Bubble - Laminar Separation and Transition to Turbulence (Www.asec.Ir)

2.1 Laminar Separation and Transition to Turbulence 43

0

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rce

nt

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Airfoil: 85052 x 302 52 x 302

3.1 f.p.s.B. L. T.

Size:Velocity:Tested:

8 12

Date: 2/25/39

Airfoil: 860

3.1 f.p.s.B. L. T.

Size:Velocity:Tested: Date: 3/4/39

16 20 24 -12 -8 -4 0 4 8 12 16 20 24

50

-5

10 20 30 40 50 60 70 80 90 100

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ent

Chor

dof

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-50

01

01

0d 0d

Figure 2.1. Low-speed aerodynamic tests reported by Brown for two airfoils [85]. The chordwas 12.7 cm and the free-stream velocity was 94 cm/s.

Hsiao et al. [91] have investigated the aerodynamic and flow structure of an airfoil,NACA 633018, for the Reynolds number between 3105 and 7.74105. Selig etal. have reported on a wide variety of airfoils with basic aerodynamic data for theReynolds number between 6104 and 3105 [88] [90] and for the Reynolds numberbetween 4104 and 3105 [89]. In the following sections, we discuss the variousaerodynamics characteristics and fluid physics for the Reynolds number between 102

and 106, with a focus on issues related to the Reynolds number of 105 or lower.

2.1 Laminar Separation and Transition to Turbulence

Figure 2.2 illustrates the aerodynamic performance and shapes of several represen-tative airfoils under steady-state free-stream. A substantial reduction in lift-to-dragratio is observed as the Reynolds number becomes lower. The observed aerody-namic characteristics are associated with the laminar-turbulent transition process.With conventional manned aircraft wings whose Reynolds numbers exceed 106, theflows surrounding them are typically turbulent, with the near-wall fluid capable ofstrengthening its momentum via energetic mixing with the free-stream. Conse-quently, flow separation is not encountered until the AoA becomes high. With lowReynolds number aerodynamics, the flow is initially laminar and is prone to sepa-rate even under a mild adverse pressure gradient. Under certain circumstances, asdiscussed next, the separated flow reattaches and forms a laminar separation bubble(LSB) while transitioning from a laminar to a turbulent state. Laminar separationcan modify the effective shape of an airfoil, and so it consequently influences theaerodynamic performance.

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44 Rigid Fixed-Wing Aerodynamics

LIEBECK L 1003

LISSAMAN 7769

EPPLER 193

PIGEONDRAGONFLY

L/D

250

200

150

100

50

0

104 105 106ReFigure 2.2. Aerodynamic characteristics of representative airfoils. Figure plotted based onthe data from Lissaman [21]. Note Re indicates the Reynolds number and L/D indicates thelift-to-drag ratio.

The first documented experimental observation of an LSB was reported byJones [92]. In general, under an adverse pressure gradient of sufficient magnitude,the laminar fluid flow tends to separate before becoming turbulent. After separation,the flow structure becomes increasingly irregular, and beyond a certain threshold,it undergoes a transition from laminar to turbulent. The turbulent mixing processbrings high-momentum fluid from the free-stream to the near-wall region, which canovercome the adverse pressure gradient, causing the flow to reattach.

The main features of an LSB are illustrated in Figure 2.3a. After separation, thelaminar flow forms a free-shear layer, which is contained between outer edge STof the viscous region and the mean dividing streamline ST. Downstream of thetransition point T, turbulence can entrain a significant amount of high-momentumfluid through diffusion [93], which enables the separated flow to reattach to the walland form a turbulent free-shear layer. The turbulent free-shear layer is containedbetween lines TR and TR. The recirculation zone is bounded by the STR andSTR.

Just downstream of the separation point, there is a dead-fluid region, wherethe recirculation velocity is significantly lower than the free-stream velocity and theflow can be considered almost stationary. Because the free-shear layer is laminarand is less effective in mixing than in turbulent flow, the flow velocity betweenthe separation and transition is virtually constant [93]. That the velocity is almost


10 0.2 0.4 0.6 0.81

0.5

0

0.5

1

1.5

x/c

C P

Re=4 104

Inviscid

Separation

TransitionReattachment

(a)

T

T

R

R

SSZ

S

FREE-STREAMFLOW

Figure 2.3. (a) Schematic flow structures illustrating the laminar-turbulent transition [93](copyright byAIAA). (b) Pressure distribution over an SD7003 airfoil, as predicted byXFOIL[96].

constant is also reflected in the pressure distribution in Figure 2.3b. The pressureplateau is a typical feature of the laminar part of the separated flow.

Thedynamics of anLSBdepends on theReynolds number, pressuredistribution,geometry, surface roughness, and free-stream turbulence. An empirical rule givenby Carmichael [94] says that theReynolds number, based on the free-stream velocityand the distance from the separation point to the reattachment point, is approxi-mately 5104. It suggests that, if the Reynolds number is less than 5104, an airfoilwill experience separationwithout reattachment; in contrast, if theReynolds numberis slightly higher than 5104, a long separation bubble will occur. This rule providesgeneral guidance to predict the reattachment, but should be used with caution. Aswe discuss later, the transition and the reattachment process are too complicated tobe described by the Reynolds number alone.

As the Reynolds number decreases, the viscous damping effect increases, andit tends to suppress the transition process or to delay reattachment. The flow willnot reattach if the Reynolds number is sufficiently low to enable the flow to com-pletely remain laminar or the pressure gradient is too strong for the flow to reattach.

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aminSticky Note 50000 . . . . . .


Thus, without reattachment, a bubble does not form and the flow is then fullyseparated.

Based on its effect on pressure and velocity distribution, theLSBcanbe classifiedas either a short or long bubble [95]. A short bubble covers a small portion ofthe airfoil and plays an insignificant rule in modifying the velocity and pressuredistributions over an airfoil. In this case, the pressure distribution closely follows itscorresponding inviscid distribution except near the bubble location, where there isa slight deviation from the inviscid distribution. In contrast, a long bubble coversa considerable portion of the airfoil and significantly modifies the inviscid pressuredistribution and velocity peak. The presence of a long bubble leads to decreased liftand increased drag.

Typically, a separation bubble has very steep gradients in the edge velocity, ue,and momentum thickness, , at the reattachment point, resulting in jumps in ueand over a short distance. For incompressible flow, the momentum thickness isdefined as

= 0

uU

(1 u

U

)d, (21)

where u is the streamwise velocity and U is the free-stream velocity. For flow over aflat plate the momentum thickness is equal to the drag force divided by U 2. If theskin friction is omitted, the correlation between these jumps can be expressed as

= (2 + H)ue

ue, (22)

whereH is the shape factor, definedas the ratio between theboundary-layer displace-ment thickness and themomentum thickness . The boundary-layer displacementis defined as

= 0

(1 u

U

)d . (23)

Due to the change in flow structures, the shape factor H increases rapidly down-stream of the separation point. Hence, according to Eq. (22), the momentumthickness jump is sensitive to the location of the transition point in the separa-tion bubble. Furthermore, because airfoil drag is directly affected by a momentumthickness jump, an accurate laminar-turbulent transitionmodel is important for dragprediction.

Figure 2.4 illustrates the behavior of an LSB in response to variations in theReynolds number. The analyses are based on the XFOIL code [96], which usesthe thin-layer fluid flow model, assuming that the transverse length scale is muchsmaller than the streamwise length scale. At a fixed AoA, four flow regimes canbe identified as the Reynolds number varies. As indicated in Figure 2.4, at theReynolds number Re = 106, on the upper surface there exists a short LSB, whichaffects the velocity distribution locally. At an intermediate Reynolds number (e.g.,Re = 4104), the short bubble bursts to form a long bubble. The peak velocity issubstantially lower than that of the inviscid flow. As the Reynolds number decreasesto, for example, Re = 2104, the velocity peak and circulation decrease further,reducing the pressure gradient after the suction peak. A weaker pressure gradientattenuates the amplification of disturbance in the laminar boundary layer, which

aminSticky Note LSB . . . .


ReReRe

SD7003

Figure 2.4. Streamwise velocity profiles over the upper surface of an SD7003 airfoil withvarying Reynolds numbers, at the AoA of 4. Inviscid as well as viscous flow solutions areshown. At the Reynolds number,Re= 106, a short bubble is observed; otherwise, the velocitydistribution largely matches that of the inviscid model. At a lower Reynolds number, Re =4104, a long bubble after bursting is observed, causing significant impact on the velocitydistribution. Finally, at the Reynolds number, Re = 2104, a complete separation with noreattachment is noticed. ue is the velocity at the boundary-layer edge parallel to the airfoilsurface, and U0 is the free-stream velocity. The results are based on computations made withXFOIL [96].

delays the transition and elongates the free-shear layer. At this Reynolds number,the separated flow no longer reattaches to the airfoil surface, and themain structuresare no longer sensitive to the exact value of the Reynolds number.

For a fixed Reynolds number, varying the AoA changes the pressure gradientaft of the suction peak and therefore changes the LSB. In this aspect, varying theAoA has the same effect on the LSB as changing the Reynolds number. Figure 2.5illustrates that, at a fixedReynolds number of 6.01104for the Eppler E374, a zigzagpattern appears in the lift-drag polar:

1. At a lower AoA, for example, 2.75, there is a long bubble on the airfoil surface,which leads to a large drag.

2. When the AoA is increased (from 4.03 to 7.82), the adverse pressure gra-dient on the upper surface grows, which intensifies the Tollmien-Schlichting(TS) wave, resulting in an expedited laminar-turbulent transition process. Ashorter LSB leads to more airfoil surface being covered by the attached tur-bulent boundary-layer flow, resulting in a lower drag. This corresponds to thelift-drag polars left turn.


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Re=60,100Re=100,400Re=200,500Re=300,700

=2.75o

=4.03o

=7.82o =8.90o

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=2.75o

=4.03o

=7.82o

=8.90o

(a)

(b)

(c)

Figure 2.5. Lift-to-drag polars of theEpplerE374 airfoil at different chordReynolds numbers.(a) Lift-to-drag polars at different Reynolds numbers; (b) pressure coefficient distribution atdifferentAoAs for theReynolds number,Re= 6.01104; (c) shape of the Eppler E374 airfoil.The results are computed with XFOIL [96].

3. When the AoA is further increased (beyond 7.82), the separated flow quicklyexperiences transition; however, with a massive separation, the turbulent diffu-sion can no longer make the flow reattach, and the drag increases substantiallywith little changes in lift.

The previously described zigzag pattern of the lift-drag polar is a noticeable featureof low Reynolds number aerodynamics. As illustrated in Figure 2.5, at a sufficientlyhigh Reynolds number, the polar exhibits the familiar C-shape.


Earlier experimental investigations on low Reynolds number aerodynamicswere reviewed by Young and Horton [97]. Carmichael [94] further reviewed the-oretical and experimental results of various airfoils with Reynolds numbers span-ning from 102 to 109. In particular, many investigators studied the near-surfaceflow and aerodynamic loads of a wing at Reynolds numbers in the range of 104

to 106. Crabtree [98] studied the formation of short and long LSBs on thin air-foils. Consistent with the preceding discussion of the two types of separation bub-bles, he suggested that the long bubble directly influences aerodynamic character-istics, whereas the short one serves as an agent for initiating a turbulent boundarylayer.

In the last two decades, numerous investigations have been reported on theinterplay between the near-wall flow structures and aerodynamic performance. Forexample, Huang et al. [99] studied the aerodynamic performance versus the surface-flow mode at different Reynolds numbers. Hillier and Cherry [100] and Kiya andSasaki [101] studied the influence of the free-stream turbulence on the separationbubble along the side of a blunt plate with right-angled corners, finding that thebubble length, sizes of vortices in the separating region, and level of the suction peakpressure can all be well correlated with the turbulence outside the shear layer andnear the separation point.

2.1.1 Navier-Stokes Equation and the Transition Model

The constant property Navier-Stokes equations adequately model the fluid physicsneeded to perform practical laminar- and turbulent-flow computations in theReynolds number range typically used by the low Reynolds number flyers:

uixi

= 0, (24)

uit

+ x j

(uiu j) = 1

pxi

+ v 2

x2j(ui), (25)

where ui are the mean flow velocities and is the kinematic viscosity. For turbulentflows, turbulent closures are needed if one is solving the ensemble-averaged form ofthe Navier-Stokes equations. Numerous closure models have been proposed in theliterature [102]. Here we present the two-equation k turbulence model [102] asan example. For clarity, the turbulence model is written in Cartesian coordinates asfollows:

kt

+ (ujk)x j

= i juix j

k + x j

[(v + vT)

kx j

], (26)

t+ (uj)

x j=

ki j

uix j

2 + x j

[(v + vT)

x j

], (27)

where

vT =k

, i j = 2vTSi j 23ki j, Si j =

12

(uix j

+ ujxi

), (28)

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= 0 + ReT/Rk1 + ReT/Rk

, = 1325

0 + ReT/Rk1 + ReT/Rk

1

, (29)

= 9100

4/15 + (ReT/R )41 + (ReT/R )4

, ReT =kv

,

= 9125

, = = 12, 0 =

13, 0 =

19, (210)

R = 8, Rk = 6, R = 2.95. (211)

For the preceding equations, k is the turbulent kinetic energy, is the dissipationrate, vT is the turbulent kinematic eddy viscosity, ReT is the turbulent Reynoldsnumber, and 0, , R, Rk, andR are model constants. To solve for the transitionfrom laminar to turbulent flow, the incompressible Navier-Stokes equations arecoupled with a transition model.

The onset of laminar-turbulent transition is sensitive to a wide variety ofdisturbances, as produced by the pressure gradient, wall roughness, free-streamturbulence, acoustic noise, and thermal environment. A comprehensive transitionmodel considering all these factors currently does not exist. Even if we limit ourfocus to free-stream turbulence, it is still a challenge to provide an accurate mathe-matical description. Overall, approaches to transition prediction can be categorizedas (i) empirical methods and those based on linear stability analysis, such as theeN method [96]; (ii) linear or non-linear parabolized stability equations [103]; and(iii) large-eddy simulation (LES) [104] or direct numerical simulation (DNS)methods [105].

Empirical methods have also been proposed to predict transition in a sepa-ration bubble. For example, Roberts [93] and Volino and Bohl [106] developedmodels based on local turbulence levels; Mayle [107], Praisner and Clark [108],and Roberts and Yaras [109] tested concepts by using the local Reynolds num-ber based on the momentum thickness. These models use only one or two localparameters to predict the transition points and hence often oversimplify the down-stream factors such as the pressure gradient, surface geometry, and surface rough-ness. For attached flow Wazzan et al. [110] proposed a model based on the shapefactor H. Their model gives a unified correlation between the transition pointand Reynolds number for a variety of problems. For separated flow, however,no similar models exist, in part because of the difficulty in estimating the shapefactor.

Among the approaches employing linear stability analysis, the eN method hasbeen widely adopted [111] [112]. It solves the Orr-Sommerfeld equation to evaluatethe local growth rate of unstable waves based on velocity and temperature profilesover a solid surface. Its successful application is exemplified in the popularity ofairfoil analysis software such as XFOIL [96]. XFOIL uses the steady Euler equa-tions to represent the inviscid flows, a two-equation integral formulation based ondissipation closure to represent boundary layers and wakes, and the eN method to

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tackle transition. Coupling the Reynolds-averaged Navier-Stokes (RANS) solverwith the eN method to predict transition has been practiced by Radespiel et al. [113],Stock and Haase [114], and He et al. [115]. An application of this approach for lowReynolds number applications can be found in the work of Yuan et al. [116] andLian and Shyy [117].

The eN method is based on the following assumptions: (i) the velocity andtemperature profiles are essentially 2D and steady, (ii) the initial disturbance isinfinitesimal, and (iii) the boundary layer is thin. Even though in practice the eN

method has been extended to study 3Dflow, strictly speaking, such flows do notmeetthe preceding conditions. Furthermore, even in 2D flow, not all these assumptionscan be satisfied [118]. Nevertheless, the eN method remains a practical and usefulapproach for engineering applications.

Advancements in turbulence modeling have made possible alternativeapproaches for transition prediction. For example, Wilcox devised a low Reynoldsnumber k turbulence model to predict transition [119]. One of his objectivesis to match the minimum critical Reynolds number beyond which the TS wavebegins forming in the Blasius boundary-layer context. However, this model fails ifthe separation-induced transition occurs before the minimum Reynolds number isreached, as frequently occurs in the separation-induced transition. Holloway et al.[120] used unsteady RANS equations to study the flow separation over a blunt bodyfor the Reynolds number range of 104 to 107. It has been observed that the predictedtransition point can be too early even for a flat-plate flow case, as illustrated by Dickand Steelant [121]. In addition, Dick and Steelant [122] and Suzen and Huang [123]incorporated the concept of an intermittency factor to model transitional flows. Onecan model this either by using conditional-averaged Navier-Stokes equations or bymultiplying the eddy viscosity by the intermittency factor. In either approach, theintermittency factor is solved based on a transport equation, aided by empirical cor-relations. Mary and Sagaut [124] studied the near-stall phenomena around an airfoilusing LES, and Yuan et al. [116] studied transition over a low Reynolds numberairfoil using LES.

2.1.2 The eN Method

In this section we offer a more detailed presentation of the eN method, because itforms the basis for low Reynolds number aerodynamics predictions and has provento be useful for engineering applications. As already mentioned, the eN method isbased on linear stability analysis, which states that transition occurs when the mostunstable TS wave in the boundary layer has been amplified by a certain factor. Givena velocity profile, one can determine the local disturbance growth rate by solving theOrr-Sommerfeld eigenvalue equations. Then, the amplification factor is calculatedby integrating the growth rate, usually the spatial growth rate, starting from the pointof neutral stability. TheTransitionAnalysis Program System (TAPS) byWazzan andco-workers [125] and the COSAL program by Malik [126] can be used to computethe growth rate for a given velocity profile. Schrauf also developed a program calledCoast3 [127]. However, it is very time consuming to solve the eigenvalue equations.An alternative approach was proposed by Gleyzes et al. [128], who found that the


integrated amplification factor n can be approximated by an empirical formula asfollows:

n = dndRe

(H)[Re Reo (H)], (212)

whereRe is themomentum thicknessReynolds number,Reo is the criticalReynoldsnumber that we define later, andH is the shape factor previously discussed.With thisapproach, one can approximate the amplification factor with a reasonably good accu-racy without solving the eigenvalue equations. For similar flows, the amplificationfactor n is determined by the following empirical formula:

dndRe

= 0.01{[2.4H 3.7 + 2.5 tanh (1.5H 4.65)]2 + 0.25}1/2. (213)

For non-similar flows (i.e., those that cannot be treated by similarity variables usingthe Falkner-Skan profile family [129]) the amplification factor with respect to thespatial coordinate is expressed as

dnd

= dndRe

12

(

ue

dued

+ 1)

ue2

ue1. (214)

An explicit expression for the integrated amplification factor then becomes

n( ) =

o

dnd

d, (215)

where 0 is the pointwhereRe = Reo , and the criticalReynolds number is expressedby the following empirical formulas:

log10 Reo =(1.415H 1 0.489

)tanh

(20

H 1 12.9)

+ 3.295H 1 + 0.44. (216)

Once the integrated growth rate reaches the threshold N, flow becomes turbulent.To incorporate the free-stream turbulence level effect, Mack [130] proposed thefollowing correlation between the free-stream intensity Ti and the threshold N:

N = 8.43 2.4 ln(Ti), 0.0007 Ti 0.0298. (217)However, care should be taken in using such a correlation. The free-stream tur-

bulence level itself is not sufficient todescribe thedisturbance, andother information,such as the distribution across the frequency spectrum, should also be considered.The so-called receptivity how the initial disturbances within the boundary layer arerelated to the outside disturbances is a critically important issue. Actually, we canonly determine theN factor if we know the effective Ti, which can be defined onlythrough a comparison of the measured transition position with calculated amplifica-tion ratios [131].

A typical procedure to predict the transition point using coupled RANS equa-tions and the eN method is as follows. First, solve the Navier-Stokes equationstogether with a turbulence model without invoking the turbulent production terms,for which the flow is essentially laminar; then integrate the amplification factor nbased on Eq. (212) along the streamwise direction; once the value reaches thethreshold N, the production terms are activated for the post-transition computa-tions. After the transition point, flow does not immediately become fully turbulent;


instead, the movement toward full turbulence is a gradual process. This process canbe described with an intermittency function, allowing the flow to be represented bya combination of laminar and turbulent structures. With the intermittency function,an effective eddy viscosity is used in the turbulence model and can be expressed asfollows:

vTe = vT, (218)where is the intermittency function and vTe is the effective eddy viscosity.

In the literature a variety of intermittency distribution functions have been pro-posed. For example, Cebeci [132] presented such a function by improving a modelpreviously proposed by Chen and Thyson [133] for the Reynolds number range of2.4105 to 2106 with an LSB. However, no model is available when the Reynoldsnumber is lower than 105. Lian and Shyy [117] suggested that, for separation-inducedtransition at such a low Reynolds number regime, the intermittency distribution islargely determined by the distance from the separation point to the transition point:the shorter the distance, the quicker the flow becomes turbulent. In addition, previ-ous work suggested that the flow property at the transition point is also important.From the available experimental data and our simulations, Lian and Shyy [117]proposed the following model:

=1 exp

([exp

(max(HT2.21,0)

20

)2 1

] (xxTxTxS

)ReT

)(x xT)

0 (x < xT), (219)

where xT is the transition onset position, xS is the separation position, HT is theshape factor at the transition point, and ReT is the Reynolds number based on themomentum thickness at the transition point.

2.1.3 Case Study: SD7003

Lian and Shyy [117] studied the Reynolds number effect with a Navier-Stokesequation solver augmented with the eN method. The lift and drag coefficients ofthe SD7003 airfoil against AoA are plotted in Figure 2.6. This figure vividly showsthe good agreement between the numerical results [117] and the experimentalmeasurements byOl et al. [134] and Selig et al. [88]. Both the simulation and themea-surement by Ol et al. [134] predict that the maximum lift coefficient happens at theAoA = 11. Close to the stall AoA, the simulations over-predict the lift coefficients.

As the AoA increases, as illustrated in Figure 2.7, the adverse pressure gradientdownstream of the point of suction peak becomes stronger and the separation pointmoves toward the leading edge. The strong pressure gradient amplifies the distur-bance in the separation zone and prompts transition. As the turbulence develops,the increased entrainment causes reattachment. At an AoA of 2, the separationposition is at around 37 p of the chord length, and transition occurs at 75 percent ofthe chord length. A long LSB forms. The plateau of the pressure distribution shownin Figure 2.7a is characteristic of an LSB. As seen in Figure 2.7b the bubble lengthdecreases with an increase in the AoA.

Lian and Shyy [117] compared computed shear stress with experimental mea-surements by Radespiel et al. [135], using the low-turbulence wind tunnel and the


1

0.8

0.6

0.4

0.2

0.25 0 5 10 15 5 0 5 10 15

0

C L C D

Angle of attack (degree) Angle of attack (degree)

Exp: Selig et al.

Exp: Selig et al.

Exp: Ol et al.CFD, N=8

CFD, N=8

(a) (b)0.180.16

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

Figure 2.6. (a) Lift and (b) drag coefficients against the AoA for an SD7003 airfoil at theReynolds number, Re = 6104 [117]. CFD refers to computational fluid dynamics.

water tunnel because of its low-turbulent nature. Radespiel et al. [135] suggestedthat large values of critical N factor should be appropriate. As shown in Figure2.8, the simulation by Lian and Shyy [117] with N = 8 shows good agreement withmeasurements in terms of the transition position, reattachment position, and vortexcore position. It should be noted that, in the experiment, the transition location isdefined as the point where the normalized Reynolds shear stress reaches 0.1 percentand demonstrates a clearly visible rise. However, the transition point in the simu-lation is defined as the point where the most unstable TS wave has amplified overa factor of eN. This definitional discrepancy may cause some problems when wecompare the transition position. In any event, simulations typically predict notice-ably lower shear-stress magnitude than the experimental measurement. Recently,LES simulations of an SD7003 rectangular wing at the Reynolds number of 6104were performed byGalbraith andVisbal [136] andUranga et al. [137]; they discussedtransitionmechanisms based on instantaneous and phase-averaged flow features and

4

3.5

3

2.5

2

1.5

1

0.5

0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

C p

= 2 = 4 = 8 = 11

0 0.2 0.4 0.6x/c

x/c

0.8 1 2 4 6 8Angle of attack (degree) 10 12

Separation positionTransition position

(a) (b)

Figure 2.7. (a) Pressure coefficients and (b) separation and transition position against theAoA for an SD7003 airfoil at Reynolds number, Re = 6104 [117].


(a)

(b)

0.03

55%=LNB, xtran

0.02 0.01 0

0.03

0.3 0.4 0.5x/c

0.6 0.7

0.02 0.01 0

U2

u v

Figure 2.8. Streamlines and turbulentshear stress for the AoA = 4. (a)Experimental measurement by Rade-spiel et al. [135] and (b) numerical sim-ulation with N = 8 by Lian and Shyy[117].

turbulence statics. Moreover, using time-resolved particle image velocimetry (PIV),Hain et al. [138] studied the transitionmechanism in the flow around an SD7003wingat Reynolds number of 6.6104; they claimed that the TS waves trigger the ampli-fication of the KH waves, which explains why the size of the separation bubble isaffected by the magnitude of the TS waves at separation.

As the AoA increases, both the separation and the transition positions moveupstream, and the bubble shrinks. The measurements at the AoAs of 8 and 11

are performed in the water tunnel with a measured free-stream turbulence intensityof 0.8 percent. At the AoA = 8 the simulation by Lian and Shyy [117] predictsthat the flow goes though transition at 15 percent of the chord, which is close tothe experimental measurement of 14 percent. The bubble covers approximately8 percent of the airfoil upper surface. The computational and experimental resultsfor the AoA of 8 are shown in Figure 2.9. With the AoA of 11, the airfoil is close tostall. The separated flow requires a greater pressure recovery in the laminar bubblefor reattachment. Lian and Shyy [117] predicted that flow separates at 5 percentof the chord and that the separated flow quickly reattaches after it experiencestransition at the 7.5 percent chord position, whereas in the experiment transitionactually occurred at 8.3 percent. This quick reattachment generally represents thetransition-forcingmechanism. Comparison shows that the computedReynolds shearstress matches the experimental measurement well (see Fig. 2.10).

For low Reynolds number airfoils, the chord Reynolds number is a key parame-ter used to characterize the overall aerodynamics. Between the separation positionand the transition position, as shown in Figure 2.11a, the shape factor H and themomentum-thickness-based Reynolds number increase with the chord Reynoldsnumber. As shown in Figure 2.11b the effective airfoil shape, which is the superim-position of the airfoil and the boundary-layer displacement thickness, has the largestcamber at the Reynolds number Re = 4104. This helps explain why the largestlift coefficient is obtained at that Reynolds number (see Fig. 2.11c). The camberdecreases significantly when the Reynolds number increases from 4104 to 6104,


0.09

0.05 0.1

(b)

0.15 0.250.2x/c

(a)

0.05 0.01

0.090.05 0.01

WUB, xtran = 14%

U2

u v

Figure 2.9. Streamlines and turbulentshear stress for theAoA= 8. (a)Exper-imental measurement by Radespielet al. [135] and (b) numerical simulationwith N = 8 by Lian and Shyy [117].

but does not show considerable changewhen theReynolds number increases further.Therefore one does not observe much increase in the lift coefficient even though theLSB length is shorter at higher Reynolds numbers. One can conclude from Figure2.11d that the enhancement of lift-to-drag ratio is mainly due to the reduction offriction drag at highReynolds numbers. As theReynolds number increases, the formdrag does not vary as much as does the friction drag.

2.2 Factors Influencing Low Reynolds Number Aerodynamics

Because of the influence of flow separation and laminar-turbulent transition, thepreferred airfoil shapes in the low Reynolds number regime are different from those

0.08 0.04

0.040.08

0.05 0.1x/c

0.15 0.2

(b)

(a)

0

0

WUB, xtran = 8.3%

U2u v

Figure 2.10. Streamlines and turbulentshear stress for the AoA = 11. (a)experimental measurement by Rade-spiel et al. [135] and (b) numerical sim-ulation with N = 8 by Lian and Shyy[117].

2.2 Factors Influencing Low Reynolds Number Aerodynamics 57

(a)

(c) (d)

(b)

10

8

6

4

2

0

40

35

30

25

20

2500

2000

1500

Re

1000

500

0

0.64

0.63

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0.61

0.6

0.02

0.01

0

0.02

0.01

0

0

50 100 150 200 40 80 120Re 103

160 200

0.25 0.5 0.75x/c

Re 103

1 0 0.2 0.4 0.6 0.8 1

Shap

e fa

ctor

C L/C

D

C L C Dp

C Df

Re=4104

Re=6104

Re=9104

Re=2105

0.4

0.3

0.2

0.1

0

SD7003Effective Shape at Ti=0.85%Effective Shape at Ti=0.25%Effective Shape at Ti=0.1%

CL/CDCL

CDfCDp

Figure 2.11. Reynolds number effect on the LSB profile and aerodynamic performance atthe AoA = 4 for an SD7003 airfoil [117]: (a) shape factor and momentum-thickness-basedReynolds number, (b) effective airfoil shape, (c) lift-to-drag ratio, and (d) drag coefficient.

in the highReynolds number regime. Furthermore, in addition to theReynolds num-ber, the airfoil camber, thickness, surface smoothness, free-stream unsteadiness, andaspect ratio all play important roles in determining the aerodynamic performanceof a low Reynolds number flyer. These factors are discussed in this section.

2.2.1 Re = 103104

Okamoto et al. [139] experimentally studied the effects of wing camber on wingperformance with Reynolds numbers as low as 103 to 104. Their experiment usedrectangular wings with an AR of 6, constructed from aluminum foil or balsa wood.Figure 2.12 illustrates the effects of camber on the aerodynamic characteristics.As the camber increases, the lift coefficient slope and the maximum lift coefficientincrease aswell. The increase in camber pushes both themaximum lift coefficient andmaximum lift-to-drag ratio to ahigherAoA.More interestingly, the 3percent camberairfoil shows a stall-resisting tendency, with the lift just leveling off above an AoAof 10. Although it has the disadvantage of a high drag coefficient, the low-camberairfoil is less sensitive to the AoA and therefore does not require sophisticatedsteering.

Sunada et al. [140] compared wing characteristics at the Reynolds number of4103 using fabricated rectangular wings with an AR of 7.25; representative wingsare shown in Figure 2.13. After testing 20 wings, they concluded that the wing

aminSticky Note 6 1000 10000 . . 3 ( 10 ) . . 4000 7.25 :

Bubble - Laminar Separation and Transition to Turbulence (Www.asec.Ir)

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