C. Lee

D.V. Roupassov

E. Moreau

J.P. Boeuf

A. Criscione

B. Frohnapfel

C. Tropea

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Preliminary Numerical Assessment of Turbulent Flow Control with Plasma Actuators

Criscione, A.

Frohnapfel, B.

Tropea, C.

KITopen

Preliminary Numerical Assessment of Turbulent FlowControl with Plasma ActuatorsA. Criscione1, B. Frohnapfel2 and C. Tropea1,21 Fachgebiet Stro¨mungslehre und Aerodynamik,2 Center of Smart Interfaces, Technische Universita¨t Darmstadt, Germanya.criscione@sla.tu-darmstadt.deSummaryA numerical investigation of damping of near-wall spanwise velocity fluctua-tions in a turbulent channel flow is carried out. Spanwise damping is realizedwith a body force which mimics the characteristic force distribution of plasmaactuators. This force distribution is implemented in a direct numerical sim-ulation. The body force is triggered by the signal of a sensor in upstreamlocation. The control loop is applied with different actuator configurations,considering a spatially continuous and discontinuous body force distributionat the wall. The influence of distributed sensors and actuators of finite sizeon the control loop performance is investigated. The results show a reductionof the skin friction at the wall with maximum power saving rates up to 20%.1 IntroductionThe control of turbulent flows is a key issue in many engineering applicationssince it is related to power savings, efficient production processes and possi-ble solutions for global environmental problems. In this regard, the plasmaactuator is an attractive actuator device due to its low mass and a rather sim-ple system integration. Flow control applications of plasma actuators includeseparation control [8], transition control [3] and wake control [6].We investigate whether it might be possible to also use plasma actuatorsfor skin friction drag reduction in turbulent flows. In the literature, attemptsof imitating spanwise wall oscillation or related predetermined active controlschemes [7] with plasma actuators have been reported [10]. We also considerthe introduction of the plasma actuator body force in the spanwise direction(x3), but in contrast to the approaches in the field of predetermined control,we focus on reactive control schemes with the goal to damp instantaneousspanwise velocity fluctuations, u3, [4, 5] in the near-wall region.We carry out direct numerical simulation of a fully developed channel flow(Reτ = 150) where the near-wall body force is introduced as an additionalterm on the right hand side of the Navier-Stokes equations. The distributionof this body force is chosen to mimic a characteristic force distribution ofthe plasma actuator. Throughout this paper the superscript ()+ indicatesnormalization with the kinematic viscosity, ν, and the wall shear velocity,uτ , of the uncontrolled channel flow.2 Body Force Introduced by a Plasma ActuatorThe Single Dielectric Barrier Discharge (SDBD) actuator, generally referredto as plasma actuator, consists of two electrodes placed parallel to each other.The lower electrode is covered with an insulating film, the so-called dielectricbarrier, while the upper electrode is exposed to the flow. This set-up is knownto produce a body force in the fluid above the actuator when a high ACvoltage is applied to the electrodes. The physics of the generation of thisbody force as well as its exact distributions in time and space are still asubject of investigation; see e.g. [1, 6] for further details and references.For our numerical experiment we consider only the time averaged forcedistribution of a plasma actuator since the characteristic frequencies of theforce modulation are significantly faster than the highest frequencies in theturbulent flow field [6]. In the literature different models are available, whichdescribe the distribution of this net body force introduced by a plasma ac-tuator. We consider two models with complementary approaches. First, anempirical model in which the force distribution above the actuator is ex-tracted from experimental velocity field data [9]. Second, an electrochemicalmodel for which the space and time evolution of the charged particle den-sities, the electrical field and the surface charges are computed numerically[1]. The distribution of the wall-parallel force component obtained with thesemodels is plotted in figure 1. Both models also provide a wall-normal compo-nent of the body force. However, this component has a much smaller integralvalue and its distribution significantly differs for different models.Although the physical dimensions of the considered actuators are signifi-cantly different, a common distribution at the location of the maximum bodyforce can be extracted. The wall normal distribution at this location is char-acterized by a strong increase of the body-force in the wall vicinity, whichis followed by an almost exponential decay with increasing wall distance. Inorder to be able to implement a body force distribution in our numerical codewhich represents a plasma actuator we describe this behavior with a fittingfunction such that bf+3 ∝ x+2 e−x+2 .3 The Control LoopIn order to damp the spanwise velocity fluctuations, u3, in the near-wallregion of a turbulent channel flow their direction and magnitude needs to beknown. However, their direct measurement cannot be realized in practice. Leeand Kim [5] showed that the spanwise wall shear stress, τw,x3 , at the actuationlocation can be successfully used as sensor input for a control scheme that[a] [b]Figure 1 Body force distribution of the empirical [a] (dielectric thickness = 1.5mm) and of the electrochemical model [b] (dielectric thickness = 0.5 mm)damps u3. However, it should be noted that in this case the near-wall regionin which u3-damping can be realized is limited to x+2 ≈ 10, because thecorrelation between τw,x3 and u3(x2) quickly deteriorates outside this region.Since a measurement of τw,x3 at the actuation location is difficult to real-ize, Frohnapfel et al. [4] suggested to use the spanwise gradient of the stream-wise wall shear stress, ∂τw,x1/∂x3, upstream of the actuator location as sensorinput. We follow that line and formulate our control loop in such a way thatthe plasma actuator body force is introduced into the numerical simulationas:bf+3 =1Φ+x+2 e−x+2 ·upstream sensing︷ ︸︸ ︷∂τw,x1∂x3(x+1 −∆x+1 ) f(x+2 ). (1)In this formulation (Φ+)−1 is the relaxation time constant which, in combi-nation with the sensor value∂τw,x1∂x3(x+1 − ∆x+1 ), determines the strength ofthe body force; f(x+2 ) is a step function which limits the wall-normal exten-sion of the region where the body force is applied to ten viscous units on thetop and bottom wall. We refer to this layer as damping layer, yd, such thatf(x+2 ≤ y+d ) = 1 and f(x+2 > y+d ) = 0. According to the results of previouswork [4] we set the distance between sensor and actuator to ∆x+1 = 50. Fig-ure 2 shows the wall-normal distribution of the time averaged body force fordifferent (Φ+)−1.For active flow control methods, the control performance can be capturedwith different indicators [2]. For a simulation with constant bulk velocity,the pumping power, P , required to drive the flow, is directly proportional tothe average streamwise wall shear stress, τw,x1 , and thus a measure for theobtained drag reduction. The additional power input per unit wall area, Pin,due to the applied control can be obtained by including the body force intothe energy equation of the flow field. For the present case it is given by:Pin =∫ H0∣∣∣∣∣∣∣P3︷ ︸︸ ︷bf3 · u3∣∣∣∣∣∣∣ dx2, (2)where H corresponds to the channel height and the overbar indicates aver-aging in time as well as in the streamwise and spanwise directions. The localpower input, P3, assumes a negative value if the body force indeed damps thespanwise velocity fluctuations. Positive values of P3 indicate that the controlloop does not work as anticipated and the body force enhances the existingvelocity fluctuations.For active flow control the drag reduction rate, R, should be replacedby the power saving rate, S, which takes into account the additional powerconsumption, Pin, of the actuator. Another important performance indicatoris the gain, G, which indicates how much power is saved per control powerinput:S =P0 − (P + Pin)P0= R− PinP0, G =P0 − PPin. (3)P0 refers to the power consumption of the uncontrolled flow. In order to obtainthe most conservative values for S and G we modify the formulation of Pinsuch that the absolute value of bf3 ·u3 is taken before the spatial and temporalaveraging is applied (|bf3 · u3| >∣∣bf3 · u3∣∣). Since we have defined Pin insuch a way that all possible energy losses associated with actuators, sensors,control circuits and so forth are neglected, G determines the lower bound ofthe system efficiency η required to obtain net power savings: η > G−1.Figure 2 Body force distributionplotted for different (Φ+)−1Figure 3 Effect of the forcingstrength (Φ+)−1 on S and G3.1 Spatially Continuous ControlMost numerical flow control research is carried out with spatially continuouscontrol where the control input is realized at every grid point assuming in-finitely small actuators that are continuously distributed along the wall. Inorder to enable comparison with other control schemes we will also considerthis case before considering the effect of finite size sensors and actuators inthe next section. Figure 3 shows the results of the applied control in termsof G and S. It can be seen that the power saving rate, S, increases with in-creasing forcing strength up to almost 20% before it starts to decrease. Thegain, G, remains almost constant for a broad range of (Φ+)−1-values beforeit decreases at the same forcing strength where reduced values of S are reg-istered. This trend can be explained when considering the local power input,P3, as given in equation 2.P3 is negative if the body force acts as a damping force and positive ifit acts as a u3-enhancing force. Figure 4 depicts the local power input dueto the applied body force as a function of wall distance. For small forcingstrengths, i.e. small (Φ+)−1, the body force acts as a damping force in theentire damping layer. For stronger forcing, positive values of P3 can be foundin the vicinity of the wall while P3 is negative in the outer region of thedamping layer. This indicates that the distribution and intensity of bf3 issuch that spanwise velocity fluctuations in the vicinity of the wall are notonly damped but forced to change their sign. For a further increase of theforcing strength this area of fluctuation enhancement increases up to the pointwhere numerical instabilities occur. This upper limit for the strength of theapplied body force arises due to the employed feed-forward control loop. Forweak control input the spatial correlations between∂τw,x1∂x3(x+1 − ∆x+1 ) and−τw,x1 on which the control is based, is not modified significantly. However,for stronger body forces the correlation does not exist any more and thus thecontrol loop breaks down. Considering the distribution of P3 in figure 4 andFigure 4 Local energy input,P+3 , plotted against the strengthfactor (Φ+)−1Figure 5 Body force distributionfor different values of the slidingfactor Γthe distribution of bf3 in figure 2, it is obvious that the peak value of bf3 inthe vicinity of the wall causes the first occurrence of positive values for P3.In order to realize higher power saving and gain rates it might therefore bedesirable to shift the location of the maximum body force further away fromthe wall. A simple shift of the location of the maximum body force can beobtained by modifying the formulation of the body force as follows:bf+3 =1Φ+(x+2 )e−Γx+2 ·upstream sensing︷ ︸︸ ︷∂τw,x1∂x3(x+1 −∆x+1 ) f(x+2 ). (4)If the sliding factor Γ is set to values smaller than one, the barycenter ofthe body force will move further away from the wall. This upward shift ofthe maximum location will also yield an increased integral value of bf3 suchthat the strength of the applied force is no longer given by (Φ+)−1 alone,but by a combination of Γ and Φ. Nevertheless, by varying Γ we are able toobtain the information whether higher values of S and G can in general beobtained with a different force distribution. The influence of varying Γ onthe distribution of bf3 for a fixed value of (Φ+)−1 is shown in figure 5.Figure 6 shows the effect of varying the sliding factor, Γ , for (Φ+)−1 =4.69 on S and G. The maximum power saving rate, S, differs only slightlyfrom the maximum value obtained for Γ = 1 in fig. 3. However, a significantincrease of G with a maximum value above 200 for Γ = 0.4 is obtained. Thisresult indicates that an optimization of the force distribution for damping ofthe spanwise velocity component within y+d = 10 is possible.Figure 6 Effect of Γ on S andG for the fitting function model at(Φ+)=4.69Figure 7 R over ∆S+ as a func-tion of the actuator length, l+, fordx+ = 5613.2 The Influence of Finite Size ActuatorsFor considerations towards a realistic control scheme, the finite actuator sizeneeds to be taken into account. Figure 8 shows an exemplary control con-figuration at the wall of the channel with defined placement parameters: l+and h+ are the length and width of the plasma actuator, respectively, dx+and dz+ refer to the streamwise and spanwise distance between the actua-tors, while ∆S+ defines the distance between sensor and upstream corner ofthe actuator. In a parameter study we find that the actuator can easily beextended in the streamwise direction up to approximately 300 viscous unitswithout significant influence on the power saving rate. However, the exten-sion in spanwise direction is rather critical and doubling the actuator widthfrom h+ = 7.5 to h+ = 15 resulted in negative power saving rates.An exemplary result of the parameter study is shown in fig. 7. In this casethe distance between two consecutive actuators, dx+, is chosen such that onlytwo actuators can be placed in the streamwise direction of the computationaldomain. These two actuators are varied in streamwise length, l+, and alsothe placement of the sensor, ∆S+ is modified. It can be seen that actuatorlengths well above 100 viscous units can be realized without significantlydecreasing the resulting power saving rate. The extension of the actuator instreamwise direction influences the optimum placement of the sensing devicewhich should be moved closer to the actuator for increasing l+.SSSSSSSSS+dxx1ActuatorActuatorActuator ActuatorActuatorActuator ActuatorActuatorActuator+DS+lflow+dz+hx3Figure 8 Discontinuous control configuration4 ConclusionIn order to obtain information about whether or not plasma actuators mightbe suitable devices for the application in reactive control schemes for turbu-lent skin friction drag reduction a numerical experiment is carried out. Theforce distribution of two fundamentally different models of the plasma ac-tuator are compared in order to extract a characteristic distribution whichcan be implemented in the numerical simulation. In the numerical experi-ment we employ a feed-forward control loop in which the spanwise gradientof the streamwise wall shear stress upstream of the actuator location servesas sensor input.We find that the force distribution extracted from the plasma actuatormodels can be successfully used in the employed reactive control loop yieldingpower saving rates of more than 20%. It is shown that an improvement ofthe control loop performance can be obtained if the barycenter of the bodyforce is moved away from the vicinity of the wall.In respect to the spatial extension of a plasma actuator it is found thatthe present feed-forward control loop can be run with actuators that have astreamwise length of up to 300 viscous units. However, an extension of the ac-tuator in the spanwise direction seems to be critical. According to the presentresults, the spanwise extent of the region in which the body force is triggeredby the same sensor signal should be limited to less than ten viscous units,which roughly corresponds to half the radius of a quasi-streamwise vortex.Finally, it should be noted that a number of additional issues, including thelimits in miniaturizing the plasma actuator, the influence of the wall normalforce component that we have neglected so far and also the influence of sensorand actuator response times, need to be addressed before a final statementabout the possible use of plasma actuators in reactive turbulent flow controlloops can be made.Acknowledgements. The authors would like to thank N. Kasagi, Y. Hasegawa(The University of Tokyo) and J. Kriegseis (TU Darmstadt) for valuablediscussions and M. Yokoo (The University of Tokyo), without his solverCHFLOW our work would not have been possible.References[1] Boeuf, J.P., Langmich, Y., Unfer, T., Callegari, T., and Pitchford, L.C. : Elec-trohydrodynamic force in dielectric barrier discharge plasma actuators. Journalof Physics D: Applied Physics (2007) 40 652-662[2] Frohnapfel, B., Hasegawa, Y., Kasagi, N., and Grosse, S. : Estimation of thespanwise wall shear stress based on upstream information for wall turbulencecontrol. Proc. 12th Euromech European Turbulence Conference (2009), 209-212.[3] Grundmann, S. : Transition control using dielectric barrier discharge actuators.Shaker Verlag (2008)[4] Kasagi, N., Hasegawa, Y., and Fukagata, K. : Toward cost-effective control ofwall turbulence for skin friction reduction. Advances in Turbulence XII (2009)189-200[5] Lee, C. and Kim, J. : Control of the viscous sublayer for drag reduction. Physicsof Fluids (2002) 14 2523-2530[6] Moreau, E. : Airflow control by non-thermal plasma actuators. Journal ofPhysics D: Applied Physics (2007) 40 605-636[7] Quadrio, M. : Drag reduction in turbulent boundary layers by in-plane wallmotion. Philosophical Transactions of the Royal Society A (2010) to appear[8] Roupassov, D.V., Nikipelov, A.A., Nudnova, M.M., and Starikovskii, A.Y. :Flow separation control by plasma actuator with nanosecond pulsed-periodicdischarge. AIAA Journal (2009) 47 168-185[9] Wilke, B. : Aerodynamische Stro¨mungssteurung mittels dielektrischenBarrierenentladungs-Plamaaktuatoren. Shaker Verlag (2009)[10] Wilkinson, S.P. : Investigation of an oscillating surface plasma for turbulentdrag reduction. AIAA 2003-1023

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Preliminary Numerical Assessment of Turbulent Flow Control with Plasma Actuators

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