal elliptic terms arising from friction in the stability analysis of such flows?
• RQ5 How do multidimensional perturbations affect the existence and uniqueness of subsonic solutions in the presence of friction?

Key findings

• A class of symmetric subsonic, supersonic, and transonic-shock solutions exists in a 3D duct with constant square cross-section and friction.
• The subsonic solutions are stable under small multidimensional perturbations of the boundary data.
• The existence of solutions is proven via a contraction mapping argument in a complete metric space of Hölder-regular functions.
• The iteration scheme involves solving a nonlocal elliptic equation due to the friction term, which couples the Bernoulli constant along the flow path.
• The contraction constant in the fixed-point argument is bounded by $ C\varepsilon $, ensuring convergence when $ \varepsilon_0 $ is sufficiently small.
• The solution is unique within a small neighborhood of the reference symmetric solution in the $ C^{2,\alpha} $ norm.