# 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 |

## References

- Tollmien, W. Über die Entstehung der Turbulenz. 1. Mitteilung. Nachr. Ges. Wiss. Gött. Math.-Phys. Klasse
**1929**, 1929, 21–44. [Google Scholar] - Schlichting, H. Zur Enstehung der Turbulenz bei der Plattenströmung. Nachr. Ges. Wiss. Gött. Math.-Phys. Klasse
**1933**, 1933, 181–208. [Google Scholar] - Schlichting, H. Boundary-Layer Theory; McGraw-Hill: New York, NY, USA, 1955. [Google Scholar]
- Baines, P.; Majumdar, S.; Mitsudera, H. The mechanics of the Tollmien-Schlichting wave. J. Fluid Mech.
**1996**, 312, 107–124. [Google Scholar] [CrossRef] [Green Version] - Kachanov, Y.S. Physical Mechanisms of Laminar-Boundary-Layer Transition. Annu. Rev. Fluid Mech.
**1994**, 26, 411–482. [Google Scholar] [CrossRef] - Sandham, N.D.; Kleiser, L. The late stages of transition to turbulence in channel flow. J. Fluid Mech.
**1992**, 245, 319–348. [Google Scholar] [CrossRef] - Härtel, C.; Kleiser, L. Subharmonic transition to turbulence in channel flow. Appl. Sci. Res.
**1993**, 51, 43–47. [Google Scholar] [CrossRef] - Germano, M.; Piomelli, U.; Moin, P.; Cabot, W.H. A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A Fluid Dyn.
**1991**, 3, 1760–1765. [Google Scholar] [CrossRef] [Green Version] - Schlatter, P.; Stolz, S.; Kleiser, L. LES of transitional flows using the approximate deconvolution model. Int. J. Heat Fluid Flow
**2004**, 25, 549–558. [Google Scholar] [CrossRef] - Schlatter, P.; Stolz, S.; Kleiser, L. Large-eddy simulation of spatial transition in plane channel flow. J. Turbulence
**2006**, 7, N33. [Google Scholar] [CrossRef] - Wu, X.; Moin, P. Forest of hairpins in a low-Reynolds-number zero-pressure-gradient flat-plate boundary layer. Phys. Fluids
**2009**, 21, 091106. [Google Scholar] [CrossRef] - Sayadi, T.; Hamman, C.W.; Moin, P. Fundamental and subharmonic transition to turbulence in zero-pressure-gradient flat-plate boundary layers. Phys. Fluids
**2012**, 24, 091104. [Google Scholar] [CrossRef] [Green Version] - Schlatter, P.; Brandt, L.; de Lange, H.C.; Henningson, D.S. On streak breakdown in bypass transition. Phys. Fluids
**2008**, 20, 101505. [Google Scholar] [CrossRef] [Green Version] - Wu, X.; Moin, P.; Hickey, J.P. Boundary layer bypass transition. Phys. Fluids
**2014**, 26, 091104. [Google Scholar] [CrossRef] [Green Version] - Sayadi, T.; Hamman, C.W.; Moin, P. Direct numerical simulation of complete H-type and K-type transitions with implications for the dynamics of turbulent boundary layers. J. Fluid Mech.
**2013**, 724, 480–509. [Google Scholar] [CrossRef] - Rowley, C.W.; Dawson, S.T. Model reduction for flow analysis and control. Annu. Rev. Fluid Mech.
**2017**, 49, 387–417. [Google Scholar] [CrossRef] [Green Version] - Taira, K.; Brunton, S.L.; Dawson, S.T.; Rowley, C.W.; Colonius, T.; McKeon, B.J.; Schmidt, O.T.; Gordeyev, S.; Theofilis, V.; Ukeiley, L.S. Modal analysis of fluid flows: An overview. Aiaa J.
**2017**, 4013–4041. [Google Scholar] [CrossRef] [Green Version] - Schmid, P.J. Dynamic Mode Decomposition of numerical and experimental data. J. Fluid Mech.
**2010**, 656, 5–28. [Google Scholar] [CrossRef] [Green Version] - Berkooz, G.; Holmes, P.; Lumley, J.L. The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech.
**1993**, 25, 539–575. [Google Scholar] [CrossRef] - Rowley, C.W. Model reduction for fluids, using balanced proper orthogonal decomposition. Int. J. Bifurc. Chaos
**2005**, 15, 997–1013. [Google Scholar] [CrossRef] [Green Version] - Schmid, P.J.; Li, L.; Juniper, M.; Pust, O. Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn.
**2011**, 25, 249–259. [Google Scholar] [CrossRef] - Seena, A.; Sung, H.J. Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations. Int. J. Heat Fluid Flow
**2011**, 32, 1098–1110. [Google Scholar] [CrossRef] - Muld, T.W.; Efraimsson, G.; Henningson, D.S. Flow structures around a high-speed train extracted using proper orthogonal decomposition and dynamic mode decomposition. Comput. Fluids
**2012**, 57, 87–97. [Google Scholar] [CrossRef] - Pan, C.; Yu, D.; Wang, J. Dynamical mode decomposition of Gurney flap wake flow. Theor. Appl. Mech. Lett.
**2011**, 1, 012002. [Google Scholar] [CrossRef] [Green Version] - Chen, K.K.; Tu, J.H.; Rowley, C.W. Variants of dynamic mode decomposition: Boundary condition, Koopman, and Fourier analyses. J. Nonlinear Sci.
**2012**, 22, 887–915. [Google Scholar] [CrossRef] - Jovanović, M.R.; Schmid, P.J.; Nichols, J.W. Sparsity-promoting dynamic mode decomposition. Phys. Fluids
**2014**, 26, 024103. [Google Scholar] [CrossRef] - Tissot, G.; Cordier, L.; Benard, N.; Noack, B.R. Model reduction using Dynamic Mode Decomposition. C. R. Mécanique
**2014**, 342, 410–416. [Google Scholar] [CrossRef] - Kou, J.; Zhang, W. An improved criterion to select dominant modes from dynamic mode decomposition. Eur. J. Mech.-B/Fluids
**2017**, 62, 109–129. [Google Scholar] [CrossRef] - Bagheri, S. Koopman-mode decomposition of the cylinder wake. J. Fluid Mech.
**2013**, 726, 596–623. [Google Scholar] [CrossRef] - Page, J.; Kerswell, R.R. Koopman mode expansions between simple invariant solutions. J. Fluid Mech.
**2019**, 879, 1–27. [Google Scholar] [CrossRef] [Green Version] - Kokkinakis, I.; Drikakis, D. Implicit Large Eddy Simulation of weakly-compressible turbulent channel flow. Comput. Methods Appl. Mech. Eng.
**2015**, 287, 229–261. [Google Scholar] [CrossRef] [Green Version] - Degani, D.; Seginer, A.; Levy, Y. Graphical visualization of vortical flows by means of helicity. AIAA J.
**1990**, 28, 1347–1352. [Google Scholar] [CrossRef] - Landahl, M.T. On sublayer streaks. J. Fluid Mech.
**1990**, 212, 593–614. [Google Scholar] [CrossRef] - Chernyshenko, S.I.; Baig, M.F. The mechanism of streak formation in near-wall turbulence. J. Fluid Mech.
**2005**, 544, 99–131. [Google Scholar] [CrossRef] [Green Version] - Brandt, L. The lift-up effect: The linear mechanism behind transition and turbulence in shear flows. Eur. J. Mech. B/Fluids
**2014**, 47, 80–96. [Google Scholar] [CrossRef] [Green Version] - Touber, E.; Leschziner, M.A. Near-wall streak modification by spanwise oscillatory wall motion and drag-reduction mechanisms. J. Fluid Mech.
**2012**, 693, 150–200. [Google Scholar] [CrossRef] [Green Version] - Lardeau, S.; Leschziner, M.A. The streamwise drag-reduction response of a boundary layer subjected to a sudden imposition of transverse oscillatory wall motion. Phys. Fluids
**2013**, 25, 075109. [Google Scholar] [CrossRef] [Green Version] - Tu, J.H.; Rowley, C.W.; Luchtenburg, D.M.; Brunton, S.L.; Kutz, J.N. On dynamic mode decomposition: Theory and applications. arXiv
**2013**, arXiv:1312.0041. [Google Scholar] - Gavish, M.; Donoho, D.L. The optimal hard threshold for singular values is 4/$\sqrt{3}$. IEEE Trans. Inf. Theory
**2014**, 60, 5040–5053. [Google Scholar] [CrossRef]

**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