-%-

....... _=.. _ _|.

--=--

-T_z _ N_ -- --
NASA Contractor Report 4306

Effect of Suction on Controlling

the Secondary Instability

of Boundary Layers

Nabil M. E1-Hady

Analytical Services & Materials, Inc.

Hampton, Virginia

Prepared for

Langley Research Center

under Contract NAS1-18599

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Division

1990

EFFECT OF SUCTION ON CONTROLLING THE SECONDARY

INSTABILITY OF BOUNDARY LAYERS

Nabil M. EI-Hady

Analytical Services & Materials, Inc., Hampton, Virginia 23666

ABSTRACT

The effect of suction on controlling the three-dimensional secondary instability is investigated for a boundary layer with pressure gradient in the presence of small but finite amplitude Tollmien-Schlichting wave. The focus is on

principal parametric resonance responsible for strong growth of subharmonics

in low disturbance environment. Calculations axe presented for the effect of

suction on the onset and amplification of the secondary instability in Blasius

and Falkner-Skan flows, as well as its effect on controlling the production of

the vortical structure.

May 21, 1990

iii

PRECEDING PAGE BLANK NOT FILMED llJ_ 11 JN'ff.NIiONALLY BLANK

EFFECT OF SUCTION ON CONTROLLING THE SECONDARY

INSTABILITY OF BOUNDARY LAYERS

Nabil M. El-Ha@

Analytical Services & Materials, Inc., Hampton, Virginia 23666

1. INTRODUCTION

Laminarization of the flow by suction, and subsequent viscous drag reduction, is the principal and most effective means used for laminar flow control (LFC) [1]. Excessive suction

increases suction drag and reduces the overall efficiency of an aircraft with LFC. Moreover,

the over-thinned boundary layer is over-sensitive to surface irregularities. Hence, it is necessary to keep the boundary layer laminar with the least possible suction. For this purpose, the

stability characteristics of the flow need to be accurately calculated, and the laminar-turbulent

transition process must be well understood. Moreover, the effect of suction control on some of

the early stages to transition need to be assessed.

The nonlinear process of the flow once initiated becomes violent and lead rapidly to transition, and it is then extremely difficult to control the flow field. Hence, suction control in

LFC systems is preferred to be in the linear range. For these systems to be efficient, some disturbance growth should be allowed, and linear primary stability theory in conjunction with the

e n method were relied on for such predictions [2]. A linear growth of a primary twodimensional (2D) Tollmien-Schlichting (TS) wave may parametrically excite a linear secondary

growth with three-dimensional (3D) character. This secondary instability may not lead to transition by itself, but as it grows, it interacts with both the mean flow and the TS wave leading

rapidly to transition. The effect of suction on the primary TS wave is well established and

known to be drastic [3]. While, in this paper, we are concerned with the effect of suction on

controlling the onset and growth of the secondary 3D instability.
-2-

Following the growth of the primary 2D wave, the flow takeson an increasinglythreedimensional(3D) behavior[4] with explosivegrowth rates. In this early stageof transition,

the 3D phenomenonis characterizedby a strongsecondaryinstability with respectto 3D disturbancesin the presenceof small but finite amplitudeprimary 2D waves [5,6]. Primary 3D

disturbancesmight be stableor very slowly growing in the absenceof the 2D waves. This

secondaryinstability hasbeenrecognizedexperimentally(Refs.7-12) andobservednumerically

(Refs. 13-18)in the uncontrolledboundarylayer (Refs. 7-15) as well as the controlledboundary layer(Refs. 16-18).

Two major typesof breakdownhave beenidentified, a fundamentalbreakdown(K-type)

and a subhannonicbreakdown(H/C-type). The experimentsindicate that the subharmonic

breakdownoccurswhenthe amplitudeof the primary TS wave is low or moderate,while the

fundamentalbreakdownoccursfor higheramplitudes.However,one typeor a mixtureof both

will appeardependinglargelyon the spectrumof the backgrounddisturbances[19]. Thelinear

secondary instability theory formulated by Herbert [20] can predict the increasingly 3D

behaviorwith largegrowth ratesthat occur in both thefundamentaland subharmonictypesof

breakdown,while Craik's resonanttriad model [21] predicts aninstability of the subharmonic

type. Craik's mechanism(referredto as C-type) is thoughtto dominateat low amplitudeof

theTS wave,while Herbert'ssubharmonicmechanism(referredto asH-type)reflectsthe situation at moderateamplitudesof the TS wave. A recentreviewon the subjectof secondaryinstability hasbeenprovidedby Herbert[22] andBayly et al. [23].

The 3D subharmonicinstability which characterizestheroad to transitioninitiated from a

low disturbancebackgroundappearsto be morerealistic in flight applications. Hence,in this

paperwe investigatethe developmentof the subharmonicsecondaryinstability in a boundary

layer with pressuregradientscontrolled by suction. Our objectiveis to evaluatethe effect of

suctioncontrolon this early stageleadingto transition. Severalquestionsneedto be answered,

Doessuctiondelaythe onsetof the secondaryinstability ? How sensitiveis the growth of the

secondaryinstability to the intensityof suction? What is the effectof theinitial amplitudeof
-3-

the primary wave on this sensitivity? Doesthe effectivenessof suctionasa methodfor delaying transitiondependon whereit is applied,or on its intensity? Finally. if the boundarylayer

would be kept laminar with the leastpossible suction,then shouldone allow for a limited

growthof the secondarydisturbance,or shouldone increasesuctionto fully stabilizethesecondary disturbance? The focus of this paper is to answerthesequestionsas well as to shed

somelight on the mechanismby which the suctioncontrol the productionof the vortical structure of the secondaryinstability.

2. ANALYSIS

2.1. The Mean Flow

We consider a 2D boundary-layer flow of an incompressible fluid with inviscid flow field

given by u=u(x) and distributed suction given by v=vw(x) at the wall. where x is the streamwise direction and y is the normal direction. The flow is governed by the nonsimilar

boundary-layer equation

f nnn+ff nn+_(1-f 2n)-Ff nn = 2_(f _f _-f _f nn)

with boundary conditions

f,(_.0)=0.

given in the Gortler variables

(1)

d_=U.dx, d_l=(U./g--2_)dy (3)

Then the velocity components u and v in terms of the new variables are

u =U, fn (4)

(5)

where the suction and pressure gradient functions F and _ respectively are defined as

r(_)=(_U,)vw (¢) (6)

f (_,0)+2_(_,0)=0, fn(_,'q_,,o)=l (2)
4

Note that negative r indicates suction, while negative f_ indicates unfavorable pressure gradient. If both the suction and pressure gradient functions are constant (equal to F0 and I_0

respectively), then f (_,rl) is a function of I1 only and we have a similar boundary-layer profile

governed by

fnaa+ffnn+_o(1-f_)-Fofnn = 0 (8)

f(O) =fn(O) = O, fn--_l asTl-_** (9)

where the condition Fo=constant demands that vw be proportional to u,/4-2_. On a flat plate, vw is

proportional to 1/_ff.

2.2. The Primary Instability

We consider the primary instability of the calculated mean flow with respect to 2D quasiparallel spatially growing TS waves. Dimensionless quantities are introduced to the governing

incompressible Navier-Stokes equations by using the reference velocity U, and the reference

length L=(v, xlU,) to2 SO that Reynolds number is given by R=(U,x/v,) lr_ where x measures the

distance from the leading edge, and v is the fluid kinematic viscosity. The primary TS wave is

assumed to take the traveling form

q (x ,y ,t)=a [ql(Y) exp(i _t_rdx-i tot)+cc. ] + O (A 2) (10)

where

A =-A(x) = A o exp(-Sct I dr), (11)

and ql stands for the velocities ul and v_, and the pressure p_ of the primary wave, and cc

denotes complex conjugate. The spatial stability analysis is chosen for being more appropriate

for this study. Hence, the wavenumber is complex and given by a=a,+io_ and the disturbance

frequency co is real. The linearized incompressible Navier-Stokes equations reduce to a

fourth-order system of ordinary differential equations with homogeneous boundary conditions
-5-

in ut and vl. These equations may be combined to yield the well known Orr-Sommerfeld

equation. The eigensolutions of the primary wave are normalized such that the amplitude a

measures directly the maximum root mean square value of the disturbance velocity in the flow

direction. For the purpose of comparison we follow Herbert [20] and let

max2[u_(y)]2 = 1 (12)

0gy_.

The linear stability theory of the primary wave provides a for a given _ and R. Then the

integration of the growth rate -ai gives the amplification factor In (A/Ao), where A0 is an arbitrary initial amplitude at the onset of the primary instability.

2.3. The Secondary Instability

We consider a basic state _(x,y,t) which consists of a 2D quasi-parallel boundary layer

with pressure gradient and suction modulated by a periodic component of the linear primary

instability problem,

_l (x ,y ,t ) = q o(x ,y )+ a q l(x ,y ,t ) (13)

To study the linear 3D instability of the basic state #(x,y,t), we superpose a small

unsteady disturbance on each flow quantity of the basic state, that is

q (x ,y ,z ,t ) = _ (x ,y ,t ) + B q 2(x ,y ,z ,t ) (14)

where q2 is a secondary disturbance eigenfunction that stands for velocities u2, v2, w2, and pressure p2, they are normalized such that the amplitude B measures the maximum root mean

square value of u2. The amplitude B is assumed small compared to the primary amplitude A,

such that the primary instability will influence the secondary instability but not vice versa.

Herbert [22] has pointed out that the 3D secondary instability occurs at small amplitudes

of the primary wave where the nonlinear distortion is weak. This instability is of vortical

nature and originates from a strong mechanism of combined tilting and stretching of the vortices [5], leading to large growth rates when compared with those for the primary wave. In
-6-

view of that, we neglect in this analysisthe nonlineardistortion of the eigensolutionsql at

finite amplitude of the primary wave. Also, the growth of the primary TS wave which occurs

on viscous scales can be considered weak, and the variation of the primary amplitude A can be

assumed locally constant.

Equation (14) is substituted into the dimensionless Navier-Stokes equations. The basic

state is subtracted, and the resulting equations are linearized with respect to the secondary

amplitude a. We end up with four coupled partial differential equations for the secondary 3D

instability. The coefficients of these equations are function of the basic state, they are independent of the coordinate z, and periodic in x and t. Hence, the z-variation can be separated, and

Floquet theory of differential equations with periodic coefficients can be applied to give a solution in the form

q2(x,y ,z,t ) = e_ e i_" _(x,y ,t )

where 13 is a real spanwise wavenumber of the secondary disturbance.

are two complex characteristic exponents, and _(x.y,t) is a periodic function of (x-_tlcO, the

same as the period of the basic state. We express _ in terms of Fourier series to obtain the following expression for q2

q2(x_y ,z,t ) = e_t e il_" _, ¢, (y ) e I'_''-_1'_ (16)

/I m.--J_

Equation (16) represent the general Floquet form for the eigenmodes of a periodic basic

state, where both the fundamental and subharmonic modes are special cases. Only two of the

four real exponents 7,,7i, a,, and cry, are determined as a solution of the eigenvalue problem,

others must be given. For the purpose of our study of the spatial instability of subharmonic

modes, we let 7, represent the growth rate of the secondary instability, a,=0 (no temporal

growth), ai=-_2 for pure subharmonic mode, and let 'ti represent the shift in the streamwise

wavenumber of the secondary disturbance with respect to the primary one. 7i---0 means that the

secondary disturbance is perfectly synchronized with the basic state.
-7-

We usethe lowest truncationof the Fourier seriesfor the subharmonicmode (n<l) in Eq

(16), to derive four coupledoi:dinarydifferential equationsfor uz, v2, w2, and Pz. These equations when supplemented with homogeneous boundary conditions, they constitute an eigenvalue

problem

7= F(a, 13,R; A)

for a given boundary-layer velocity profile with pressure gradient and suction.

(17)

The eigenvalue

problem (17) of the secondary instability provides _ for a given 13and R. Then the integration

of the growth rate "tr gives the amplification factor In (B/Bo), where B0 is an arbitrary initial

amplitude at the onset of the secondary instability.

3. NUMERICAL TREATMENT

Similar boundary layer profiles were calculated using Eqs (8)-(9). These equations were

numerically integrated by using a shooting technique with fourth-order Runge-Kutta and

Adams-Moulton integrator. In cases of continuous suction, where vw= constant, similar solutions do not exist, and Eqs (1)-(2) were numerically integrated by using a step by step procedure in the streamwise direction. A three-point implicit finite difference technique was used

to reduce them to a set of simultaneous tridiagonal equations. These equations were linearized

and then solved using the algorithm of Thomas. The method of solution closely parallel that

of Price and Harris [24].

The primary instability which modulate the 2D boundary layer is governed by a fourthorder system of equations. While the eigenvalue problem (17) describing the secondary disturbance, is governed by a sixth-order system of equations. Both can be written as complex system of linear first-order ordinar.'y differential equations in the form

Pnma_

4

DZt,,-_.,a,_Zlm = 0, n=1,2..4 (18)

m=l
-8-

where zl. arc defined by

Secondary

where z_, axe defined by

Z,1 = Z13 = 0 at y--O (19)

ZH, Z13 --> 0 as y--_ (20)

ZII=ut, ZI2=Dut, Zl3=Vl, ZI4=Pl

6 6

oz , - Zb..z . =a (21)

m=l m--I

Z21= Zzs = Zzs = 0 at y =0 (22)

Z21, Z23, Z2s -->0 as y-._ (23)

Z21 = u2, Zz2 = Du2, Z23 = v2,

Z24=P2, Z2s=w2, Z26=Dw2

and the overbar denotes a complex conjugate. The a,, are the elements of 4x4 variable

coefficient matrix of the primary eigenvalue problem, while b,, and c,, axe the elements of

6x6 variable coefficient matrices of the secondary eigenvalue problem. These elements are

given in the Appendix. Note that in case of no modulation of the mean flow by a primary

wave (i.e. A=O), then the system of equations (21)-(23) will govern the stability of a primary

subharmonic 3D wave.

Both the primary and secondary system of equations are numerically integrated as initial

value problem using a freestream solution as initial condition. For the secondary eigenvalue

problem, we assume that the amplitude of the primary vanishes far in the freestream at y > y,

(e denotes the edge of the boundary layer). Then the system (21) will have constant

coefficients and can be solved analytically producing three linearly independent exponentially

decaying solutions to conform with the boundary condition (23). With the freestream solution

as initial condition, Eqs (21) are integrated from y=y, to y=0 at the wall, using a variable step-
-9-

size algorithm [25], basedon the Runge-Kutta-Fehlburgfifth-order formulas. The solution is

orthonormalizedat a preselectedset of points using a modified Gram-Schmidtprocedure. A

Newton-Raphsontechniqueis usedto iterateon the eigenvalueto satisfythe last wall boundary

condition within a specifiedaccuracyof O (10-5).

4. RESULTS AND DISCUSSION

For the Blasius and Falkner-Skan profiles, our results were checked with that of Bertolotti

[26], and found to be in full agreement.

All results reported here are for the nondimensional frequency F=106¢o/R=60, that remains

fixed as a wave with fixed physical frequency travels downstream. For Blasius flow, a primary

TS wave with this frequency grows between R1=554 and Rn=1052 (first and second neutral stability points), reaching a maximum amplification factor of A/Ao=42. Also in this region, primary

3D subharmonic waves are subject to amplification for a broad band of spanwise

wavenumbers, but the time and length scales of these instabilities are too small to compare

with experimentally observed transition. A strong growth of secondary 3D subharmonics can

be due to parametric excitation by the finite amplitude primary TS wave [20].

4.1. Fiat Plate Boundary Layer Controlled by Suction

At R=1050 ( almost at R# for Blasius flow), Fig (1) shows the growth rates of the secondary 3D subharmonic disturbances as function of the spanwise wavenumber parameter b and

for various amplitudes A of the primary wave. The parameter b defined as b=103_/R

represents a fixed physical spanwise wavenumber for a wave traveling downstream. Figure (1)

compares the results of a flat plate boundary layer with suction (r'0=-.1) and with no suction.

At fixed I'0, the figure indicates a stabilizing effect on the secondary instability as the primary

amplitude A decreases. When the amplitude is very small, considerable growth rates exist in a

small band of spanwise wavenumbers. The maximum growth occurs for a wave with slightly

lower spanwise wavenumber as the amplitude A decreases. This shift to lower spanwise
- 10-

wavenumberis more noticeable with the increase of suction parameter. As Reynolds number

increases, the growth rates of the secondary disturbances increase at fixed F and A.

The application of active control by suction in LFC systems restrains both the mean-flow

profile and the primary wave. We find that the prime influence of suction on the secondary

instability is due to the appreciable decay of the amplitude of the primary wave. This conclusion is explained next with the use of Fig. (2). In Blasius flow, a primary wave with initial

amplitude A0 =0.00025 at Rt grows such that at R =1050 downstream its amplitude reaches a

value of 0.01. Figure (2), curve a, shows a wide band of 3D subharmonics amplifying at that

location with the most unstable at b = 0.17. The influence of a modified mean flow can be

demonstrated by applying suction at R = 1050 with fixed A=0.01. This decreases the growth rate

of the subharmonic and slightly limit the band of amplified spanwise wavenumbers (curves b

and c), while the most unstable wavenumber is not affected by suction. When suction is

applied ( F0---0.05) starting upstream at R_, the mean flow is modified and the growth of the

primary wave is slowed down, such that at R = 1050 downstream the amplitude of the primary

a value of 0.0015. Calculations, curve d, indicate a strong stabilizing effect on

instability with the most unstable 3D subharmonic occurring at a lower spanwise

wave reaches

the secondary

wavenumber.

Previous results may be viewed as local, they only reflect the stability characteristics of

the secondary subharmonic disturbance at a fixed Reynolds number. To model the experiment

and evaluate the overall effect of suction on the onset, growth rate, and amplification factor of

the secondary 3D subharmonic, we should combine the effect of increasing the amplitude A of

the primary TS wave as well as increasing R as the disturbance moves downstream. In Fig

(3), we do that and show the variation with R of the growth rates of the secondary subharmonic wave. In these calculations, the initial amplitude of the primary wave is assumed

A0=.001, and the spanwise wavenumber b=.15 which is an average value of the most unstable

wavenumbers for the parameters under investigation. The growth rates of the corresponding

primary waves are also included in the figure for comparison.
-11-

The micro-eventswhich lead to the secondary 3D instability go as follows. Initially, the

primary instability sets in at RI on the primary neutral curve, and -oq starts to grow. At a certain value of the primary amplitude, a secondary 3D mode is induced and sets in at g_ on a

secondary neutral curve. _'r starts tO grow strongly due to the increase in both A and g. Ultimately, -ai begins to decay, and the primary amplitude reaches a maximum when -t_=0.

Shortly downstream, "t, reaches a maximum and then starts to decay with the secondary amplitude reaching its maximum when _,,=0. The overall effect of small suction rates is to delay the

onset of the secondary instability (occurs later downstream), and to decrease significantly its

growth rate. Calculated amplification factors for the secondary disturbance (not shown) indicate 71% reduction due to the increase of r0 from 0 to -.05. Increasing r0 to -.1 dampened

completely the subharmonic secondary disturbance in spite of considerable growth shown by

the primary wave. We know from the experiment [9] that the secondary instability depends on

the primary wave amplitude as well as on the wave fetch. In a fixed disturbance environment,

small suction rates may not affect the initial amplitude of the primary wave but certainly will

influence its growth. Our calculations in Fig (3) indicate a delay in the onset of the secondary

instability due to suction. This delay is accompanied by a slight decrease in the primary threshold amplitude (equal to .0029 at r0=0 and .0024 at to=--.05). However, at to=--.1, the primary

amplitude reaches a maximum of only .0014 which is apparently below the value needed to

induce a secondary subharmonic instability. Notice that the initial amplitude of the primary

wave is fixed in all cases and equal to .001.

Figure (4), a case of suction rate r'0=--.05, shows a primary instability that sets in at R :650.

The onset of the subharmonic instability occurs at R=850 when A0=.00Â, at R=775 when A0=.002,

and at R:635 when A0=.0066 (which is well before the onset of the primary wave), reaching

maximum amplification factors of 8, 15, and 30 respectively. Note that when the initial amplitude A0 is large enough, the induced instability can be so strong and secondary instability

occurs directly by-passing the usual growth of the primary wave. In a situation like this, transition prediction schemes based on linear primary theory fail completely.
- 12-

In boundary-layerflows, the onsetof the secondary3D instability is known to be an

importantfeatureof the early stagesleadingto transition. For LFC purposes,one might try to

completely avoid or delay this instability by using suction. Then, one faces questionslike

whereto apply suction,and whetherit shouldstartbeforeor after the onsetof the secondary

instability, and whatis theoptimum suctionneededto keepthe flow laminar. To answerthese

questions,calculationswere madeto comparethe stability characteristicsof different casesof

Blasiusflow with continuouslyappliedsuctionstartingat five differentlocations. Theseresults

arereportedin Fig (5) which showsthe variation with R of the growth rates of both the primary and secondary disturbances at fixed initial amplitude of the primary wave (A_--0.001).

Calculations show that in order to control the secondary 3D instability, suction should be

applied further upstream near Rt of the primary wave and not to apply it near the onset of the

secondary instability. While investigating the effect of suction on primary TS waves, Reed and

Nayfeh [27] and Saric and Reed [28] reached similar conclusions that suction should be concentrated not in the region of maximum growth but near its first neutral stability.

Figure (6) shows the variation with suction parameter r'0 of the maximum growth rates

with respect to R of both the primary wave and secondary subharmonic disturbance for A0=.001

and b=.15. Point P indicates a suction level that completely stabilizes the primary wave, while

point S indicates a suction level that allows for a limited primary growth but completely stabilizes the secondary disturbance ( point S is extrapolated due to doubtful Floquet theory results

when the secondary growth rates are small). An optimum suction requirements for an LFC

system may be located somewhere upstream of point S allowing for a considerable growth of

the most unstable primary wave, as well as a limited growth of the secondary disturbance as

long as the primary amplitude is small and the induced secondary disturbance is not strong

enough for nonlinear self- and cross-interaction with the primary wave. As A0 increases, point

S is expected to move towards point P and cross it over for high enough primary initial amplitudes. In such situation, suction level required to fully stabilize the primary wave may not

control the amplification of the secondary disturbance, and very high suction levels are needed
to control the flow.

- 13-

4.2. Falkner-Skan Flow Controlled by Suction

The need to control the secondary instability in a boundary layer with pressure gradient is

more realistic such as the case of flow on airfoils. Herbert and Bertolotti [29] studied the

secondary instability of Falkner-Skan flows and found that favorable pressure gradient limits

the band of unstable spanwise wavenumbers, while small adverse pressure gradient is strongly

destabilizing. This means that larger suction rates are needed to control the boundary layer in

these cases. For a boundary layer with pressure gradient and suction, the variation of the

growth rates of the secondary 3D subharmonic with the spanwise wavenumber parameter b

exhibits the same features given before in Fig(l).

Figure (7) gives an overall view of the effect of both pressure gradient and suction parameters on the secondary 3D subharmonic at fixed A0 using the maximum amplification factor as

a basis for comparison. As adverse pressure gradient increases, the secondary subharrnonic

disturbance becomes more unstable. With the suction parameter increasing, the curves converge rapidly to lower amplification factors indicating that the secondary instability is more

sensitive to suction as adverse pressure gradient increases. Figure (7) shows also the increase

in the maximum amplification factors for different values of A0 at [_o=0. The sensitivity of the

secondary instability to suction appears to be higher with the increase in adverse pressure gradient than the increase in A 0.

4.3. Mean Profile, Mode Shape, and Vortical Structure

Modifications to the Blasius profile due to suction and adverse pressure gradient are

shown in Fig (8). Suction leads to a fuller mean u-velocity profile and decrease in the magnitude of the V-velocity. While adverse pressure gradient makes the U-velocity profile more

inflectional. At fixed amplitude of the primary wave, these modifications have moderate effect

on the secondary instability, see Fig (2) for the case of suction, and Ref. 27 for the case of
- 14-

pressure gradient.

Figure (9) gives a comparison of the mode shapes of the u-, v-, and w-velocity components of the secondary disturbance at different suction and pressure gradient parameters.

The v-component is small and is slightly affected by suction or pressure gradient. The

corresponding mode shapes of the u, and v-velocity components of the primary wave are

shown in Fig (10) for comparison. Suction tends to move the critical layer closer to the wall

with the maximum of the primary wave velocity components and the maximum of the secondary disturbance velocity components following it. The location of the critical layer on the 11-

axis is marked in both figures. By increasing suction, the thickness of the boundary layer

decreases, and the disturbance is confined to a region closer to the wall where dissipative

action is strong, thereby increasing the stability of the flow. As adverse pressure gradient

increases, the opposite occurs and a change over from viscous to inflectional instability takes

place [29].

A more detailed description of the physics of the process of suction control can be

obtained from contour plots of the vorticity components. Figures (11)-(14) give contours of

vorticity components in a wave-fixed coordinate system for different cases of suction and pressure gradients at R=1050. In each figure, frames (a)-(d) show the effect of the mean-flow

modification on the vorticity contours (a=0.04 and B=I are fixed), while frame (e) shows the

total effect of the suction on the vorticity contours due to modifications in the mean flow and

the reduction in the primary amplitude (A =0.OO6 and 8 =0.005). These values of ,4 and a are calculated at R=1050 assuming a0=0.001 at the onset of the primary wave, and Bo=lxl0 "_ at the

onset of the secondary subharmonic. Comparison of frames (a) and (d) in each figure shows

that suction with re=--.2 applied to an inflectional profile with _=.19 qualitatively produces the

same vorticity contour plots as for a blasius profile.

An array of streamwise-periodic concentrations of vorticity is established by the primary

viscous instability. The strength of the vorticity intensifies with the increase of the amplitude of

the primary wave. Figure (11) shows plots of the initial 21) vorticity contours of the basic
- 15-

flow ( only spanwise component exist), which peaks near the wall, and extends to the critical

layer. The location of the critical layer is defined by tick marks on the Tl-axis.

With the onset of the secondary instability, 3D vortical structure is induced by the deformarion of the initial 2D vorticity. Figure (12) shows plots of the spanwise vorticity L-contours

in the x-y plane at z=0 for different cases of suction and pressure gradient. The figure that is

plotted over four primary wavelengths (L_=2_a), indicates that regions of concentrated spanwise vorticity are convected downstream, pulled towards the wall as suction increases, and

lifted away from the wall as adverse pressure gradient increases. In both situations, the concentrated vorticity follows the critical layer as the suction and/or pressure gradient changes.

The spanwise-velocity variations produce streamwise vorticity (_ that is contoured in Fig

(13). The figure is plotted in the z-y plane at x=0 over two spanwise wavelengths of the

secondary disturbance (_=2_13). It shows an array of counter-rotating vortices extending away

from the wall as adverse pressure gradient increases, and pulled towards the wall as suction

increases.

Following the onset of the secondary instability, spanwise and streamwise vortices experience a process of stretching and tilting as they move downstream. Together, they form a

large scale 3D structure (A-shaped) which was observed experimentally and numerically. For

different suction and pressure gradient parameters, Fig (14) shows the deformation in the total

vorticity (-_'_) in an x-z plane almost at the _-location of the critical layer.

5. CONCLUDING REMARKS

Calculations show that the secondary 3D subharmonic instability is very sensitive to and

can be controlled by suction. Meaning, the onset of the instability is delayed, the growth rates

and amplification factors are reduced, the unstable band of spanwise wavelengths is limited,

and the vortical structure is closer to the wall. As adverse pressure gradient increases, the sensitivity to suction increases. For higher initial amplitude of the primary wave, suction is less

effective in controlling the secondary 3D subharmonic instability.
- 16-

Calculationsshowthat eliminationof the primary wave by suction at an arbitrary stage of

its growth might not delay or eliminate the secondary instability specially if it has already

taken place. The suction need be applied near Rz of the primary wave to gain more control on

the secondary growth. An optimum suction amount may be lower or higher than the suction

needed to fully eliminate the primary wave. This is very much dependent on the initial amplitude of the primary wave.

Application of suction influences both the mean profile and the primary wave. The effect

of a modified mean profile on the secondary 3D instability is moderate, while the main effect

is due to enhancing the growth of the primary wave. By applying suction and including both

effects, calculations indicate that the most unstable secondary subharmonic disturbance occurs

at lower spanwise wavenumber. Then, at certain downstream location, an observed flow structure of the H-type might be altered to C-type with larger spanwise wavelength or might completely disappear as suction increases.

Evaluation of the effect of suction on the subharmonic secondary instability is a step

towards the goal of optimizing an LFC system. The idea of monotoring the 3D A-shaped

structure as its spanwise wavelength changes with suction may be used for that purpose.

ACKNOWLEDGEMENT

This work was supported by NASA Langley Research Center, under contract NAS1-18599.

APPENDIX : Nonzero Elements of the Coefficient Matrices a,_, b,_, and c_ of Eqs (18)

and (21) :

a12 = 1, a21 =/g (t_U-_)q-0_ 2, a23 = R DU , a24 = iotR ,

a31 = -i Ct, a42 = -i WR , a43 = -i (otU-f.o)-(x2/R

b12 = 1 , b21 = 1 , b23 = RDU ,

1

b26 = (V+_itxr)R $
- 17-

b3x =-bEd R , b34 = -_ , b45 = 1

1 R 1. 2

b54 = ('l+ 2 i ctr )U--_i 03) - ()q-_t a,. )

b56 = -R _ , b62 = b31/R , b63 = -b54/R ,

+ 82

b65 = -_/R ,

,

C21 = ('yq--_tO_r)Ru I , C22=RVl ,

3.

css = R vl , c61 = ('r--z_a,)v_ ,

Z.,

c23 = R Du 1 ,c54 = (Y"-2iCtr)Rul ,

c 63 = -c 54/R -Dv I , c 64 = _v I

REFERENCES

1. W. D. Harvey and J. D. Pride, The NASA Langley Laminar Flow Control Experiment.

AIAA Paper 82-0567 (1982).

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AIAA Paper 79-1493(1979).

3. W. S. Saric, Laminar Flow Control with Suction: Theory and Experiment. AGARD Report

No. 723 (1985).

4. I. Tani, Three-Dimensional Aspects of Boundary-Layer Transition. In Proceedings of Indian

Academy of Science 4, 219 (1981).

5. S. A. Orszag and A. T. Patera, Secondary Instability of Wall-bounded

Shear Flows. J. Fluid Mech. 128, 347 (1983).

6. Th. Herbert, Secondary Instability of Plane Channel Flow to Subharmonic ThreeDimensional Disturbances. Phys. Fluid 26, 871 (1983).

7. P. S. Klebanoff, K. D. Tidstrom, and L. M. Sargent, The Three Dimensional

Nature of Boundary Layer Instability. J. Fluid Mech. 12, 1 (1962).

8. L. S. G. Kovasznay, H. Komoda, and B. R. Vasudeva, Detailed Flow Field in Transition. In

Proc. Heat Transfer and Fluid Mech. Institute, Stanford University Press, 1-26 (1962).
- 18-

9. Yu. S. Kachanov and V. Ya Levchenko, Resonant Interactions of Disturbances in Transition

to Turbulence in a Boundary Layer. J. Fluid Mech. 138, 209 (1984).

10. W. S. Saric and A. S. W. Thomas, Experiments on the Subharmonic Route to Transition.

In Turbulence and Chaotic Phenomenon in Fluids, ed. T. Tatsumi, North Holland (1984).

11. W. S. Saric, V. V. Koslov, and V. Ya. Levchenko, Forced and Unforced

Subharmonic Resonance in Boundary-Layer Transition. AIAA Paper 84-0007 (1984).

12. A. S. W. Thomas, Experiments on Secondary Instabilities in Boundary Layers. In Proc. of

10th US Nail. Congr. Appl. Mech., Austin, Tex, 436-444 (1987).

13. A. Wray and M. Y. Hussaini, Numerical Experiments in Boundary Layer

Stability. Proc. Roy. Soc. London A, 392, 373 (1984).

14. T. A. Zang and M. Y. Hussaini, Numerical Experiments on Subcritical transition Mechanisms. AIAA Paper 85-0296 (1983).

15. P. R. Spalart and K. S. Yang, Numerical Study of Ribbon-Induced Transition in Blasius

Flow. J. Fluid Mech. 178, 345 (1987).

16. H. F. Fasel, U. Rist, and U. Konzelmann, Numerical Investigation of the ThreeDimensional Development in Boundary Layer Transition. AIAA Paper 87-1203 (1987).

17. T. A. Zang and M. Y. Hussaini, Numerical Experiments on the Stability of Controlled

Shear Flows. AIAA Paper 85-1698 (1985).

18. L. Kleiser and E. Laurien, Numerical Investigation of Interactive Transition Control. AIAA

Paper 85-0566 (1985).

19. B. Siger, H. L. Reed, and J. H. Ferziger, Effect of Streamwise Vortices on Transition in

Plane-Channel Flow. Phys. Fluids A, 1, 1960 (1989).

20. Th. Herbert, Analysis of the Subharmonic Route to Transition in

Boundary Layers. AIAA Paper 84-0009 (1984).

21. A. D. D. Craik, Nonlinear Resonant Instability in Boundary Layers. J. Fluid Mech. 50, 393

(1971).
- 19-

22. Th. Herbert, SecondaryInstability of BoundaryLayers.Ann. Rev. Fluid Mech. 20, 487

(1988).

23. B. J. Bayly, S. A. Orszag,andTh. Herbert,Instability Mechanismsin Shear-FlowTransition. Ann. Rev.Fluid Mech.20, 359(1988).

24. J. M. Price and J. E. Harris, ComputerProgramfor Solving CompressibleNonsimilar

Boundary-Layer Equations for Laminar, Transitional, or Turbulent Flows of a Perfect Gas.

NASA TM X-2458 (1972).

25. M. R. Scott and H. A. Watts, Computational Solutions of Two-Point Boundary-Value Problems Via Orthonormalization. SIAM J. Num. Analy. 14, 40 (1977).

26. F. P. Bertolotti, Temporal and Spatial Growth of Subharmonic Disturbance in Falkner-Skan

Flow. M.S Thesis, VPI&SU, 1985.

27. H. L. Reed and A. H. Nayfeh, Stability of Flow Over Plates with Porous Suction. AIAA

Paper 81-1280 (1981).

28. W. S. Saric and H. L. Reed, Effect of Suction and Blowing on Boundary Layer Transition.

AIAA Paper 83-0043 (1983).

29. Th. Herbert, and F. P. Bertolotti, Effect of Pressure Gradients on the Growth of Subharmonic Disturbances in Boundary Layers. In Proc. on Low Reynolds Number Airfoil Aerodynamics, Univ. of Notre Dame, ed. T. J. Mueller (1985).
- 20 -

.O3O

.O25

.020

.015

_'r .010

.OO5

0

Fo=O

FO=-I

b

Fig. 1 Effect of the amplitude a of the primary wave on the growth rate of the

secondary 3D subharmonic at different suction parameters, R = 1050, and F=60.
-2,1 -

l

C)

00

O

O

I i I I t i I I I _,

I..r) O _ O 1./')

O,1 (X,l -r- -r-- C)

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e,t O

°*,-4

cJ II
- 22 -

Yr .02

Fo=O

Secondary

Primary

0

-.01 -.10-_

-.O2 I I I

500 700 900 1100 1300 1500

R

Fig. 3 Effect of suction parameter on the onset and growth rates of the

primary wave and secondary 3D subharmonic, Ao=.O01, b=.lS, and F=60.
- 23 -

30 A0=

25

In B/B 0

20

15

10

A 0 = .002

A0= .001

0

t tt

-5 I I

500 700 900 1100

R

..//-- Primary

%

%

I I

1300 1500

Fig. 4 Effect of the initial amplitude Ao of the primary wave on the onset and

amplification factor of the secondary 3D subharmonic, dotted arrow: onset of the primary, solid arrow: onset of the secondary subharmonic, 1"0=-.05, b=.lS, and F=60.
-24-

growth

rate,

X 103

45-

40

Suction

starts at

35-

1

30- 2

3

4

25- 5

N

20-

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10_ i 2

Secondary

Primary

R

5OO

55O

65O

712

79O

No suction

5

4

3

90O

R

5

4

3

2

-1

!

1300

Fig. 5 Continuous suction starting at different locations and its effect on the onset and

growth rate of both the primary wave and the secondary 3D subharmonic, Ao=.001,

b=.15, and constant suction level (corresponding to ['o =- .05 at R = 1000).
-25-

Max

growth

rate

xlo 

25

2O

15

10

5

0

-5

-10

Secondary

Primary

Over suction

i I i I I I

0 .05 .10 .15

-r o

Fig. 6 Variation of maximum growth rate with the suction parameter for both the primary wave and the secondary 3D subharmonic, Ao=.001, b=.lS, and F=60.
-26-

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Report Documentation Page r_LtlL_%_ _rL_,aul_ _ arkl

1. Report No,

NASA CR-4306

2. Government Accession No,

4, Title and Subtitle

Effect of Suction on Controlling

Instability of Boundary Layers

the Secondary

7. AuthorIs)

Nabil M. EI-Hady

[nc.

9. Performing Organization Name and Address

Analytical Services & Materials,

Hampton, VA 23666

12. Sponsoring Agency Name and Address

Space Administration

3. Recipient's Catalog No.

5. Report Date

July 1990

6. Performing Organization Code

8, Performing Organization Report No.

National Aeronautics and

Langley Research Center

Hampton, VA 23665-5225

10, Work Unit No.

505-60-21-06

11. Contract or Grant No.

NA51-18599

13. Type of Report and Period Covered

Contractor Report

14. Sponsoring &gency Code

15, Supplementa_ Notes

Lanqley Technical

Final Report

Monitor: Ajay Kumar

16. Abstract

The effect of suction on controlling the three-dimensional secondary instability

is investigated for a boundary layer with pressure gradient in the presence of

small but finite amplitude Tollmien-Schlichting wave. The focus is on principal

parametric resonance responsible for strong growth of subharmonics in low disturbance

environment. Calculations are presented for the effect of suction on the onset

and amplification of the secondary instability in Blasius and Falkner-Skan flows,

as well as its effect on controlling the production of the vortical structure.

17. Key Words (Suggested by Author(s))

Boundary Layer

Secondary Instability

Suction Control

18. Distribution Statement

Unclassified-unlimited

Subject Category 34

19. SecuriW Cla_if. (of this report)

unclassified

20. Security Classif. (of this page)

unclassified

;21, No. of pages

40

22. Price

A03

NASA FORM 1626 OCT 86

For sale by the National Technical Information Service, Springfield, Virginia 22161-2171

NASA-Langley, 1990