AI AA-93-3053

Studies of Boundary-Layer Receptivity with Parabolized Stability Equations

Thorwald Herbert, The Ohio State University, and Nay Lin, DynaFlow, Inc. Columbus, Ohio

AIAA 24th Fluid Dynamics Conference July 6-9, 1993 / Orlando, FL

For permission to copy or republlsh, contact the American instlute of Aeronautics and Astronautics 370 L‘Enfant Promenade, S.W., Washington, D.C. 20024

STUDIES OF BOUNDARY-LAYER RECEPTIVITY WITH PARABOLIZED STABILITY EQUATIONS

Thorwald Herbert' Department of Mechanical Engineering

The Ohio State University Columbus, Ohio 43210, USA

Nay Lint DynaFlow, Inc.

Columbus, Ohio 43221, USA

Abstract fication of input models characteristic of the disturbance environment.

Transition prediction with DNS or PSE requires spec- ~~~i~~ the past years, the PSE approach has been ification of a model environment that affects the tram extended to compressible flow, three- sition process through initial and boundary conditions. dimensional boundary layers, and realistic geometries Usually, this environment is specified in terms of nor- with full account for curvature effects4. As projected mal modes at. an initial position. This Specification is by D. Bushnell at the 29th Aerospace Sciences Meeting, based on empiricism and is often inappropriate. To free the PSE code is mature enough to the ,N codes the transition analysis from empiricism, it is necessary based on approximations to the local linear stability the- to specify the environment in more physical terms and ory. Transition studies on various idealized and practical to incorporate local and area-distributed receptivity as configurations have been performed to further validate part of the analysis. In principle, the PSE are capable to the concept and to explore the utility as an engineer- L/ deal with linear and nonlinear receptivity mechanisms. ing method for advanced design. investigation of the We demonstrate this capability by analyzing the origin effects of turbulence level and curvature on transition of Klebanoff modes, their connection to Gortler vortices, in the flow over a turbine blades has yielded the first and the forcing of cross-flow vortices in swept Hiemenz prediction of the transition point in realistic geometry flow. Our study serves to refine the PSE and to extend on purely analytical grounds. The formulation and ap- their scope as a tool for studies on receptivity, stability, plication of the input model in this study has revealed, and transition. however, that deeper understanding and improved ana-

lytical capabilities for receptivity mechanisms are highly

1 Introduction desirable. Traditionally, numerical studies of transition specify

The basic elements of the transition process in bound. the initial data at the inflow boundary of the COmpUta- ary layers and avenues to improve engineering methods tional domain in the form of normal modes provided by for transition analysis have been discussed by Herbert', linear stability analysis or as random noise. T O shorten Based on detailed results for the flat-plate boundary the run time Of DNS, the flow is typically unstable at layer2, 3, he concluded that the parabolized stability the inflow boundary. Moreover, most of these studies are equations (PSE) show good potential as an efficient re. guided by experiments such that type and magnitude of search and engineering method for transition analysis the initial data, stability characteristics, and transition and prediction under given initial and boundary location can be considered known. In the PSE analysis tions, this framework, emphasis shifts from the speci. of realistic configurations, the computational domain is fication ofN factors as a transition criterion to the speci- the boundary layer downstream of some initial position

In absence of experimen- tal guidance, only the stability characteristics can be analyzed. Stability theory can be employed t o vaguely identify most likely scenarios and the type and scales of initial data that will govern transition. The magnitude of the initial disturbances is uncertain, however, and de-

as sketched in figure 1. *Professor of Mechanical Engineering

tResearch Scientist, Member A I A A

Copyright 0 1993by Th. Herbert. Published by the American

President, DynaFlow, Inc., Senior Member A I A A

Institute for Aeronautics and Astronautics, Inc. with permis- v sion.

1

pends on the receptivity of the flow to environmental disturbances in the free stream and on the wall up to the starting position &. An additional problem is the streakiness of boundary-layer flows a t higher turbulence levels6, 7, which affects the transition process through secondary instabilityg yet can neither be explained nor specified by normal modes.

A second major shortcoming of the initial-dataspecifi- cation was first observed by Rai & Moin” in their transi- tion simulation for a flat plate with an inflow boundary upstream of the leading edge. Random noise of 7.6% was necessary t o obtain a 2.4% turbulence level at the leading edge that further decayed before transition oc- curred. In our analysis of the flow over a turbine blade5, the strong stability of the boundary layer in the accel- erated flow region on the suction side reduced the am- plitude of initial disturbances by eight orders of mag- nitude before they started to grow as the flow deceler- ated. Since in this case the flow near the edge of the boundary layer maintains a turbulence level of about 0.5%, such decay of eigenmodes is unrealistic, and larger disturbances would be generated by area-distributed re- ceptivity. The concept of area-distributed receptivity as sketched in figure 2 has been employed by Kendall” to explain some key observations. Crouch12 h a s theo- retically studied the distributed or non-localized recep- tivity of the boundary layer flow over a wavy wall to sound. In this case, distributed receptivity arises form disturbances both at the wall and the edge of the bound- ary layer in absence of inhomogeneous initial conditions. The receptivity along the boundaries (ukec) can obvi- ously maintain disturbances even if initial disturbances (u1) decay.

In contrast to the local stability analysis which ex- ploits the homogeneity of the problem for eigenvalue calculations, the PSE permit inhomogeneous boundary conditions. Thus, it is possible to implement known re- ceptivity mechanisms by proper models for initial and boundary conditions. I t is also desirable to extend the catalog of known mechanisms to better account for the various environmental influences in figure 1. Existing transition data can be utilized to adjust the few param- eters in these models to characterize “standard environ- ments” similar t o the standard atmosphere.

For Yeceptivity t o occur, external disturbances should provide energy at frequencies and length scales rele- vant to instability and transition mechanisms. The length scales are provided by spatial variations of the disturbances and/or the basic-state boundary layer flow. Earlier l4 therefore formulated initial- boundary-value problems for situations where the ex- ternal forcing provides energy directly at instabil- ity wavenumbers. Studies based on high-Reynolds number expansion^'^, ‘ 6 17, finite-Reynolds number lo- cal approaches”, ’’, experiment^'^, ’”, and numerical simulations” give insight into the receptivity to long-

2

wave disturbances and the scattering of sound waves by streamwise variations.

has been first addressed by Fedorov2’. He studied the

L l excitation of steady cross-flow vortices in subsonic com- pressible boundary layers over swept wings and found that micro-roughness or weak, spanwise periodic suc- tionfblowing can excite cross-flow vortices of substan- tial amplitudes. Radeztsky et aLZ3 performed low-speed experiments on a swept wing and report a strong effect of small roughness elements on transition. Crouchz4 ex- tended his local nonlinear approach to the receptivity of the Falkner-Skan-Cooke flow to steady and unsteady cross-flow vortices and obtained results consistent with the experiments of Muller & BippesZ5. Receptivity to cross-flow vortices is an important aspect for the swept wings of airplanes where cross-flow instability may be the dominant cause of transition.

The receptivity studies so far were focussed on the generation of normal modes associated with the eigen- values of the stability equations. Recent studies in plane channel flow and other strictly parallel flowsz6, ’’, ”, ” revealed that strong transient energy growth can occur in these flows although they are stable according to lin- ear stability theory. This energy growth is linked to the known bypass-transition phenomena in plane Couette and channel flow and favors steady, spanwise periodic disturbances in twc-dimensional flows. For a parallel Blasius boundary layer at fixed Reynolds number, But- ler & Farrel” have shown optimal energy growth for streamwise vortices by more than three orders of mag- ‘d nitude. Key to this mechanism is the non-self-adjoint nature of the three-dimensional stability problem and the evolution of its solution in time.

In the following we report on some PSE studies on receptivity for unsteady and steady disturbances in twc- and three-dimensional boundary layers. Emphasis is on the appearance of Klebanoff modes owing to transient energy growth and their connection to Gortler vortices. As a related topic, we analyze the generation of steady crossflow vortices by roughness elements or spanwise periodic suction/blowing.

The receptivity of three-dimensional boundary layers , ,

2 PSE in Natural Variables We consider the receptivity and stability of steady, lam- inar, incompressible boundary-layer flows over the plane y = 0 with edge velocities U.(I) and We(z ) in the I and z directions, respectively. We decompose the total flow field v , p into the steady laminar basic flow V, P and disturbances v’ = (u’, u’, tu’), p’:

Owing to the uniformity in L, the basic flow takes the form v = ( U ( ~ , y ) , V ( z , y ) , W ( z , y ) ) . For boundary -

layers, V , U,, and W, are small and their derivatives Us,, V,, W,, with respect to x are negligible. Of particu- lar interest for theoretical studies are the exact similarity solutions such as the two-dimensional Blasius, Falkner- Skan, or Hiemenz flows or the three-dimensional Falkner- Skan-Cooke or swept Hiemenz flows. Little modification is required to extend the list of basic flows to mixing layers, two-dimensional or axisymmetric jets and wakes, or to numerically generated nonsimilar solutions of the boundary-layer equations.

For nonparallel flows, the normal-mode solutions are not valid since the coefficients of the stability equations depend weakly on I. Following a WKBJ analysis, we decompose the disturbances q' = (u', u', w' , p') into an amplitude function q and a wave function x :

d ( " , Y , Z , t ) = q ( z , Y ) x ( Z , z , t ) , (2)

group contains the streamwise derivatives of the ampli- tude functions (u ,u , w , p ) . One of these derivatives also appears in the continuity equation (4a). The last group originates from the streamwise changes of the wavenum- ber a.

For a proper function a(.), the streamwise variation of q' is governed by the wave function x while the deriva- tives q. and az are small. The PSE approximation as- sumes the variation of q and a as sufficiently small to neglect the terms vzz on the right hand side of eqs. (4).

It is convenient to rewrite the parabolized eqs. (4) in the compact form

( 5 ) aq da ax d x

L q + M- + - N q = 0 ,

where the operators L , M , and N act only in y and

(6) aL 1 a M i a 2 L aa 2 aa 2 da2 '

x = exp(i6'(z) + ipz - iwt) , 9, = a(.) . (3) M=-i - , N = - - = - - - The exponential factor x with wavenumbers a and /3 in x and z , respectively, and frequency w describes the wave nature of the solution.

Substituting eqs. ( 1 ) to (3) into the Navier-Stokes equations and extracting the linearized disturbance equations provides in nondimensional form

i a u + Du + ipw + uz = 0 (4a)

+ {Uzu + V D u ] U

1 1 {zr 1 Re 2ia { Re + (U - --)ut + p z - --u = -uzr (46)

For the study of instabilities in a semi-infinite domain, we apply homogeneous boundary conditions

u = O , u = O , w = O a t y = O (7)

u - 0 , u - 0 , w - O a s y - 0 0 (8)

since any inhomogeneous conditions are satisfied by the basic flow. Inhomogeneous boundary conditions are key to the study of forced oscillations and receptivity prob- lems.

Eqs. ( 5 ) are inherently nonlinear since a and da/dx are a priori unknown. Similar to the parabolized Navier- Stokes equations (PNS), the PSE exhibit a small residual ellipticity3'. The equations can be solved with marching

I .

techniques as an ill-posed initial-boundary-value prob- lem provided the step size is sufficiently large to "skip" over the weak and rapidly decaying upstream propa- gation of information. Alternatively, additional small terms (like p z in eq. (4b)) can be neglected to arrive at a well-posed parabolic problem that can be solved without restriction on the size of the marching step.

Solving eqs. ( 5 ) requires an additional condition to remove the ambiguity in the partition (2) which would permit the incorporation of streamwise changes of the wave function x into q. In absence of any information

--[LIZ - (a2 + /3')]u + i (aU +OW - w ) u { ;e

+ D p + {DVu + VDu}

1 } - { gu} = ,u,,

1 { Re (4c)

2ia + ( U - - )us

- (a2 + P2)]w + i (aU + pW - w ) w

on streamwise changes, we can impose the restriction +DWu + iPp) + { W,u + V D w }

2ia 1 (4d) and absorb all streamwise changes into q(z,y). Once a . . .

first approximation on streamwise changes is known, we , impose a streamwise independent norm on q. We con- sider the complex wave number given by the logarithmic derivative

(10) -i(ln v ' ) ~ = a - z- ,

where D = a / a y . The left-hand side of eqs. (4b) to (4d) contains four groups of terms enclosed in braces. The first group are the terms retained by the normal-mode equations for parallel flow. The second group originates from the streamwise changes of the basic flow. The third

. Vr \c/ V

3

where v causes in general a dependence of the local wave number on y. To remove this dependency, we multi- ply eq. (10) with the weight Iv12, integrate over the do- main R in y, and divide by the kinetic energy integral Sn Iv12dy:

where the dagger denotes the complex conjugate. We now choose

to normalize v and consider eq. (11) the definition of

Equations (5) together with the norm (12) and suit- able initial and boundary conditions are the parabolized stability equations (PSE) which permit calculation of a(.) and q(z, y) in a streamwise marching procedure.

2.1 A Note on Results Similar to DNS, the solution of the PSE refers to fixed scales U; and 6; = 6:(20) for velocity and length, where 6: (2) = ( V * Z * / U ; ) * / ~ and z = 2’ /6f ; (the star denotes dimensional quantities). The Reynolds number Reo = U ; ~ ; / V ’ is fixed. The position 20 is not necessarily the initial position zI. In contrast, local stability analyses commonlyuse the local length scale 6: and the Reynolds numbers Re, = V:X*/V*, Re = The frequency w in the PSE describes a wave of fixed physical frequency, and there is no need to refer to the usual nondimensional frequency F = & / R e , where we use the caret to indicate local scaling with 6:(z). Hence

4.).

Since 2 is scaled with 60. we have for Blasius flow the - . relations

Re, = zReo , Re = m. In nonparallel flow, additional corrections to & may be necessary to account for specific quantities and/or posi- tions y.

In general discussions, we report the amplitude A = (Sn lvl2dy)’/*. For comparison with experiments, we convert to the local magnitude of velocity components (for steady modes) or the rms value (for oscillatory modes).

3 Klebanoff Modes Klebanoff & Tidstrom3I used a low-turbulence wind tun- nel to study the evolution of transition from ribbon- induced T S waves in a flat-plate boundary layer. They found the experiments biased by the occurrence of a

nearly periodic spanwise modulation of the mean flow with amplitudes up to 5% and a wavelength A, % 56 as shown in figure 3. The modulation was confined to the boundary layer and not discernible in the free stream. Tedious search ultimately tracked this distortion to be influenced by the damping screens in the settling cham- ber far upstream.

In a later experiment with new damping screens, Kle- banoff et aL3* obtained a mean flow with barely de- tectable variations. Nevertheless, the ribbon-induced TS waves revealed spanwise variations of the same wave- length A, as before. For detailed studies, the spanwise mean-flow modulation was then enhanced by placing strips of cellophane, A,/2 long and A,/2 apart, on the surface.

Investigating the effect of free-stream turbulence, Klebanoffd found the spanwise modulations of the mean flow “composed predominantly of low frequencies which are stable according t o stability theory.” This descrip- tion is clearly supported by spectra like the one shown in figure 4. Klebanoff attributed the modulations to a thickening and thinning of the boundary layer.

“Klebanoff modes” were also found in other boundary- layer experiments. Kendall7, l 1 has performed a series of studies on pre-transitional phenomena. He finds Kle- banoff modes with A, % 26 that grow with z1l2 to am- plitudes of 5% before the onset of T S instability. Similar modulations have been observed in channel flow 33 where the amplitude varies slowly. The modulations in channel flow are associated with the very weakly damped longi- tudinal vortices and u-modes 34 that are eigensolutions of the Orr-Sommerfeld and Squire equation, respectively, for 01 = 0, p # 0. In the Blasius flow, these modes are hidden in the continuous spectrum and the local analysis cannot provide initial data for the marching technique.

3.1 PSE Analysis

With proper boundary conditions, the PSE analysis for steady modes, w = 0, can be performed with zero initial conditions. To introduce u-modes, which in a parallel flow have u = w = 0, disturbances can be generated by a small, spanwise periodic variation of the wall-shear stress, Du = e at y = 0, 2 = z I . PSE runs show the rapid formation of a streamwise u component and weaker u and w components that shortly downstream decay with oii = U(10-4). Owing to the nonparallelism of the basic flow and the nonzero derivative ur , pure u- modes cannot appear in the Blasius flow. Since neither the streamwise amplitude variation nor the u profile is consistent with observations, the spanwise modulations are likely associated with other mechanisms.

To study the generation and evolution of longitudi- nal vortices, disturbances can be generated by a small, spanwise periodic variation of the normal velocity com- ponent at the wall, u(0 ) = 6 at 2 = I.. Figure 5

W

shows the evolution of the amplitude for three initial positions, z. = 25,100,400, with @ = 0.14, Reo = 400, and 6 = The spanwise wavenumber is larger than the value p x 0.105 (b ~ t ! 0.25 at Re = 941) observed by Klebanoff e t al.32 and about half the value @ x 0.33 ( b x 0.6 at Re = 717) reported by Kendal17a ' I . The small initial disturbances monotonically grow to consid- erable amplitudes. The growth rate increases with z.. The amplitude evolution of disturbances with different spanwise wave numbers is given in Figure 6 for z6 = 25 and the previous values of Reo and c. The curves for the larger wavenumbers show that the initial growth phase ultimately changes into decay. The maximum amplitude is higher and appears at lower z as 0 increases.

The single disturbance with @ = 0.3 introduced at za = 25 starts t o decay near the position I = 1285, Re = 717, of Kendall's detailed measurements. The largest amplitude occurs for p ~ t ! 0.5, in the same range as in the experiments. The value of @ associated with the largest amplitude depends on the initial position z. and the Reynolds number. The profile of the calculated u disturbance shown in figure 7 with its maximum near 11 = yReo/Re = 2.2 is similar to the observed distri- bution (crosses) in figure 8. The accompanying curve would result from a spanwise thickening and thinning of the boundary layer6. The velocity profiles of the distur- bances closely resemble those of Gortler vortices. The u component is most pronounced and v and w are about two orders of magnitude smaller.

While the strongest response is for steady modes, the mechanism covers a broad band of low frequencies. Figure 9 compares the maximum rms fluctuation of me- andering Klebanoff modes for Reo = 400, @ = 0.35, zJ = 25, and a range of frequencies with experimental data6 that do not discriminate different spanwise wave numbers. Within a linear framework, the rms fluctua- tions can be adjusted by an arbitrary factor. Neverthe- less, our results exhibit a similar drop in receptivity as the experiments, although we neglect the effects of the starting position I, and the distributed forcing.

The results for oscillatory Klebanoff modes also shed light on the lack of any coupling with T S waves. For the highest frequency shown in figure 9, w = 0.036, we obtain 01 = 0.0707 + i0.0078 and a phase speed of u/a, = 0.51, higher than for T S waves. The velocity distribution is quite distinct from TS waves as shown for the streamwise component u in figure 10. The stream- wise rms profile resembles the profile of the steady mode in figure 7. The disturbance is tightly confined inside the boundary layer.

The significant transient amplitude (or energy) growth of steady and low-frequency disturbances in absence of linear instability is an example of the bypass mecha- nism described by Gustavssonz6 and HenningsonZ7 for channel flow. The relatively broad range of spanwise wavenumbers with significant growth of Klebanoff modes

v

w

is consistent with the observation of different wave- lengths in different experiments. The results of Butler & FarreIz8 for optimal temporal growth in a parallel Bla- sius flow at fixed Re are consistent with our findings. The spatial evolution in the PSE analysis accounts for the additional effects of boundary-layer growth and the associated change of the Reynolds number.

In the previous results, the disturbances have been introduced at a single fixed position similar to the spac- ers used by Klebanoff et al.32. Besides through u ( O ) , the Klebanoff modes can be introduced through inho- mogeneous conditions on Dw at the wall, and most ef- ficiently, through small spanwise variations of the wall pressure p ( 0 ) . For a smooth plate, generation of Kle- banoff modes by spanwise pressure variations associated with free-stream turbulence is the most likely scenario. Turbulent structures of the size of the observed span- wise wave lengths are necessarily outside the boundary layer and convected at freestream velocity. Individual structures produce a wall-pressure streak during their turnover time that becomes both stronger and longer as the turbulence level increases. The more coherent forc- ing over some streamwise distance likely causes a steeper increase in the amplitude of Klebanoff modes than the increase in the turbulence level. To reproduce the effect of free-stream turbulence, proper inhomogeneous bound- ary conditions in some range of z 2 I. can be applied in the PSE analysis.

3.2 Effect of Curvature

The similarity between the velocity distributions of K l e banoff modes and Gortler vortices suggest that they may be strongly affected by curvature. Also, the occurrence of Gortler vortices on concave walls may be related to the same receptivity mechanisms. To analyze the effect of curvature, the PSE have been derived from the Navier- Stokes equations for a surface-oriented coordinate sys- tem with streamwise varying radius of curvature R(z). While these equations permit interesting studies on the disturbance evolution on wavy walls or airfoils, we con- sider here walls with a constant (dimensional) radius of curvature and K = aO/R(z) = const. with respect to the reference length a t Reo = 400. As before, we introduce disturbances through a small, spanwise periodic wall ve- locity v(0) =

Figure 11 compares the growth curves for values of K . lo6 = -25(10)25 together with the result for the flat plate, K = 0, for p = 0.36. Concave walls are character- ized by K < 0. After initial growth, the amplitudes for K > -5 . decrease with increasing I, the amplitude for K = 5.10V6 remains nearly constant, while the ampli- tudes for larger concave curvature increase with varying rate.

The growth rates as a function of K are given in figure 12 for 2: = 1600. The data from PSE runs are shown by

at z3 = 25.

5

the circles connected by aspline. The growth rate varies monotonically as K increases. After rescaling the data for K = -25. to the local conditions a t Re = 800, the growth rate &; = -0.002386 agrees surprisingly well with the value &; = -0.002382 of the local analysis of a Gortler vortex at the proper parameters = 0.72, it = 5. Some of this agreement is incidental since a PSE analysis for locally parallel flow, which is more con- sistent with the local analysis, provides & j = -0.002056. Growth characteristics and comparison of velocit,y p r u files leave no doubt that the forced Klebanoff modes for K = -25. are Gortler vortices. The PSE analy- sis clearly shows the existence of these vortices in the form of Klebanoff modes on a flat plate and increas- ingly damped modes on convex walls of increasing cur- vature. I t seems appropriate to maintain the different names since Gortler vortices are inherently associated with centrifugal instability.

I, Swept Hiemenz Flow

The parabolized equations can be directly applied to three-dimensional boundary layers in Cartesian coordi- nates, such as the Falkner-Skan-Cooke flows over a swept wedge or the swept Hiemenz flow along and towards the sides of a stagnation line.

The originally strong interest in the stability of the boundary layer over a rotating disk as a prototype of boundary layers on realistic swept wings has faded since the stability properties are in fact more complex than in swept-wing flows. Falkner-Skan-Cooke and swept Hiemenz flow are more closely related to different as- pects of swept-wing flows and we present only a few re- sults for the latter case. Based on eqs. (4), the marching variable is I. There is no need to march in curvilinear coordinate systems that adapt one axis to the inviscid streamline direction, as it is common in traditional N factor calculations. There is also no problem with the controversial in-plane curvature terms which have been the topic of various papers 35, 36. Given the wavenumber /3 in the spanwise z direction, the streamwise wavenum- ber a, adapts properly to the local orientation of the disturbances. Since the basic flow is independent of z , a must be constant during the run 37.

Figure 13 shows the amplitude growth of steady cross- flow vortices in the swept Hiemenz flow with Re = 500, /3 = -0.4, a case studied with DNS by Spalart 38. Re is formed with the spanwise velocity W, and the refer- ence length ( V / A ) ’ / ~ , where U,(z) = Az. The march- ing variable I is the Reynolds number formed with Urn. In general, the linear stability characteristics provided by the local theory are in good agreement with the PSE results. Accounting for the variation of the basic state causes some differences but no dramatic change. We note, however, that these observations cannot br gener- alized to other, more realistic flows.

The wave numbers a, and growth rates ai obtained by PSE analysis are shown in figure 15. Similar results have been presented by Malik & Li 39.

U . l 3.4 Receptivity to Roughness LJ

While the PSE approach allows a more efficient, accu- rate, and reliable analysis of growth rates and N factors, it also provides new capabilities for analyzing and under- standing the receptivity of three-dimensional boundary layers. Fedorov” studied the excitation of cross-flow vortices in subsonic boundary layers over swept wings and found that micro-roughness of the height of 1% of the displacement thickness or spanwise periodic suc- tion/injection with u(0) = l o V 4 can excite cross-flow vortices with an initial amplitude of 0.1%. Radeztsky et al.23, zo report a strong effect of small roughness ele- ments on transition in swept-wing flows. Theoretical re- sults on the receptivity of the Falkner-Skan-Cooke flow to steady and unsteady cross-flow vortices have been ob- tained by Crouchz4.

The important aspects of these studies can be quali- tatively reproduced by modeling the roughness elements or spanwise periodic suction/blowing by a disturbance of the u component of the velocity normal to the wall. Fig- ure 15 shows the amplitude of forced cross-flow vortices of = -0.4 a t I = 500 depending on the position I$ of the u disturbance. In all cases, u ( 0 ) = loV4 at zs. The response is strongest if the disturbance (or the rough- ness in the experiments) is placed shortly upstream of the neutral point for the onset of instability. This result u’ is consistent with the experiment^^^ and the conclusion of Crouch24 that receptivity is strongest in the neighbor- hood of the neutral point.

The small spanwise irregularity in the wall velocity is amplified by a factor 7000 which is the result of subcrit- ical transient growth combined with subsequent growth due to instability. The amplitude ratio at z = 500 is AIAI = 654 (figure 13).

5 j f Conclusions

The parabolized stability equations have been success- fully applied to shed light on the mysterious Klebanoff modes that influence transition yet were unaccessible to prior analysis. While steady Klebanoff modes are diffi- cult to measure, the properties of low-frequency modes are in good agreement with experimental data. Our re- sults confirm Klebanoffs view that T S waves and the low-frequency longitudinal-vortex modes are “two phe- nomena which exist independently although they may both contribute to the intensity of a given frequency.” We also demonstrate the analytical connection between steady Klebanoff modes and Gortler vortices on curved walls. Both types of disturbances are governed by the -

same receptivity to spanwise periodic wall waviness, suc- tion, or unmeasurably small wall-pressure variations. We consider the pressure variations key to the effect of free-stream turbulence on boundary-layer transition.

For swept Hiemenz flow, similar receptivity mecha- nisms near the attachment line combined with strong instability further downstream cause strong cross-flow vortices as a precursor t o transition. The strong effect of roughness is consistent with other theoretical, numer- ical, and experimental studies.

Incorporation of receptivity processes into the PSE analysis of transition enables establishing prototype en- vironments to characterize free-flight or wind-tunnel conditions and to account for detailed surface conditions of aerodynamic bodies. Our results are a further step on the long way to understand, predict, and control transi- tion in boundary layers.

k4

Acknowledgment Stimulatingdiscussions with L. M. Mack, J. M. Kendall, and J. D. Crouch were beneficial to this work. W. S. Saric provided data for comparison and continuation of the work for actual geometries. The work of T H is sup- ported by the Air Force Office of Scientific Research un- der Contracts F49620-93-1-0135 and F49620-92-J-0271 and by the Office of Naval Research under Contract N00014-90-J-1520.

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l7 Kerschen, E. J . 1989 “Boundary layer receptivity,” AIAA-89-1109.

’* Choudhari, M. and Street, C. L. 1992 “A fi- nite Reynolds-number approach for the prediction of boundary-layer receptivity in localized regions,” Phys Fluids A , Vol. 4 , pp. 1495-2514.

l9 Nishioka, M. and Morkovin, M. V. 1986 “Boundary- layer receptivity to unsteady pressure gradients: ex- periments and overview,” J . Fluid Mech., Vol. 171, pp. 219-261.

* O Saric, W. S. 1993 “Physical description of boundary- layer transition: experimental evidence,” AGARD Report 793.

21 Lin, N . 1992 “Receptivity of boundary layers over a flat plate with different leading-edge geometries: Navier-Stokes simulations,” Ph.D. Thesis, Arizona State University, Tempe, Arizona.

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23 Radeztsky, R. H., Reibert, M. S., Saric, W . S., and Takagi, S. 1993 “Effect of micron-sized roughness on transition in swept-wing flows,” AIAA 93-0076.

24 Crouch, J . D. 1993 “Receptivity of three-dimensional boundary layers,” AIAA-93-0074.

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27 Henningson, D. S . and Schmid, P. J . 1992 “Vector eigenfunction expansions for plane channel flows,” Stud. A p p l . Math., to appear.

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*’ Trefethen, L. N., Trefethen, A. E., and Reddy, S. C. 1992 “Pseudospectra of the linear Navier-Stokes evo- lution operator and instability of plane Poiseuille and Couette flows,” Techn. Report TR 92-1291, Dept. Computer Science, Cornell Univ.

30 Haj-Hariri, H. 1992 “Characteristics analysis of the parabolized stability equations,” Stud. Appl . Math., submitted for publication.

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32 Klebanoff, P. S., Tidstrom, K. D., & Sargent, L. M. 1962 “The three-dimensional nature of boundary- layer instability,” J. Fluid Mech., Vol. 12, p p . 1-34.

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v free-stream conditions fluctuatlons, sound, paftides. ... \

outer boundary conditions

instabilities receptivity

initial conditions

wall boundary conditions waviness, roughness. vlbrations, ...

Figure 1: Schematic of the computational domain for transition analysis.

Figure 2: Area-distributed receptivity includes contribu- tionsfrom both initial and boundary conditions (Kendall 1992).

Y Ut

Figure 3: Spanwise waviness of the boundary-layer flow in the wind tunnel (Klebanoff & Tidstrom 1959).

Figure 4: Spectrum showing low-frequency fluctuations inside the boundary layer. The lines mark the range of TS instability (Klebanoff 1971).

9

0.015 - - x.-a --- x.=100

0.010 - I - 3 -

0.005 -

0.0 500.0 1000.0 1500.0 2000.0 x

0.05

0.04

Figure 5: Growth of longitudinal vortices forced at dif- ferent initial positions.

L

- p~0.14 B~0.30 -- 8.0.45

-

rl

Figure 7: Calculated Velocity distribution of the Kle- banolf mode.

Figure 6: Growth curves of forced longitudinal vortices for different spanwise wavenumbers Figure 8: Measured Velocity distribution of the Kle-

banoff mode.

10

PSE - A Experiment

A A A

A A - 10' 10

lo-'

10-

0 500 1000 1500 2000 25W

Figure 9: Spectrum of Klebanoff modes at Re, = 2.06. lo5. Reo = 400, p =.35, zs = 25.

Figure 11: Growth curves of forced longitudinal vortices for different wall curvature in comparison with the result for a flat wall (dashed).

0 2 4 6 8 10 11

Figure 10: Velocity distribution of an oscillating Kle- banoff mode with w = 0.036 at Re, = 2.06 . lo5. Reo = 400, p =,35, 2, = 25.

0.002

0.001 8 li .. ZJ 0.000

4.001

-20-05 - l e 0 5 Oe*oo 19-05 2e05 K

Figure 12: Growth rates of forced longitudinal vortices vs. curvature a t I = 1600.

11

- nonparal J parallel

U." 400.0 800.0 1 X

:A 100.0 150.0 200.0 250.0 300.0

I .o 1.0

Figure 13: Amplitude growth curves (A' factors) for 0.8

swept Hiemenz flow. PSE results with and without ac- counting for nonparallel terms. 8 0.6

I x I

0 2

0.03

0.02

0.01

0- 0.00

-0.01

-0.02

-0.03 0 200 400 600 800 1004

X

0.0

x.

Figure 15: Amplitude of croscflow vortices at I = 500 in response to forcing at different upstream positions.

'4

Figure 14: Growth rate of steady cross-flow vortices in swept Hiemenz flow.

12