Anatoli Tumin

University of Arizona, Tucson, Arizona

David E. Ashpis

Glenn Research Center, Cleveland, Ohio

Transient Growth Theory Prediction of Optimal

Placing of Passive and Active Flow Control

Devices for Separation Delay in LPT Airfoils

NASA/TM—2003-212228

May 2003

https://ntrs.nasa.gov/search.jsp?R=20030062088 2018-05-15T18:32:47+00:00Z
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Anatoli Tumin

University of Arizona, Tucson, Arizona

David E. Ashpis

Glenn Research Center, Cleveland, Ohio

Transient Growth Theory Prediction of Optimal

Placing of Passive and Active Flow Control

Devices for Separation Delay in LPT Airfoils

NASA/TM—2003-212228

May 2003

National Aeronautics and

Space Administration

Glenn Research Center
Acknowledgments

The authors appreciate useful comments given by Dr. Stuart Leib of The Ohio Aerospace Institute.

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Available electronically at http://gltrs.grc.nasa.gov
NASA/TMó2003-212228 1 

TRANSIENT GROWTH THEORY PREDICTION OF OPTIMAL PLACING OF 

PASSIVE AND ACTIVE FLOW CONTROL DEVICES FOR SEPARATION 

DELAY IN LPT AIRFOILS 

Anatoli Tumin

The University of Arizona 

Tucson, Arizona 85721 

David E. Ashpis 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract

 An analysis of the non-modal growth of perturbations in a boundary layer in the 

presence of a streamwise pressure gradient is presented. The analysis is based on PSE 

equations for an incompressible fluid. Examples with Falkner-Skan profiles indicate that 

a favorable pressure gradient decreases the non-modal growth while an unfavorable 

pressure gradient leads to an increase of the amplification. It is suggested that the 

transient growth mechanism be utilized to choose optimal parameters of tripping 

elements on a low-pressure turbine (LPT) airfoil. As an example, a boundary-layer flow 

with a streamwise pressure gradient corresponding to the pressure distribution over a LPT 

airfoil is considered. It is shown that there is an optimal spacing of the tripping elements 

and that the transient growth effect depends on the starting point. At very low Reynolds 

numbers, there is a possibility to enhance the transient energy growth by means of wall 

cooling. 

Nomenclature

E = energy norm 

G = Eout/Ein energy ratio 

H = shape factor 

HL = / ν L UeL

HLs = / ν L U s exit

Href = scale in y- and z- directions 

L = length along the streamwise direction 

Lref = reference length 

Ls = surface length 

Re = Reynolds number 

ReL = / U L eL ν

T T w ad / = temperature factor (ratio of the wall temperature to the temperature of the 

 adiabatic wall) 

U = streamwise velocity component of the mean-flow velocity 
NASA/TMó2003-212228 2 

UeL = free-stream velocity at x = L

Uref = reference velocity 

V = normal-to-the wall velocity component of the mean-flow velocity 

u = streamwise velocity disturbance 

v = normal velocity disturbance 

w = spanwise velocity disturbance 

x = streamwise coordinate 

y = coordinate normal to the wall 

z = spanwise coordinate 

β = spanwise wave number 

βH = Hartree parameter 

ε = / ref ref ν ∞ U L

ν = kinematic viscosity 

θ = momentum thickness 

π = pressure disturbance 

Superscripts 

T = transposed 

Subscripts 

e = free stream 

exit = exit conditions

in = starting point 

out = ending point 

1. Introduction

 Laminar-turbulent transition in shear flows is still an enigma in the area of fluid 

mechanics. The conventional explanation of the phenomenon is based on the instability 

of the shear flow with respect to infinitesimal disturbances. The conventional 

hydrodynamic stability theory deals with the analysis of normal modes that might be 

unstable. The latter circumstance is accompanied by an exponential growth of the 

disturbances that might lead to laminar-turbulent transition. Nevertheless, in many cases, 

the transition scenario bypasses the stage of the exponential growth associated with the 

normal modes. This type of transition is called as bypass transition. Observations of 

laminar-turbulent transition in plane Couette flow, in circular pipe flow, in boundary 

layers at relatively low Reynolds numbers, etc., can serve as examples of bypass 

transition. An understanding of the phenomenon has eluded us to this day. One 

possibility is that bypass transition is associated with so-called algebraic growth of 

disturbances in shear flows.1ñ3 

 The phenomenon of algebraic growth of disturbances in shear flows has been of great 

interest during the last two decades as it may be associated with the bypass transition 

mechanism. Ellingsen and Palm1

 considered, in the inviscid case, an initial disturbance 

independent of the streamwise coordinate and found that the streamwise disturbance 

amplitude may grow in time, even though the basic flow does not posses an inflection 
NASA/TMó2003-212228 3 

point. Landahl2 showed that all parallel inviscid shear flows are unstable to a wide class 

of three-dimensional disturbances. The result is independent of whether or not the shear 

flow is unstable to an exponential growth of wavelike disturbances. This type of 

instability that is not related to exponential growth is also referred to as ìnon-modal 

growth.î Mathematically, the effect of non-modal growth is associated with nonnormality of the linearized Navier-Stokes operator and non-orthogonality of the 

eigenfunctions. In simple words, the essence of non-modal growth is the possibility of 

combining the exponentially decaying modes in such way that their sum will possess 

transient growth. One can find a vast bibliography on the topic in the Otto Laporte Award 

Lecture by Reshotko3

 and in a monograph by Schmid and Henningson.4 

 Numerical analysis of spatial non-modal growth within the scope of the linearized 

boundary-layer equations for an incompressible flow over a flat plate was carried out in 

Refs. 5 and 6. Spatial analysis within the scope of the linearized Navier-Stokes equations 

(quasi-parallel approximation of compressible and incompressible flows) was presented 

in Refs. 7 to 9. The main results of these theoretical models are as follows: 

• A system of counter-rotating streamwise vortices, which are periodic in the 

spanwise direction, provides the strongest growth of the disturbance. 

• There is an optimal spacing of the streamwise vortices, leading to the strongest 

effect. 

 The effect of pressure gradients on the transient growth mechanism was considered 

within the scope of temporal theory by Corbett and Bottaro10 and within the scope of 

spatial theory by Tumin and Reshotko.9

 These results were based on the quasi-parallel 

flow assumption. Tumin11 analyzed the pressure gradient effect for the Falkner-Skan 

profile within the scope of an analytical model when the spanwise wave number is very 

small. The pressure-gradient effect within the scope of spatial theory with nonparallel 

base flow and finite spanwise wave numbers has not been considered, yet. 

 Another motivation for the present work stems from separation flow control on lowpressure turbines (LPT). The performance of LPTs is strongly affected by the flow 

separation. There is a possibility of delaying the boundary-layer separation by tripping 

the boundary layer with the help of roughness elements or other devices. Usually, trialand-error method is used to determine an appropriate placement of the control elements. 

This approach is time consuming and expensive. A recent investigation by Reshotko and 

Tumin12 demonstrated that roughness-induced transition might be related to the transient 

growth mechanism. 

 Periodically spaced in the spanwise direction, roughness elements generate a system 

of counter-rotating streamwise vortices. Due to a secondary instability mechanism, the 

streamwise vortices can lead to earlier transition to turbulence. They also provide a 

mixing enhancement due to redistribution of the streamwise momentum. Consequently, 

optimization of the streamwise vortices for maximum energy growth leads to maximizing 

of the flow control effectiveness. In the present work, analysis of the optimal 

disturbances/streamwise vortices associated with the transient growth mechanism will be 

performed for boundary layers in the presence of a streamwise pressure gradient. The 

theory will provide the optimal spacing of the control elements in the spanwise direction 

and their placement in the streamwise direction. 
NASA/TMó2003-212228 4 

2. Governing Equations

 In the present work, we utilize the method developed in Ref. 19. Because the flows of 

interest have relatively low Mach numbers, we consider steady three-dimensional 

disturbances in an incompressible two-dimensional boundary layer. We choose the 

streamwise coordinate x along the surface. The coordinate y will measure distance from 

the wall. We define a small parameter / ref ref ε ν = U L , where ν , ref U , and ref L are 

viscosity, reference velocity, and reference length, respectively. The streamwise 

coordinate is scaled with ref L while the vertical coordinate y and spanwise coordinate 

z are scaled with / ref ref ν L U . The following scaling is assumed for the velocity 

disturbances uv w , , and , and the pressure π : 

 2 , , , ref ref ref ref uU v U w U p U ∼∼ ∼ ∼ ε ε ε ρ (1) 

 This scaling of the linearized Navier-Stokes equations and neglecting of the curvature 

effects lead to the governing equations for Gˆrtler instability with the Gˆrtler number 

equal to zero. We look for a periodic solution in the spanwise direction with the 

corresponding wave number β. The governing equations for the amplitude functions can 

be written in dimensionless form as follows:5,6 

 0 u v

w

x y

β ∂ ∂

++ =

∂ ∂ (2) 

 ( )

2

2

2

uUu Uu V v u

x yy y

β ∂ ∂ ∂∂

+ + =−

∂ ∂∂ ∂

 (3) 

 ( ) ( )

2

2

2 2 p v uV vU Vv Vw v

xy y y

β β ∂ ∂ ∂∂

+ + + += −

∂∂ ∂ ∂

 (4) 

 ( ) ( )

2

2

2

w Uw Vw p w

x y y

β β ∂∂ ∂

+ −= −

∂ ∂ ∂

 (5) 

where U xy V xy () () , and , are the streamwise and normal velocity components of the 

base flow, respectively (in addition, the latter is divided by ε). The following boundary 

conditions are applied to the solutions: 

 0 : 0 y uvw = == = (6a) 

 : , , 0 y uw →∞ →π (6b) 

The equations (2) to (5) can be solved subject to boundary conditions (6a) and (6b) with 

prescribed initial velocity perturbations at x = x0. 
NASA/TMó2003-212228 5 

 The governing equations can be recast as follows: 

 ( ) 01 2 y yy x Af B f B f B f =+ + (7) 

where , 01 2 AB B B , , and are 4 4 × matrices (one can find them in the Appendix; see 

also Ref. 5) and ( ) ,, , T f = uvwp . The superscript ìTî stands for ìtransposed,î and the 

subscripts ìxî and ìyî denote differentiation with respect to x and y, respectively . 

3. Optimization of Energy Growth

 The authors of Refs. 5 and 6 employed an iterative procedure to find the optimal 

disturbances in terms of the maximum of the energy growth ratio / GE E = out in , where 

 and Ein out E stand for the input and output energy norms. Andersson et al.5

 used the 

same definitions of Ein and Eout as for the disturbance energy 

 ( ) max 2 22 2 2

0

y

E = ++ u v w dy ε ε ∫ (8) 

whereas Luchini6

 employed the knowledge that the optimal disturbances are represented 

by streamwise vortices with corresponding output as the streamwise velocity streaks, 

 ( ) max 2 22

0

y

Ein = + ε v w dy ∫ (9a) 

 

max 2

0

y

Eout = u dy ∫ (9b) 

 

( )

max

max

2

2 0

2 2

0

y

out

y in

u dy

E G

E

v w dy

ε

− = =

+

∫

∫

 (9c) 

 As was shown in Ref. 5, the two definitions of the optimal disturbances lead to the 

same results at Reynolds numbers of 104

 and higher. Because the iteration procedure 

based on the optimization of ratio (9c) provides significant simplification, we adopt it for 

the following analysis. Because Eqs. (2) to (5) are independent of ε , the value of 2 ε G is 

invariant with respect to the Reynolds number. 
NASA/TMó2003-212228 6 

4. Numerical Results

4.1 Falkner-Skan Base Flow 

We consider a Falkner-Skan family of boundary-layer profiles with free-stream velocity 

distribution m U Cx e = and corresponding Hartree parameter β H = + 2/ 1 m m( ) . For the 

purpose of convenience, we have used the velocity scale m U U CL ref eL = = and the 

length scale L Lm ref = + / 1 ( ) . The latter allowed the use of the conventional scaling of 

boundary-layer solutions with H Lm U L U ref eL ref ref = += ν ν /1 / ( ) . 

 Figure 1 shows the scaled the energy ratio versus spanwise wave number β for three 

Hartree parameters, β H = ñ0.1, 0.0, and 0.1. The starting and the ending points, / in x L

and / out x L , are equal to 0.2 and 1.0, respectively. The Reynolds number ReL in Fig. 1 

and what follows is defined as / U L eL ν . One can see that an unfavorable pressure 

gradient ( ) β H < 0 leads to an increase in the energy growth while a favorable pressure 

gradient ( ) β H > 0 leads to suppression of the transient growth mechanism. The latter is 

consistent with results obtained within the scope of parallel flow approximation.5

 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.000

0.001

0.002

0.003

0.004

0.005

0.006

G/

R

eL

βHref

βH=0.0

 +0.1

 -0.1

Fig. 1. Effects of the spanwise wave number β and the Hartree parameter β H on 

transient growth (starting point / 0.2 in x L = ).
NASA/TMó2003-212228 7 

 Figures 2 and 3 show similar results, but the starting points are / in x L = 0.4 and 0.6, 

respectively. respectively. A comparison of Figs. 1 to 3 indicates that there is a spanwise 

wave number, 0.5 0.6 β Href = − , and a starting point, / in x L , that maximize the energy 

growth. These parameters correspond to optimal spanwise spacing and streamwise 

placing of perturbators for maximum flow control effectiveness. Figure 4 demonstrates 

the energy ratio versus the downstream coordinate x / L at 0.5 β Href = and / in x L = 

0.2. 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.000

0.001

0.002

0.003

0.004

0.005

0.006

βH=0.0

 +0.1

 -0.1

G/

R

eL

βHref

Fig. 2. The same as Fig. 1, with starting point / 0.4. in x L =
NASA/TMó2003-212228 8 

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0000

0.0005

0.0010

0.0015

0.0020

0.0025

0.0030

0.0035

0.0040

0.0045

βH=0.0

 +0.1

 -0.1

βHref

G/R

eL Fig. 3. The same as Fig. 1, with starting point / 0.6. in x L =

01234

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

G/

R

eL

X/L

βH=0.0

 +0.1

 -0.1

Fig. 4. Optimal energy ratio versus the downstream coordinate at three Hartree 

parameters ( / 0.2 in x L = , 0.5 β HL = ). 
NASA/TMó2003-212228 9 

4.2 Example No. 1 of LPT Conditions 

 Volino13 simulated low-pressure turbine (LPT) airfoil conditions in a low-speed wind 

tunnel. The test section was designed as a passage between two airfoils. The local freestream velocity at a favorable pressure-gradient region was closely approximated by the 

following equation: 

 

0.214

1.48 e

exit s

U x

U L

  =    

 (10) 

where Ls is the suction surface length and Uexit is the nominal exit free-stream velocity 

based on the inviscid solution. The distribution (10) corresponds to the Falkner-Skan 

flow with the Hartree parameter 0.353 β H = . 

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

0.0009

0.0010

G/

R

e

βHLs

 X/Ls=0.111

 0.155

 0.200

 0.244

 0.289

Fig. 5. Effects of the spanwise wave number β and the starting point / in x L on 

transient growth at conditions of the experiment in Ref. 13 ( β H = 0.353 , 

/ out s x L =0.444).

 Figure 5 demonstrates the energy ratio scaled with the Reynolds number Reexit = 

/ U L exit s ν versus the spanwise wave number scaled with / H LU Ls s exit = ν . The 

ending point was prescribed at / 0.444 out s x L = while the starting points varied from 

0.111 to 0.289. The streamwise velocity perturbation at / 0.444 out s x L = , 
NASA/TMó2003-212228 10 

/ 0.111 in s x L = , and β HLs = 0.925 is shown in Fig. 6, and the corresponding optimal 

profiles of v and w are presented in Fig. 7. 

 The results indicate that we are dealing with a very strong favorable pressure gradient 

that suppresses the transient growth mechanism. If the Reynolds number is about 50,000, 

the transient growth will provide an energy amplification of less that 50. If we take into 

account that the real perturber will not produce the optimal inflow field, there is little 

likelihood that the bypass transition via a secondary instability of the velocity streaks will 

prevent the flow separation. 

0 5 10

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

y/Hout

u/

U

e, out

Fig. 6. The streamwise velocity perturbation at the ending point / out s x L = 0.444. 

The parameters correspond to the experimental conditions in Ref. 13 ( / in s x L = 

0.111, β HLs = 0.925). 
NASA/TMó2003-212228 11 

0 5 10 15 20

0.0

0.1

0.2

0.3

0.4

0.5

|v|/Ue, out, |w|/Ue, out

y/Hout

 |v|

 |w|

Fig. 7. The optimal velocity perturbations at / in s x L =0.111 and β HLs = 0.925 

corresponding to the streamwise velocity perturbation at / out s x L = 0.444 shown in 

Fig. 6. 

There is a possibility of enhancing the transient growth mechanism by means of wall 

cooling. The effect of wall cooling was investigated by Tumin and Reshotko9

 within the 

scope of a parallel flow approximation. In order to estimate possible increases of the 

energy ratio on a cold wall at a high favorable pressure gradient, we utilize the method of 

Ref. 9 for a compressible flow with local Mach number 0.5 and Hartree parameter 0.353. 

The results are shown in Fig. 8. One can see that cooling of the wall might provide a 

tenfold increase in the energy ratio. 
NASA/TMó2003-212228 12 

0.0 0.5 1.0 1.5 2.0 2.5

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

 Tw/Tad = 1.00

 0.5

 0.25

x/Ls

G/ReL

Fig. 8. Effect of the temperature factor on energy growth at the experimental 

conditions of Ref. 13. 

4.3 Example No. 2 of LPT Conditions 

 Experiments on a flat plate with a pressure gradient corresponding to a modern lowpressure airfoil were carried out by Volino and Hultgren14 at the NASA Glenn Research 

Center. Figure 9 shows a comparison of the theoretical and experimental free-stream 

velocity distributions along the flat plate (one can find more details in Ref. 14). Although 

there is a discrepancy between the theoretical and experimental data for the region of the 

favorable pressure gradient, we utilize a theoretical distribution in the following 

evaluations. The numerical methods for solving the boundary layer equations are 

described elsewhere.15,16 Figures 10 and 11 show comparisons of the theoretical and 

experimental shape factors and the momentum thickness, respectively. One can see that 

the theoretical distribution of the free-stream velocity provides reasonable agreement 

between the theoretical and experimental data for the boundary-layer parameters. 
NASA/TMó2003-212228 13 

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Ue/Uref

x/Lref

 Experiment

 Theory

Fig. 9. Comparison of theoretical and experimental14 free-stream velocity 

distribution along the plate.

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

0

3

6

9

 ReLref=100,000

 25,000

 Experiment

H

x/Lref

Fig. 10. Comparison of theoretical shape factors at two Reynolds numbers with the 

experimental data in Ref. 14. 
NASA/TMó2003-212228 14 

0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60

0.0

0.4

0.8

 ReLref=100,000

 ExperimentI

θ sqrt(ReLref)/Lref

x/Lref

Fig. 11. Comparison of the theoretical momentum thickness with the experimental 

data in Ref. 14. 

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

U/Uref

y/sqrt(νLref/Uref)

experiment

 theory

x/Lref = 0.266

Fig. 12. Comparison of the theoretical and experimental14 velocity profiles 

at / ref x L = 0.266. 

Comparisons of the theoretical and experimental velocity profiles are shown in Figs. 12 

and 13. Because there are some discrepancies between the theoretical and experimental 

data, one should consider the following transient growth analysis as an estimation under 

the experimental conditions. 
NASA/TMó2003-212228 15 

0.0 0.5 1.0 1.5 2.0 2.5 3.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x/Lref = 0.51

y/sqrt(νLref/Uref)

U/Uref

experiment

 theory

Fig. 13. Comparison of the theoretical and experimental14 velocity profiles 

at / ref x L = 0.51. 

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0000

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

0.0007

0.0008

G/Re

βHref

 xin/Lref=0.125

 xin/Lref=0.250

 xin/Lref=0.375

Fig. 14. Effects of the spanwise wave number, β , and the starting point, 

/ in ref x L , on the energy growth at the experimental conditions of Ref. 14. 

 Figure 14 demonstrates the energy ratio versus the spanwise wave number at 

/ out s x L =0.5 and three starting points, / in s x L = 0.125, 0.250 and 0.375. One can see 

that the results are similar to the previous case with a Hartree parameter of 0.353. 
NASA/TMó2003-212228 16 

5. Summary

 The results for the transient growth phenomenon within the scope of the linearized 

boundary-layer equations in the presence of a streamwise pressure gradient are consistent 

with previous results obtained within the scope of the parallel flow approximation and 

linearized Navier-Stokes equations.9

 A favorable pressure gradient decreases the nonmodal growth while an unfavorable pressure gradient leads to an increase of the 

amplification. 

 The example of a Falkner-Skan flow with a Hartree parameter β H = 0.353 

corresponds to the experimental data13 and simulates the flow over a low-pressure turbine 

airfoil upstream of the separation point. At this pressure gradient, the transient growth 

mechanism appears to be weak, and the latter, together with the low Reynolds number 

allows us to speculate that tripping of the boundary layer via a secondary instability of 

streaky structures is likely to fail. This conclusion should be verified by further work. 

Estimation of the transient growth at experimental conditions on a flat plate with a 

streamwise pressure gradient14 shows that the results are similar to the example of 

Falkner-Skan flow at 0.353 β H = . The theory of the transient growth mechanism 

predicts the possibility of enhancing the energy growth by means of wall cooling. The 

example within the scope of the parallel flow theory9

 demonstrates that cooling of the 

wall might provide a tenfold increase in the energy ratio. Future experiments on boundary 

layer tripping accompanied by wall cooling will contribute to our understanding of the 

bypass transition mechanism. 

 Theoretical analysis of the optimal disturbances predicts that there is an optimal 

spacing between perturbers and their optimal location from the leading edge. The latter 

results can be utilized in future experiments with tripping of the boundary layer over the 

LPT airfoil. This type of experiment can be carried out at the NASA Glenn Research 

Center,14 at the US Naval Academy,13 and at the University of Notre Dame.17 

 Consideration of the optimal velocity perturbations in Fig. 7 indicates that they are 

spreading across the boundary layer. This means that an array of generators localized on 

the wall will not provide excitation of the optimal disturbances. Therefore, the question 

of realizability of the optimal disturbances arises. For example, one can solve the 

receptivity problem for an array of generators on the wall and evaluate their shapes (or 

other parameters), providing velocity disturbance profiles closest to the optimal ones. 

Another option is to solve the receptivity problem for distributed generators upstream 

from the starting point, in x , to find their distribution, leading to the optimal disturbances. 

The next option is to design a disturbance generator that provides impact on the flow 

inside the boundary layer instead of a perturbation of the near-wall region only. For 

example, it might be a focused laser beam proceeding from the wall, where it could be 

delivered by a fiber-optic system.18 These fundamental issues should be addressed in 

future research programs on the application of bypass transition mechanisms to 

separation flow control at low Reynolds numbers. 
NASA/TMó2003-212228 17 

Appendix

 For purpose of consistency of the formulation, we repeat the main features of the 

numerical scheme described in Ref. 5. Matrices , 01 2 AB B B , , and in Eq. (7) are as 

follows: 

2

0 2

2

00 0

1 0 00

0 0 0 00 , 0 0 02 0

00 0 0 0

y

y

y

U U

V U V V

U V

β

β

β β

β β

  −       − −   = =   −− −     − −  

A B

1 2

0 1 0 0 0000

0 0 0 1000 , 0 2 0 1 0100

0 0 0 0010

V

B B

V

V

   −    − = = − −

   −

The scheme utilized for Eq. (7) in the streamwise direction is 

( )( ) ( ) ( )( ) 1 0 1 1 1

01 2 y yy xf f f   − =∆ + +     Af Af B B B

() () ( ) ( ) ()( ) 1 1 1 1 1

01 2

3 1 2 , 1 2 2

n nn n n n

y yy x f f fn + −   + + +

− + =∆ + + ≥     Af Af Af B B B

where n stands for the step number along the coordinate x. At each streamwise position, 

the one-dimensional boundary-value problem is solved using a spectral collocation 

method based on Chebyshev polynomials. Usually, we used 100 intervals along the 

coordinate x and 100 Chebyshev polynomials for the solution approximation. 

 To find the optimal perturbations corresponding to the maximum of the energy ratio 

in (9c), the forward solution of Eq. (7) is accompanied by the backward solution of the 

adjoint problem.5,6 In the present work, the adjoint system was discretized, i.e., the 

discretized adjoint equations were employed. Another approach is to utilize the adjoint 

form of the discretized forward equations (see discussion in Ref. 5). An arrangement of 

the iterations is described elsewhere.5,6 Usually, 2 to 3 iterations were enough to achieve 

convergence at the 0.1 percent level. 
NASA/TMó2003-212228 18 

References

1. Ellingsen, T., and Palm, E., ìStability of Linear Flow,î Phys. Fluids, Vol. 18, 

1975, pp. 487ñ488. 

2. Landahl, M. T. ìA Note on an Algebraic Instability of Inviscid Parallel Shear 

Flows,î J. Fluid Mech., Vol. 98, 1980, pp. 243ñ251. 

3. Reshotko, E., ìTransient Growth: A Factor in Bypass Transition,î Phys. 

Fluids, Vol. 13, 2001, pp. 1067ñ1075. 

4. Schmid, P.J., and Hennigson, D.S., Stability and Transition in Shear Flows, 

Springer, New York, 2001. 

5. Andersson, P., Berggren, M., and Henningson, D., ìOptimal Disturbances and 

Bypass Transition in Boundary Layers,î Phys. Fluids, Vol. 11, 1999, 

pp. 134ñ150. 

6. Luchini, P., ìReynolds Number Independent Instability of the Boundary Layer 

Over a Flat Surface, Part 2: Optimal Perturbations,î J. Fluid Mech., Vol. 404, 

2000, pp. 289ñ309. 

7. Reshotko, E., and Tumin, A., ìThe Blunt Body Paradox ñ A Case for Transient 

Growth,î Laminar-Turbulent Transition, edited by H.F. Fasel and W.S. Saric, 

Springer, New York, 2000, pp. 403ñ408. 

8. Reshotko, E., and Tumin, A., ìSpatial Theory of Optimal Disturbances in a 

Circular Pipe Flow,î Physics of Fluids, Vol. 13, 2001, pp. 991ñ996. 

9. Tumin, A., and Reshotko, E., ìSpatial Theory of Optimal Disturbances in 

Boundary Layers,î Physics of Fluids, Vol. 13, 2001, pp. 2097ñ2104. 

10. Corbett, P., and Bottaro, A., ìOptimal Perturbations for Boundary Layers 

Subjected to Stream-Wise Pressure Gradient,î Phys. Fluids, Vol. 12, 2000, 

pp. 120ñ130. 

11. Tumin, A., ìA Model of Spatial Algebraic Growth in a Boundary Layer 

Subjected to a Streamwise Pressure Gradient,î Phys. Fluids, Vol. 13, 2001, 

pp. 1521ñ1523. 

12. Reshotko, E., and A. Tumin, ìInvestigation of the Role of Transient Growth in 

Roughness-Induced Transition,î AIAA Paper No. 2002ñ2850, June 2002. 

13. Volino, R.J., ìSeparated Flow Transition Under Simulated Low-Pressure 

Turbine Airfoil Conditions: Part 1 ñ Mean Flow and Turbulence Statistics,î 

Proceedings of ASME TURBO EXPO 2002 Conference, Paper GTñ2002ñ

30236. 

14. Volino, R.J., and Hultgren, L.S., ìMeasurements in Separated and Transitional 

Boundary Layers Under Low-Pressure Turbine Airfoil Conditions,î 

Proceedings of ASME TURBO EXPO 2000 Conference, Paper 2000ñGTñ0260. 

15. Cebeci, T., and Bradshow, P., Physical and Computational Aspects of 

Convective Heat Transfer, Springer-Verlag, New York, 1984. 

16. Cebeci, T., An Engineering Approach to the Calculation of Aerodynamic 

Flows, Springer, N.Y., and Horizons Publishing, Long Beach CA, 1999. 
NASA/TMó2003-212228 19 

17. Huang, J., Corke, T.C., Thomas, F.O., ìSeparation Control Over Low Pressure 

Turbine Blades,î Bulletin of the American Physical Society, Vol. 47, No. 10, 

Nov. 2002, p. 167. 

18. Enikov, E., Private communication, University of Arizona, Tucson, 2002. 

19. Tumin, A., and Reshotko, E., ìOptimal Disturbances in Compressible 

Boundary Layersî, AIAA Paper No. 2003ñ0792, January 2003. 
This publication is available from the NASA Center for AeroSpace Information, 301–621–0390.

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E–13848

WBS–22–274–00–0206

25

Transient Growth Theory Prediction of Optimal Placing of Passive and Active

Flow Control Devices for Separation Delay in LPT Airfoils

Anatoli Tumin and David E. Ashpis

Hydrodynamic stability; Vortex generators; Flow control; Turbulence; Transition;

Turbomachinery; Turbines; Separation

Unclassified -Unlimited

Subject Categories: 02, 07, and 34 Distribution: Nonstandard

Anatoli Tumin, University of Arizona, Tucson, Arizona 85721 (work funded under NCC3–991); David E. Ashpis,

NASA Glenn Research Center. Responsible person, David E. Ashpis, organization code 5820, 216–433–8317.

An analysis of the non-modal growth of perturbations in a boundary layer in the presence of a streamwise pressure

gradient is presented. The analysis is based on PSE equations for an incompressible fluid. Examples with Falkner-Skan

profiles indicate that a favorable pressure gradient decreases the non-modal growth while an unfavorable pressure

gradient leads to an increase of the amplification. It is suggested that the transient growth mechanism be utilized to

choose optimal parameters of tripping elements on a low-pressure turbine (LPT) airfoil. As an example, a boundarylayer flow with a streamwise pressure gradient corresponding to the pressure distribution over a LPT airfoil is considered. It is shown that there is an optimal spacing of the tripping elements and that the transient growth effect depends on

the starting point. At very low Reynolds numbers, there is a possibility to enhance the transient energy growth by means

of wall cooling.