# Analysis of Transition for a Flow in a Channel via Reduced Basis Methods

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## Abstract

**:**

## 1. Introduction

## 2. Transition in a Planar Channel Flow

#### 2.1. Streamwise Streaks

#### 2.2. Flowfield Structure

## 3. Feature Extraction and Dynamic Mode Decomposition

## 4. Onset of Transition and Dynamic Mode Decomposition (DMD) Characterization

#### Modes Selection Approaches: Temporal Envelope vs. DMD Energy

- 1
- The time dynamics of all the DMD modes, namely the functions ${\alpha}_{i}{e}^{{\omega}_{i}t}$, are evaluated at the sampling points and their amplitude is collected in a matrix ${\mathit{T}}_{dyn}$, where on each row there are the time amplitudes of the corresponding DMD mode;
- 2
- Since the DMD modes ${\varphi}_{i}$ are normalized, these functions represent the actual contribution of each mode to the resulting flow field; therefore the maxima for each column of the matrix ${\mathit{T}}_{dyn}$ are computed and the corresponding modes are selected;
- 3
- Once these modes have been selected, the positions of the maxima in the matrix ${\mathit{T}}_{dyn}$ are set to zero, the new maxima are computed and the corresponding modes selected;
- 4
- The procedure can be iterated for any levels of selection, depending on how much finer the user wants to resolve coherent structures in the flow fields and how many information he means to extract.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

POD | Proper Orthogonal Decomposition |

DMD | Dynamic Mode Decomposition |

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**Figure 1.**Q-criterion isosurfaces coloured by the velocity magnitude at four instants of time and depicting the bypass transition process. From top-left to bottom right: $t=1450\phantom{\rule{0.277778em}{0ex}}{L}_{h}{\mathrm{u}}_{b}^{-1}$ ($Q={10}^{-4}$), laminar; $t=1480\phantom{\rule{0.277778em}{0ex}}{L}_{h}{\mathrm{u}}_{b}^{-1}$ ($Q={10}^{-4}$) and $t=1530\phantom{\rule{0.277778em}{0ex}}{L}_{h}{\mathrm{u}}_{b}^{-1}$ ($Q=5\times {10}^{-4}$), transition; $t=1700\phantom{\rule{0.277778em}{0ex}}{L}_{h}{\mathrm{u}}_{b}^{-1}$ ($Q=0.05$), fully developed.

**Figure 5.**Wall normal profiles of various streamwise velocity functions during transition. (

**a**) Stream-wise velocity profile; (

**b**) Wall normal derivative; (

**c**) Wall normal derivative times streamwise velocity; (

**d**) Second order wall normal derivative.

**Figure 6.**Shear stress evolution for the time interval used in Dynamic Mode Decomposition (DMD) (a). Full time window from transition to a fully developed turbulent flow (b).

**Figure 8.**DMD modes selection using t-envelope method (

**a**) and energy spectrum of all the extracted DMD modes (

**b**): the red circles highlight the modes selected using the energy norm, the black circles the ones selected using the t-envelope method with eight levels of selection.

**Figure 9.**DMD spectrum ($\omega $): (a,c) whole spectrum and (b,d) detailed view on the selected modes. Red circles highlight the DMD modes selected on the DMD spectrum using the t-envelope method (first row) and the energy method (second row).

**Figure 10.**Amplitude of $\alpha $ coefficients. Red circles highlight the DMD modes selected using the t-envelope method (a) and the energy method (b).

**Figure 11.**Selected modes via the t-envelope approach coloured by $\mathbf{\varphi}$ magnitude. Modes are ordered from left-to-right from top-to-bottom according to the corresponding energy content.

**Figure 12.**Selected modes via the energy approach coloured by $\mathbf{\varphi}$ magnitude. Modes are ordered from left-to-right from top-to-bottom according to the corresponding energy content.

**Figure 13.**Comparison of the onset of transition described through numerical simulation (first row), DMD t-envelope (second row), DMD energy (third row); $t=1430{L}_{h}{\mathrm{u}}_{b}^{-1}$ (first column), $t=1480{L}_{h}{\mathrm{u}}_{b}^{-1}$ (first column), $t=1530{L}_{h}{\mathrm{u}}_{b}^{-1}$ (third column).

**Figure 14.**Relative error of the Energy and t-envelope method with respect to simulations for the velocity vector: velocity magnitude and velocity components.

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**MDPI and ACS Style**

Pascarella, G.; Kokkinakis, I.; Fossati, M. Analysis of Transition for a Flow in a Channel via Reduced Basis Methods. *Fluids* **2019**, *4*, 202.
https://doi.org/10.3390/fluids4040202

**AMA Style**

Pascarella G, Kokkinakis I, Fossati M. Analysis of Transition for a Flow in a Channel via Reduced Basis Methods. *Fluids*. 2019; 4(4):202.
https://doi.org/10.3390/fluids4040202

**Chicago/Turabian Style**

Pascarella, Gaetano, Ioannis Kokkinakis, and Marco Fossati. 2019. "Analysis of Transition for a Flow in a Channel via Reduced Basis Methods" *Fluids* 4, no. 4: 202.
https://doi.org/10.3390/fluids4040202