## singular value decomposition

The maximum amplitude of output for a given input amplitude?

M的作用如图所示，对圆盘进行拉伸和旋转

SVD截断

A least squares solution x0 that minimises $|M x-b|$.

## 全局稳定性分析方法

N 是 Navier-Stokes operator.

The temporal evolution of perturbations q’

the linearized Navier-Stokes operator about the base flow, also called the Jacobian of the system

A global linear stability analysis consists of searching modes of the form

reduces N-S equation into the eigenproblem:

The corresponding eigenvalues characterize the stability of $q_b$: if an eigenvalue $\lambda_i$ has a positive real part $\delta_i = real(\lambda_i)$ (positive growth rate), then the base flow is unstable. Imaganry part means the oscillating mode at a frequency $w_i$.

## Resolvent analysis for the of pseudo-resonance

Resolvent analysis最早应用是在稳定性和转捩中的nonmetal analysis中运用。例如在寻找线性方程组的矩阵时间指数的范数，对transient growth的研究，e-pseudospectra, 以及定义线性NS在对外部forcing最大反馈的比例。

R 被称为resolvent operator, 或者叫放大因子。

The optimal forcing which maximizes the energy gain G

## Mean flow stability vs base flow stability analysis

In some cases (e.g. the cylinder flow), the frequency selection process is strongly driven by nonlinear effects. 因此，线性基流稳定性分析不能正确地预测流体的动力学行为，而这产生不满足RPIF(((real part positive) with a frequency (the imaginary part))属性的情况.

## 整体程序框架

### 流体域：

$\sigma$ 为Cauchy 应力张量，跟流体的形变和雷诺数又关系。

• 固体域使用Lagrangian框架、流体域使用Eulerian框架，这样的好处是表达更加自然美观。

## 弱解形式，应用FEM求解

freefem++ 参考代码：https://github.com/ERC-AEROFLEX/ALE

2.1.2公式转化为稳定性问题：

## Linear Stability Analysis 更深一层次的理解

$\mathcal{A}$ 为Jacobian matrix of F.

## Non-modal Stability Analysis 更深一层次的理解

$\mathcal{A}^{\dagger}$为伴随矩阵，它具有非常好的性质。关于伴随矩阵和逆矩阵的关系，和其性质，参考链接

As a result of this non-normality, the eigenvectors of A do not form an orthonormal set of vectors。

## Resolvent Analysis

the response of the system to the forcing f(t) 表现为卷积的形式：

Consider linear stable systems，influence of the forcing on the current state decays exponentially according to the least stable eigenvalue。

## 线化不可压N-S方程

fluaction equation:

RANS equation: (w=0)

pressure Poisson equation

mean component

frequency component

McKeon & Sharma (2010), 做的是在管道里面。

## 预解算子的边界条件

### 对于软壁面边界条件的思路🤔：

Impose boundary conditions for Navier-Stokes Resolvent This accounts for a simple compliant surface model, which essentially leads to a harmonic-motion relationship between pressure and velocity。

possion equation:

boundary condition at the wall:

## 加强筋边界-channel

Solid obstructions within the fluid domain—riblets in our case—are modeled as a spatially varying permeability function Kx; y; z,

which appears as an additional body force in the momentum equations. 3

【方程1】

【方程2】

【方程3】

【方程4】

【方程5】

【方程6】

【方程7】

【方程8】

【方程9】

## SVD分解

f_k仅仅于速度有关系，因此也可以写成：

svd分解：

## 整体目标：把握信号的特点和机理的关系，构建模型

2d，周期性变换，均匀流，环形

### 验证模型3:

https://su2code.github.io/tutorials/Inc_Turbulent_NACA0012/

http://www.cemeai.icmc.usp.br/projetos/item/965-simulacoes-numericas-de-turbulencia-em-asas-de-aeronaves

## 可压的resolvent anlaysis N-S model

### 可压LES model （Mach > 0.2，Mach < 6）

#### 推导

viscous dissipation

### LES的滤波概念：尺度分离

non-resolved part：

## LES 方程

【方程1】

【方程1滤波形式】：方程整体滤波，卷积核函数操作，不可避免引入Commutation Error，通常密度最稳定，分离误差较小。

【方程2-焓的形式】

viscous dissipation

【方程2-焓的滤波形式】：

【方程2-温度的形式】

【方程2-温度的滤波形式】

【方程2-压力的形式】

【方程2-压力的滤波形式】

【方程2-熵的形式】

【方程2-熵的滤波形式】

【方程3-状态方程】

【方程3-状态方程的滤波形式】

【模型】：还有一些变参数模型，不具体介绍

【方程4-Total Energy】

【方程4-Total Energy滤波形式】

【方程5-动量方程】

【方程5-动量方程滤波形式】

## Resolvent analysis of compressible N-S equation

strain and stress tensor：

steady version of compressible Navier- Stokes equations：

## Weak form of incompressible N-S equation

### Compressible N-S equation

baseflow solution：

eigenvalue：

### 额外兴趣发散：交大的廖世俊团队4

1. On the structure and origin of pressureuctuations in wall turbulence: predictionsbased on the resolvent analysis
2. A framework for studying the e ect of compliant surfaces on wall turbulence
3. Resolvent Analysis for Turbulent Channel Flow with Riblets
4. On the existence of steady-state resonant waves in experiments
5. Elements of resolvent methods in uid mechanics: notes for an introductory short course v0.3
6. Large Eddy Simulation for Compressible Flows
7. phd thesis: Fluid dynamic instabilities in complexflow systems
8. Computation of the blu -body sound generation by a self-consistent mean flow formulation