modelling turbulent skin-friction control using linearised

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Modelling turbulent skin-friction control usinglinearised Navier-Stokes equationsTo cite this article C A Duque et al 2011 J Phys Conf Ser 318 042026

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Modelling turbulent skin-friction control using

linearised Navier-Stokes equations

C A Duque1 M F Baig1 D A Lockerby1 S I Chernyshenko2

and C Davies3

1 School of Engineering University of Warwick Coventry CV4 7AL UK2 Department of Aeronautics Imperial College London SZ2 4AZ UK3 School of Mathematics Cardiff University Cardiff CF24 4AG UK

E-mail fmsBaigwarwickacuk caDuque-dazawarwickacuk

duncanlockerbywarwickacuk SChernyshenkoimperialacuk

DaviesC9cardiffacuk

Abstract A linear model has been used to make qualitative predictions of turbulent drag-reduction resulting from open-loop control using streamwise travelling waves of spanwise wallvelocity There is very good agreement over a wide parameter space between our qualitativeprediction of drag decreaseincrease and the DNS results of Quadrio et al (2009) atReτ = 200

1 Introduction

In the last two decades numerous studies have shown that spanwise (cross-flow) wall oscillationsand streamwise-travelling waves of spanwise velocity bring about a substantial reduction inturbulent skin-friction drag It has been found that these wall-oscillation control approacheshave a strongly disruptiveinhibitive effect on the near-wall streak structures (Jung etal 1992Karnidakis amp Choi 2003 Ricco amp Quadrio 2008) Given the commonly-held belief that thenear-wall streak structures play a major role in the turbulent regeneration mechanism (Kline atal 1967) we might expect that an explanation for the efficacy of such control methods would bereadily forthcoming However there is still a lack of clarity in our understanding of the physicalmechanism(s) underpinning this type of control strategy and this stands in the way of efficientprediction and engineering application

Choi et al (2002) demonstrated experimentally drag reduction of up to 45 using purewall oscillations in the spanwise direction An optimum period of around T+ = 100 minus 125 wasshown to be independent of the wall-oscillation amplitude (here lsquo+rsquo denotes viscous wall units)It was also found that the location of vortices and low-speed streaks relative to the oscillatoryStokes layer was closely connected to the amount of drag reductionincrease observed Thus inthis study the physical mechanism for the skin-friction change was attributed to displacementof the vortices and streaks relative to each other

Quadrio et al (2009) investigated numerically streamwise-travelling waves of spanwisevelocity (STWSV) generated with a wall velocity of the type Vw = A sin(κxxminusωt) where A is thewall motionrsquos amplitude κx is a streamwise wavenumber and ω is the oscillation frequency of thewave This form of forcing generates waves of spanwise velocity moving (backward or forward)

13th European Turbulence Conference (ETC13) IOP PublishingJournal of Physics Conference Series 318 (2011) 042026 doi1010881742-65963184042026

Published under licence by IOP Publishing Ltd 1

in the streamwise direction with a phase speed c = ωκx It represents a generalized type ofspanwise-wall actuation that includes the two extreme cases pure spanwise oscillation (κx = 0)and stationary streamwise waves of spanwise velocity (ω = 0) Quadrio et al (2009) found arange of actuation parameter combinations (κx -ω) for which drag reduction was of the orderof 47-48 (greater than that achievable using pure spanwise oscillations or standing streamwisewaves of spanwise velocity) They also found that in the κx -ω parameter space there existsa region where drag increase occured this region is associated with a constant phase speedc = 05 in inner units c+ asymp 10 Recently Auteri et al (2010) performed experiments toconfirm the numerical results of Quadrio et al (2009) and showed that turbulent skin-frictionreduction up to 33 is achievable for slow velocity forward-travelling waves in a turbulent pipeflow Moreover they also report that backward-traveling waves lead to drag-reduction and thatthere is a regime of κx -ω where drag-increase occurs all in consonance with the numericalfindings

The majority of the numerical studies on drag reduction using spanwise wall motion havebeen performed using direct numerical simulation (DNS) As such computational limitationshave restricted the studies to relatively low Reynolds numbers In this paper we investigate thebehaviour of near-wall streaks using linearised Navier-Stokes (LNS) equations with a combinedbase flow of STWSV and a mean turbulent velocity profile Due to the high computationalefficiency of our LNS solver we can investigate high Re flow simulations corresponding to cruiseflight conditions far beyond the current capabilities of DNS and LES By comparing predictionsusing the linear approach with full DNS data we can also aim to shed light on the fundamentalphysical mechanism(s) that are responsible for the observed changes in turbulent skin-frictionusing spanwise wall motion in general

The idea of using LNS equations to predict aspects of nonlinear turbulent flows stems from thework of Landahl (1989) who argued that near-wall turbulence is a linearly-driven system andsuggested that turbulent forcing is produced by spatially and temporally intermittent vorticitybursts originating in the viscous sublayer The generation of near-wall streaks in turbulent flowsusing a linearised approach was first performed by Butler amp Farrell (1993) who investigated therelationship between optimal perturbations and near-wall coherent structures The response ofthe linearised Navier-Stokes equations with respect to the generation of near-wall streaks wasalso analysed by Chernyshenko and Baig (2005) and by the low-order model of Lockerby et al(2005)

The paper is organised as follows Section 2 describes the mathematical formulation andthe numerical scheme employed for the investigation Section 3 describes the generation ofnear-wall streaks in the linear model (via transient growth of an initial disturbance) and themeasures used to quantify the response of the linearised system Section 4 is devoted to thenumerical implementation of the open-loop control in the form of STWSV in Section 5 wepresent results showing how average streak energy is affected by control parameters and comparethis to published DNS results of drag-reduction Finally in Section 6 a brief summary of thepaper is presented

2 Mathematical model

For the present study a velocity-vorticity formulation of the Navier-Stokes equations originallyproposed by Davies amp Carpenter (2001) has been adopted A brief overview of the formulationis given here but the interested reader is referred to Davies amp Carpenter (2001) for a detailedexposition

The formulation assumes a known base flow solution represented by the velocity and vorticityfields Ub and Ωb respectively The total velocity and vorticity fields can then be decomposed asUb + u and Ωb + ω where u = (uvw) and ω = (ωx ωy ωz) represent perturbations from the


2

prescribed base-flow Here uv and w are the velocity components in the streamwise (x) spanwise(y) and wall-normal (z) directions For the known base-flow corresponding to a turbulent meanvelocity profile we make the reasonable approximation of constant boundary-layer thickness

The components of the perturbation flow-fields u and ω may be divided into two distinctsets The components (ωx ωy w) are referred to as primary variables while the remainingcomponents (u v ωz) are secondary variables In this formulation the evolution of the threeprimary variables (ωx ωy w) can be determined using just three equations

partωxpartt

+partNz

partyminus partNy

partz=

1Rnabla2ωx (1)

partωypartt

+partNx

partzminus partNz

partx=

1Rnabla2ωy (2)

nabla2w =partωxpartyminus partωy

partx (3)

where R is the Reynolds number based on outer-flow variables and the convective term isN = Ωb times u + ω timesUb + ω times u Linearity of the system of equations is attained by droppingω times u from the convective term Using relationships between primary and secondary variablesthe above three equations need be integrated for evolution of the primary perturbations

21 Numerical scheme and discretisationThe streamwise and wall-normal directions are discretised using a second-order finite differenceand mapped-domain Chebyshev polynomial expansion respectively In the spanwise directionwe perform a Fourier decomposition and as the equations are linear the individual modes canbe solved independently (ie as separate two-dimensional calculations) Perturbations thushave the form

u = (u v w) eiβy

ω = (ωx ωy ωz) eiβy (4)

where β is the non-dimensional spanwise wavenumber The fluid is bounded by a rigid wall atz = 0 where the no-slip condition is applied and all of the perturbation variables vanish asz rarr infin At the upstream inlet of the computational domain all the primary variables are setto zero At the outflow of the domain the second-derivatives of the primary variables are set tozero to enforce non-reflecting wave-like conditions (though in this paper simulation times arekept short enough for no significant disturbance to reach the downstream boundary)

3 Generating near-wall streaks

Transient growth (characterised by inviscid algebraic growth followed by viscous decay) is amechanism whereby infinitesimal perturbations can interact with the underlying shear profilecreating much more energetic disturbances which can take the form of streamwise near-wallstreaks (Butler amp Farrell 1992 Henningson et al 1993 and Hultgren amp Gustavsson 1981)As demonstrated by Butler amp Farrell (1993) the generation of turbulent near-wall streaks isalso governed by a transient growth mechanism allowing the possibility of modelling turbulentstreaks using linearised equations (see also eg Chernyshenko and Baig (2005) Lockerby etal (2005) Cossu et al (2009))

31 The initial conditionIn order to generate near-wall streaks in our LNS simulations we introduce a near-wall andhighly-localised initial perturbation (bearing no resemblance to the near-wall streaks) and allow


3

transient growth to act on this and thus streaks to develop naturally This is a simplification tothe low-order model (LOM) of Lockerby et al (2005) which uses a parameterised body forcingto find a near-optimal streak response using a coarse-grain optimisation Here instead a singleinitial condition is chosen that excites a broad range of spatial and temporal modes from whichthe natural selectivity of the LNS amplifies certain streak scales ie a near-optimal response isobtained without the need for an optimisation procedure or for any empirical input The initialcondition we have selected approximates a spatio-temporal impulse at the wall The selectionof initial condition has been done with an intention of not imposing any preferred wall-normaland spanwise length scales in the flow The initial velocity field we prescribe which satisfiescontinuity is a numerical approximation to

u = 0 (5)v = δ(xminus xf ) δ(y minus yf ) δz(z minus zf ) (6)w = minusδ(xminus xf ) δy(y minus yf ) δ(z minus zf ) (7)

where δ is the Dirac delta function and a subscript denotes its derivative (since this is a linearsimulation there is no necessity to provide an amplitude other than for normalization) Theparameters xf yf and zf are locations in the flow domain where the spatio-temporal impulse isapplied The following numerical approximations are used

δ(x) asymp eminusax2 δ(z) asymp eminusbz2 δz(z) asymp minus2beminusbz

2 (8)

δ(y) asympNsumj=1

eiβjy δy(y) asympNsumj=1

iβjeiβjy (9)

where a and b are positive parameters which regulate the range of spatial scales being activatedin streamwise and wall-normal directions respectively The number of Fourier modes used inthe spanwise approximation to the delta function is N=24 where the j-th mode is such thatβj = jβ1 We have typically taken β+

1 = 2π1000 so that the modes cover a range of spanwisewavelengths with the largest being 1000 wall-units and the smallest being asymp 40 wall-units

The parameter zf is made very small (z+f = 1) ie approaching the wall beneath a certain

value (lt2-3 wall units) the results are insensitive to changes in this parameter The otherlocation parameters xf and yf have no material significance provided the base flow solution istranslationally invariant in the x and y directions For the STWSV simulations the x-variationof the travelling wave means that the response to the initial condition must be evaluated fora range of phase angles The results are independent of the choice of the parameters a and bgiven they are sufficiently large which is the case for the simulations presented here

We have performed numerical simulations using the stated initial condition for a turbulentchannel flow at Reτ = 200 The mean streamwise steady turbulent flow profile U(z) used hasbeen generated using the parametric composite form valid from the wall to the centreline of theflow It deploys a Musker profile from wall to the overlap region while the outer part makes useof a Coles wake function (see Nagib amp Chauhan (2008) for further details)

Figure 1 shows the evolution in time of the total streamwise kinetic energy of the disturbancegenerated from the initial condition The transient growth behaviour is apparent an initialgrowth period is followed by an exponential viscous decay There is no empirical input to theinitial conditions thus the time scale exhibited which is of an appropriate order compared toobserved near-wall time scales is an outcome of the LNS and base velocity profile alone

32 Quantification of the responseA key element in the study of short-term disturbances is the appropriate selection of a lsquomeasurersquothat quantifies the size of the disturbance (Schmid 2007) The appropriate choice largely


4

depends on the information that we require to extract from the response and in the currentstudy we consider two distinct measures

The first is a measure of the total streamwise momentum of the response evaluated at aparticular wall-normal plane for a particular spanwise mode The purpose of such as measureis for comparison with streak visualisation performed in DNS and experiment streaks withthe most momentum (rather than energy) are those seen in planes of u in DNS and thosehighlighted by the transport of passive scalars in experiment (eg by hydrogen bubbles inwater) The mathematical definition of this measure is

νm =int tf

0

int Lx

0|u(x z tβ)| dxdt (10)

where |u(x z tβ)| is the magnitude of the streamwise-velocity perturbation for thespanwise wavenumber considered (β) z is the wall-normal plane at which the measure isevaluatedvisualised Lx is the streamwise length of the domain and integration in time isperformed until tf corresponding to when the total kinetic energy has reduced to 5 of its peakvalue Here the measure is time integrated since total response rather than an instantaneousmaximum is arguably more appropriate for identifying time-persistent structures (ie streaks)

The benefit of evaluating νm at a particular wall-normal plane (z) and for a specifiedspanwise mode (β) (as in Chernyshenko and Baig 2005b) is that it allows the identification oftwo experimentally-observable streak characteristics 1) the dominant spanwise streak-spacing(λ=2πβ) and 2) the typical wall-normal streak location A contour plot of νm as a functionof z+ and λ+ is shown in Figure 2 for the initial condition introduced in sect31 The dominantresponse has a spanwise wavelength of approximately λ+ = 100 minus 120 and occurs at a wall-normal location of approximately z+ = 10minus 15 this is in consonance with experimental resultsreported by Smith (1983) among others Another feature worthy of note is that the spanwisewavelength of the dominant response increases with wall-normal position which has also beenobserved experimentally (Smith (1983))

The second measure we consider relates to the total energy of the response which is arguablymore appropriate for evaluating the effect on turbulent energy and thus drag of an appliedcontrol Here simply total streamwise kinetic energy is calculated again integrated over timeGiven that the control we are considering is time-dependent this again is more appropriate thanfinding an instantaneous maximum The energy measure is defined as follows

νe =1N

Nsumj=1

int tf

0

int infin0

int Lx

0u2j dxdzdt (11)

where u2j = |u(x z tβj)|2 contributes the streamwise kinetic energy of the perturbation due to

the jth spanwise wavenumber (βj) Note that the instantaneous kinetic energy ν displayed inFigure 1 is defined in a similar way but without any time integration

4 Streamwise-travelling waves of spanwise velocity (STWSV)

It has been shown by Quadrio et al (2009) in DNS of plane channel turbulent flow and alsoexperimentally by Auteri et al (2010) that STWSV generate a thin unsteady streamwise-modulated transversal Stokes layer which can bring about both drag reduction and drag increasedepending upon the actuation parameters κx and ω (defined earlier) This spanwise oscillatoryflow can be viewed as a generalized Stokes layer (Quadrio amp Ricco (2004))

In order to derive a numerical approximate solution for the generalised Stokes layer weconsider plane Poiseuille channel flow with the walls subjected to spanwise velocity in the form


5

Figure 1 Transient growth of streamwisekinetic energy ν Values normalised usingthe global maximum

Figure 2 Contours of νm showingresponse of the LNS in terms of thespanwise wavelength λ+ and the wall-normalvisualization plane z+

of streamwise-travelling waves given as Vw = A lt(ei(κxxminusωt)

) As in Quadrio amp Ricco (2004)

the governing Navier-Stokes equations can be simplified substantially if we assume that termsinvolving y-derivatives are null (as there is no variation in the spanwise direction) and that thewall-normal pressure-gradient is negligible The y-momentum equation can then be solved forthe spanwise velocity component v independently of the streamwise and wall-normal velocitycomponents For travelling waves of small enough amplitude it reduces to the form

partv

partt+ U

partv

partx=

1R

(part2v

partx2+part2v

partz2

) (12)

where R is the appropriate Reynolds number and the convective term is responsible for thecoupling between the turbulent flow profile U(z) and the spanwise flow By introducing thevariable ζ = xminus ct where c = ωκx so that part

partx = partpartζ and part

partt = minusc partpartζ we get

(U minus c)partvpartζ

=1R

(part2v

partζ2+part2v

partz2

) (13)

Further assuming the spanwise velocity v takes the mathematical form v = A lt(eiκxζV (z)

)

where V (z) is a complex-valued profile function the above equation reduces to((U minus c)iκxR+ κ2

x

)V (z) = V

primeprime(z) (14)

This equation with the boundary conditions V (0) = 1 and limzrarrinfin V (z) = 0 can be cast asa boundary-value problem and solved numerically using finite-difference discretisation of thesecond derivative We have used a LU decomposition of the complex matrix to determine thesolution V (z) In conjunction with the form for v defined above this allows us to calculate thespanwise velocity at each grid point as a function of time (for particular ω and κx) The Stokessolution thus obtained uses a laminar flow assumption but it is acceptable for turbulent flowsbecause the dominant viscous terms are expected to be much larger in magnitude compared tothe Reynolds stress terms as explained in Ricco amp Quadrio (2008)

5 Results and Discussion

We have performed numerical simulations of LNS equations with the open-loop control ofSTWSV for a turbulent channel flow at Reτ = 200 this corresponds directly to the DNS


6

simulations performed by Quadrio et al (2009) Again the mean streamwise steadyturbulent flow profile U(z) used has been generated using the parametric composite form (Nagibamp Chauhan 2008) The streamwise-modulated unsteady spanwise velocity profile v(x z t) asdiscussed above has been used to incorporate the open-loop control within the model Thephase angle φ of the streamwise travelling waves with respect to the axial location xf of theinitial localised perturbation has been varied from 0 to π in multiple steps (ten equally-spacedincrements were taken) and the response of the LNS (in terms of the scalar measure νe) has beenphase-averaged As the measure νe is related to the energy of time-persistent flow structures(ie structures like streaks) an increasedecrease of the measure is likely to be closely relatedto an increasedecrease of the wall shear stresses and therefore to an increasedecrease of theskin-friction drag

In order to compare our results directly with the DNS of Quadrio et al (2009) we haveperformed a total of 1875 three-dimensional simulations to cover the parametric ω -κx spacefrom minus3 to 3 and from 0 to 5 respectively for ten phases angles The amplitude of the waveshas been kept constant corresponding to the value A+ = 12 used in Quadriorsquos work For everypoint in the parametric space the incremental change ε in the energy measure νe is calculated

ε =νenc minus νec

νenc (15)

where νec and νenc are the phase-averaged values for the cases with and without controlrespectively A parameter-space map of ε is plotted in Figure 3 for comparison we havereproduced the map of drag-reduction (see Figure 4) from the data of Quadrio et al (2009)There are a number of remarkable similarities between the two figures Firstly for all negativephase velocities (ω lt 0) there is a positive ε (a streak energy reduction) and this correspondsto drag reduction in the DNS data Furthermore in both plots this becomes weaker as ωbecomes more negative Secondly there is a region of negative ε (a blue region correspondingto streak energy increase) along a diagonal band corresponding to a constant phase velocityof approximately c+ = 10 this finding is very much in agreement with the results of Quadrioet al (2009) who found that for spanwise waves having phase velocities approaching thatof the convection velocity of the near-wall coherent structures ie c+ asymp 10 there occurs alarge increase in turbulent skin-friction drag At a slightly steeper angle but still in the regionof positive phase velocity there exists a global maximum of ε which roughly correlates to thelocation of the global maximum in the DNS data There is also a local maximum in both plotsat approximately κx = 0 ω = 05 with a similar relative magnitude

51 Observation of a correlation between |V primeprime | and drag-reductionFor the purposes of simplicity we assume here a linear near-wall velocity profile ie U+ = z+Equation (14) (expressed in inner-units) can then be solved explicitly to obtain an analyticalexpression for the spanwise velocity profile function V (z+) which involves the Airy functionof the first kind Because the wall-normal second-derivative of the spanwise velocity (again ininner-units) is given by v

primeprime(x+ z+ t+) = A+ lt(eiκ

+x ζ

+V

primeprime(z+)) the maximum absolute values

of Vprimeprime(z) determine the maximum absolute values of vprimeprime globally both along the streamwise

direction x+ and over time as well as in the wall-normal directionContours of the analytically derived values of |V primeprime

(z+ = 10)| have been plotted as a functionof κx and ω in Figure 5 The plot displays a striking resemblance to the drag-reduction map ofQuadrio et al (2009) and seemingly suggests that zones of larger magnitudes of |V primeprime | resultin a significant decrease of streamwise kinetic-energy perturbations leading to turbulent drag-reduction The green corridor in the map represents κx -ω combinations where |V primeprime | asymp 0 Itslocation merits the speculation that there could be an inflection point type of instability whichcorresponds to the corridor of drag-increase in Quadriorsquos map


7

Figure 3 Contour plot of ε on a κx -ω mapat Reτ = 200

Figure 4 Contour plot of DNS results fromQuadrio et al (2009) at Reτ = 200 showingpercentage change of skin-friction drag on aκx -ω map

Figure 5 Contour plot showing variation of |V primeprime(z+ = 10)| on a κx -ω map

6 Summary

This work has investigated the linear response of perturbations interacting with a turbulentmean-flow profile and a control flow generated by streamwise travelling-waves of spanwise wallvelocity The effects of the wavenumber κx and oscillation frequency ω of the travelling waveson the turbulent channel flow at a Reynolds number Reτ = 200 have been studied showingthat both backward- as well as forward-travelling waves lead to a significant reduction of themeasure νe except for waves having phase velocity c+ asymp 10 which correspond to a corridorwhere νe increases These findings for the variation of νe are in excellent qualitative agreementwith the DNS results for drag-reduction obtained by Quadrio et al 2009 Preliminary resultsat a higher flight scale Reynolds number show a qualitatively similar pattern of variation forνe but ongoing LNS investigations still need to be completed in order to sample a sufficientlylarge set of κx -ω combinations

Acknowledgments

The authors would like to acknowledge the financial support from EPSRC through grantEPG0602151 together with Airbus Operations Ltd and EADS UK Ltd


8

References

Auteri F Baron A Belan M Campanardi Gamp Quadrio M 2010 Experimentalassessment of drag-reduction by traveling waves in a turbulent pipe flow Phys of Fluids 22115103

Butler KM and Farrell B F 1992 Three-dimensional Optimal perturbations in aviscous shear flow Phys of Fluids 48 1637ndash1650

Butler KM and Farrell B F 1993 Optimal perturbations and streak spacing in wall-bounded turbulent shear flows Phys of Fluids 43 774ndash777

Chernyshenko SI and Baig MF 2005 The mechanism of streak formation in near-wallturbulence J Fluid Mech 544 99ndash131

Chernyshenko SI and Baig MF 2005 Streaks and Vortices in Near-Wall TurbulencePhil Trans of Roy SocMath Phy and Engg Sciences 363 1097ndash1107

Choi JI Xu CX amp Sung HJ 2002 Drag-reduction by spanwise wall-oscillation in wall-bounded flows AIAA J 40 842ndash850

Cossu C Pujals G and Depardon S 2009 Optimal transient growth and very largescalestructures in turbulent boundary layers J Fluid Mech 619 79-94

Davies C and Carpenter P W 2001 A novel velocity-vorticity formulation of the Navier-Stokes equations with application to boundary layer disturbance evolution J Comp Phys172 119ndash165

Henningson DS Lundbladh A and Johansson AV 1993 A Mechanism for bypass-transition from localized disturbances in wall-bounded shear flows J Fluid Mech 250 169ndash207

Hultgren LS and Gustavsson LH 1981 Algebraic growth of disturbances in a laminarboundary layer Phys of Fluids 24 No6 1000ndash1004

Jung WJ Mangiavacchi N and Akhavan R 1992 Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations Phys of Fluids A 4 1605ndash1607

Karnidakis GE amp Choi K-S 2003 Mechanisms on transverse motions in turbulent wall-flows Ann Rev Fluid Mech 35 45ndash62

KlineSJ Reynolds WC Schraun FA and Runstadler PW 1967 The structureof turbulent boundary layers J Fluid Mech 30 741-773

Landahl M T 1989 Boundary layer turbulence regarded as a driven linear system PhysicaD 37 11-19

Lockerby D A Carpenter P W and Davies C 2005 Control of sublayer streaksusing microjet actuators AIAA J 43 1878ndash1886

Nagib HM and Chauhan KA 2008 Variations of Von-Karman coefficient in canonicalflows Phys of Fluids 20 101518

Quadrio M and Ricco P 2004 Critical assessment of turbulent drag-reduction throughspanwise wall-oscillations J Fluid Mech 521 251ndash271

Quadrio M Ricco P and Viotti C 2009 Streamwise travelling waves of spanwise wall-velocity for turbulent drag-reduction J Fluid Mech 627 161ndash178

Quadrio M and Ricco P 2010 The laminar generalized Stokes layer and turbulent drag-reduction arXiv10083981v 1-29

Ricco P and Quadrio M 2008 Wall-oscillation conditions for drag-reduction in turbulentchannel flow Int J Heat and Fluid Flow 29 601-612

Schmid PJ 2007 Nonmodal stability theory Ann Rev Fluid Mech 39 129-162


9

Smith C R and Metzler S P 1983 The characteristics of low-speed streaks in thenear-wall region of a turbulent boundary layer J Fluid Mech 129 2754

Viotti C Quadrio M and Luchini P 2009 Streamwise oscillation of spanwise-velocityat the wall of a channel for turbulent drag-reductionPhys Fluids 21 115109


10

Modelling turbulent skin-friction control using

linearised Navier-Stokes equations

C A Duque1 M F Baig1 D A Lockerby1 S I Chernyshenko2

and C Davies3

1 School of Engineering University of Warwick Coventry CV4 7AL UK2 Department of Aeronautics Imperial College London SZ2 4AZ UK3 School of Mathematics Cardiff University Cardiff CF24 4AG UK

E-mail fmsBaigwarwickacuk caDuque-dazawarwickacuk

duncanlockerbywarwickacuk SChernyshenkoimperialacuk

DaviesC9cardiffacuk

Abstract A linear model has been used to make qualitative predictions of turbulent drag-reduction resulting from open-loop control using streamwise travelling waves of spanwise wallvelocity There is very good agreement over a wide parameter space between our qualitativeprediction of drag decreaseincrease and the DNS results of Quadrio et al (2009) atReτ = 200

1 Introduction

In the last two decades numerous studies have shown that spanwise (cross-flow) wall oscillationsand streamwise-travelling waves of spanwise velocity bring about a substantial reduction inturbulent skin-friction drag It has been found that these wall-oscillation control approacheshave a strongly disruptiveinhibitive effect on the near-wall streak structures (Jung etal 1992Karnidakis amp Choi 2003 Ricco amp Quadrio 2008) Given the commonly-held belief that thenear-wall streak structures play a major role in the turbulent regeneration mechanism (Kline atal 1967) we might expect that an explanation for the efficacy of such control methods would bereadily forthcoming However there is still a lack of clarity in our understanding of the physicalmechanism(s) underpinning this type of control strategy and this stands in the way of efficientprediction and engineering application

Choi et al (2002) demonstrated experimentally drag reduction of up to 45 using purewall oscillations in the spanwise direction An optimum period of around T+ = 100 minus 125 wasshown to be independent of the wall-oscillation amplitude (here lsquo+rsquo denotes viscous wall units)It was also found that the location of vortices and low-speed streaks relative to the oscillatoryStokes layer was closely connected to the amount of drag reductionincrease observed Thus inthis study the physical mechanism for the skin-friction change was attributed to displacementof the vortices and streaks relative to each other

Quadrio et al (2009) investigated numerically streamwise-travelling waves of spanwisevelocity (STWSV) generated with a wall velocity of the type Vw = A sin(κxxminusωt) where A is thewall motionrsquos amplitude κx is a streamwise wavenumber and ω is the oscillation frequency of thewave This form of forcing generates waves of spanwise velocity moving (backward or forward)


Published under licence by IOP Publishing Ltd 1









2



partωxpartt

+partNz

partyminus partNy

partz=

1Rnabla2ωx (1)

partωypartt

+partNx

partzminus partNz

partx=

1Rnabla2ωy (2)


partx (3)



u = (u v w) eiβy







3





2 (8)

δ(y) asympNsumj=1


iβjeiβjy (9)









4



νm =int tf

0

int Lx





νe =1N

Nsumj=1

int tf

0

int infin0

int Lx

0u2j dxdzdt (11)







5






partv

partt+ U

partv

partx=

1R

(part2v

partx2+part2v

partz2

) (12)





=1R

(part2v

partζ2+part2v

partz2

) (13)


)


x

)V (z) = V

primeprime(z) (14)





6




νenc (15)




+x ζ

+V






7




6 Summary


Acknowledgments



8

References























9




10









2



partωxpartt

+partNz

partyminus partNy

partz=

1Rnabla2ωx (1)

partωypartt

+partNx

partzminus partNz

partx=

1Rnabla2ωy (2)


partx (3)



u = (u v w) eiβy







3





2 (8)

δ(y) asympNsumj=1


iβjeiβjy (9)









4



νm =int tf

0

int Lx





νe =1N

Nsumj=1

int tf

0

int infin0

int Lx

0u2j dxdzdt (11)







5






partv

partt+ U

partv

partx=

1R

(part2v

partx2+part2v

partz2

) (12)





=1R

(part2v

partζ2+part2v

partz2

) (13)


)


x

)V (z) = V

primeprime(z) (14)





6




νenc (15)




+x ζ

+V






7




6 Summary


Acknowledgments



8

References























9




10



partωxpartt

+partNz

partyminus partNy

partz=

1Rnabla2ωx (1)

partωypartt

+partNx

partzminus partNz

partx=

1Rnabla2ωy (2)


partx (3)



u = (u v w) eiβy







3





2 (8)

δ(y) asympNsumj=1


iβjeiβjy (9)









4



νm =int tf

0

int Lx





νe =1N

Nsumj=1

int tf

0

int infin0

int Lx

0u2j dxdzdt (11)







5






partv

partt+ U

partv

partx=

1R

(part2v

partx2+part2v

partz2

) (12)





=1R

(part2v

partζ2+part2v

partz2

) (13)


)


x

)V (z) = V

primeprime(z) (14)





6




νenc (15)




+x ζ

+V






7




6 Summary


Acknowledgments



8

References























9




10





2 (8)

δ(y) asympNsumj=1


iβjeiβjy (9)









4



νm =int tf

0

int Lx





νe =1N

Nsumj=1

int tf

0

int infin0

int Lx

0u2j dxdzdt (11)







5






partv

partt+ U

partv

partx=

1R

(part2v

partx2+part2v

partz2

) (12)





=1R

(part2v

partζ2+part2v

partz2

) (13)


)


x

)V (z) = V

primeprime(z) (14)





6




νenc (15)




+x ζ

+V






7




6 Summary


Acknowledgments



8

References























9




10



νm =int tf

0

int Lx





νe =1N

Nsumj=1

int tf

0

int infin0

int Lx

0u2j dxdzdt (11)







5






partv

partt+ U

partv

partx=

1R

(part2v

partx2+part2v

partz2

) (12)





=1R

(part2v

partζ2+part2v

partz2

) (13)


)


x

)V (z) = V

primeprime(z) (14)





6




νenc (15)




+x ζ

+V






7




6 Summary


Acknowledgments



8

References























9




10






partv

partt+ U

partv

partx=

1R

(part2v

partx2+part2v

partz2

) (12)





=1R

(part2v

partζ2+part2v

partz2

) (13)


)


x

)V (z) = V

primeprime(z) (14)





6




νenc (15)




+x ζ

+V






7




6 Summary


Acknowledgments



8

References























9




10




νenc (15)




+x ζ

+V






7




6 Summary


Acknowledgments



8

References























9




10




6 Summary


Acknowledgments



8

References























9




10

References























9




10




10

modelling turbulent skin-friction control using linearised

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