1、NASA Contractor Report 191413ICASE Report No. 92-72ICASEt/3ySIMULATION OF BOUNDARY-LAYER TRANSITION:RECEPTIVITY TO SPIKE STAGEFabio P. BertolottiJeffrey D. CrouchNASA Contract Nos. NAS1-18605 and NAS1-19480December 1992Institute for Computer Applications in Science and EngineeringNASA Langley Resear
2、ch CenterHampton, Virginia 23681-0001Operated by the Universities Space Research AssociationNational Aeronautics andSpace AdministrationLangley Research CenterHampton, Virginia 23681-0001a0N!t_o,ZI,LQZO-! F-+Jw 0mw,1“O“O,_30,!“c_Provided by IHSNot for ResaleNo reproduction or networking permitted wi
3、thout license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-SIMULATION OF BOUNDARY-LAYER TRANSITION:RECEPTIVITY TO SPIKE STAGEFabio P. Bcrtolotti 1Institute for Computer Applications in Science and EngineeringNASA Langley Research Cent
4、erHampton, VA 23681-0001an dJ_ffrcy D. Crouch 2Boeing Commercial Airplane GroupSeattle, WA 98124.ABSTRACTThe transition to turbulence in a boundary layer over a fiat plate with mild surfaceundulations is simulated using the parabolized stability equations (PSE). The simulationsincorporate the recept
5、ivity, the linear growth, and the nonlinear interactions leading tobreakdown. The nonlocalized receptivity couples acoustic perturbations in the free-streamwith disturbances generated by the surface undulations to activate a resonance with thenatural eigenmodes of the boundary layer. The nonlinear s
6、imulations display the influenceof the receptivity inputs on transition. Results show the transition location to be highlysensitive to the anaplitudes of 1)oth the acoustic disturbance and the surface waviness._This research was supported by the National Aeronautics and Space Administration under NA
7、SA Con-tract Nos. NAS1-18605 and NAS1-19480 while the author was in residence at, the Institute for ComputerApplications in Science and Engineering (IC,ASE), NASA Langley Research Center, Hampton, VA 23681-0001. This author gratefully acknowledges the assistance and support provided by S.A. Orszag u
8、nder DA RPAcontract N00014-86-K-0759 during the initial part of this work.2part of this work was conducted while this author was at the Naval Research Laboratory in Washington,DC, supported by an ONT Postdoctoral Fellowship.Provided by IHSNot for ResaleNo reproduction or networking permitted without
9、 license from IHS-,-,-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I. IntroductionIn the absenceof significant crossflow or curvature, transition to turbulence generallyresults from the amplification of traveling eigenmodessuch as Tollmien-Schlich
10、ting (TS)wavesor Squiresmodes. Thesetraveling modesare generatedthrough the processcalledreceptivity. While surfacevibrations and atmosphericturbulence can activate receptivitymechanisms,experimentshave shown that the laminar-turbulent transition over a wingsurfaceis strongly affectedby the acoustic
11、field generatedby the enginesand the turbulentboundary layer on the fuselage 1.The interaction of the acoustic field with a single bump has been the focus of severalinvestigations in the past, e.g. Goldstein we include both the receptivity and the subsequent linear and nonlinearevolutions. The relat
12、ive efficiency of the PSE simulations accommodates the investigation ofthe effects of “receptivity input parameters“ on the path to transition. These investigationsdisplay the strong effect which small, i.e. O(1), changes in the wall roughness height andgeometry have on the transition process.II. Wa
13、ll geometry and acoustic fieldWe consider a fiat plate immersed in an incompressible flow field with acoustic noise.The plate surface is covered by small amplitude irregularities with length scales comparableto TS waves.We employ a Cartesian coordinate system with the average plate surface in the ,r
14、*-z*plane, x* measuring streamwise distance, and y* the distance normal to the plate (theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-symbol * denotesa dimensionalquantity). We nondimensionalizequantities using the free-* _/ . T. * being the locat
15、ion wherethe.stream velocity U_ and the length 8_ = te ocT/boo , with zTReynolds number Rx = U_ox_/z.* equals one million. Results are presented in terms of the.usual stability parameters, R = _ (distance) and F = lOS(co*/2w),.,*/(U2o) 2 (frequency).The ;r-z-periodic function H(z, z) describes the s
16、urface undulations. We represent H byits Fourier series representation,kF_-OO S_-OOwhere the coefficients /V_,_ and 14/_,_ are complex conjugates. Additionally, we imposesymmetry in z, thus W_,_ = 14/,.,_. We use values of lwl of the order of 0.002 (see Tabh, l)which translate into a height to wavel
17、ength ratio of order 1/1000. At STP and/77_ = 10 m/sthe peak-to-peak surface variation is in the range of 30 mierolneters, while at/F_ = 100 m/sthe variation is about 10 micrometers.The flee-stream acoustic field is of the formC_OAssociated with each discrete acoustic frequency Al is a velocity fiel
18、d having a Stokes layerat the wall satisfying the no slip boundary condition, and matching the acoustic field in thefree-stream. The values of co in (2), and of a and fl in (1) represent the lowest common divisorof the set of frequencies and wave numbers present. In case the wall spectrum is dominat
19、edby sharp peaks but the acoustic spectrum is not, we choose co such that for each wall mode(no:, l,fl), the triplet (leo, ha, kfl) is as close to branch I of the TS wave neutral stabil:_ty curveas possible. In this way we focus our attention on the temporal-spatial combination thatwill feed the gre
20、atest amount of energy into the eigenmodes. Conversely, if a flat spatialspectrum is present and isolated peaks exist in the acoustic spectrum, we select the valuesof a and fl that yiehl triplets close to branch I.III. Receptivity mechanismAcoustic disturbances in the free-stream generate Stokes mod
21、es within the boundarylayer. In the incompressible limit these modes have only temporal modulation,vl = l(y)c it_t q- c.c. , 1 ,-2,- l, 1,2, (3)Meanwhile, the mean flow over a wavy surface produces steady wall modes,v,_,k = %,k(z, y)c i_+ik_: + c.c. ,2Provided by IHSNot for ResaleNo reproduction or
22、networking permitted without license from IHS-,-,-Y (a)zFigure 1: Rendition of surface undulations used in present study. The normal coordinate isstretched. (a) Case “high“ and “low“, (b) Case “riblet“.,-2,-1, 1,2, (4)These modes are standing waves with wave numbers given directly by“ the surface. S
23、ingle-handedly, neither tile acoustic nor the wall velocity fields can directly energize a travel-ing eigenmode since these fields lack the necessary spatial or temporal variation, respec-tively. However, the simultaneous presence of both fields produces traveling waves due tothe quadratic nonlinear
24、ity of the Navier-Stokes equations,Vl,n,k = l,z,k(X, !t)e -ilwt+ikflz+inc_x, (5)whose form is identical to that of the natural eigenmode of the boundary layer, except thevalue of the exponent c, may not match that of the eigenmode of the boundary layer stabilityequations, which has the form,Vl,n,k _
25、 Vl,n,k(X M C Zo ,a_ = % + ic_,The receptivity mechanism is illustrated in Figure 2.(6)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-The key ingredient to the nontocalizedreceptivity processis a resonancewhich resultswhenthe wall wavenumber c_ appr
26、oaches the eigenmode wave number a_ = % +/eee. Here,7 is the growth rate. Near branch I the growth rate is small, and, for an appropriate valueof ee, tile difference la_ - c_I may be small. The resulting response of the boundary layer,under tile forcing provided by the traveling wave may be quite la
27、rge. The detuning la_ - a IAcoustic WallForced EigenmodeFigure 2: Components of the nonlocalized receptivity model (Spa nwise wavenurnber fie. notshown).changes as the modes are convected downstream. During the large response near branch I,energy is transferred into the eigenmode of the boundary lay
28、er. The nonlocalized receptivitymodel 5,7 shows that tile rate of energy transfer between the forced wave and the eigenmodeis proportional to the rate of variation of the forced wave response. Farther downstream,tile eigenmode undergoes the typical exponential growth characteristic of tile linear re
29、gime.Receptivity results from the net energy transfer into tile eigenmode. The superposition ofthe forced wave and the eigenmode, with their appropriate wave numbers, provides tile total(physical) traveling-wave disturbance.IV. The PSE formulationAs a consequence of the basic flow being independent
30、of the spanwise coordinate z, weCall reduce the number of unknowns from velocity components u, v, w (along x, y, and z,respectively) and pressure p to only u and v. We eliminate pressure by taking the curl ofthe momentum equation, and w using the continuity equation. The boundary conditions are4Prov
31、ided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-zero velocity at the wall, and (except for the acoustic modes and the mean-flow distortion)vanishing velocity far from the plate.The parabolized stability equations, commonly abbreviated to PSE, were develo
32、ped byHerbert 2 .During every streamwise step, the nonlinear algebraic system is solved iterativey bymodifying tile values of “)p until the normalization conditions (9) are satisfied to a pre-determine level of accuracy.A further element affecting accuracy is the presence of (11) in _rp. For small a
33、mounts ofseparation a_ - al, the difference in wavenumber is well captured by the streamwise changeof _,p. For larger amounts, however, the PSE results loose accuracy. A more detail studyof this issue, as well as the effect of step-size on accuracy, can be found in reference 14. Itsuffices here to s
34、ay that a difference in ce_ - a/_ 10-32.0 10 -30.01.0 x 10 -30.01.0 x 10 -35.0 X 10 -30.0 0.0 1.25 10 -3The geometry between “high“ and “low“ differs only in the peak-to-peak variation of tileundulations. Tile “riblet“ geometry contains two additional modes which describe stream-wise aligned undulat
35、ions, similar to the streamwise “riblets“ used in turbulence drag reduc-tion, only that our “riblets“ are not sharp peaked, and have a very small height.V.1. Effect of roughness heightTo study the effect of surface roughness height on transition we consider tile “high“ and“low“ geometries. The acous
36、tic modes have a u peak-to peak variation of 0.0010 Uo, (i.e.41 : 42 = 0.0005 in equation 2) while tile peak-to-peak variation of the wall modes (Table 1)differ amongst each other by at most a factor of 2, hence no bias exists towards one particularmode. This scenario contrasts that of ribbon-induce
37、d transition where a two-dimensionalmode dominates in amplitude prior to the onset of secondary instability.Figure 3 displays the amplitude evolution of modes in the “high“ roughness case. Thetransition process follows the well known subharmonic route. We focus our attention on twomodes:, the two-di
38、mensionM mode (2,2,0), which develops into a TS wave, and the (1,1,1)mode, which develops into tile subharmonic mode. Initially each mode is composed solelyof a forced traveling wave. As the modes propagate downstream the nonlocalized receptivityprocess pumps energy into the eigenmodes. Consequently
39、, the (2,2,0) and (1,1,1) modes areof nearly the same amplitude and exhibit similar growth rates during their early evolution.At R = 589, the (2,2,0) mode passes through branch I. The eigensolution component of themode undergoes exponential growth, while the forced traveling wave diminishes downstre
40、am.The development of the subharmonic mode, (1,1,1), can be analyzed in two parts, the firstone at Iocations below, say R = 1050 (including the sharp dip in amplitude), and the secondone for locations above R = 1050. The first part is dominated by the process of receptivity,while the second part is
41、dominated by the parametric resonance with the (2,2,0) mode. The9Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-I0-I10-20-30-410-5_EI2,2,0 e 2,2,1 +- 0,0,1_- 1,1,1 _ 1,1,0 . 0,0,2! ! i : i i i :. “ _ _;-c4-. : . : : . -. , . jz._ _. ._1.I *- -:*- “
42、t._- : -_ : ; “ I : : :l e :/m“, : V _ “ : :I I I I 1 I 1 I400 800 1200 1600 2000RFigure 3: Root mean square of maximum u velocity versus R for the “high“ roughness case.dip in amplitude near R = 1000 is caused by a change in phase of 180 degrees in the complexvelocity field v(1,1,1). This change is
43、 due to :r dependent changes in the coefficients of theequations which govern the interaction between the forced mode and the eigenmode. Athigher levels of acoustic forcing the receptivity and parametric regions overlap, eliminatingthis sharp dip in amplitude.The (2,2,1) mode does not lead to fundam
44、ental (i.e. K-type) dynamics. The “porpoising“in amplitude seen in Figure 3 persists even with a four-fold increase in the acoustic amplitude,which increases by an equal amount the amplitude of the traveling modes, but only slightlythe amplitude of the steady modes. The reason for this lack of K-typ
45、e resonance, thus,cannot be explained simply in terms of a threshold amplitude of the (2,2,0) mode. Since K-type resonance involves the triad interaction between the (2,2,0), (2,2,1) and (0,0,1) modes,one may suspect an unfavorable phase relation in the triad to be quenching of the resonance.A four-
46、fold increase in wall mode amplitudes, on the other hand, increases the amplitudeof all modes, including the (0,0,1) mode, and the flow displays a mixed II-type and K-typetransition.Additional insight into the energy transfer between traveling modes and eigenmodes, aswell as phase cancellation givin
47、g rive to the “porpoising“ amplitudes, can be obtained fromthe perturbation theory 5 wherein the forced traveling wave and the eigensolution are keptdistinct. In the PSE formulation one cannot easily separate the sohition into the sum of thetwo components.Lowering the amplitude of the wall undulatio
48、ns to the “low“ level (see table i) resultsin the dynamics shown in Figure 4. Between R = 980 and R = 1200 the subharmonic10Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10-110-210 -_E10-410-5m- 2,2,0 e 2,2,1 + - 0,0,1_- 1,1,1 e- 0,0,0 . 0,0,2: i.-,! ;. ,: i,7.:,“ ._: xi !“ I I I I I I I I400 800 1200 1600 2000RFigure 4: Root mean square of maxinmnl u velocity versus R for the “low“ roughness case.undergoes rapid growth as in fig 3, however the amplitudes of the (2,2,0) an
