# Feedback Control for Transition Suppression in Direct Numerical Simulations of Channel Flow

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Approaches

#### 2.1. Direct Numerical Simulation

#### 2.2. Linearized Navier–Stokes Equations and Feedback Control Design

## 3. Results and Discussions

#### 3.1. Transient Energy Growth of Optimal Disturbance

#### 3.1.1. Oblique Disturbance $(\alpha ,\beta )=(1,1)$

#### 3.1.2. Streamwise Wave Disturbance $(\alpha ,\beta )=(1,0)$

#### 3.1.3. Spanwise Wave Disturbance $(\alpha ,\beta )=(0,2)$

#### 3.2. Suppression of Laminar-to-Turbulent Transition

#### 3.2.1. Reduction in Transient Energy Growth

#### 3.2.2. Controlled Flow with Oblique or Streamwise Wave Disturbances

#### 3.2.3. Controlled Flow of Spanwise Wave Disturbances

#### 3.2.4. Increase in Perturbation Threshold for Laminar-to-Turbulent Transition

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic of plane Poiseuille flow with streamwise wave disturbance (not to scale), both with and without wall actuation.

**Figure 2.**Transient energy growth (TEG) of optimal disturbance with a range of initial kinetic energy density ${E}_{0}$ from linear (red dots) and nonlinear (solid lines) simulations for uncontrolled flows.

**Figure 3.**Time evolution of flow field to illustrate the laminar-to-turbulent transition process for the case $(\alpha ,\beta )=(1,1)$: The flow is initialized by optimal disturbance with kinetic energy density of ${E}_{0}=0.5\times {10}^{-4}$. Iso-surfaces of $\mathcal{Q}$-criterion [35] ($\mathcal{Q}{(h/{\overline{u}}_{c})}^{2}=0.001$) colored by streamwise vorticity ${\tilde{\omega}}_{x}h/{\overline{u}}_{c}$ are visualized. Only the upper half domain ($y\ge 0$) is displayed for clarity.

**Figure 4.**Time evolution of kinetic energy density E of case $(\alpha ,\beta )=(1,1)$ initialized by different amplitudes of optimal disturbance ${E}_{0}$: Red lines represent transition cases, and grey lines denote that the flow remains laminar. Dashed line indicates that the flow is in turbulent state. (

**a**) Kinetic energy density normalized by its initial value $E/{E}_{0}$ and (

**b**) kinetic energy density E.

**Figure 5.**Time evolution of friction velocity ${u}^{*}$ of case $(\alpha ,\beta )=(1,1)$ initialized by different amplitudes of optimal disturbance ${E}_{0}$: Red lines represent transition cases, and grey lines denote that the flow remains laminar. Dashed line indicates that the flow is in turbulent state.

**Figure 6.**Time evolution of flow field to illustrate the laminar-to-turbulent transition process for the case $(\alpha ,\beta )=(1,0)$: The flow is initialized by optimal disturbance with kinetic energy density of ${E}_{0}=1.0\times {10}^{-4}$. Iso-surfaces of $\mathcal{Q}$-criterion ($\mathcal{Q}{(h/{\overline{u}}_{c})}^{2}=0.005$) colored by streamwise vorticity ${\tilde{\omega}}_{x}h/{\overline{u}}_{c}$ are visualized. Only the upper half domain ($y\ge 0$) is displayed for clarity.

**Figure 7.**Time evolution of friction velocity ${u}^{*}$ of case $(\alpha ,\beta )=(1,0)$ initialized by different amplitudes of optimal disturbance ${E}_{0}$: Red lines represent transition cases, and grey lines denote that the flow remains laminar. Dashed line indicates that the flow is in turbulent state.

**Figure 8.**Time evolution of kinetic energy density E of case $(\alpha ,\beta )=(1,0)$ initialized by different amplitudes of optimal disturbance ${E}_{0}$: Red lines represent transition cases, and grey lines denote that the flow remains laminar. Dashed line indicates that the flow is in turbulent state. (

**a**) Kinetic energy density normalized by its initial value $E/{E}_{0}$ and (

**b**) kinetic energy density E.

**Figure 9.**Time evolution of flow field to illustrate the laminar-to-turbulent transition process for the case $(\alpha ,\beta )=(0,2)$: The flow is initialized by optimal disturbance with kinetic energy density of ${E}_{0}=1.0\times {10}^{-4}$. Iso-surfaces of $\mathcal{Q}$-criterion ($\mathcal{Q}{(h/{\overline{u}}_{c})}^{2}=0.0004$) colored by streamwise vorticity ${\tilde{\omega}}_{x}h/{\overline{u}}_{c}$ are visualized.

**Figure 10.**Time evolution of kinetic energy density E of case $(\alpha ,\beta )=(0,2)$ initialized by different amplitudes of optimal disturbance ${E}_{0}$: Red lines represent transition cases, and grey lines denote that the flow remains laminar. Dashed line indicates that the flow is in turbulent state. (

**a**) Kinetic energy density normalized by its initial value $E/{E}_{0}$ and (

**b**) kinetic energy density E.

**Figure 11.**Time evolution of friction velocity ${u}^{*}$ of case $(\alpha ,\beta )=(0,2)$ initialized by different amplitudes of optimal disturbance ${E}_{0}$: Red lines represent transition cases, and grey lines denote that the flow remains laminar. Dashed line indicates that the flow is in turbulent state.

**Figure 12.**Transient energy growth (TEG) of optimal disturbance with a range of initial kinetic energy densities ${E}_{0}$ from linear (red dots) and nonlinear (solid lines) simulations for controlled flows: Red dashed line is the kinetic energy density initialized from the optimal disturbance (OD) associated with the uncontrolled system.

**Figure 13.**A comparison of maximum amplified transient energy density ${(E/{E}_{0})}_{\mathrm{max}}$ of a range of amplitude of initial optimal disturbances between uncontrolled (solid lines) and controlled (dashed lines) cases.

**Figure 14.**Time history of normalized wall-normal velocity ${v}_{\mathrm{bc}}/\sqrt{{E}_{0}}$ at the lower wall of the case $(\alpha ,\beta )=(1,1)$ when actuation is turned on.

**Figure 15.**A comparison of instantaneous streamwise vorticity ${\tilde{\omega}}_{x}h/{\overline{u}}_{c}$ flow fields at slice $z/h=0$ between uncontrolled and controlled cases with $(\alpha ,\beta )=(1,1)$ and ${E}_{0}=0.5\times {10}^{-4}$: Black contour lines denote $\mathcal{Q}$-criterion in a range of $0.01\le \mathcal{Q}{(h/{\overline{u}}_{c})}^{2}\le 0.05$.

**Figure 16.**A comparison of instantaneous flow field at slices $z/h=0$ and $t{u}_{c}/h=12$ between uncontrolled and controlled flow with $(\alpha ,\beta )=(1,1)$ and ${E}_{0}=0.5\times {10}^{-4}$: Contours are wall-normal velocity $\tilde{v}$, and black contour lines denotes $\mathcal{Q}$-criterion in a range from 0.01 to 0.05 with increment of 0.01.

**Figure 17.**Time history of normalized wall-normal velocity ${v}_{\mathrm{bc}}/\sqrt{{E}_{0}}$ at the lower wall of case $(\alpha ,\beta )=(0,2)$ when actuation is turned on.

**Figure 18.**Modification of instantaneous streamwise velocity gradient in spanwise direction $\partial \tilde{u}/\partial z$ at slice $x/h=0$ and $t{\overline{u}}_{c}/h=50$ in (

**a**) uncontrolled flow compared to (

**b**) controlled flow: Actuation velocity is denoted by black arrows.

**Figure 19.**Secondary instabilities induced by the actuation in the flow field at $t{\overline{u}}_{c}/h=66$ of the controlled case with $(\alpha ,\beta )=(0,2)$ and ${E}_{0}=1.0\times {10}^{-4}$: (

**a**) Wall-normal vorticity at slice of $x/h=0$, (

**b**) spanwise vorticity at slice of $x/h=0$, and (

**c**) iso-surfaces of $\mathcal{Q}$ colored by streamwise vorticity.

**Figure 20.**Summary of cases considered in the present work: Solid circle represents that laminar-to- turbulent transition occurs; open circle denotes that the flow remains laminar. U and C denote uncontrolled and controlled cases, respectively.

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**MDPI and ACS Style**

Sun, Y.; Hemati, M.S.
Feedback Control for Transition Suppression in Direct Numerical Simulations of Channel Flow. *Energies* **2019**, *12*, 4127.
https://doi.org/10.3390/en12214127

**AMA Style**

Sun Y, Hemati MS.
Feedback Control for Transition Suppression in Direct Numerical Simulations of Channel Flow. *Energies*. 2019; 12(21):4127.
https://doi.org/10.3390/en12214127

**Chicago/Turabian Style**

Sun, Yiyang, and Maziar S. Hemati.
2019. "Feedback Control for Transition Suppression in Direct Numerical Simulations of Channel Flow" *Energies* 12, no. 21: 4127.
https://doi.org/10.3390/en12214127