[American Institute of Aeronautics and Astronautics 28th Fluid Dynamics Conference - Snowmass...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

Boundary-Layer Transition Prediction ToolkitMujeeb R. Malik*

High Technology CorporationHampton, VA

Abstract

Laminar-turbulent transition on aerodynamic surfacesoccurs between often widely apart limits, namely the upperlimit relevant in extremely low-disturbance environmentsand the lower limit given by the "bypass transition." The eN

method only provides the upper bound on the transitionlocation, while experimental information or direct Navier-Stokes simulations are required to determine the lowerbound. Computational tools (receptivity theory/linearNavier-Stokes, nonlinear parabolized stability equations,etc.) are now available to estimate the departure from theupper bound due to factors such as surface roughness andacoustic disturbances but await validation against experi-ments. In this paper, progress made in boundary-layerstability and transition is reviewed and description is givenfor the computational tools developed to ease the determi-nation of the upper limit via N-factor calculations. Thisincludes a suction optimization code for the efficient designof laminar flow control wings. A brief discussion of someof the physical issues associated with boundary-layer transi-tion is given and some key results, including those pertain-ing to the question of absolute instability, are presented.

1. Introduction

Boundary-layer transition is a subject which at firstsight appears narrow but, in fact, is vast and offers manychallenges for a fluid dynamicist. It has drawn a great dealof attention in the past two decades because of its relevanceto laminar flow control (LFC) technology and aerothermaldesign of aerospace vehicles. Here, progress made in transi-tion prediction methodology during the past two decades isbriefly reviewed. The review is not meant to be exhaustiveand mostly covers the work of the author, his colleagues,and that of other associates, but reference is made to otherworks where appropriate. Some recent more comprehensivereviews on this subject have been given by Reshotko (1994),Arnal et al. (1995) and Reed et al. (1996).

My involvement in the area of stability and transitionbegan in 1979 and I presented my first paper on the subject(Malik and Orszag, 1980) in Snowmass at the 1980 Fluidand Plasma Dynamics Conference. The seventeen yearssince then have seen tremendous progress in various areasassociated with laminar-turbulent transition in boundary-layers. This includes development of accurate and robustmethods for the hydrodynamic stability problem withparticular emphasis on compressible flow (Malik, 1982;Malik et al., 1982; Macaraeg et al., 1988; Malik, 1990),

advances in the understanding of boundary-layer receptivity(Goldstein, 1983, 1985; Ruban, 1985; Kerschen et al., 1990;Choudhari and Streett, 1994; Crouch, 1994; Saric et al.,1994; Denier et al., 1991), discovery of secondary instability(Orszag and Patera, 1983; Herbert, 1983, 1988), applicationof boundary-layer stability computations to complex three-dimensional (3D) flows (cf. lyer, 1996), development ofparabolic and parabolized stability equations (PSE) methods(Hall, 1983; Herbert, 1991; Bertolotti et al., 1992; Chang, etal., 1991; Li and Malik, 1997), and direct numerical simula-tion (DNS) of transition (Wray and Hussaini, 1984; Kleiserand Zang, 1991; Pruett and Chang 1995). These advanceshave encompassed a wide speed regime and have helped usunderstand some of the ways in which stream disturbancesare internalized in the boundary-layers as instability modes[Tollmien-Schlichting (TS), Gortler, crossflow, Mackmodes]; new numerical techniques have enabled accuratedescription of linear and nonlinear growth of these instabili-ties, as well as the process of their breakdown to turbulence.Some numerical simulations have been carried through thetransitional regime all the way to turbulence (Rai and Moin,1993). Experiments (Saric and Thomas, 1984; Kachanovand Levchenko, 1984; Wilkinson and Malik, 1985; Bippes,1991; Wlezien, 1994) have helped a great deal in developingthis understanding as well as validating the theoretical andcomputational tools.

Despite all this advancement and our ability to computereadily receptivity, linear and nonlinear growth of primaryand the ensuing secondary instabilities, one thing hasremained the same during these 17 years: the method ofchoice for boundary-layer transition prediction (to be moreprecise, estimation), at least for high Reynolds numberengineering applications, remains to be the eN method(Smith, 1952; Smith and Gamberoni, 1956; Van Ingen,1956). This is because, except for some carefully controlledexperiments, we generally do not know the disturbance fieldor the "external" forcing(s) which is(are) the root cause oflaminar-turbulent transition. Transition is influenced by anumber of parameters (cf. Bushnell and Malik, 1987;Bushnell, 1990) including stream turbulence (vorticity fluc-tuations, temperature spottiness, noise), particulates, rough-ness, acoustic radiation from other vehicle components and,possibly, electrostatics. This influence can be either at thereceptivity stage or the instability growth stage or both.Various stages of the laminar-turbulent transition have beenwell illustrated by Morkovin (1985). There are, at least, twoknown routes to transition. The one which is better under-stood, and occurs in low-disturbance environments,

President, Associate Fellow AIAA 1Copyright © 1997 by High Technology Corporation. Published by the American Institute of Aeronautics & Astronautics, Inc. with permission.


involves: receptivity (internalization of stream/wall distur-bances in the boundary-layer), linear/nonlinear growth ofboundary-layer instability modes, secondary instabilities,and breakdown to turbulence. The second, which occurs inhigh-disturbance environments, involves "bypass" of theabove linear process. Morkovin (1993) gives a discussionof the various categories of bypasses and their underlyingphysical mechanisms. Of some relevance, in this context, isTrefethern et al.'s (1993) transient or nonmodal growthwhich is an extension of Landahl's (1980) algebraicinstability concept.

Laminar-turbulent transition is a very complex anddelicate phenomena due to the preponderance of parametersthat influence the outcome (e.g., Reshotko, 1976); nonethe-less, its prediction is a deterministic problem. In otherwords, transition location is computable if one knows all thenecessary details about parameters which influence transi-tion in a given case. This is a big if, however, and one isoften forced to estimate the location of transition onset usingapproximate methods or empirical correlations. The e^method only provides one bound on the transition location,i.e., the "extremely" low-disturbance limit. The other limitis the "bypass" transition limit (ReQmin, Morkovin, 1985)which we only empirically know for various flows, althoughDNS can play an important role in determining this limit (cf.Spalart, 1988). These two limits are often wide apart andtransition can occur anywhere in between. Thus, in practice,transition prediction is a sorting out process where one firstdetermines these two limits for a given flow configurationand then determines where it may actually occur dependingupon the sensitivity to surface finish, steps/gaps/waviness,wind tunnel and flight environments, surface temperaturedistribution, pressure gradients, suction-hole distributionand hole Reynolds numbers (for LFC). It is well-knownthat while suction in general stabilizes boundary-layer insta-bilities, strong suction may trip the flow or a particularsuction distribution may be more advantageous than othersbecause of their effects not only on stability but also onreceptivity (suction holes could constitute potential rough-ness sites adversely affecting transition). Parametric sensi-tivity to departure from the upper limit can then be obtainedeither by using empirical correlations [e.g., Mack's (1977)N-factor correlation for effect of "turbulence" on two-dimensional (2D) boundary-layer transition] or throughmore sophisticated tools such as receptivity theory, nonlin-ear PSE, and DNS.

In this paper, we first describe briefly various computa-tional tools used in linear stability analysis (eN) as well asfor nonlinear evolution of disturbances. We then give someexamples of the use of the eN method for estimating transi-tion onset in flows where primary instability modes consti-tute Gortler, TS (first mode) and crossflow disturbances and

also point out instances were these predictions are rendered"useless" due to surface roughness or free-stream noise.The physical mechanisms of transition, particularly incrossflow dominated flows, are also discussed. Other topicsof interest in boundary-layer transition or of potentialconcern to an LFC designer are briefly described.

2. Computational Tools for Studies in Stability andTransition

2.1 Graphical Transition Prediction Toolkit CGTPT™')*

Various boundary-layer and stability codes have beendeveloped for computing 2D and three-dimensional (3D)boundary-layers and the associated instability modes. Theusefulness (and the limitations) of the linear theory or PSE-based eN method are well understood. For a designer usingthe eN method, it is important that the computational toolsare robust and packaged in an easy to use fashion. This isthe goal of the Graphical Transition Prediction Toolkit. Thegraphical toolkit, GTPT™, is designed to provide a conve-nient platform for use of these codes by engineers and anal-ysis of the results from each code. The boundary-layer andstability codes which are all based upon compressible gov-erning equations are described below.

BLQ3D: This is a quasi-3D boundary-layer code whichsolves 2D/axisymmetric and quasi~3D (infinite swept-wing,infinite swept tapered wing with conical similarity)boundary-layer equations. Pressure distribution along onestreamwise cut of the airfoil is required, along with suctiondistribution, if needed, and wall temperature distribution fornon-adiabatic wall cases. The generated solution files aresaved to be used as inputs to the stability codes.

BL3D: This is a fully three-dimensional boundary-layercode developed for finite swept-wings. It requires theprescription of inviscid flow-field and wall suction and tem-perature distribution if needed.

LST3D: Unlike the COSAL code (Malik, 1982), this isa fully three-dimensional linear stability theory (LST) codewhich solves the spatial eigenvalue problem and can treatboth crossflow and TS disturbances under quasi-parallelapproximation. In the quasi-parallel approximation, theeffect of boundary-layer growth is assumed to be of a lowerorder and, hence, ignored. In other words, the mean flow isa function of the wall-normal coordinate y only. If x and zrepresent the coordinates in the plane parallel to the wall andt is time, then a disturbance 0 can be represented as

<t>(x,y,z,t) = <j>(y)e'(ca+^~ox' + complex conjugate (1)

* GTPT is a trademark of High Technology Corporation (HTC).


Where a, ft are the x, z wavenumbers and ft) is the distur-bance frequency. Substituting (1) in linearized Navier-Stokes equations results in homogeneous equations whichalong with homogeneous boundary conditions, constitute aneigenvalue problem described by the complex dispersionrelation

a = a((o, (2)which is solved numerically. For a general wave packet a,ft, and co can all be complex (e.g., see §5.0). For a convec-tive monochromatic disturbance, however, one can usespatial theory where CO is real and the wavenumbers arecomplex. This option is used in LST3D. The following N-factor strategies can then be employed:

1. The saddle-point method: This option requires that thegroup velocity ratio be real, i.e., daj/dftr=Q. Anadditional condition is provided by either requiring thatthe disturbance growth rate be locally maximized or byfixing fir (the real part of ft). The former has been usedby Cebeci and Stewartson (1980) and Cebeci et al.(1988). The latter more physical approach, suggestedby Nayfeh (1980), requires repetitive calculations forvarious f t r . Both these approaches allow the growthrate direction to automatically switch from theattachment-line to the chordwise direction furtherdownstream. It is the attachment-line region, however,where nonparallel effects are strong and are ignored inthis quasi-parallel calculation.

2. Fixed wave-orientation: N-factor calculations are per-formed for a disturbance with fixed wave angle withrespect to the streamline.

3. Fixed wavelength: The dimensional spanwise wave-length of the disturbance is kept constant during the N-factor computation. For an infinite swept-wing, the"spanwise" direction is clearly defined. For a finiteswept tapered wing, however, this could either be thedirection parallel to the leading-edge or someprescribed direction such as conical lines for a wing.

PSE3D: This is a parabolized stability equations (PSE)based code for three-dimensional boundary-layers over infi-nite or finite swept wings. Mean flow variation and surfacecurvature effects can be accounted for by using this code.Hence, this code allows consideration of all the "known"convective instability modes of two- and three-dimensionalboundary-layer flows. The PSE code is also used to studynonlinear evolution of disturbances.

In order to demonstrate the capability of the three-dimensional PSE code, we consider the case of an infinite-swept wing boundary-layer subject to crossflow instability.Wall suction is provided near the leading-edge for laminar

flow control. In order to construct a fully 3D mean flow, weset the suction level to zero in a small area (0.79x1.18m)near x = Ift in Fig. l(a) where the chordwise velocitycomponent of the 3D basic flow, computed by BL3D, isshown. Spanwise variation of the basic flow is evident anda wake region trailing the suction "blocker" is clearlyvisible. Note that the assumed Cp distribution does not varyin the spanwise direction and any spanwise mean flow varia-tion is caused by the suction blocker only. Evolution of astationary crossflow disturbance was computed using the 3DPSE approach as shown in Fig. l(b). The figure shows thecontour plot of N-factors over and in the wake of the suctionblocker. The basic flow pattern is provided as a referenceusing the contour lines (dashed lines) of the chordwisevelocity component at a height of 12% of the boundary-layer thickness. If transition were to occur at a constant N-factor, the transition front will appear as a wedge behind thesuction blocker. In the regions away from the suctionblocker, the infinite-swept wing solution is recovered.

20.3172 39.6096 68.9019 78.1943 97.4667 116.779 136.071 155.364

x(ft)

Fig. l(a). Contour plot of the chordwise velocitycomponent of the basic flow over a "suction blocker" in aninfinite swept-wing.

0.0 0.5 1.0 • 1.5 2.0 2.5 3.0x(fl)

Fig. l(b). Computed N-factors using 3D PSE for stationarycrossflow disturbances. The background (dotted lines) showthe basic flow variation.


2.2 Suction Optimization Code

For practical design, suction optimization is needed tominimize suction system requirements (both the totalsuction and the perforated area). Some early work reportedin open literature is from Reed and Nayfeh (1981) who con-sidered the TS problem. More recent work is reported byBalakumar and Hall (1996). The challenge in suctionoptimization for complex swept-wing flows, as with anyother optimization, is to develop a code which is robustenough to be used by an aircraft designer. Suction optimiza-tion code developed at HTC can utilize either LST3D orPSE3D in the design process and will be incorporated inGTPT™. An example calculation for a swept-tapered wingis shown in Fig. 2 where the "envelope" of all the N-factorcurves with and without suction are shown along with thepredicted suction distribution. The objective function usedin the optimization was to minimize N-factor at xlc = .6 fora given amount of total suction. Other options, e.g., mini-mization of suction for transition (as determined by aprescribed N-factor) at a fixed xlc, have also been incorpo-rated. Figure 2 shows two solutions of the suction optimiza-tion. The first one (broken line) only minimizes the N-factor at the prescribed xlc. Nevertheless, it is seen thatthere is a peak in the N-factor of about 8 at xlc ~ .06. Thisearly peak should be a reason for concern in view of theuncertainties associated with the e^ method of transitionestimation. Therefore, a multipoint optimization schemehas been used to obtain the second solution (solid line)where an additional constraint requiring N < 5 for xlc = .08has been imposed. This solution is better than the formerfrom the viewpoint of LFC design.

without suctionwith optimal suctionoptimal suction (with N<5

atx/c=0.08)

Fig. 2. Two solutions of suction optimization for a sweptwing. The first solution (broken line) only minimizes the N-factor at xlc ~ .6 for a given total suction. An additionalconstraint (N < 5 at xlc = .08) is imposed for the secondsolution (solid line) using same total suction.

2.3 2D Eigenvalue Code

There is a class of problems which is governed byhomogeneous governing equations and boundary conditionsbut, since the mean flow varies strongly in two dimensions,the resulting eigenvalue problem is governed by partialdifferential equations (PDE's). We call this a 2D eigenvalueproblem, in contrast with the "ID" eigenvalue problemgoverned by a system of ordinary differential equations(e.g., the Orr-Sommerfeld equation). In the 2D eigenvalueproblem, one must discretize the PDE's in two spacevariables resulting in a large matrix eigenvalue problem.Examples of physical problems which require the use of 2Deigenvalue techniques include corner layer, attachment-lineboundary-layer and secondary instability studies. Two-dimensional eigenvalue techniques have been developedboth for incompressible and compressible flows.

2.4 Linearized Navier-Stokes

The Linearized Navier-Stokes (LNS) approach has beenused by Streett (1995), and more recently by Guo et al.(1997), for computation of instability waves in 2D and 3Dboundary-layers. In this approach, the solution is assumedto have only one prescribed frequency and, thus, the govern-ing equations are rendered time-independent which can besolved much more efficiently for problems where only atime-asymptotic solution is desired. Of course, it is morecostly to solve this elliptic set of equations than the parabo-lized stability equations but the LNS approach offers variousadvantages over the PSE approach. For example, the LNSapproach can be used in receptivity studies (Choudhari andStreett, 1994) and in flows which are rapidly varying or fortracking multiple instability modes. PSE would tend to failfor such applications. LNS is of particular relevance instudying high Mach number flows at low Reynoldsnumbers.

2.5 A Strategy for Transition Prediction

Direct numerical simulations (DNS) of Navier-Stokesequations have been successfully used to compute somebypass transition cases (Rai and Moin, 1993; Spalart, 1988).Its use in high Reynolds number complex flows, however, isprohibitively expansive for present-day computers particu-larly when transition occurs in a low-disturbance environ-ment. An alternative strategy can be constructed byconsidering various stages of the transition process depictedin Fig. 3.

Once the external disturbance field has been modeled/specified, appropriate computational tools can be used tocarry through the various stages of transition as depicted inFig. 4. This approach should prove more efficient than abrute-force application of DNS. If nonlinear PSE (NPSE) is


used to capture secondary instabilities, the 2D eigenvaluecalculations are not required. A good estimate of transitiononset can be obtained by unsteady NPSE calculations andDNS is only required if one needs to compute through thetransition zone. Pruett and Chang (1995) used combinedNPSE and DNS to compute through the transition zone for aMach 8 boundary-layer.

Initialdisturbance

field— +>

Mode Selection/Receptivity

"Strongly"nonlinear evolutionand breakdown to

turbulence*——

jnear and "weakly"nonlinear growth &

secondaryinstabilities

Fig. 3. Stages in the transition process in low-disturbanceenvironments.

Specify/Modeldisturbance

field— *• LNS

DNS •* ——NPSE

and2D eigenvalue

code

Fig. 4. Computational tools for transition prediction.

When nothing is known about the disturbance field andonly the "upper" bound for the location of onset of transitionis required, the eN method can be employed. In "extremely"low-disturbance environments, this "upper" bound turns outto be quite close to the actual location of transition onset.Some such examples are provided in the next section.

3. Some Practical Applications

3.1 Gortler and First Mode Disturbances: The Case of the"Quiet" Tunnel

In order to appreciate the level of care needed to maketransition "predictable" by methods such as the eN method,we consider the Langley Mach 3.5 low-disturbance ("quiet")tunnel, Fig 5. Several filters have been installed in the airsupply system and the settling chamber to remove moisture/particulates and to reduce acoustic disturbance level(Beckwith and Bushnell, 1988). As a result the rms pressurefluctuation in the settling chamber reduces to about .005percent of stagnation pressure and particulate size is

dropped to less than \\irn. A honeycomb and several turbu-lence screens are used in the settling chamber to controlvorticity fluctuations. These fluctuations are further reducedbecause of the large velocity expansion ratios in supersonic,as well as hypersonic, nozzles. Boundary-layer removalslots, which remove the turbulent boundary-layers develop-ing along the settling chamber walls, are provided upstreamof the nozzle throat. Therefore, new laminar boundary-layers develop along the four nozzle walls. The nozzleshown is of rectangular cross-section with two contouredwalls and two flat walls (the side walls). The flat side wallsbecome transitional in the throat region via crossflow insta-bility due to the crossflow induced by the curvature of theinviscid streamlines which in turn is induced by thecontoured walls. In addition, vortices present in the fourcorners become highly unstable and contribute anotherpossible cause of transition in the corner layers. Noiseradiation from the side walls to the center region of the testsection is not a factor for rapid expansion nozzles withrelatively large width-to-height ratio. Noise radiated fromthe turbulent boundary-layers along the contoured walls isthe main limiting factor for the length of the "quiet" testcore. Transition along the contoured walls results due toamplification of Gortler vortices and eN calculations (Chenet al., 1985) indicated that the transition onset location couldbe well-correlated with an N-factor of about 9-11 for arange of unit Reynolds numbers (3xl06/ft— 18xl06/ft). Ofcourse, this could only be achieved if nozzle surfaces werewell polished and the roughness Reynolds number R^ (R^ =Uk k/vk where k is the height of the roughness, t/& thestreamwise velocity at the height k in the absence of rough-ness and Vjt the kinematic viscosity at that location) valuesin the throat region were maintained to be less than 12 (thecritical value of /?£ for low-speed zero-pressure gradientcase is about 400) which corresponded to a maximum peak-to-valley defect of .5 micron (Beckwith and Bushnell,1988). With such a care, it was possible to maintain laminarboundary-layers on the nozzle walls and as a result, theradiated normalized rms pressure fluctuations on the nozzlecenterline were measured to be no more than .05 percent. Asharp cone model placed in this low-disturbance test coreyields transition Reynolds numbers of 7xl06 to 9xl06 (Fig.6), depending upon unit Reynolds number and surfacefinish, which compares well with adiabatic flight data(Beckwith and Bushnell, 1988; Beckwith et al., 1983).

When the nozzle wall boundary-layers are turbulent, theradiated noise level in the test section is increased by nearlytwo-orders of magnitude and the transition Reynoldsnumber on the cone models is decreased and falls into therange of that found in conventional wind tunnels. A moredramatic shift in the transition Reynolds number due tonoise occurs in the case of flat-plate boundary-layer (Fig. 7)such that the ratio of cone/flat-plate transition Reynolds


number switches from about .65 to 2.5 due to the noise. The"low noise" transition Reynolds number are well correlatedby the eN method using N ~ 10 both for the case of flat-plateand cone. No method currently exists for predicting theeffect of noise on transition; transition in conventionalsupersonic nozzles is not predictable by currently availableand exercised computational techniques.

X Isolation valvesressure regulator valves

^-Stainless-steel pipe

Exhaust tovacuum spheresor atmosphere

(a) Schematic of air supply system.

Transition

Flow

Radiatednoise

Dim. In.Exit 6 x 10Ax 5 to 10Ay 1.5 to 3.0Az 7.5

(b) Quiet test core size and shapeFig. 5. Langley Mach 3.5 "quiet" tunnel. The figure givesan idea about the care needed to yield boundary-layertransition on the nozzle wall predictable by the eN method.

108

Re,T 6

4

2-

Xs. cm Noise

D 12.7 Lowest* 20.3 LowA 12.7 Highv 20.3 Highest

Flight data -̂ .Me = 1.6to 1.8 j^

Conventional windtunnel data

M e = 2.5 to 4.4

Flight dataMe =1.4 to 4.6

106 8

107Re/m

6 8 108

Fig. 6. Measured transition Reynolds numbers on sharpcones (quiet and conventional tunnels).

Modelo Conea Flat plate

—— Flat plate- - Flat plate

Mach no.3.513.5 i3.03.7

Wind tunnelLaRC low noise

AEDCJPL20in.

2r Flat plate prediction

107

86

ReT4

2

106

jy u-

——— (JP CPQ%- —————

Cone prediction/ ̂ ^QoforN = 10 -/

,-Tunnel A (40"x40")^-— "/—JPL20"

1 X J-**1*' 1 1 T 1

105 2 4 6 8106 2Re/in.

Fig. 7. Comparison of transition onset Reynolds numberson cone and flat-plate in "quiet" and "noisy" tunnels.

In the above example, even though the forward parts ofthe cone and flat-plate models were in the low-noise region,transition actually occurs in the high-incident noise region.It was therefore of interest to extend the length of the"quiet" test core so that transition on aerodynamic modelscan be obtained in a completely low-noise environment.This required delaying transition on the nozzle walls, i.e.,controlling Gortler instability. Gortler instability, being acurvature-induced instability, is difficult to control with wallsuction or cooling. The latter control, in fact, tends todestabilize this instability and could also aggravate theroughness sensitivity due to the thinning of the boundary-layer. Crossflow (a swept-nozzle!) can tend to stabilizeGortler vortices (Hall, 1985), but then crossflow instabilitycould introduce its own associated instability which isfurther enhanced by concave curvature (Zurigat and Malik,1995). Therefore, the only effective way to control Gortlerinstability is to reduce the concave curvature. Beckwith etal. (1988) designed a new long slow expansion nozzle whichcould yield much longer quiet test core resulting in higherRe fa. The longer the nozzle, however, the less the width ofthe quiet test rhombus due to the noise radiated from theside walls and corner vortices. Hence, even though a long,quiet test section can be designed considering the contouredwalls only, its width becomes so small that such a designbecomes unattractive. Therefore, the concept was applied todesign an axisymmetric nozzle and such a nozzle was fabri-cated and tested (Chen et al., 1990).

Figure 8 shows R^ (Reynolds number based on thelength of the quiet test case) from test data for two rapidexpansion (R.E.) pilot quiet nozzles and the new AdvancedMach 3.5 axisymmetric quiet nozzle (Adv.) over the unitReynolds number range tested. The predicted (using GortlerN = 9) values of R^ are included for comparison. For eachnozzle, the measured maximum surface roughness, k, in the


throat region is also listed to show the effect of surfacefinish on the performance. The data for the pilot nozzle andthe advanced nozzle indicate an increasing trend of R^x withunit Reynolds number, R^, except for the large values of k.This unit Reynolds number effect is believed to be causedby the increasing local favorable pressure gradients, thatsuppresses the growth of Gortler vortices, as transitionmoves upstream along the nozzle wall with increasing R^.The maximum value of R^ obtained from the advancednozzle is about 1.4xl07. Except for the problem of surfacefinish, values of R^ for the advanced nozzle (designedusing NG = 9) are more than double the value of the pilotquiet nozzles. It is unfortunate that this nozzle was neverused to test cone models to see if the transition Reynoldsnumber remains to be 9xl06 (the highest obtained in thepilot tunnels) or if higher numbers are achieved. It may benoted that the Gortler N-factor calculations given in thefigure were performed using the quasi-parallel approach.Recent calculations performed at HTC using PSE show thatnonparallel effect is small for the conditions relevant forthese nozzles (see Fig. 9).

4

2

'"'a6

tx 4

2

106

6

I S- /!& %p '- & QOO^ o

jfik ^3 *-*y o ooA

f Q A- O -*•" l l l l l l I A I !? M III

4 6 8 , 2 4 6 8 f l107 108

tech Nozzleno._ type33.53.53.53.53.53.53.5

Axis2-D2-D2-D2-DAxisAxisAxis

R.E.R.E.R.E.R.E.R.E.Adv.Adv.Adv.

MaxkUrn-.52-5 \Original1 .0 / blocks5-01 New0.5 J blocks

2.52.0(Theory torGortler N = 9)

Fig. 8. Quiet test core Reynolds numbers in the advancedMach 3.5 axisymmetric quiet nozzle (Adv.) and pilot quietnozzles (Rapid Expansion, R.E.).

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Fig. 9. Comparison of quasi-parallel and PSE results forGortler instability in the Mach 3.5 nozzle.

The effect of surface finish (in the throat region) can beclearly seen (in Fig. 8). With k = 2.5/Jm, laminar flow is lostfor Roalm > 2.2; when k - 2/J.m, laminar flow can beextended up to about 3.5xlO^//w and beyond that it suddenlydisappears. The /fy = 12 criterion (Beckwith et al., 1988)indicates that the maximum value of k should be less than1.2/Jtn in order to achieve the design values of R^ at highervalues of /?„. This example demonstrates how "lots" oflaminar flow is entirely lost by very "little" roughnessmaking the prediction of the eN method completely useless.Similarly, the effect of noise on the first mode is evident inFig. 7.

3.2 Crossflow Mode: The Case of ASU Experiment

The influence of micron-sized roughness on crossflow-induced transition has been clearly brought out in theArizona State University (ASU) experiment (Radeztsky etal., 1993) where transition location is influenced by rough-ness as small as .5/MI (R^ ~ .001), provided roughness islocated near the leading edge. There are, however, twodifferences between this case and the R^ = 12 criterion forthe "quiet" nozzle. In the swept-wing case, roughness has adirect "input" into the stationary crossflow instability andthe roughness with the quoted values of 7?^ of up to about4.5 moves the location of transition but does not trip theboundary-layer. In the case of the nozzle, roughness islocated in the throat region where boundary-layer is stable(linearly) to centrifugal instability and therefore any station-ary perturbation has a large region of damping before theybegin to amplify in the downstream concave curvatureregion. The boundary-layer in the throat region is alsostable to TS instability owing to the favorable pressuregradients. Therefore, the loss of laminar flow with a rough-ness of Rk ~ 12 is a complete "bypass" of the usual transi-tion process. This should serve as a challenge problem forDNS of transition or, perhaps, an opportunity to demonstratethe usefulness of the transient growth approach (Henningsonet al., 1995).

Figure 10 shows N-factor calculations using nonlinearPSE for the ASU experiment (a = -4°, Rec = 2.4xlO6).Calculations are shown for the most amplified stationarydisturbance with span wise wavelength of 12mm, with threedifferent initial amplitudes. When the leading edge waspolished to yield surface finish of .25/M, transition occurredat xlc ~ .1 for a slightly higher Rec of 2.6x 106 (Radeztsky etal., 1993). At this location, the linear N-factor for stationarydisturbances is about 8.8 (it will be somewhat higher forRec =2.6xl06 and even higher if traveling disturbancesare considered). Thus, linear theory does well in predictingthis crossflow-induced transition as it does in the case ofGortler-induced transition (Fig. 8) and oblique first mode-induced transition (Fig. 7).


Fig. 10. Effect of initial amplitude on the growth ofstationary crossflow disturbance with /^ = 12mm for theASU wing (nonlinear PSE).

It is clear from the above example that the crossflowcase is no exception to the usefulness of the linear theorybased eN method for estimation of transition. As notedabove, the e^ method provides an "upper limit" on thetransition Reynolds number which can be achieved withextreme care. It first appears quite surprising that a rough-ness of height .5/Jm (Re^- .001) begins to influence transi-tion location, but the fact that a roughness directly "inputs"into the stationary crossflow disturbance explains thisextreme sensitivity. If one assumes that amplitudes ofO(20%) (see §4.2) are needed for crossflow-induced transi-tion and that e9 amplification precedes this stage, it resultsin the initial disturbance amplitude of about .0025 percent.While receptivity calculations are not yet available, it is notunlikely that a wall perturbation of .5/J.m in a .8mm thickboundary-layer would produce a mean flow disturbance of.0025 percent; a 9f^m [the painted surface case of Dagenhartet al. (1989)] disturbance should produce a much biggerresponse. Therefore, the crossflow problem is much moresensitive to the roughness particularly when it is locatednear the critical point (onset of instability). The sameshould hold true for stationary Gortler vortices.

3.3 Crossflow in the Supersonic Regime: Mach 3.5 Delta-Wing Model

Experiments in the Langley Mach 3.5 "quiet" tunnelwere performed on a 15-inch long symmetric delta wingmodel (leading-edge sweep = 77.1°) with (Cattafesta, 1996)and without (Cattafesta et al., 1994) wall suction. Figure 11shows the transition front measured on the suction modelusing temperature sensitive paint (TSP) technique. Linearstability calculations were also performed and the values ofthe N-factor which cross the transition front lie between 8 to10.5. Earlier calculations for the solid model (with nosuction holes) indicated that an N-factor of about 14("envelope" approach) correlates the transition onset front

quite well over a range of Reynolds numbers and angles ofattack. Thus the N-factors at transition are lower in the caseof suction due probably to the adverse effect of suctionholes which induce roughness effect and enhancedreceptivity.

Transition Front(TSP)

N=10.5

X(ft)

Fig. 11. Measured transition front using TSP on the suctionmodel (Moo = 3.5), along with some N-factor results.

Constant N-factor contours (both with and withoutsuction) indicated that the transition front is parallel to theleading-edge on one side and to the line of symmetry on theother and that it turns around in the middle. Earlier experi-ments (Cattafesta et al., 1994) using the thermocoupletechnique for transition detection did not support the calcu-lations as it indicated that the transition front on the insidewas not parallel to the line of symmetry, rather perpendicu-lar to it. TSP results, however, turned out to be in agree-ment with the eN calculations.

The observed and computed transition front can also bediscerned from the contours of crossflow Reynolds numbersRcf (Rcf =|wmax|5j Iv, the usual definition used in low-speed experiments); however, the values of R^ are muchhigher (-600) than those found in the low-speed experi-ments. This was also found to be the case in the experi-ments of King (1991) on a cone at incidence in the sametunnel. Malik et al. (1991) performed computations for aMach 8 cone at incidence and found that at the locationswhere the N-factor reaches 10, crossflow Reynolds numberreaches values of about 2000. Malik et al. (1991) suggestedthat the higher values of crossflow Reynolds number resultdue to ~M^ growth in the boundary-layer thickness (i.e., thecompressibility effect) which can be scaled out by defininga new crossflow Reynolds number (see also Malik andBalakumar, 1992) Rcf where

Rcf ='R.cf

1 + 7-1

The value of Rcf at transition (measured or predicted usingN ~ 10) was found to lie in the neighborhood of 200 (i.e.,same range as the low speed case) for low supersonic aswell as hypersonic flows. Reed and Haynes (1994)


presented a more formal derivation of the compressibilityeffect and also accounted for nonadiabatic wall temperature.

3.4 The Attachment-Line Boundarv-Laver

The attachment-line boundary-layer on a swept-wingrequires careful consideration since, in general, when thisboundary-layer becomes turbulent, laminar flow is lost onthe entire wing. It was found by Hall et al. (1984) that theattachment-line is subject to small-amplitude travelingdisturbances above a Reynolds number, R, of 583. Thefrequencies of these viscous TS-like disturbances agreedwell with the experimental results of Pfenninger and Bacon(1969) and Poll (1979,1980). More recently, Lin and Malik(1996) used the 2D eigenvalue approach to solve for theattachment-line instability. Their results agreed with theHMP mode found by Hall, Malik and Poll (1984), but thisnew approach allowed them to discover additional instabil-ity modes and to account for leading-edge curvature,ignored by Hall et al. This effect was found to be small (Linand Malik, 1997) for practical subsonic wings. N-factorcalculations performed for the HMP mode correlates wellwith the available experimental data (Fig. 12). Lin andMalik (1995) also considered the supersonic attachment-lineproblem using the 2D eigenvalue approach and found thenonparallel effect to be significantly destabilizing becausethe critical Reynolds number decreased from R - 573 toR = 349.

450

400

350

300

T250

200

150

100

50

OV.o.xp Experiments (see.Poll, 1979).. . : i. 2D stability theory • !

8 104 2

s/e8 105

Fig. 12. Comparison of e^ results (HMP mode) andexperiments for the attachment-line boundary-layer.

Experiments (Pfenninger and Bacon, 1969; Poll,1979,1980) have shown that the attachment-line boundary-layer can sustain turbulent flow at Reynolds numbers muchbelow the critical value of 583 quoted above. Large ampli-tude disturbances are known to damp out only at R < 245[e.g., Poll (1980)]. Hall and Malik (1986) used the weaklynonlinear theory, as well as direct numerical simulations, to

show that the attachment-line boundary-layer is subject tosubcritical instability which allows large amplitude distur-bances to amplify above a critical Reynolds number. Thevalue of this critical Reynolds number was found to beabout 535 below which all large disturbances pertaining tothe HMP mode were damped. Later calculations byJimenez et al. (1990) failed to find the subcritical instability,but the recent direct Navier-Stokes simulations by Joslin(1994,1995) supported the conclusion drawn by Hall andMalik (1986). In any case, neither the results of Hall andMalik nor those of Joslin fill the wide gap between 583 and245. DNS results of Spalart (1988) showed that the turbu-lence could not be sustained in the attachment-line bound-ary-layer below R < 245, thus agreeing with the experi-ments. He was further able to show that the "turbulent"boundary-layer at R > 245 could be "laminarized" withwall suction; this practically significant result has also beenobtained experimentally by Poll and Danks (1995). Theirexperimental results showed that the boundary-layer couldbe laminarized up to R of 600 with sufficient wall suction.

3.5 Streamline Curvature Mode

Malik and Poll (1985) used orthogonal curvilinearcoordinates to study the effect of (in-plane) streamlinecurvature on crossflow instability. Their results were latercorrected in Malik and Balakumar (1993). Itoh (1994),using similar streamline-aligned coordinates, showed thatin-plane streamline curvature introduced instability of itsown. He later formulated (Itoh, 1996) the same problem in"natural" coordinates for the swept leading-edge flow andfound the same mode of instability. Guo et al. (1997) usedLNS to find instability in the parameter regime where Itohfound the "streamline curvature" mode. It seems that thisinstability cannot be captured by the method of multiple-scales as a perturbation solution to the Orr-Sommerfeldequation to account for nonparallel effects but PSE shouldbe able to capture it. In any case, the growth rate for thismode is smaller than the usual crossflow mode, so an eN

calculation will consider this mode to be insignificant exceptthat it has been found (Guo et al., 1997) that this curvature-induced instability is much less sensitive to wall suction ascompared to the crossflow instability. Therefore, dependingupon flow parameters, this mode deserves careful attentionin LFC design using wall suction.

3.6 Corner Flow Instability

The corner viscous layer is relevant in various aerody-namic configurations such as wing body junction flows,engine inlets and wind tunnels of rectangular or squarecross-section. Experiments by Zamir (1981) indicated thatthe viscous layer in a streamwise corner, formed by theintersection of two semi-infinite flat plates at 90° to each


other, becomes unstable at a Reynolds number of about 10^as compared to the critical Reynolds number ( UQX I v) ofabout 9x 104 for the Blasius flow. Balachandar and Malik(1995) performed an inviscid instability analysis of the simi-larity mean flow based on the formulation of Rubin andGrossman (1971). Their 2D eigenvalue analysis showedthat the corner layer is inviscidly unstable. If this inviscidinstability were to persist at small Reynolds numbers, thiswould corroborate with Zamir's experiments. Lin et al.(1996), however, performed viscous stability analysis of themean flow computed by using steady Navier-Stokes equa-tions. This analysis showed that the inviscid mode is stableat low Reynolds numbers. The corner layer was found to beunstable to viscous instabilty but the critical Reynoldsnumber was no lower than that for the Blasius case. Atpresent, no clear explanation is available for the discrepancybetween the experiment and the computations. It is likelythat the experimental observation was related to some sortof nonlinear phenomenon or that the experiments were notcarefully performed and need to be repeated.

Lin et al. (1996a) and Wang et al. (1997) have alsoperformed 2D eigenvalue calculations for corner flow in asupersonic nozzle of square cross-section. For this flow, itis found that the corner layer develops vortical structureswhich are highly unstable.

3.7 Wavv Wall

Although modern manufacturing techniques canprovide smooth surfaces that are compatible with laminarflow, manufacturing tolerance criteria are needed forunavoidable surface imperfections. These imperfectionsinclude waviness, bulges, steps and gaps as well as three-dimensional roughness elements. Experimental studieswhich attempt to provide these criteria for humps and wavywalls are those of Page (1943) and Carmichael andPfenninger(1959).

Wie and Malik (1997) used interacting boundary-layer(IBL) and linear PSE to provide such a criterion for transi-tion in 2D boundary-layers. They computed AN (increase inthe N-factor due to waviness) for various parameters (e.g.,see Fig. 13) and obtained a correlation for the change in N-factor due to waviness, AW, as

nk2Re

where n is the number of waves, k the total wave height, Athe wavelength and Re the unit Reynolds number. Theconstant c depends upon the stream pressure gradient andhas a value of .14 for waviness on a flat plate. The resultsshowed that the critical size of waviness below which wavi-ness has no influence on TS instability does not exist, pro-

vided that the waviness is located on the right of the lowerbranch of the neutral curve.

x(ft)

Fig. 13. Effect of number of waves on the N-factor in awavy-wall boundary-layer.

Calculations based on IBL and quasi-parallel stabilitywere earlier made by Nayfeh et al. (1988), Cebeci and Egan(1988) and Masad and lyer (1994) for a hump and by Masadand Nayfeh (1994) for backward- and forward-facing steps.

3.8 Hypersonic Boundary-Layer Transition

The pioneering work of Mack (1969,1984) and Kendall(1967, 1975) laid out the foundation for hypersonic bound-ary-layer stability and transition. Both the theory and exper-iments showed the existence of second-mode disturbances.The acoustic second-mode disturbance, now called theMack mode, was later verified in a series of experiments byStetson et al. (1983, 1984). This mode which is destabilizedby wall cooling, becomes dominant at a Mach number ofabout 4. Calculations by Malik (1987,1989), however,showed that, in an eN sense, the switch between the first andsecond mode takes place at a Mach number of about 7 foradiabatic wall conditions. For cold walls, this switch overoccurs at lower Mach numbers since wall cooling stabilizesthe first mode and destabilizes the second mode. Recentexperiments (Lachowicz et al., 1996) performed in a Mach 6"quiet" tunnel, designed using the concept described inBeckwith et al. (1988), provided support for the predictedeffect of adverse pressure-gradient on second-mode distur-bances. Figure 14 [from Balakumar and Malik (1994)]shows that the N-factor at transition onset is about 8.5 forthe experiment performed on a cone flare model in this tun-nel. The predicted most amplified frequency (~220kHz) wasalso observed to be the most amplified frequency in theexperiments of Lachowicz et al. (1996).

An important issue in hypersonic transition is the effectof nose bluntness. Calculations by Malik et al. (1990)

10


agreed with the experimental findings of Stetson et al.(1984) in that small nose bluntness stabilizes the boundary-layer. The critical Reynolds number increased by almost afactor of 15 due to bluntness (for the nose Reynolds numberof 31250), although the predicted (N = 10) transitionReynolds number only increased by about 25%. The ratioof "blunt" to "sharp" transition Reynolds number increaseswith decreasing N-factor (i.e., increasing free-stream distur-bance). Some experiments indicate that the trend switches(stabilizing to destabilizing) for large bluntness, but no satis-factory theoretical explanation is as yet available.

WALL TEMP (°F)

275

M = 6 T0 = 810°R P0 = 130psi.

MEASURED WALL TEMPERATURE

\

Fig. 14. Comparison between measured and computed(laminar) wall temperatures and the second mode N-factorcalculation for the most amplified frequency (f = 220kHz)for a cone-flare model.

The effect of real gas has been studied by Malik(1989a,1990), Malik and Anderson (1991), Stuckert andReed (1994) and recently by Hudson et al. (1996). Malik etal. (1989) performed stability calculations using equilibriumassumption for re-entry F cone (Wright and Zoby, 1977) andobtained an N-factor of 7.5 at the initiation of transition forthis Mach 20 flight experiment. One issue that becameprominent during the National Aerospace Plane (NASP)transition program was the role of supersonic modes inhypersonic transition (Malik and Macaraeg, 1993). Thedominant second-mode disturbances in the perfect gasexamples cited above are subsonic, i.e., their phase speedcr > 1 -\/Me . Supersonic modes with cr<\-l/Me havebeen discussed by Mack (1969,1984), but are known to beeither stable or only weakly unstable relative to the subsonicmode in wall-bounded flows (for supersonic mixing-layersand jets, the significance of the supersonic modes is well-known, cf. Tarn and Hu, 1989; Macaraeg and Streett, 1989,1991; Malik and Chang, 1997). Stability calculations usingchemically reacting equilibrium gas assumption showed thatsupersonic modes gain a new significance in the eN sense.Predicted (N - 10) transition location was several feet aparton a hypersonic body, depending upon the perfect and equi-librium gas assumption. Recently, Chang et al. (1997) have

used PSE to study the effect of finite-rate chemistry andchemical equilibrium. For an example calculation (with Mx= 20), they find that N-factor results using finite-rate chem-istry lie in between the results obtained by perfect gas andequilibrium gas assumptions. In such a situation, one mustemploy finite-rate chemistry for transition estimation sincethe error introduced by using equilibrium gas assumption (orthe perfect gas) is rather large.

4. Transition Mechanisms

No attention is paid to actual transition mechanismswhen eN is used for estimation of location of transition.This is true of the low-speed TS case as well as the Gortler,first-mode, second-mode and crossflow cases discussedabove. While significant progress has been made in under-standing the breakdown mechanisms in two- and three-dimensional boundary-layers, the use of this capability todate in transition prediction has been limited due to the lackof precise knowledge of the initial conditions. This situationwill change as progress is made towards developingappropriate models for the environmental disturbances andtheir imprint in the boundary-layer. Carefully performedreceptivity experiments with detailed characterization ofenvironmental disturbances are in dire need.

Here, we briefly review some of the transition mecha-nisms relevant in 2D and 3D boundary-layers. These,coupled with the yet to obtain receptivity information,would form the basis for a more rational approach for transi-tion prediction.

4.1 TS and Mack Modes

There are various breakdown mechanisms for the low-speed TS case and the most obvious ones are:

1. Fundamental resonance or Klebanoff-type;2. Subharmonic resonance or Craik- or Herbert-type;3. Combination resonance;4. Oblique mode interaction.

These mechanisms have been discussed by Craik (1971),Herbert (1988), Coarke and Mangano (1989), Kachanov etal. (1984) and Henningson et al. (1995).

Compressible stability calculations for the cone andflat-plate suggest that transition occurs due to oblique first-mode disturbances. No experiments have been performed inthe "quiet" tunnel to actually understand the transitionmechanism or even to see if first-mode disturbances arenaturally present. Other experiments performed in noisytunnels do show the presence of oblique modes but theywere excited by surface glow discharge (Kosinov et al.,1990). Naturally occurring second-mode disturbances have

11


been observed in the Mach 6 "quiet" tunnel (Lachowicz etal., 1996).

The most likely scenario for the actual mechanism oftransition for the flat-plate/cone case is the oblique-modebreakdown (Bestek et al., 1992; Chang and Malik, 1994). Inthis case, a pair of oblique modes interact to produce astream wise vortex. The mutual and self-interaction of thestreamwise vortex and the oblique modes results in the rapidgrowth of other harmonic waves and transition soonfollows. The r.m.s. amplitude of the streamwise velocitycomponent is found to be on the order of 4%—5% at thetransition onset location marked by the rise in mean wallshear. Chang and Malik found that, for this mechanism,initial amplitude of the oblique modes has to be no morethan -.001% for transition Reynolds numbers observed in"quiet" tunnels. Chang and Malik (1993) found that thesame oblique mode mechanism is operative in hypersonicboundary-layers.

Kosinov et al. (1994) found subharmonic resonance intheir flat-plate experiment at Mach 2. Nonlinear PSE allowsone to efficiently construct various scenarios for transition,provided initial conditions are prescribed.

4.2 Pseudo-Saturation of Crossflow Disturbances

Malik et al. (1994) used nonlinear PSE to study theevolution of crossflow disturbances in a model 3Dboundary-layer. They observed that nonlinearity suppressesthe fundamental whose growth rate eventually begins tooscillate about a small value. This saturation or pseudo-saturation was earlier observed by Malik (1986) for rotating-disk flow and by Meyer and Kleiser (1988) for the Falkner-Skan-Cooke boundary-layer. One of the solutions obtainedby Gajjar (1996) also shows oscillatory behavior for thenonlinear crossflow vortex. Clearly secondary instabilitieswould soon destroy this pseudo-equilibrium state resultingin loss of laminar flow. For the model boundary-layer,Malik et al. (1994) found that the nonlinear growth ratebegins to depart from the linear growth rate when the cross-flow disturbance amplitude reaches about 4%. In Fig. 10,for the ASU wing, the amplitude of the disturbance at x/c =.45 (where nonlinear and linear calculations begin to departfor MO = .0001) is found to be 4.13%.

Figure 15 shows nonlinear PSE results with threedifferent initial amplitudes to compare with the experimentsof Reibert (1996). In all three cases, the amplitude of thefundamental reaches the range of 17-18% at x/c ~ .5 (wheretransition is observed in the experiment). Secondary insta-bility calculations, of the type described in §4.3 below, areunderway. It is hoped that these calculations will shed somelight on the apparent insensitivity of transition location to

the change in roughness height k which varied from 6/zw to

0.15

Amp.

0.10

Comp. Exp. k

Fig. 15. Nonlinear evolution of crossflow disturbances(fundamental mode) and comparison with experiments(Reibert, 1996) using three different roughness heights.

4.3 High-Frequency Secondary Instability

There are several possible scenarios for transition incrossflow-dominated 3D boundary-layers, particularly onswept wings. The one in which the "quasi-saturated"stationary crossflow disturbances develop high-frequencysecondary instability was observed by Kohama et al. (1991)in the ASU experiment and computed for a modelboundary-layer by Malik et al. (1994). The more recentexperimental work can be found in Kohama et al. (1996).Earlier, this same phenomena was found by Kohama (1984)and theoretically computed by Balachandar et al. (1992) forrotating disk flow. Poll (1985) observed a disturbance with11.5kHz frequency in his swept-cylinder experiment.Secondary instability computations by Malik et al. (1996)showed that the disturbance with the highest growth rate hasa frequency of about 17kHz, thus agreeing with Poll's result.

Computational results by Malik et al. (1996) show thatthe modulated mean flow is subject to two types ofsecondary modes where one mode correlates with thespanwise shear and the other with the vertical shear. This issimilar to the case in the Gortler vortex problem (Li andMalik, 1995) where these modes are referred to as sinuousand varicose (horse-show vortex) modes (Swearingen andBlackwelder, 1987; Hall and Horseman, 1991). Thus, apartfrom the role of traveling disturbances, mechanisms forbreakdown of Gortler and crossflow vortices are quitesimilar (Malik and Li, 1993). For the supersonic nozzlewhere Gortler instability is present, transition likely occursvia a similar secondary instability mechanism.

12


4.4 Traveling and Stationary Disturbance Interactions

Although no direct measurements were made,Cattafesta et al. (1994) suggested that traveling disturbancesmust have been dominant in his Mach 3.5 delta-wing. Therole of traveling vs. stationary disturbances has been a sub-ject of some controversy in low-speed experiments but nowit is widely accepted that this is a receptivity issue pertainingto the relative importance of surface finish and the environ-mental disturbances (turbulence, etc.). Thus, stationarydisturbances are relevant on a wing with "roughness" butotherwise low-disturbance environment (e.g., Dagenhart etal., 1989) while traveling disturbances would dominate in anexperiment performed in relatively high turbulence envi-ronment (cf. Muller and Bippes, 1988). While Poll (1985)observed both stationary and traveling disturbances, Takagiand Itoh (1994) only observed traveling disturbances whenthe experiment was performed on a "large" diameter sweptcylinder, diminishing the roughness effect.

A simple case of the stationary/traveling wave interac-tion was studied by Malik et al. (1994). When the initialamplitude of stationary vortex is large compared to thetraveling mode, the stationary vortex dominates most of thedownstream development. Eventually, however, the travel-ing mode becomes of the same order as the stationary mode.The situation changes when the initial amplitudes of thetraveling and the stationary modes are the same. Owing toits higher growth rate, the traveling mode dominates most ofthe downstream development and the growth of the station-ary mode is suppressed. Some of the features observed byMueller and Bippes (1988) were captured qualitatively bythe results computed by Malik et al. (1994).

The rotating-disk experiment of Corke and Knasiak(1996) suggested a triad resonance between traveling modesof two frequencies and stationary modes. This resonanceled to "difference" interactions providing energy to lowwavenumber stationary modes which became dominant neartransition to turbulence. Steady and traveling wave interac-tions have also been studied by Herbert and Schrauf (1996)in a swept-wing boundary-layer. Clearly, nonlinear PSEprovides an efficient tool for studying such interactionsprovided initial amplitudes can be prescribed from receptiv-ity studies.

5.0 Spatio-Temporal Growth and Absolute Instability

We have so far only considered convective instabilities,i.e., those for which an impulse response would decay tozero for large time since the induced instability is "swept-away" from the source with its group velocity. The spatiallinear stability theory, PSE and the harmonic LNS of §2.4can be used to describe the behavior of such instabilities at

all points. On the other hand, if the impulse response isunbounded for large time, then the flow is called absolutelyunstable. In this case, the energy is trapped in the neighbor-hood of the source and grows in time. A stability theorycapable of describing space-time evolution of instabilities(or time-dependent LNS or DNS) must be used to describeit.

Using complex Fourier and Laplace transform tech-niques, Briggs (1964) and Bers (1975) derived generalconditions for convective and absolute instabilities. If a and0) are the disturbance wavenumber and frequency, respec-tively, then it is required that a saddle point singularitybetween two spatial branches of the dispersion relation mustexist in the a plane for Im(co)>0. Such a singularity isknown as a "pinch point." The "group velocity" deal da = 0at such a point. If the saddle point results due to thecollision of two spatial branches originating in the same halfof the a-plane, then it constitutes a second-order poleresulting only in transient growth. For the pinch point toresult in absolute instability, the collision must occurbetween two spatial branches originating in two distincthalves of the a-plane.

Absolute instability has been studied, for example, byHuerre and Monkewitz (1985) in free-shear layers and byKoch (1985) for wake flows; Huerre and Monkewitz (1990)provide a review on the subject. The results of Balakumarand Malik (1990) for rotating-disk flow provided a hint thatthis flow might be absolutely unstable but this possibilitywas not explored because it appeared that the instabilitymight occur at too high a Reynolds number. Recently, how-ever, Lingwood (1995, 1996) showed that this boundary-layer becomes absolutely unstable at a Reynolds number of507, i.e., just below the Reynolds numbers where disk flowbecomes transitional. Lingwood (1996a) then performed anexperiment in an attempt to show that this instability can berealized in an experimental set up. While wall roughnessmay dominate the crossflow problem resulting in the transi-tion mechanism mentioned in §4.3, the result of Lingwood(1995) has opened a new dimension to the boundary-layertransition prediction issue particularly for ultra-smoothsurfaces. Therefore, the possibility of absolute instabilityneeds to be investigated for other boundary-layers.

Lin et al. (1997) recently studied the stability of the 3Dboundary-layer formed on an infinite-swept cylinder. Theproblem they considered was of an impulse response of aspanwise periodic disturbance with real wavenumber ft.They were able to find pinch points in the chordwise planefor this flow near the leading-edge at Reynolds numbersslightly below the Reynolds number at which theattachment-line becomes unstable to infinitesimal distur-bances.

13


Figure 16 shows some results from their calculation.Figure 16(a)_shows the neutral^ curves for absolute instabilityfor various R. The critical R for the 60° swept cylinder isfound to be 540.5. Figure 16(b) shows the two spatialbranches at R = 632 in the a-plane. At the pinch point(denoted by the intersection of curves C+ and C~), /3 = .24and a> - (.68412, .00106). It should be noted that for a 3Dwave packet, pinch points are required to occur simultane-ously in the a and J3 planes for flow to be absolutely unsta-ble. In the present calculation, the problem that is studied isthe impulse response to a disturbance imposed all along theinfinitely long cylinder. Under such a scenario, the distur-bances are likely to become unbounded near the leading-edge provided R > 540.5. In current LFC design, R isgenerally maintained at a much lower level to avoid attach-ment-line contamination. This instability, however, may berelevant for future applications of LFC to large aircraftwhere contamination may be suppressed by other means.

0.36

0.32

P0.28

0.24

0.20

0.16

R=632

540.5

2 3 4 5 6 7 8 9 1 06(deg.)

(a) neutral curves; 6 is the angle measured from theattachment-line.

0.1 r

(b) pinch point in the a (chordwise) planeFig. 16. Spatio-temporal instability in the swept-cylinderboundary-layer (sweep angle 60°).

6. Concluding Remarks

This paper reviews recent progress made in the predic-tion (estimation) of boundary-layer transition. For a givenflow geometry, transition onset constitutes two often widelyapart limits: one for the "ultra-smooth" surface/low-distur-bance case and the other for the "rough" surface/high-dis-turbance case. The former is determined by the eN methodwhile the latter requires experimental information or directNavier-Stokes simulation. Transition prediction is a sortingout process where one first determines the upper limit andthen estimates its onset by considering the effects of specificsurface quality and free-stream environment. Computa-tional tools for determining the upper limit for fully 3Dboundary-layers have been developed, with particular em-phasis for use by less-experienced users. This includes asuction optimization code for laminar flow control design.Computational tools have also been developed to accountfor surface quality effects but await validation againstexperiments.

A brief discussion of some of the physical issues asso-ciated with boundary-layer transition such as wall rough-ness, waviness and disturbance nonlinearity as well astransition breakdown mechanisms is given. The issue ofabsolute instability in a swept leading-edge boundary-layeris also addressed.

This review only considers application of stabilitytheory relevant to laminar flow control technology or transi-tion in low-disturbance environments. Transition in highly-disturbed environments (e.g., gas turbine blades) or otherspecific applications such as transition in high-lift systems isnot considered here although the latter could be studied by acombination of computational tools described herein forlinear and nonlinear evolution of disturbance.

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Balachandar, S., Streett, C. L. & Malik, M. R., 1992"Secondary Instability in a Rotating Disk Flow," J. FluidMech., Vol. 242, pp. 323-347.

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Bertolotti, F. P., Herbert, Th. & Spalart, P. R., 1992 "Linearand Nonlinear Stability of the Blasius Boundary Layer," J.FluidMech., Vol. 242, pp. 441^74.

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• Analysis

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