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Upper bound of transient growth in accelerating and decelerating wall-driven flows using the Lyapunov method

Zhengyang Wei, Weichen Zhao, Chang Liu

Abstract

This work analyzes accelerating and decelerating wall-driven flows by quantifying the upper bound of transient energy growth using a Lyapunov-type approach. By formulating the linearized Navier-Stokes equations as a linear time-varying system and constructing a time-dependent Lyapunov function, we obtain an upper bound on transient energy growth by solving linear matrix inequalities. This Lyapunov method can obtain the upper bound of transient energy growth that closely matches transient growth computed via the singular value decomposition of the state-transition matrix of linear time-varying systems. Our analysis captures that decelerating base flows exhibit significantly larger transient growth compared with accelerating flows. Our Lyapunov method offers the advantages of providing a certificate of uniform stability and an invariant set to bound the solution trajectory.

Upper bound of transient growth in accelerating and decelerating wall-driven flows using the Lyapunov method

Abstract

This work analyzes accelerating and decelerating wall-driven flows by quantifying the upper bound of transient energy growth using a Lyapunov-type approach. By formulating the linearized Navier-Stokes equations as a linear time-varying system and constructing a time-dependent Lyapunov function, we obtain an upper bound on transient energy growth by solving linear matrix inequalities. This Lyapunov method can obtain the upper bound of transient energy growth that closely matches transient growth computed via the singular value decomposition of the state-transition matrix of linear time-varying systems. Our analysis captures that decelerating base flows exhibit significantly larger transient growth compared with accelerating flows. Our Lyapunov method offers the advantages of providing a certificate of uniform stability and an invariant set to bound the solution trajectory.

Paper Structure

This paper contains 6 sections, 1 theorem, 24 equations, 8 figures, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setup and Laminar Base Flows
  3. Transient Growth of Accelerating and Decelerating Flows
  4. Upper Bound of Transient Growth Using Lyapunov Method
  5. Results and Discussions
  6. Conclusions

Key Result

Theorem 1

Given a linear time-varying system in eq:discrtizedformlinearsystem. If we can find a continuously differentiable Hermitian matrix $\boldsymbol{P}(t)$ and $\overline{G}>0$ by solving where $(\cdot)\preceq0$ means negative semi-definiteness. Then, the equilibrium $\boldsymbol{x}=\boldsymbol{0}$ of system eq:discrtizedformlinearsystem is uniformly stable and the transient energy growth $G(t)$ is bo

Figures (8)

  • Figure 1: Diagram of wall-driven channel flow, with an example snapshot of the laminar flow for exponentially decaying wall motion.
  • Figure 2: The transient growth $G(t)$ and its upper bound $\overline{G}$ for decelerating WDF at $\text{Re} = 500$, $\kappa = 0.1$, and $[k_x, k_z] = [1.2, 0]$. Each color corresponds to a different initial time at $t_0 = [0, 10, 20, 40, 60, 80, 100]$.
  • Figure 3: The transient growth $G(t)$ and its upper bound $\overline{G}$ for decelerating WDF at $\text{Re} = 500$, $\kappa = 0.1$, $t_0 = 20$, and $[k_x, k_z] = [1.2, 0]$. Panel (a) shows results with $\Delta t=1$ and $M=$24, 32, and 48 (colorbar). Panel (b) shows results with $M=32$ and $\Delta t = 1$, 0.5, and 0.2 (colorbar).
  • Figure 4: The transient growth $G(t)$ and its upper bound $\overline{G}$ for decelerating WDF at $\text{Re} = 500$, $\kappa = 0.1$, and $[k_x, k_z] = [0, 1.6]$. Each color corresponds to a different initial time at $t_0 = [0, 10, 20, 40, 60, 80, 100]$.
  • Figure 5: The transient growth $G(t)$ and its upper bound $\overline{G}$ for accelerating WDF at $\text{Re} = 500$, $\kappa = 0.1$, and $[k_x, k_z] = [1.2, 0]$. Each color corresponds to a different initial time at $t_0 = [0, 10, 20, 40, 60, 80, 100]$.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Definition 1: Definition 4.4 in khalil2002nonlinear
  • Theorem 1
  • proof