Aircraft Laminar Flow Control

29 Pages·1998·0.28 MB·English

by  Joslin, Ronald D

Annual Review of Fluid Mechanics 1998: vol. 30: 1-29 1
AIRCRAFT LAMINAR FLOW CONTROL
y
Ronald D. Joslin
FluidMechanicsandAcousticsDivision, NASALangleyResearchCenter,Hampton,
VA23681-0001; email: R.D.Joslin @larc.nasa.gov
KEY WORDS: boundary-layer control; LFC design tools; insect contamination;
suction LFC;thermal LFC;manufacturing tolerance
ABSTRACT
Aircraft laminar (cid:13)ow control (LFC) fromthe 1930’s through the 1990’s is reviewed
and the current status of the technology is assessed. Examples are provided to
demonstrate thebene(cid:12)ts ofLFCforsubsonic andsupersonic aircraft. Early studies
related to the laminar boundary-layer (cid:13)ow physics, manufacturing tolerances for
laminar (cid:13)ow,andinsect-contamination avoidance arediscussed. LFCconcept stud-
ies in wind-tunnel and (cid:13)ight experiments are the major focus of the paper. LFC
design tools are brie(cid:13)y outlined for completeness.
1. INTRODUCTION
This overview will review the current state-of-the-art in laminar(cid:13)ow control
(LFC). LFC research began in the 1930’s and (cid:13)ourished through the 1960’s
when itwasde-emphasized due toreprioritizationofnationalpriorities. Dur-
ingthe 1970’swhen the OPEC embargocaused a fuel shortage andhigh-cost
fuel, LFC research became importantagain because of the aerodynamic per-
formance bene(cid:12)ts it could potentially produce for commercial aircraft. The
next 20 yrs of research had resulted in numerous signi(cid:12)cant achievements in
both the wind tunnel and (cid:13)ight in the United States and Europe.
1.1. What is Laminar Flow Control?
WhatisLFC?LFCisanactiveboundary-layer(cid:13)owcontroltechnique(usually
steady suction) employed to maintain the laminar (cid:13)ow (LF) state at chord
yThe United States Government has the right to retain anonexclusive,
royalty-free license in and to any copyright covering this paper.
AIRCRAFT LAMINAR FLOW CONTROL 2
Reynoldsnumbersbeyondthatwhichisnormallycharacterized asbeingtran-
sitional or turbulent in the absence of control. Understanding this de(cid:12)nition
is an important (cid:12)rst step toward understanding the goals of the technology.
Often, one mistakenly assumes that LFC implies the re-laminarization of a
turbulent (cid:13)ow. These are twodi(cid:11)erent (cid:13)owphysics phenomenaandalthough
the same control system may be employed for both problems, the energy
requirements for re-laminarization could typically be an order of magnitude
greater than that required for LFC. Finally, LFC is a capability that is de-
signed to bene(cid:12)t the aircraft during cruise.
A signi(cid:12)cant advancement made in the development of LFC technology is
theconcept ofhybridlaminar(cid:13)owcontrol(HLFC).HLFCintegrates thecon-
cepts ofnaturallaminar(cid:13)ow(NLF)withLFCtoreduce suctionrequirements
and reduce system complexity (Figure 1). NLF employs favorable pressure
gradient to delay the transition process, is sweep limited, and usually has
poor o(cid:11)-design cruise aerodynamic performance. LFC is complex, involving
suction (and ducts, (cid:13)utes, and pump power) over the whole wing chord (or
engine nacelle). The key features of HLFC are (a) suction is required only
in the leading-edge region ahead of the front spar, (b) NLF is obtained over
the wing (aft of front spar) through proper tailoring of the surface geome-
try (favorable pressure gradient), and (c) the HLFC wing design has good
performance in the turbulent mode.
1.2. What are the bene(cid:12)ts of Laminar Flow Control?
Thibertetal(1990)haveshownthatskin-frictiondragcouldamounttoabout
50%of the total drag for a subsonic transport aircraft, which is a function of
the aircraft size. Since laminarskin friction can be as muchas 90%less than
turbulent skin friction at the same Reynolds number, it is obvious that LF
would be more desirable than turbulent (cid:13)ow for reducing the drag of aero-
dynamicvehicles (except in recovery regions where the pressure dragpenalty
of boundary-layer separation is severe). Unfortunately, achieving LF over
the entire con(cid:12)guration is impractical because of the sensitivity of the LF to
external and vehicle disturbances. However, drag reduction due to LF over
select portions of a vehicle are achievable.
For an aircraft, the wings, engine nacelles, fuselage nose, and horizontal
andvertical tailare candidates for achievingLF. Althoughthe summationof
the drag reduction from each individual component would indicate a bene(cid:12)t
due to LF, the maximum/optimal bene(cid:12)ts of LFC are achieved by utilizing
the bene(cid:12)ts of LF to reduce the size of the aircraft. These LF bene(cid:12)ts in-
clude reductions in take-o(cid:11) gross weight (TOGW), operating empty weight
(OEW),block fuel(BF) foragiven missionand increase incruise lift-to-drag
ratio (L/D). Associated bene(cid:12)ts may include reductions in both emissions
(pollution)and noise.
AIRCRAFT LAMINAR FLOW CONTROL 3
For an advanced subsonic transport, Arcara et al (1991) projected 1995
technology improvements into a HLFC design. With 50% chord LF on the
upper wing surfaces and horizontal and vertical tails and 40% on the engine
nacelles,theLFCaircraftwouldhaveprojectedreductions inTOGWof9.9%,
OEW of 5.7%, and BF of 18.2% and an increase in L/D of 14.7% compared
to the turbulent baseline. Robert (1992) discussed the potential bene(cid:12)ts of
HLFC applied to Airbus A320 and A340 subsonic transports. For the A320
with a range of 500 nm (nautical miles), cruise represented only 35% of the
totalfuel burn while for the A340 with a 3000 nm,cruise represented 80% of
the total fuel burn.
Because LFC is a cruise technology,the A340wouldbene(cid:12)t morefromthe
application of LFC than the A320. Using HLFC over the (cid:12)rst 15% to 20%
chordfortheA340,apotentialdragreductionof14%couldbeobtainedusing
LF concepts on the wing, horizontal tail,vertical tail,and nacelles.
Boeing (1990) and McDonnell Douglas (Powell et al, 1989) assessed the
bene(cid:12)ts of using supersonic laminar (cid:13)ow control (SLFC) for an advanced
supersonic transport designed to carry 247 passengers at Mach 2.4 over a
distanceof5000nm. WithLFcovering40%ofthe wingwetted areaand20%
of the empennage, reductions in TOGW, OEW, and BF of 8.5%, 6.2%, and
12.0%,respectively, were projected with the LFC transport compared to the
turbulent supersonic aircraft (Boeing: NASA CR 181917,1990).
Aerodynamic performance bene(cid:12)ts bought by skin-friction drag reduction
can translate into reduced operating costs of an aircraft. Arcara et al (1991)
havebrie(cid:13)ydiscussed theimpactoffuelcostontheLFCbene(cid:12)ts. Essentially,
aprojectedreductionindirectoperatingcostof5.8%wasachievedwithHLFC
for fuel costing $0.65 per gallon, while HLFC led to an 8.8% reduction in
operating cost for higher priced fuel at $2 per gallon. In addition to fuel
savingsandreduced emissionsand noise, LFCleads toa reduced aircraft size
because of the reduction in fuel; this should lead to reductions in the total
aircraft operating costs.
1.3. Content of This Paper
The goal of this paper is to review the status of LFC technology. Early
studies related to boundary-layer (cid:13)ow physics, manufacturing tolerances for
LF,insect-contaminationavoidance,andslot-,porous-,andperforatedsuction
LFC studies in wind-tunnel and (cid:13)ight experiments are discussed. Tools for
LFC aerodynamic design are included for completeness. More attention is
given to recent wind-tunnel and (cid:13)ight tests that demonstrate multiplefacets
of LFC.
Because of the brevity of this article and the rich nearly 60-yr history of
LFCandLFC-relatedresearch,theliteraturecitedhereissubjectivelylimited.
For a thorough review of research related to LFC, refer to bibliographies by
AIRCRAFT LAMINAR FLOW CONTROL 4
Bushnell & Tuttle (1979) and Tuttle & Maddalon(1993), to overview papers
by Hefner (1992), Wagner et al (1988), and Collier (1993) and to a historical
perspective of LFC (Joslin 1998).
2. LAMINAR FLOW CONTROL DESIGN
METHODOLOGY
For an LFC design (a wing for example), the analysis begins by de(cid:12)ning an
initial wing geometry. With wing geometry de(cid:12)ned, the wing pressures and
velocities can be obtained using transonic wing theory and/or computational
(cid:13)uid dynamics(CFD). The inverse approach of prescribing a target pressure
distributionand solvingfor the wing geometryis then utilized. After obtain-
ingthe external (cid:13)ow-(cid:12)eldforthe (cid:12)nalgeometry,boundary-layerand stability
theory calculations are used to determine the suction (cid:13)ow rates and distribu-
tionforthedesired transitionlocation. Withthesuction(cid:13)owratedetermined
from boundary-layer stability considerations, the pressure drop through the
skinmustbesettoobtainareasonablesub-surface compartmentationscheme
andperforation spacing distributionfor the desired suction distribution. The
process isiterativeuntilanacceptable designisobtained. Finally,the suction
system ducting and compressor speci(cid:12)cations are prescribed.
Besides the design procedure, other key information that must be under-
stood for LFC design include: (a) the physics associated with the laminarto
turbulentboundary-layertransitionprocess, (b)impactofsurface tolerances{
roughness, waviness, steps, and gaps{onLF extent (required for manufactur-
ing), (c) slot suction, porous, and perforated suction and thermal LFC, and
(d) issues relating to manufacturing LFC articles, and (e) the methodology
andlimitationsoftransitionprediction(determiningLF extent forprojecting
bene(cid:12)ts to aircraft). These issues are covered in this section.
2.1. Boundary-Layer Flow Physics
A brief summary of the boundary-layer (cid:13)ow physics follows. For more com-
prehensivediscussions ofviscous(cid:13)owphysicsrefer toBaylyetal(1988),Reed
& Saric (1989),and Kachanov (1994).
The (cid:12)rst major theoretical contributions to the study of boundary-layer
transition were made in the late 1800s and early 1900s. After these early
contributions, Tollmien (1926) and Schlichting (1932) discovered convective
traveling-wave instabilities now termed Tollmien-Schlichting (TS) instabili-
ties,andLiepmann(1943)andSchubauer&Skramstad(1947)experimentally
con(cid:12)rmed the existence of these TS waves.
InadditiontoTS-disturbancedominatedtransition,LFCapplicationsmust
account for a dynamic instability, termed the cross(cid:13)ow (CF) disturbance.
AIRCRAFT LAMINAR FLOW CONTROL 5
The (cid:13)owbecomes unstable due toan in(cid:13)ectionpointin the spanwise velocity
component. As shown by Anscombe & Illingworth (1952), CF-disturbances
are characterized by corotating vortices. At a critical speed (or Reynolds
number), \striations" in the surface (cid:13)ow visualizations appear and as the
speed of the free-stream continues to increase the transition location moves
forward.
The presence of TS and CF disturbances in the boundary-layer (cid:13)ow is de-
pendent on the pressure gradient and on the wing sweep angle. Based on
numerous investigations (e.g., Anscombe & Illingworth, 1952), our under-
standing of the role of sweep and pressure gradient on transition indicates
o
that TS disturbances dominate transition for wings with sweeps 0 through
o
about 25 . If the design pressure gradient is more favorable (accelerating
(cid:13)ow), longer runs of LF can be realized because the growth of the TS dis-
turbance is suppressed, while the opposite e(cid:11)ect is realized with an adverse
o
pressure gradient. For wing sweep angles greater than 25 , both TS and CF
disturbancescandominatetransition,andmuchofthe(cid:13)owphysicsassociated
with the nonlinear interaction of these modes is unknown. For greater than
o o
30 to35 wingsweeps, CF disturbances dominatetransition{oftenvery near
the wingleadingedge. Therefore, wingdesign should minimizethe growth of
these disturbances to enable long runs of LF.
In addition to TS and CF disturbances which lead to transition over the
wing chord, attachment-line instabilities are possible and can cause transi-
tion in the leading-edge region. As the many(cid:13)ight experiments have shown,
maintainingNLFontheattachmentlineisattainablebykeepingtheReynolds
number below some critical value; the Reynolds number can be lowered by
reducing sweep, leading-edge radius, or unit Reynolds number. Decreasing
the leading-edge radius has a compounded bene(cid:12)t of decreasing the chord-
wise extent ofthe cross(cid:13)ow regionand providinga morerapid acceleration of
the (cid:13)ow over the wing. Turbulence (or attachment-line contamination)from
thefuselageturbulentboundary-layer(cid:13)owcansweepoutontotheattachment
line and cause the entire wing to be engulfed in turbulent (cid:13)ow. However, a
turbulence diverter such as Gaster’s bump (1965) can be e(cid:11)ectively used to
establishaLF attachmentline,allowingthe potentialforLFonthe wing. As
an alternative, strong suction can be used at the fuselage/wing juncture to
re-laminarizethe (cid:13)owon the attachment line(Pfenninger 1965). Someof the
major(cid:13)ight experiments have shown the necessity for one of these devices to
achieve LF.
Additionally, a turbulent wedge, originating at the fuselage/wing leading-
edgejuncture,cansweepoutoveraportionofthewingrootregiontodecrease
the LF bene(cid:12)ts. Clearly, one would attempt to optimize the fuselage/wing
juncture point to cause this wedge to cling to the fuselage as much as pos-
sible, thereby enabling LF in a region close to the fuselage. No study which
AIRCRAFT LAMINAR FLOW CONTROL 6
has investigated the potential instabilityofthe interface between a turbulent
wedge and LF over a wing could be found for this review.
2.2. Manufacturing Tolerances
As discussed by Jones (1938), investigations in the 1930s recognized that
smallroughness andwaviness at highReynoldsnumberscould reduce the LF
extent. The stringent surface smoothness and waviness criteria (tolerances)
forLFposed a majorchallenge forresearch inthe 1950sand1960s. Apartial
explanationforthe de-scope ofsubsonicLFCinthe1950swasattributableto
the severe surface manufacturingtolerances required to achieve LF. However
the1990smanufacturingtechnologieshavematuredtothepointthatsurface-
de(cid:12)nitiontolerances arereadilyachievable. Athoroughreviewofthe surface-
tolerance issues is described by Carmichael (1979) and Holmes et al (1985);
the current understanding is summarized in this section. For LFC design,
Braslow et al (1990) gives formulaefor manufacturingtolerances.
Flight and wind-tunnel tests have provided our current understanding of
themechanismswhichcause transitiontomoveforwardduetosurface imper-
fections. The impact of a surface imperfection (such as a rivet head) on the
transition location can be viewed either by looking at the transition location
asafunctionofimperfectionsizefora(cid:12)xedunitReynoldsnumberorbykeep-
ing the size of the imperfection (cid:12)xed and looking at transition location as a
functionofunit Reynoldsnumber. In either case, the imperfectionstimulates
eigenmodes in the boundary layer; the linear stability of the (cid:13)ow dictates
whether these modes will grow or decay as they evolve in the (cid:13)ow. However,
as the height of the imperfection or unit Reynolds number increases, a point
is reached when (cid:13)ow separation occurs due to the surface imperfection. At
thispoint,inviscidinstabilityarisingfromthe in(cid:13)ectionalvelocitypro(cid:12)le can
grow and induce transition. Or if the imperfection is su(cid:14)ciently large, lin-
ear instabilityampli(cid:12)cationis "by-passed" and transitionfollowsby-way-ofa
nonlinearprocess. Finally,ourcurrentunderstandingofimperfectionssuggest
that larger critical step heights can be realized using rounded steps because
a reduced region ofseparation and reduced in(cid:13)ectionalinstabilitygrowth are
encountered in the experiments.
Based onour understanding ofthe (cid:13)owphysics associated a surface imper-
fection, we summarize that (a) a local separation region due to the surface
imperfection can induce transition, (b) the local adverse pressure gradient
due to the surface imperfection could cause ampli(cid:12)cationof TS disturbances
and decrease the LF extent, (c) the bene(cid:12)cial stabilizing in(cid:13)uence of com-
pressibilityon TS disturbances overcomes the the detrimentalincrease in the
amplitudeofthe pressure disturbance caused bycompressibilitynear the sur-
faceimperfection,(d)thecriticalwaveheightdecreaseswithincreasednumber
AIRCRAFT LAMINAR FLOW CONTROL 7
of waves, waves, and (e) forward-facing rounded steps can increase the the
critical step height compared with forward-facing square steps.
This section ends with a brief mention of innovative tools which are used
to estimate the in(cid:13)uence of a surface imperfection on LF extent. Masad
(1996)has shown that interacting boundary-layertheory (which accounts for
viscous-inviscidinteraction)coupledwithlinearstabilitytheorycanbeusedto
parameterize allowablesurface imperfections for maximizingLF extent while
minimizingconstraints put on manufacturing.
2.3. Suction and Thermal Laminar Flow Control
Investigations to obtain LF with the slot-suction LFC concept began in the
late 1930s and continued into the 1980s. Zalovcik et al (1944) with a B-18,
Groth et al (1957) with a F-94, Whites et al (1966) and Fowell & Antonatos
(1965)withtheX-21andMaddalon&Braslow(1990)withtheJetstarshowed
that large regions of LF could be realized in (cid:13)ight using slot-suction LFC.
Pfenninger et al (1957) and Bobbitt et al (1992) showed that full-chord LF
could be achieved in the wind-tunnel using slot-suction.
In the late 1950s and early 1960s, a series of supersonic LFC wind-tunnel
experimentsbyGrossetal(1964)andGrothetal(1965)reportedtheachieve-
6
ment of full-chord LF up to chord Reynolds numbers of 20 to 29 10 using
(cid:2)
slot-suction. For example, Groth et al (1965) showed that full-chord LF on
o o
36 and 50 swept wing slot-suction models led to skin-friction and drag re-
ductions using LFC.
Finally,Gregory (1961)reviewed the status ofsuction surfaces forLFC ap-
plicationsandpointedoutthatasawingwassweptalossofslote(cid:11)ectiveness
occurred, especiallynear the leadingedge region. The advantagesofa wholly
porous(orperforated)suctionsurface becamepronouncedforLFCappliedto
swept wings.
In the 1940s, full-chord LF was achieved using porous-suction on a wing
model as described by Burrows et al (1949) and Braslow & Visconti (1950).
Headetal(1955)obtainedfull-chordLFin(cid:13)ightusingporoussuctionlocated
from6%-98%chordonaVampireIIIaircraftwing. Intheseearlywind-tunnel
and (cid:13)ight tests, suction was achieved through a porous-material skin which
covered the model.
As noted in the discussion of perforated-suction LFC by Saric (1985),
threshold design parameters of size, spacing, geometry, and inclination of
theholesandthesuction levelanddistributioncan introducestreamwisevor-
ticity in a boundary layer, potentially inducing transition. Although there is
no design tool available today which completely accounts for the impact of
each parameter, progress is being made to understand the in(cid:13)uence of each
parameter on the (cid:13)ow.
One of (cid:12)rst fundamental experimental studies on critical suction through
AIRCRAFT LAMINAR FLOW CONTROL 8
isolated holes and a row of holes was reported by Goldsmith (1955, 1957).
Minimumandmaximumsuctionvaluesweredetermined;beloworabovethese
critical values, the (cid:13)ow was turbulent. The critical suction was a(cid:11)ected by
the hole diameter,the holespacing, andthe boundary-layerthickness. In the
later report it was noted that the action of vortices from holes in di(cid:11)erent
rows could lead to horse-shoe vortices, which then led to turbulence. This
undesirable (cid:13)ow phenomena can happen with lower suction compared with
the isolated hole or row of holes.
Meitz & Fasel (1994) used a Navier-Stokes solver to study the unsteady
(cid:13)ow (cid:12)eld adjacent and downstream of suction holes. In agreement with the
Goldsmithexperiments, the simulationsshowed that lowsuction through the
holes generated a pair of vortices which decayed with downstream distance.
As the suction increased to somecriticalvalue,the vortices becameunstable.
Larger suction led to vortex shedding at the suction hole location.
MacManus&Eaton(1996)performedthree-dimensional(3D)Navier-Stokes
simulationsofsuctionthroughholestostudysingleandmultiplerowsofholes.
The simulationsshowed that (a) irregularities of the hole shape had minimal
e(cid:11)ect onthe induced (cid:13)ow,(b)it wasundesirable to haveholes inclinedto the
surface,(c)the(cid:13)ow(cid:12)eldattheholeinletwashighly3D,(d)thepressure drop
andmass-(cid:13)owrate were insensitivetothe holeinletshape, and(e)signi(cid:12)cant
inter-hole (cid:13)ow (cid:12)eld e(cid:11)ects existed for multiple rows of holes. In companion
experiments,MacManusetal(1996)usedLDVtoshowthatthe suctionhole-
induced(cid:13)owwashighly3Dandcouldleadtoprematuretransitionwhenlarge
suction was used.
Besides the suppression of disturbance growth by suction, cooling (in air)
can be used to suppress disturbances (Dunn & Lin, 1953). As shown by
Boeing (1990) cooling has a larger impact on TS disturbances and only a
subtle in(cid:13)uence on CF disturbances; hence, cooling would not be used in the
leading-edge region of swept wings for CF stabilization.
LFCconcepts ofthefuturemusttrade-o(cid:11)thebene(cid:12)ts ofvariousboundary-
layercontrolconcepts,includingHLFCsuctionontheleadingedgewitheither
strips of perforated-suction control or thermal control on the wing roof-top
region. Furthermore, reducing the costs associated with manufacturing(and
maintaining)these systems will be a primary driver in the the acceptance of
LFC on a future aircraft.
2.4. Manufacturing Capabilities
In the early years of airplanes, thin metal skins, multiple spanwise stringers,
and countless fasteners (e.g., rivets) on the surface prevented achieving LF
without hand-(cid:12)lling and sanding the surface. With the advent of bonded-
sandwichconstructionmethods,theproductionsurface structure becamesuf-
(cid:12)cientlysti(cid:11) thatunder loads adequate waviness criteria should be maintain-
AIRCRAFT LAMINAR FLOW CONTROL 9
able (in subsonic/transonic aircraft). Hence, the production capabilityin the
1990shas solved (inprinciple)the task ofmanufacturingLF-qualitysurfaces.
For the manufacturing process, Schwab (1992) discussed electron-beam
drilling for creating holes in a surface for suction LFC. Using this method,
some 3000 holes per second could be generated in stainless steel sheets 0.5
and 1.0 mm thick with hole diameters as small as 0.04 mm and 0.06 mm,
respectively. Cross sections ofthe drilledholesindicated that the upper most
part of the drilled holes were 2-2.5 times larger in diameter than the exit di-
ameter, with the exit of the hole being absolutely burr-fee and round. The
Jetstar (Powell 1987) used this technique for generating a perforated-suction
surface.
Laser-drilled holes are an alternative to the electron-beam process. The
B-757 (Collier, 1993) and F-16XL (Norris 1994) LFC (cid:13)ight test articles have
been generated with lasers. The F-16XL SLFC glove had a titanium skin
with laser-drilled holes; the laser produced holes with characteristics similar
tothe electron-beam(Norris 1994). However, Pollet al(1992)and Buxbaum
& H(cid:127)ohne (1996) observed that laser-drilled holes had a random variety of
shapes with no distinct characteristic diameter.
Insummary,LFCslot-,porous-,andperforated-suctionconcepts havebeen
demonstrated in wind-tunnels and (cid:13)ight experiments and thermal LFC was
been shown e(cid:11)ective in computations. State-of-the-art manufacturing pro-
cesses (e.g., electron-beam or laser drilling) have been demonstrated for the
fabrication of perforated-suction surfaces. With the exception of the Jetstar
LFC (cid:13)ight test, little understanding of the potential degradation of the per-
forated structure with time is available in the literature. The Jetstar LFC
(cid:13)ightexperiment showednodegradationofthe systemafter nearlyfouryears
of operations (Maddalon & Braslow 1990).
2.5. Transition Prediction Design Tool Methodology
TheLFCdesignofanaircraftrequiresanaccuratepredictionoftheamountof
LF(the transitionlocation)or the accurate prediction ofsuction distribution
for a giventarget transition location. This section describes the conventional
and advanced transition prediction tools,some ofwhich include prediction of
perturbations to the laminar boundary layer, the spectrum and amplitudes
of these perturbations, and the linear and nonlinear propagation of these
perturbations which ultimatelyleads to transition.
GRANVILLE CRITERION: Granville (1953) developed an empirical criterion
fordeterminingthe transition locationfor 2D TS-dominated(cid:13)ows. Granville
showedthat avarietyof(cid:13)ightandlow-turbulence wind-tunneldatacollapsed
intocriterion(curve)basedonthedi(cid:11)erencebetweenthemomentumthickness
Reynolds number at transition and at the neutral point correlated with an
average pressure gradient parameter. As shown by Holmes et al (1983), this
AIRCRAFT LAMINAR FLOW CONTROL 10
transition-predictioncorrelationisquiteaccurateforTS-dominatedtransition
on a 2D NLF wing.
C1 AND C2 CRITERIA: Along the approach of the Granville criterion, Arnal
et al (1984) developed transition criterion referred to as C1 and C2. The
C1 criteria involves a correlation of transition onset integral values of the
cross(cid:13)ow Reynolds numberand the streamwise shape factor. The C2 criteria
is a correlation of transition onset with a Reynolds number computed in the
direction of the most unstable wave, the streamwise shape factor, and the
free-stream turbulence level.
LINEAR STABILITY THEORY: The ability to predict transition based on
the disturbance ampli(cid:12)cation came in the 1950s with the empirical method
by Smith (1956), Smith & Gamberoni (1956), and van Ingen (1956). This
N
transition-prediction method{called e or N-factor method{correlates the
predicted disturbance growth with the measured transition location. Hence,
althoughthe N-factormethodislimitedtothecalibrationofavailableexperi-
mentaldata,themethodisthemaintoolinusethroughthe1990s. Theuseof
linearstabilitytheoryissummarizedintherecent reviewbyReedetal(1996)
and additional discussion focusing on the theoretical and computational as-
pects of transition prediction and LFC are provided by Cousteix (1992) and
Arnal (1994).
According to the conventionalnormalmode assumptionused to derive the
Orr-Sommerfeldequation, the eigensolutions take the form.
(cid:8)=(cid:8)^(y)ei((cid:11)x+(cid:12)z(cid:0)!t) (1)
wherei=p 1and(cid:8)^ isthedisturbancepro(cid:12)le. Withthemean(cid:13)owavailable,
(cid:0)
a stability problem has to determine 6 unknowns: (cid:11)r;(cid:11)i;(cid:12)r;(cid:12)i;!r; and !i,
whicharethestreamwisewavenumberandgrowthrate,spanwisewavenumber
andgrowthrate,andwavefrequencyandtemporalgrowthrate. Eitherspatial
ortemporalstabilityanalysismaybeperformed,wherebythetemporalanaly-
sisislessexpensiveandthespatialanalysisismorephysical. Forthetemporal
formulation,(cid:11) and (cid:12) are real numbers and ! is a complex number which is
determined by solving the stability equations. The temporal approach was
used by Srokowski & Orszag (1977) in the SALLY code and later by Malik
(1982) in the COSAL code. The spatial formulation is more representative
of the real boundary-layer (cid:13)ow; the frequency (!r) is speci(cid:12)ed, !i = 0, and
(cid:11)r;(cid:11)i;(cid:12)r;(cid:12)i are parameters to be determined. While an eigenvalue anal-
f g
ysis will provide two of these values, additional criteria must be speci(cid:12)ed to
determine the remainingtwo parameters. Arnal (1994) discussed approaches
availableto determine the two free parameters for 3D disturbances.
Upon solving the eigenvalue problem,integrations are made fromthe neu-
tral point (s0) to the transition point (s1) where the N-factors are correlated
withtheexperimentaldatabase. TheN-factorparameterofinterest isde(cid:12)ned