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On the classical solution for the steady triple-deck equations

Ming Dong, Chao Wang, Qin Wu, Zhifei Zhang

Abstract

This paper establishes the existence and uniqueness of classical solutions to the steady Triple-Deck equations, which describe incompressible boundary layer flow over localized roughness at high Reynolds numbers. The triple-deck theory was developed to overcome the Goldstein singularity in classical Prandtl boundary layer theory, capturing the interaction between the viscous sublayer, main layer, and upper layer when surface roughness of height $O({\rm Re}^{-\frac58})$ is present. The key ingredients of this paper include: (1) A decomposition separating roughness effects from nonlinear difficulties; (2) A novel Green's function using Airy functions that overcomes low-frequency singularities via the non-vanishing property of $\sqrt{3}Ai(z)+Bi(z)$; (3) The introduction of weighted Sobolev norms $\norm{|\p_x|^{\frac{1}{18}}y^{\frac16}ω}_{L^2}$ of the vorticity yielding $M$-independent estimates for displacement $A$. As a byproduct, local uniqueness of Couette flow is established when $F=0$.

On the classical solution for the steady triple-deck equations

Abstract

This paper establishes the existence and uniqueness of classical solutions to the steady Triple-Deck equations, which describe incompressible boundary layer flow over localized roughness at high Reynolds numbers. The triple-deck theory was developed to overcome the Goldstein singularity in classical Prandtl boundary layer theory, capturing the interaction between the viscous sublayer, main layer, and upper layer when surface roughness of height is present. The key ingredients of this paper include: (1) A decomposition separating roughness effects from nonlinear difficulties; (2) A novel Green's function using Airy functions that overcomes low-frequency singularities via the non-vanishing property of ; (3) The introduction of weighted Sobolev norms of the vorticity yielding -independent estimates for displacement . As a byproduct, local uniqueness of Couette flow is established when .

Paper Structure

This paper contains 9 sections, 10 theorems, 191 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Triple-Deck equations
  3. Main results
  4. Main difficulties and Key ideas
  5. Notations and Organization of the paper
  6. Functional spaces and nonlinear estimates
  7. The solvability of the linear system
  8. Existence and uniqueness of classical solution
  9. The physical derivation of triple-deck equation

Key Result

Theorem 1.1

Let $\alpha\in(0,\frac{7}{3}]$. The linear system Eq for omega0 admits a solution $(\omega_0, A_0)\in{\mathcal{X}}_{\alpha,M}$ satisfying the estimate where $C>0$ is independent of $\alpha,M$.

Figures (3)

  • Figure 1: Demonstration of the triple-deck structure (not to scale): I Viscous Sublayer; II Main Layer; III Upper Layer; $\delta$ Thickness of Prandtl layer.
  • Figure 2: Analysis on $m(\xi)$.
  • Figure 3: Demonstration of the triple-deck structure (not to scale): I Viscous Sublayer; II Main Layer; III Upper Layer; $\delta$ Thickness of Prandtl layer; $\tilde{U}_0$ Streamwise velocity.

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.1
  • ...and 15 more