# Control at the wall of turbulent flows

## Object

Control of turbulence represents one of the major directions of current fluids research. Turbulent flows are characterized by complex dynamics and a large range of scales, which makes them difficult to compute. A wide variety of applications exists for this theoretically challenging issue, ranging from enhanced mixing in the pharmaceutical industry to drag reduction over aircrafts and ships. The focus of this group is to build on our experience with numerical simulations (both direct and LES) of turbulent flows to experiment with flow control techniques. Numerical simulation provides a unique, idealized setting frame in which 1) a wide range of control parameters can be explored easily 2) the effect of a given control on the flow can be entirely determined in space and time. Simulating manipulated flows therefore constitutes a first step towards the validation of control. The insight obtained with artificial control in numerical simulations can then be confronted with experiments to develop a realistic control strategy. This approach has been carried out at LIMSI for both free-shear (wake of bluff bodies) and wall-bounded flows. In all cases the flow-dependent manipulation technique consisted in blowing and sucking at the flow boundary.

### Turbulence at the wall

Our goal is to develop new control strategies for drag reduction in a turbulent channel. Specifically, the idea is to determine instantaneous key features of wall turbulence using accessible data such as wall shear measurements. Our approach is based on the POD technique coupled with neural networks. Neural networks are excellent tools to estimate a non-linear dynamical process from observable data. The method is first tested in a turbulent channel of relatively small horizontal dimensions, then in a larger channel. In both cases estimation and control are performed on subdomains of relatively small dimensions.

### Bluff bodies

The turbulent separated flows past several bluff bodies, i.e. a circular cylinder, the Malavard-Cousteau's Turbosail and a NACA0012 airfoil, are computed by using Large Eddy Simulations (LES). In order to modify the aerodynamic coefficients of the circular cylinder, a boundary layer suction is applied at the wall of the body. Moderate Reynolds numbers less than $Re = 100,000$ are considered in this study, corresponding to the subcritical regime (laminar separation of the boundary layer). Comparisons with experimental studies carried on in LEA/ENSMA (Poitiers, France) are shown for the flow around a circular cylinder, at a high Reynolds number $Re = 100,000$. The control efficiency on lift enhancement and drag reduction is analyzed for the turbosail and NACA profiles as function of the angle of incidence. The suction modifies also the topology of the flow.

## Blowing and suction at the wall of a turbulent channel using neural networks

### Control strategy

The control strategy is similar to the one described by Lee et al. (1997). This neural network aims to find a correlation between wall-shear stress and wall actuations from given data sets. We only use the transverse wall-shear stress $\frac{\partial w}{\partial y}$ as the adding of $\frac{\partial u}{\partial y}$ was not relevant for the neural network performance. The desired wall actuations is the one recommended by Choi et al. [2] i.e. the opposite of the velocity at $y^{+}=10$ (Fig. 1). This strategy of blowing and suction at the wall brings about a drag reduction of 25% by reducing the sweep and ejection events in the wall-layer.

 Figure 1: Opposition control.

### Neural network description

We use a standard two-layer feedforward network with hyperbolic tangent hidden units and a linear output unit. So the function describing the neural network output is:

$v_{jk}^{net}=W_{a}\tanh\left(\sum_{i=-(N-1)/2}^{(N-1)/2}W_{i}\left.\frac{\partial w}{\partial y}\right\vert _{j,k+i}-W_{b}\right)-W_{c}$ (1)

with ${1 \leq j \leq N_{x}}$ and $\displaystyle{1 \leq k \leq N_{z}}$

$N$ is the number of neighbouring points in the spanwise direction. $N_{x}$ and $N_{z}$ are respectively the number of gridpoints in the streamwise direction and in the spanwise direction. The $W's$ are the weights of the neural network that are established during a learning process. This learning process is based on a simplex algorithm used to train the neural network obtaining the desired Choi actuation by minimising the error defined in (2), where $v_{jk}^{des}$ represents sets of Choi's wall velocity field.

$error=\frac{1}{2}\sum_{j}\sum_{k}e^{\lambda\vert v_{jk}^{des}\vert}(v_{jk}^{des}-v_{jk}^{net})^{2 }$ (2)

The training data consist of several samples of 2D fields of velocity at $y^{+}=10$ and wall shear stress at different time. Those data sets were carried out with a 3D direct numerical simulation of a fully turbulent channel flow. The size of the computational box has been chosen to include in its width two pairs of low/high-speed streaks on statistical average. We used then a computational domain $(Lx,Ly,Lz)=(4\pi/3,2,4\pi/3)h$ , and a grid resolution of (96,65,48). The Reynolds number based on the channel half-height $h$ and center line velocity is 4000 and the one based on $h$ and friction velocity $u_{*}$ is about 140.

### Results and prospects

We have tested the validity of the neural networks by implementing the new boundary conditions at the bottom wall in the channel flow simulation, we obtained 20% drag reduction (Fig.2) which is the reduction obtained by Lee et al. The visualisation of the resulting flow shows us that as regards streamwise vorticity the mean flow remains unchanged unlike the near-wall structures which have less intensity.

To improve the efficiency of the control, we have used an additional tool: the Proper Orthogonal Decomposition. POD is an appropriate tool in the study of channel flow owing to the presence of coherent structures in the wall-layer. As we project the velocity field on the first eigenfunction ($n=1$), we observe the correlation between the extracted velocity and wall shear stress. Let us call $\hat{v}$ the projection of normal velocity onto the first eigenfunction of the uncontrolled flow. The resulting curves revealed that $\hat{v^{*}}$ and $\frac{\partial \hat{w}}{\partial y}$ were well correlated in the uncontrolled flow. The relation can be written as:

$\hat{v^{*}} \simeq iC \frac{\partial\hat{w}}{\partial{y}}_{y=0}$ (3)

This control law is very similar to the one suggested by Lee et al, but this time we consider the projection of the velocity field on the first mode. C was found to be about 10 outer units. As we controlled the flow with Choi's actuation and projected the resulting normal velocity on the same uncontrolled eigenmode, the wall shear and the projection of the velocity field remain well correlated. This suggests that even though POD eigenfunctions are modified when control is applied, uncontrolled POD eigenfunctions are still relevant to describe some of the flow dynamics. The wall-shear stress is then a good indicator of drag reduction as his time-evolution is directly linked with the behaviour of the most energetic structure that develop itself in all the flow.

We have used a neural network to estimate the first eigenmode given by POD decomposition of normal velocity, we obtain then a drag reduction of 22% (Fig. 2). This shows that there is a strong link between turbulence production mechanism and wall shear measurements.

## Control around bluff bodies using Large Eddy Simulations

### Overview of the numerical methods

The LES code used has been developed in LIMSI for a decade [1]. It solves the spatially-filtered incompressible Navier-Stokes equations written in their velocity-vorticity formulation. The LES technique is based on the fact that most of the turbulent kinetic energy is contained in the largest structures of the flow. Therefore, in LES, only these structures are directly computed. The smallest scales, typically smaller than the size of the grid cell, are modeled using a subgrid-scale model. The mixed scale model [2], developed jointly in LIMSI and ONERA which considers both energetic and dynamical aspects of the interaction between the large eddies and the smallest structure, is used in this study.

### Results and prospects

#### Comparison LES-experiment for a circular cylinder

The flow around a circular cylinder at a high Reynolds number $Re = 100,000$ is first considered, without and with control by boundary layer suction. In order to show the ability of the LES to compute high Reynolds numbers separated turbulent flows, the results are compared with experimental results obtained by J. TENSI and S. BOURGOIS (LEA/ENSMA). The mean pressure distribution are presented on Figure 1 without control (left) and with control by suction (right). Despite the high Reynolds number and the related turbulent behavior with a large spectrum of spatial scales, a good agreement is recovered between the two different approaches, either for the uncontrolled or the controlled flow, proving that LES is a good tool for studying turbulent flows.

 Figure 1.a: Comparison LES-experiment without control. Mean pressure distribution around a circular cylinder, at $Re = 100,000$ : present results (solid lines); experimental results from LEA/ENSMA (dots). Figure 1.b: Comparison LES-experiment with control by boundary layer suction at the wall of the cylinder.

#### Turbosail and NACA profiles

The second part of this study deals with the control of flow around the Turbosail at $Re = 20,000$. This bluff body was created by Malavard and Cousteau in order to create a new wind-powered boat, named Alcyone, to replace the old Calypso. The Turbosail, an ovoid-section cylinder acts exactly like a sail, the lift force being created by a huge fan, located in the cylinder, and sucking the flow through a permeable wall (Figure 2). In this study, the suction velocity is ten times smaller than the upstream velocity and the suction surface, beginning at about 45 % of the profile chord, spreads over 26 % of the chord.

When the suction is not applied, the Turbosail has really poor aerodynamical performances as a sail. Indeed, a high drag force appears and no lift force is observed, for the three considered angles of incidence 0°, 10° and 20°. However, when the suction is switched on, and whatever the angle of incidence is, the performances are sharply improved and the Turbosail may then be considered as a very efficient way to move a boat. The influence of the suction on aerodynamic coefficients can be shown on Figure 3.

 Figure 2: Inside view of the Malavard-Cousteau's Turbosail. Figure 3: Aerodynamic coefficients as function of the angle of incidence. - uncontrolled flow and -- control by boundary layer suction.

Finally, the flow around a thin wing profile (NACA0012) for $Re = 10,000$ is considered. The suction is here applied on a surface of 12 % of the chord, starting close to the upstream stagnation point, with a suction velocity ten times smaller than the upstream velocity. On the opposite of the Turbosail, the NACA0012 has a great lift-on-drag ratio even when no suction occurs. Nevertheless, the suction is able to raise this ratio up to very high values, proving that this kind of control techniques is not only able to create a lift force, but also to enhance it (Figure 4).

 Figure 4: In the case of a NACA0012 profile, influence of a boundary layer suction on the aerodynamic coefficients as function of the angle of incidence. The uncontrolled flow is represented by solid lines whereas the dotted-dashed pattern stands for the controlled flow.

In addition to the efficiency improvement, the suction can also greatly modify the flow patterns, such as the vortex shedding. Indeed, it can be observed on Figure 5 that when a suction is applied, the shed vortex are smaller than they would be in the uncontrolled case. The main reason is that the boundary layer separation is delayed by the suction. The wake thickness and consequently the typical size of the vortex are then reduced.

The control influences then also the vortex frequency. In the case of the uncontrolled flow, two kind of vortices are generated, at both attack (boundary layer separation) and leading edges (Figure 5.a). The resulting shed vortex has a mixed frequency. In the case of control, vortices are only created at the leading edge of the profile (Figure 5.b). The resulting frequency corresponds then to the leading edge frequency, higher than the initial one.

 Figure 5.a: Uncontrolled flow around a NACA0012 profile for an angle of incidence of 10°. The boundary layer separation generates vortices. Figure 5.b: Control by boundary layer suction at the wall of a NACA0012 profile for an angle of incidence of 10°. The suction delayes the separation.

## Discussion and conclusion

### Neural networks and turbulence at the wall

In this work, we have tried to control the flow near the wall of a boundary-layer by the means of two approaches: Neural networks and POD analysis. By combining the two approaches, we hope we could develop new strategies for drag minimization using realistic wall measurements. These results underline the complex influence of a wide variety of organized motions in the turbulence generation process. Practical impediments need to be adressed as well as theoretical issues to make a control strategy successful. Up to 13% reduction was obtained when only wall information was used, while full flow information resulted in 20%. The performance of Fourier based neural networks is comparable to those derived in physical space (15% for Lee, Babcock and Goodman), while requiring handling very little information.

### Bluff bodies and LES at high Reynolds number

Two main conclusions can be drawn about this study. First, the ability of Large Eddy Simulations to predict high Reynolds number separated turbulent flows has been proved by means of comparisons with experimental results (LEA/ENSMA), even if a control by suction is applied. The control by boundary layer suction has then been shown to be of great interest to modify the aerodynamic coefficients, either for a bluff body or for an airfoil. Moreover, the suction can also strongly modify the vortex shedding pattern and appear to be a good method to reduce Vortex-induced Vibrations on airplane wings for instance.

Several other means of control have to be checked in the future, mainly synthetic jets which are known to be rather efficient as a flow control technique, consisting of a half-period suction followed by a half-period blowing to ensure that there will be no net mass injection in the flow. In addition, because of the high Reynolds number considered here, the influence of the suction in the spanwise direction might also be investigated.

This study is part of Guillaume Fournier's PhD Thesis (2005).

## References

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