[american institute of aeronautics and astronautics 32nd aiaa fluid dynamics conference and exhibit...

__________________ *Kent H. Smith Professor Emeritus of Engineering, Department of Mechanical and Aerospace Engineering, Fellow AIAA. †Assistant Professor, Department of Aerospace and Mechanical Engineering; (Associate Professor on leave from Tel-Aviv University), Senior Member AIAA.

1 American Institute of Aeronautics and Astronautics

INVESTIGATION OF THE ROLE OF TRANSIENT GROWTH IN ROUGHNESS-INDUCED TRANSITION

Eli Reshotko*

Case Western Reserve University, Cleveland, OH 44106

Anatoli Tumin† The University of Arizona, Tucson, AZ 85721

Abstract Surface roughness can have a profound effect on boundary layer transition. However, the mechanisms responsible for transition with three-dimensional distributed roughness have been elusive. Various T-S based mechanisms have been investigated in the past but have been shown not to be applicable. During the past year, the applicability of transient growth theory to roughness induced transition has been studied. A model for roughness-induced transition is developed that makes use of computational results based on the spatial transient growth theory pioneered by the present authors. For zero pressure gradient flows, the resulting transition relation is reminiscent of the transition correlations found for slender hypersonic vehicles. For nosetip transition, the resulting transition relations reproduce the trends of the Reda and PANT data and carefully account for the separate roles of roughness and surface temperature level on the transition behavior. .

Nomenclature , ,A B C = matrices 5 5×

D = d

dy

G = Energy ratio parameter K = height of the roughness elements M = Mach number M = matrix in the energy norm N = number of eigenmodes used in

the optimization procedure p = pressure

NR = the nose radius

Re = Reynolds number T-S = Tollmien-Schlichting

T = temperature t = time u =streamvise velocity disturbance v = transverse velocity disturbance w =spanwise velocity disturbance x = streamwise coordinate y = distance from the wall

z = spanwise coordinate α = streamwise wave-number β = spanwise wave-number

Hβ = Hartree parameter

γ = specific heat ratio

ν = kinematic viscosity θ = momentum thickness ρ = density

ω = angular frequency Superscripts T = transposed Subscripts and other symbols aw = adiabatic wall e =edge of the boundary layer in = input value s = mean flow parameter tr = at the transition point w = wall condition ̂ = perturbation of the parameter

Introduction Surface roughness can have a profound effect on boundary layer transition. The mechanisms associated with single roughness elements are only partially understood while those responsible for transition with distributed roughness are not yet

32nd AIAA Fluid Dynamics Conference and Exhibit24-26 June 2002, St. Louis, Missouri

AIAA 2002-2850

Copyright © 2002 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.


known. This has led to a large body of empirical information in the literature that is not fully consistent. These correlations are generally based on

two dimensional parameters such as Rek , */k δ ,

/k δ whereas distributed roughness is inherently three-dimensional. The three-dimensionality is usually introduced by providing separate curves in the correlations for each three-dimensional shape and distribution. Nevertheless, these correlations are still the operative data base for incorporating the effects of distributed roughness in design. Because the fundamental mechanisms are not known, there is considerable uncertainty in the reliability and extrapolability of such correlations. The first author is one of a number of investigators who in the past have pursued a T-S explanation for the effects of roughness, both discrete and distributed. Only discrete two-dimensional roughness elements yield to a T-S explanation (Klebanoff & Tidstrom1). The attempts to find a T-S explanation for three-dimensional roughness, both discrete and distributed have failed. These attempts have been documented in Reshotko2 and Morkovin3. Experimental studies by Reshotko & Leventhal4, Corke, Bar-Sever & Morkovin5 and Tadjfar et al6 have presented some of the physical observations of flow over distributed roughness. It is generally agreed that roughness displaces the mean flow outward affecting the profiles only within the roughness height. Subcritical amplification is observed principally at low frequencies and the growth can easily reach nonlinear levels quickly. It is suspected that in common with single 3D roughness elements, the distributed roughness gives rise to vortex structures emanating from the elements. These vortices are primarily streamwise. Papers summarizing these observations are by Reshotko2 and Morkovin3,7. A possible unifying explanation for these observations lies in the mechanism of transient growth. Transient growth arises through the coupling between slightly damped, highly oblique (nearly streamwise) Orr-Sommerfeld and Squire modes leading to algebraic growth followed by exponential decay outside the T-S neutral curve. A weak transient growth can also occur for two-dimensional or axisymmetric modes since the Orr-Sommerfeld operator (also its compressible counterpart) is not self-adjoint, therefore its eigenfunctions are not strictly orthogonal. In recent papers the authors have shown the potential importance of transient growth in explaining the long standing blunt body paradox (Reshotko & Tumin8) and have completed papers on

the role of transient growth in Poiseuille pipe flow (Reshotko & Tumin9) and boundary layer flow (Tumin & Reshotko10). These three papers are the first to use a spatial growth formulation of these transient growth problems rather than the earlier temporal growth approach. Because transient growth factors can be extremely large in flows that are T-S stable or in parameter ranges that are T-S stable, transient growth is an attractive mechanism to consider with respect to distributed roughness effects. Recent works by Luchini11 and by Tumin & Reshotko10 show that for Blasius flow, maximum transient growth factors are for streamwise (stationary, zero frequency) disturbances. As the frequency increases, the growth factors are reduced.

Spatial Theory of Optimal Disturbances Spatial formulation of the transient growth theory within the scope of linearized Navier-Stokes equations was introduced in Ref. [8-10]. For the sake of consistency, we recapitulate briefly the key elements of the theoretical model. The linearized Navier-Stokes equations for three-

dimensional disturbances ( )exp i x z tα β ω + −∼ of

prescribed frequency ω and transverse wave number β are reduced to a system of ordinary differential

equations in the following form

( )2 0AD BD C+ + Φ = (1)

where ( ) ( ) ( ) ( ) ( )ˆˆ ˆ ˆ ˆ, , , ,T

u y v y p y T y w yΦ = is a

five-element vector comprised of the complex amplitude functions for the ,x y− − components of

velocity, pressure, temperature and z -component of velocity (superscript T stands for the vector transpose). , ,A B C are 5 5× matrices (nonzero elements are given in the Appendix of Ref. [10]) and

/D d dy= . The boundary conditions for equation (1)

are

51 2 40 : 0y = Φ = Φ = Φ = Φ = (2)

j: | | y → ∞ Φ < ∞

The boundary condition (2) outside the boundary layer allows decaying eigenmodes

( 51 2 4: , , , 0y → ∞ Φ Φ Φ Φ → ) that represent the

discrete spectrum, and non-growing, non-decaying modes as well, that represent the continuous spectrum.


The signaling problem suggests that at the moment of time 0t = a localized disturbance source is switched on. Analysis of the linearized Navier-Stokes equations under the assumption that the downstream and upstream boundaries are at ±∞ leads to the conclusion that only branches corresponding to the decaying modes of the continuous spectrum apply to the solution downstream of the source. Following from the signaling problem, the flowfield downstream from a disturbance source may be represented as an expansion in eigenfunctions including the decaying modes of the continuous and discrete spectrum plus any growing discrete modes if present. Having selected the downstream eigenmodes, one can continue the analysis as an initial value problem spanned by the eigenmodes, and the method of analysis developed by Schmid & Henningson12 for temporal stability theory may be applied. We introduce a vector-function

ˆˆˆ ˆ ˆ( , , , , )Tu v T wρ=q and a scalar product

1 2 2 1 20

( , ) H dy∞

= ∫q q q Mq (3)

where the matrix M is

( )( )( )

2

2]

[ , , / ,

/ 1 ,

es s s s

es s s

diag T M

T M

ρ ρ γρ

ρ γ γ ρ

=

−

M (4)

The definition of the scalar product (3) leads to the energy norm introduced by Mack13

22

0( , ) H dy

∞= = ∫q q q q Mq (5)

Expanding the vector-function q into N decaying

eigenmodes (they might belong to the discrete spectrum or to the numerical discretization of the continuous spectrum), we obtain

( )

1ˆ( , , , ) ( ) k

N i xi z tk k

kt x y z e y e

αβ ω κ−

== ∑q q (6)

The vector-function ˆ ˆ ˆ ˆ( , , )k k k ku v w=q is comprised

of three velocity components corresponding to k -th

eigenfunction. The coefficients kκ are optimized to

achieve the maximum energy growth at the specific downstream coordinate

2

20

( )( , , Re, ) max

xG xβ ω =

q

q (7)

where 2

0q corresponds to the energy at the initial

location. For purpose of consideration of the curvature effect, the governing equations were written in spherical coordinates. The latter led to appearance the

ratio / Ny R in the coefficients. The curvature-

associated corrections of the coefficients were neglected in the viscous terms of the equations. One can find details of the numerical procedure in Ref. [10].

Transient growth calculations

a) Zero pressure gradient (flat plates or cones) Spatial transient growth calculations have been performed for zero pressure gradient boundary layers

(Hartree parameter 0Hβ = ) at Mach numbers from

1.5 to 6 and for a range of surface temperatures from

0.25 < /w awT T < 1.0. The calculations are all for

stationary disturbances (ω = 0, α = 0). These results are summarized in Fig. 1. Consistent with expectations from the transient growth theory, the growth factors scale with length Reynolds number or the square of a thickness Reynolds number. For 0.75

< /w awT T < 1.0, the growth factors are essentially

independent of Mach number while for colder walls, the lower the Mach number the higher the growth rate. The corresponding optimal spanwise wavenumbers are shown in Fig. 2. For all cases, the optimal spanwise wavelengths are from 3 to 3.5 boundary layer thicknesses. For the cold wall case

( /w awT T = 0.25), the optimal transient growth factors

are very sensitive to Mach number and can be summarized by the following relation:

1/ 2( / Re ) 0.0411 0.0601Re / eLG Mθ= + (8)

This relation can be later used to estimate the effect of roughness on transition for flat plates or cones.


b) Stagnation point flow Extensive calculations have also been carried out

for axisymmetric stagnation point flows ( 0.5Hβ = ).

These are relevant to the hypersonic sphere-cone nosetip configurations for which there is an extensive experimental data base and significant transition correlations (Batt & Legner14,15, Reda16). The

calculations were first performed for /w awT T = 0.25.

However since most of the experimental data base is

centered around /w awT T = 0.5, the results for this

latter case will be emphasized here. Further, in contrast to the flat plate, curvature is a significant factor for the sphere. Curvature is included in our calculations and in the results to be presented.

For surface temperatures in the vicinity of /w awT T =

0.5, the transient growth results are shown in Figure 3. The optimal spanwise wavenumber is essentially

constant over the Mach number range at optβ θ =

0.28 which corresponds to about 3.2 boundary layer thicknesses. The effect of curvature is shown in Fig. 4. The larger the curvature (smaller nose radius), the smaller the growth factor. Since most of the experimental runs had surface temperature level variations during the run, and since the growth factors are sensitive to surface temperature level, it has further been determined that G1/2 varies as

( ) 0.77/w awT T

− in the vicinity of /w awT T = 0.5 (see

Ref. [8]). The above can be summarized by:

-0.77

0.9441/ 2

Re 20.102

1 14.9 Re e

TwTM aw

RN

G

θ

θ

θ

=+

(9)

where NR is the nose radius. The factor in the

denominator on the right side is the “stabilizing” effect of curvature.

Influence of roughness on transition

The influence of roughness on transition can be modeled in a manner similar to that used by Andersson et al17 for free-stream turbulence effects on transition. We assume that an energy norm at transition is related to an input energy through the transient growth factor G.

tr inE GE= (10)

The input energy is taken to resemble a mass-flow fluctuation squared

2

inkw

eE

ρρ θ

= (11)

which for a boundary layer can be approximated

2

inT keTw

Eθ

= (12)

Again, the growth factor G scales with the square of a thickness Reynolds number or with length Reynolds number to the one power. Thus we can write

1/ 21/ 2

Re( ) Re

G kTeTw

Etr θ θθ= (13)

where 1/ 2/ ReG θ is obtained from the transient

growth results for the particular geometry and flow

parameters. Transition is assumed to occur when trE

reaches a specific value, here taken as a constant. a) Zero pressure gradient (flat plates or cones) Using the result of Eq. (8), the transition relation

for zero pressure gradient flows with /w awT T = 0.25

would be:

,

,

(0.0619 0.0905 Re / )

Re

etr

trkTeTw

M

const

θ

θ θ

+ ×

= (14)

For small ,Re trθ :

-1

,Re ~ ( / ) e wtr kT Tθ θ (14a)

For , /Re tr eMθ large:

1/ 2 -1/ 2

,Re /( ) ~ ( / )tr e e wM kT Tθ θ (14b)

This latter relation is somewhat reminiscent of transition correlations found for slender hypersonic vehicles.


b) Stagnation point flow According to Eq. (13), we have to extract the

factor 1/ 2 / ReG θ from the transient growth results.

The calculations summarized by Eq. (9) are for

parallel flows (e

M = const). However, for the

stagnation point flow, the edge Mach number varies almost linearly with angle from the stagnation point. From Eq. (9), it is seen that

1/ 2 0.944/ Re ~ 1/ eG Mθ (15)

Thus the growth factor is largest near the stagnation point and diminishes rapidly as the edge Mach number increases. An integration of the differential growth factors from the stagnation point to any downstream location shows that the integrated

growth factor is essentially constant beyond Me ~ 0.1

and therefore 1/ 2,/ Re trG θ is essentially independent

of the local Mach number at the transition location. Thus, for the stagnation point in the vicinity of

/w awT T = 0.5, the relation is:

-0.77

,Re ( / ) ( / 2 )

(1 14.9 Re )

e w w etr

NR

kT T T Tconst

θ

θ

θθ =

+ (16)

The last relation shows the trends of transition Reynolds number with roughness height and surface temperature level. For constant surface temperature

level, ,Re trθ should vary as ( ) 1/k θ

−. This is

consistent with Reda’s ballistic range data16 as shown in Fig.5. The PANT data14,15 shown in Figs. 6a and 6b display this trend as well. In addition, some of the

PANT data were taken for constant ( )/k θ but with

varying surface temperature about /w awT T = 0.5. For

this case the last relation shows that Reθ,tr should vary

as ( ) 1.77/w awT T

−. This again is supported by the

PANT data as shown in Figs. 6a and 6b. To be noted is that all of the nosetip transitions in the PANT data base take place well within the sonic point on the

sphere (0.2 < ,e tr

M < 0.8) and with 20 < ,Re trθ <

120. In treating these data, only the revised data are used here. The revised data are based on the Batt & Legner protocol of identifying transition location.

The present summary relation for the PANT data base is

-1 -0.77,Re 350 / (2 / ( ) )e w e wtr kT T T Tθ θ= (17)

The curvature factor is ignored as it varies only within a narrow range for the whole data base. The

numerical factor of 350 for /w awT T = 0.5 comes

from averaging in only those points for which 0.45 <

/w awT T < 0.55.

It will now be shown that the Reda correlation (Fig. 5) and the present PANT correlation (Eq. 17) are quantitatively comparable. Since θ appears in the numerator of both sides of Eq. 17, this relation can be rewritten as

1.77/ 350(2 / )e e w eU k T Tν = (18)

which for /w awT T = 0.255 gives Reda’s value of 106.

Reda18 estimates his surface temperature level to have been about 0.3.

Summary and Conclusions

Under parallel flow assumptions, transient growth calculations have been performed for zero pressure gradient and stagnation point flows over Mach number and surface temperature ranges that are relevant to slender and blunt hypersonic configurations. In addition, a model for roughness-induced transition has been developed that makes use of the transient growth results. For the zero pressure gradient flows, the resulting transition relation is reminiscent of the transition correlations found for slender hypersonic vehicles. For nosetip transition, the dominant transient growth takes place in the near vicinity of the stagnation point so that the resulting correlation is independent of the local Mach number at the transition location. This yields transition relations that reproduce the trends of the Reda and PANT data, but with exponents for roughness and surface temperature effects obtained from the transition modeling and the transient growth theory. The authors therefore believe that transient growth is a very promising explanation for early transition due to distributed roughness.

Acknowledgment This work is supported by the U.S. Air Force Office of Scientific Research.


References 1. Klebanoff, P.S., and Tidstrom, K.D., 1972:

“Mechanism by which a Two-Dimensional Roughness Element Induces Boundary-Layer Transition,” Phys. Fluids, 15, pp.1173-1188.

2. Reshotko, E., 1984: “Disturbances in a Laminar Boundary Layer due to Distributed Surface Roughness,” in Tatsumi, T. ed., Turbulence and Chaotic Phenomena in Fluids, Elsevier, pp. 39-46.

3. Morkovin, M.V., 1990a: “Panel Summary on Roughness,” in M.Y. Hussaini, R.G. Voight, eds, Instability and Transition, Vol. I, Springer-Verlag, pp. 265-271.

4. Reshotko, E., and Leventhal, L., 1981: “Preliminary Experimental Study of Disturbances in a Laminar Boundary-Layer due to Distributed Surface Roughness,” AIAA Paper 81-1224.

5. Corke, T.C., Bar-Sever, A., and Morkovin, M.V., 1986: “Experiments on transition enhancement by distributed roughness,” Phys. Fluids, 29 pp. 3199-3213.

6. Tadjfar, M., Reshotko, E., Dybbs, A., and Edwards, R.V. 1985: “Velocity Measurements Within Boundary Layer Roughness Elements Using Index Matching,” in A. Dybbs, E. Pfund, eds, International Symposium on Laser Anemometry, FED Vol.33, ASME Press (Book #G00328), pp. 59-73.

7. Morkovin, M.V., 1990b: “On Roughness-Induced Transition: Facts, Views & Speculation,” in M.Y. Hussaini, R.G. Voight, eds, Instability and Transition, Vol. I, pp. 281-295.

8. Reshotko, E., and Tumin, A., 2000: “The Blunt Body Paradox – A Case for Transient Growth,” in H.F. Fasel, W.S. Saric, eds, Laminar-Turbulent Transition, Springer, pp. 403-408. (Paper presented at 5th IUTAM Symposium on Laminar-Turbulent Transition, Sedona, AZ, U.S.A., 1999

9. Reshotko, E., and Tumin, A.: 2001: “Spatial Theory of Optimal Disturbances in a Circular Pipe Flow,” Physics of Fluids, 13, pp.991-996.

10. Tumin, A., and Reshotko, E., 2001: “Spatial Theory of Optimal Disturbances in Boundary Layers,” Physics of Fluids, 13, pp. 2097-2104.

11. Luchini, P. 2000: “Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations,” J. Fluid Mech, 404, pp. 289-309.

12. Schmid, P. J., and Henningson, D. S., 1994: “Optimal energy density growth in Hagen-

Poiseuille flow”, J. Fluid Mech., 277, pp. 197-225.

13. Mack, L. M., 1969: “Boundary layer stability theory”, JPL Report 900-277.

14. Batt, R.G., and Legner, H.L. 1980: “A Review and Evaluation of Ground Test Data on Roughness Induced Nosetip Transition,” Report BMD-TR-81-58.

15. Batt, R.G., and Legner, H.L. 1981: “A Review of Roughness Induced Nosetip Transition,” AIAA Paper 81-1223.

16. Reda, D. C. 2001: “Roughness-Dominated Transition on Nosetips, Attachment Lines and Lifting-Entry Vehicles,” AIAA Paper 2001-0205.

17. Andersson, P., Berggren, M., and Henningson, D., 1999: “Optimal disturbances and bypass transition in boundary layers,” Phys. Fluids, 11, pp. 135-150.

18. Reda, D.C. 2002: Private communication

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.0025

0.0030

0.0035

0.0040

0.0045

0.0050

0.0055

0.0060

0.0065

0.0070 M=1.5, T0=333K

2.0 2.5 3.0 4.0, T

0=405K

6.0, T0=791K

G/R

e L

Tw/T

ad

Figure 1. Optimal growth factors for zero

pressure gradient, 4Re 9 10L = ×


0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

M=1.5, T0=333K

2.0 2.5 3.0 4.0, T

0=405K

6.0, T0=791K

β opt(ν

eX /

Ue)1/

2

Tw/T

ad

Figure 2. Optimal spanwise wavenumber for zero

pressure gradient, 4Re 9 10L = ×

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.982/(1+14.9 × Rθθ /R

N)

(G/G

0)1/

2

Rθ (θ / R

N)

Rθ= 82, 123 & 410

Fit

Figure 4. Effect of curvature on optimal growth factor for axisymmetric stagnation point flow

0.8; / 0.5w aweM T T= =

1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.102/Me

0.944

Sqr

t(G

)/R

θ

Reciprocal edge Mach number, 1/Me

Theory Fit

Figure 3. Optimal growth factors for axisymmetric stagnation point flow, / 0.5w awT T =

Figure 5. Nosetip transition data from ballistic-range experiments; 3D distributed roughness [16]


Figure 6a. Transition correlation in PANT coordinates (PANT Series A). Adapted from [14,15].

Figure 6b. Transition correlation in PANT coordinates (PANT Series J). Adapted from [14,15].

[american institute of aeronautics and astronautics 32nd aiaa fluid dynamics conference and exhibit...

Documents