Applying the Boundary-Layer Independence Principle to Turbulent Flows

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  • Applying the Boundary-Layer IndependencePrinciple to Turbulent Flows

    Israel Wygnanski, Philipp Tewes, and Lutz Taubert

    University of Arizona, Tucson, Arizona 85721

    DOI: 10.2514/1.C032206

    Velocities measured in turbulent boundary layers over yawed flat plates confirmed that the mean velocity profiles

    normal to the leading edge are proportional to the velocity profiles parallel to it, with a proportionality constant

    depending on the yaw angle. This turned out to be the necessary and sufficient condition to make the wall stress

    components normal and parallel to the leading edge also proportional in the same manner, thus reaffirming the

    boundary-layer independence principle for turbulent and laminar flows alike. Reinterpretation of old experiments

    thus changed themantra stating, the independence principle does not apply to turbulent flow, thus providing a new

    insight into three-dimensional boundary-layer flows on yawed, high-aspect-ratio wings. It explains the prevalence of

    attached spanwise flow near the trailing edges of such wings, and it provides a rationale for turbulence modeling on

    them. Furthermore, it indicates the direction along which active separation control should take place.

    Nomenclature

    CD = sectional/wing-drag coefficientCL = sectional/wing-lift coefficientCf = skin-friction coefficientCp = pressure coefficientC = momentum coefficientc = chord normal to leading edgec = chord in freestream directionD = drag forceG = function describing skin friction of yH = boundary-layer shape parameterk, = constantsMa = Mach numberp = static pressureRe = Reynolds numberUe, Ve,We = velocities at the edge of the boundary layer in x, y,

    and z directionsU, V, W = friction velocities in x, y, and z directionsU = freestream velocityu, y = dimensionless velocity, dimensionless wall

    coordinateu, v, w = velocities inside the boundary layer in x, y, and z

    directionsx, y, z = body-fixed coordinate system, y, = coordinate system in direction of stream = angle of attack = boundary-layer thicknessf = flap deflection = displacement thickness = self-similarity coordinate = yaw angle = momentum thickness = dynamic viscosity = kinematic viscosity = air density

    = total shear stress = flow angle on surface = stream function

    Introduction

    R ECENT observations in a turbulent mixing layer emanatingfrom a skewed trailing edge indicated that the boundary-layerequations governing the evolution of this flow in the direction normalto the trailing edge are independent of the equations governing thespanwise flow along it [1]. Thus, the applicability of the indepen-dence principle to this highly turbulent flow triggered a renewedinterest in the old measurements (e.g., [2,3]) that refuted its validityto turbulent boundary layers thus changing the mantra theindependence principle does not apply to turbulent flow [4].The boundary-layer equations representing incompressible flow

    over a yawed flat plate are

    ux vy 0 (1)

    uux vuy xyy (2)

    uwx vwy zyy (3)

    Where u and w are components of velocity in the directions normal(x), and parallel (z) to the leading edge of the plate, thus at y : u Ue U cos andw We U sin where is theyaw angle (Fig. 1). All of the derivatives z 0 due the infiniteaspect ratio of the yawed plate. The somewhat unusual representationof the momentum equations results from the addition of the conti-nuity equation to the equations of momentum after multiplying it bythe appropriate velocity component.In laminar flow, xy uy and zy wy; thus, the two

    momentum equations are identical, as are their respective boundaryconditions. Therefore, the solutions are proportional to one anotherwith the proportionality constant depending on . Each one of themomentum equations provides a balance between the inertia termsand the stress terms; thus, knowledge of one determines the other andvice versa, regardless of the state of the flow. Hence, if in turbulentflow, ux; y is proportional to wx; y, the shear stresses should beproportional to one another as well (i.e., xy zy, and this includesthe different Reynolds stress terms as well, and in particular, u 0v 0 w 0v 0 because these are the dominant terms outside the viscoussublayer). It permits one to solve Eqs. (1) and (2) independently ofEq. (3) because the two first equations do not contain w terms. Thistype of boundary-layer solutions is referred to in the literature as the

    Received 26 November 2012; revision received 1 July 2013; accepted forpublication 10 July 2013; published online 24 January 2014. Copyright 2013 by the American Institute of Aeronautics and Astronautics, Inc. Allrights reserved. Copies of this paper may be made for personal or internal use,on condition that the copier pay the $10.00 per-copy fee to the CopyrightClearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; includethe code 1542-3868/14 and $10.00 in correspondence with the CCC.

    *Professor, Department ofAerospace andMechanical Engineering. FellowAIAA.

    Research Associate, Department of Aerospace and MechanicalEngineering. Student Member AIAA.

    Research Assistant Professor, Department of Aerospace and MechanicalEngineering. Member AIAA.

    175

    JOURNAL OF AIRCRAFTVol. 51, No. 1, JanuaryFebruary 2014

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    http://dx.doi.org/10.2514/1.C032206

  • independence principle. It applies to laminar boundary layers but itis considered wrong wherever turbulence is involved because thefluctuations v 0, might have been affected by the Reynolds stressw 0v 0, thus coupling the two equations of motion.Streamwise velocity profiles measured in turbulent boundary

    layers over yawed plates by Ashkenas and Riddell [2] and lateragain by Ashkenas [3] are shown in Fig. 2. Although the plateswere yawed at 0, 30, and 45 deg to the oncoming stream, thevelocity profiles representing u2 w20.5, when normalized by thefreestream velocity and a representative displacement thickness, areidentical irrespective of yaw angle and Reynolds number. Theseresults suggest thatw u at all Reynolds numbers considered. Thus,xy is expected to be proportional to zy according to Eqs. (2) and (3).At the time that Ashkenas and Riddell did their experiment [2],

    hot-wire anemometry was at its infancy, and Coless law of the wall[5] was not formally published. Therefore, the velocity profilesshown in Fig. 1 were measured with flattened pitot tubes and weresusceptible to integration errors near the surface even though thetubes were flattened. Three years later, Ashkenas [3] repeated themeasurements using hot wires to measure the Reynolds stresses butretained the pitot tubes for mean velocity measurements. He con-firmed and expanded on the previous experimental results for 0,30, and 45 deg, yet hewas unable to explain the reasons for the failureof the independence principle in turbulent flow. We shall try to showthat these experimental results prove the validity of the independenceprinciple when one recalls that skin friction drag is a vector that isrelated to the vectorial sum of the two momentum thicknessesassociated with the chordwise and spanwise boundary layers.The proportionality of the mean velocity profiles and the univer-

    sality of the law of the wall for a smooth surface in the absence ofpressure gradient suggest that the combined mean velocity distribu-tion near the surface depends only on the distance from the surface(y), on 2x;0 2z;00.5, and on the character of the fluid prescribed byits density and kinematic viscosity .Using Eqs. (2) and (3) and applying simple dimensional

    considerations yield

    uu Fuy

    (4a)

    where

    u x;0

    p(4b)

    and

    ww Fwy

    (4c)

    where

    w z;0

    p(4d)

    Equations (4a)(4d) lead to linear velocity distributions within theviscous sublayer and logarithmic ones further away from the surface.Assuming that the logarithmic portion of the law of the wall iscorrectly measured and its universality holds for all the sweep anglesconsidered in [2], one may use the well known distribution u 10.41 ln y 5 to obtain u and w and normalize the velocitiesmeasured by these quantities. The logarithmic portion of thusnormalized velocity distribution holds for 30 < y < 300 as it doesfor two-dimensional turbulent boundary layers over smooth flatplates (Fig. 3). The velocities corresponding to y < 20 are scattered,probably because of the type of probe used in the experiment.Ashkenas [3] also used Coless law to estimate the wall shear stressand used the momentum equation to calculate the Reynolds stressdistribution in the boundary layer and compare it with the x-wire data.The comparison was reasonably good, confirming the reliability ofthe experimental results. Following Rottas [6] and Townsends [7]approaches, the velocity defect law was also recovered from thesemeasurements, but the results are not presented because they do notdiscover anything new.Thevelocity profiles shown in Fig. 2 are also self-similarwithin the

    Reynolds number range investigated. The self-similarity of the inertiaterms, the left-hand side of Eqs. (2) and (3), results in similarity ofReynolds stresses xy and zy that may extend over the bulk of theboundary layer, suggesting the possibility of testing Townsendsself-preservation approach [7] for the bulk of the turbulentboundary layer:

    Gxy xyU cos 2 (5)

    where Gx scales with in the same manner as the mean velocityprofiles do. Assuming a chordwise flow component stream functionof the form Uf cos , where y, taking the

    Fig. 1 Plan view defining the coordinate system.

    0 1 2 3 4 5 6 7 80

    0.1

    0.2

    0.3

    0.4

    0.5

    0.6

    0.7

    0.8

    0.9

    1

    y /

    (u2

    + w

    2 )0.

    5 / U

    =12in, = 0 deg, Re

    =1,500

    =24in, = 0 deg, Re

    =2,335

    =27in, = 0 deg, Re

    =2,315

    =36in, = 0 deg, Re

    =2,865

    =38in, = 0 deg, Re

    =2,920

    =47in, = 0 deg, Re

    =3,460

    =16in, =30 deg, Re

    =1,875

    =32in, =30 deg, Re

    =2,690

    =48in, =30 deg, Re

    =3,320

    = 8in, =45 deg, Re

    =1,460

    =18in, =45 deg, Re

    =1,855

    =24in, =45 deg, Re

    =2,270

    =36in, =45 deg, Re

    =3,230

    =44in, =45 deg, Re

    =3,595

    =54in, =45 deg, Re

    =4,270

    =66in, =45 deg, Re

    =4,960

    Fig. 2 Turbulent velocity profiles at various streamwise distances andangles [2,3].

    100

    101

    102

    103

    6

    8

    10

    12

    14

    16

    18

    20

    22

    24

    26

    y+ = y ( x,0

    + z,0

    )0.5 / ( 0.5)

    u+ =

    [(

    u2

    + w

    2 ) /

    ( x

    ,0 +

    z,0

    )]0.

    5

    u+ = 1/0.41 (y+) + 5

    Fig. 3 Universal character of the turbulent velocity profiles according tothe law of the wall.

    176 WYGNANSKI, TEWES, AND TAUBERT

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  • appropriate derivatives and substituting them in Eq. (2) together withGx results in

    G 0x ddxff 0 0 0 (6)

    For self-similarity to exist, Eq. (6) will have to depend only on asingle variable (), and ddx needs to be constant, suggestingthat x cos .The boundary-layer thicknesses reported in [2,3,8] were all

    measured in the direction of streaming irrespective of , (0 45 deg). Therefore, the measured is proportional to cos . Thethicknesses shown in [2,3,8] confirm this result, providingdd cos 0.0013, as may be deduced from Fig. 4.Young and Booth [8] suggested that, if one assumes a power-lawvelocity profile, its self-similarity with respect to yaw will not beprecise and the ratio Re;Re; 0 cos0.2 that is weaklydependent on . Perhaps for this reason there is a trend in the scatterof the results seen in Fig. 4. Although the self-similarity argument iscompelling, it is less rigorous than the proportionality argument thatwas shown to exist in Fig. 2, because the flow very close to the solidsurface may not have been self-similar.The applicability of the independence principle becomes clear

    when one considers the integral momentum equations because theyrelate the skin friction drag to momentum thickness regardless of thestate of the flow (i.e., whether the flow is laminar or turbulent). Onehas to remember that the two independent boundary layers contributeto drag depending on the sweep angle and that drag is a vector.FromEqs. (1) and (2), one obtains that dxdx x;0U2e, where

    x 0 uUe 1 uUe dy; the upper limit represents the boundary-

    layer thickness, where u Ue & uy 0, and x;0 represents thewall shear stress in the x direction (Fig. 1).Integrating Eq. (3) while using Eq. (1) to substitute for the normal

    velocity component at the edge of the boundary layer yieldsdzdx z;0w2e, where z 0 uWe 1

    wWe dy. Because Ue

    U cos andWe U sin , and recalling thatw u, yields thedrag components in the chordwise and spanwise directions,

    Dx Zc

    0

    x;0 dx U2eZc

    0

    dxdx

    dx U2ex;c (7)

    Dz Zc

    0

    z;0 dx W2eZc

    0

    dzdx

    dx W2ez;c W2e cot x;c (8)

    The resultant drag in the direction of the freestream is

    DDx cosDz sinU2ex;ccos and DxDcos(9)

    This proves that the wall stresses x;0 & z;0 are proportional to eachother in the samemanner that the inertia terms are in either laminar orin turbulent flows. Thus, when integrated across the boundary layer,the independence principle applies to turbulent and laminar flowsalike. The drag coefficient CD

    D0.5U2c

    2x;c cos c. If thechord is defined in the direction of streaming (c c cos ), seeFig. 1, then CD 2x;cc. This corresponds to the conventionaldefinition of drag coefficient in two-dimensional steady flow.The validity of the result may be demonstrated by using the

    Blasius solution for a laminar boundary layer and power-lawapproximation for a turbulent boundary layer. In the former, kU0.5 kRe0.5 , where k 0.664, while in thelatter 0.036U0.2 0.036Re0.2 . When transformedto the coordinates of Fig. 1 (Ue U cos and x cos ), themomentum thickness in the laminar case yields x kxUe0.5,which is independent of the sweep angle , while the transformedturbulent momentum thickness x 0.036Ue0.2x0.8 cos0.6 depends on the sweep. By using Eq. (9), the laminar drag D U2ck cos cU0.5 U2ckRe0.5c cos , where Rec cos is aReynolds number based on the freestreamvelocity and a chord lengthmeasured along the freestream direction (c cos ). Thus,DxD cos , although Dx does not contain . The drag coefficientin the laminar case is CD D0.5U2c 2kRe0.5c cos 2c cos c cos . Using the same procedure for the turbu-lent boundary layer yields D U2c0.036Re0.2c cos ; DxD cos ; and CD D0.5U2c 2kRe0.2c cos 2c cos c cos . This example proves again that the ratio of themomentum thickness at the trailing edge of a yawed flat plate to itschord is equal to the drag coefficient regardless of the state of the flow.Thus, the independence of the transformedmomentum thickness (x)of in laminar flow is a coincidence that led Ashkenas and Riddell[2] astray.Figure 5 is a reproduction of Fig. 22b in [2] showing the empirically

    obtained growth law of the displacement thickness. It clearly indicatesthat cos0.4 , relative to the freestream coordinate system,instead of being cos0.6 , as was anticipated by transformingthe coordinates and violating the integral momentum equation. Theratio between the anticipated and measured displacement thicknesses(or momentum thicknesses for that matter) is DxD...

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