Linear and nonlinear development of perturbations in the asymptotic suction boundary layer

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LICENTIATE T H E S I S

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering Division of Fluid Mechanics

:

Linear and Nonliner Development of Perturbations in the

Asymptotic Suction Boundary Layer

Niklas Davidsson

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Linear and Nonlinear Development of Perturbations in the

Asymptotic Suction Boundary Layer

by

Niklas Davidsson

Division of Fluid Mechanics

Department of Applied Physics and Mechanical Engineering Lule˚ a University of Technology

SE-971 87 Lule˚ a Sweden

Lule˚ a, February 2005

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Preface

This work has been carried out under supervision of Professor H˚ akan Gustavs- son at the Division of Fluid Mechanics, Department of Applied Physics and Mechanical Engineering, Lule˚ a University of Technology, Sweden during the years 2002-2005. The research is a part of the program for Energy Related Fluid Mechanics founded by the Swedish Energy Agency.

I would like to thank my supervisor H˚ akan Gustavsson for the support and guidance. I would also like to express my gratitude to Hans ˚ Akerstedt, my co- supervisor, and the staff at the Division of Fluid Mechanics for providing friendly atmosphere. Ori Levin and Dan Henningson, KTH Mechanics, is gratefully acknowledged for valuable cooperation and guidance.

Niklas Davidsson

Lule˚ a, February 2005

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Summary

Turbulent processes play an important role in most flow systems. To opti- mize and control these flows, knowledge about the mechanisms that lead to and maintain turbulence is crucial. Boundary layers with wall suction are known to delay/prevent transition to turbulence as well as separation. Therefore it is a promising example of passive flow control which may reduce losses in many industrial energy conversion systems. Of particular interest is the asymptotic suction boundary layer (ASBL). It is reached downstream of a Blasius bound- ary layer (BBL) if spatially uniform and steady suction is applied over a large area. The flow is parallel, has an analytically well-defined velocity profile and hence serves as an ideal model flow to evaluate transition mechanisms. Ex- perimentally, a common observation for boundary layers subject to free stream turbulence is that streamwise elongated structures (streaks) pre-date transition to turbulence. Moreover, streaks appear also well into the turbulent regime.

In this perspective, a study on the formation, growth and instability of streaks should provide insight on the stabilising effects of wall suction.

This licentiate thesis is devoted to theoretical studies of the ASBL. The fo- cus lies in investigating the mechanisms involved in disturbance growth and transition, but also on the comparison with the corresponding suction-free flow (i.e. the BBL). The thesis consists of three papers.

In the first paper, the transient growth of streaks is studied. The streak is created by the lift-up of a localized initial disturbance given by a delta function, and the Orr-Sommerfeld/Squire system of equations is solved as an initial value problem for both the ASBL and the BBL. An analytical solution is obtained for the ASBL, for which disturbances initiated in the free-stream are found to move towards the wall with the suction velocity. A parameter study shows that the most amplified disturbances are obtained when placed inside the boundary layer, and that the overall largest growth is obtained for the BBL. For disturbances placed in the free stream, however, the relation is opposite; the ASBL convects the disturbance towards the wall where it quickly experiences shear and becomes larger than for the BBL.

In the second paper, the nonlinear evolution of a localized model disturbance is followed and compared for the two flow cases. A methodology for determining non-linear terms is presented. The model prescribes an identical wall-normal disturbance velocity in the two flows, thus the slight difference obtained originate from the Squire equation. The results show a complex behaviour of the non linearities, with a double peak response in the wave number plane. Whereas both these peaks are damped out in the ASBL, one develop a slight initial growth in the BBL. To get substantial growth larger Reynolds numbers must be considered.

In the third paper, the transition process in the ASBL is studied by means of

temporal direct numerical simulations. Three scenarios are considered: growth

and breakdown of streaks initiated by streamwise vortices, oblique transition

where an oblique wave pair is used to trigger the streamwise vortices, and three-

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dimensional random noise. The different stages of transition are identified and found to be in accordance with previous work on channel flows and boundary layers. Most competitive in terms of transition for the lowest initial energy is the oblique transition scenario. It also has the steepest relation to the Reynolds number, indicating that this scenario will dominate even more as the Reynolds number is increased.

Appended papers

• Paper A: Davidsson, E. N. and L. Gustavsson, L. H., Elementary solutions for streaky structures in boundary layers with and without suction, to be submitted .

• Paper B: Davidsson, E. N. and L. Gustavsson, L. H., Nonlinear growth of a model disturbance in boundary layers with and without suction, to be submitted .

• Paper B: Davidsson, E. N., Levin, O. and Henningson, D. S., Transition

thresholds in the asymptotic suction boundary layer, to be submitted.

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Paper A

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Elementary solutions for streaky structures in boundary layers with and without suction

E. Niklas Davidsson L. H˚ akan Gustavsson

Division of Fluid Mechanics, Lule˚ a University of Technology, SE-97187 Lule˚ a, Sweden

Abstract

The behaviour of small, streamwise elongated disturbances in the as- ymptotic suction boundary layer (ASBL) and the Blasius boundary layer (BBL) are compared. In particular, initial perturbations localized (δ- functions) in the wall-normal direction are studied. Analytical solutions are presented for the wall-normal and streamwise velocities in the ASBL case whereas both analytical and numerical methods are used for the BBL case. The initial position of the perturbation and its spanwise wave number are varied in a parameter study. We present results of maxi- mum amplitudes obtained, the time to reach them, their position and optimal spanwise scales. Free-stream disturbances are shown to migrate towards the wall and reach their (negative) optimum inside the boundary layer. The migration is faster for the ASBL case and a larger amplitude is reached than for the BBL. For perturbations originating inside the bound- ary layer the amplitudes are overall larger and show the phenomenon of overshoot, i.e. positive amplitudes moving out of the boundary layer. The overshoot is largest for the ASBL case and has a complex inner structure for certain spanwise scales. The results for streak growth obtained do not give a clean-cut indication that the ASBL’s observed stabilizing behaviour is related to transient growth.

1 Introduction

Streamwise elongated structures (streaks) is a prominent feature in boundary

layer flows close to turbulent transition and also well into the turbulent regime

(Johansson et al., 1991). In linear stability analysis, the generation of streaks

is inherent to the mechanism of transient growth since structures of optimal

energy growth have in general zero (or small) streamwise wave numbers (see

for instance Butler and Farrell 1992 for temporal and Andersson et al. 1999 for

spatial calculations). Non-linearly, streaks may be generated in the oblique tran-

sition model (Schmid and Henningson, 1992) where they result from non-linear

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interaction of oblique waves. Since streamwise elongated structures (α = 0, or

∂

∂x

= 0) do not exhibit non-linear self-interaction on the wall-normal component (Gustavsson, 1991), secondary effects operating on a streaky background have been of considerable recent interest, both to explain the sudden growth of sec- ondary perturbations observed at transition (see e.g. Reddy et al. 1998; Brandt and Henningson 2002; Wundrow and Goldstein 2001, for theory, and Matsubara et al. 2000; Asai et al. 2002, for experiments), and to model structures observed in the fully turbulent regime (Schoppa and Hussain, 2002). In this perspective, the formation and growth of streaks should be a first indicator of the tendency for a flow to become unstable (in the bypass sense) and should therefore be of help in designing strategies for flow control. Comparisons between different flows may then give insights about the mechanisms involved in the transition process.

Of particular interest are cases where experiments show distinct differences on the stability of the flow. One such case is offered by a boundary layer with wall suction which is known to delay/prevent turbulent transition (see Fransson and Alfredsson 2003 for experiments on free-steam turbulence) as well as separation.

Suction is known to decrease growth of (2D) Tollmien-Schlichting waves and a critical Reynolds number of 54000 is reported (see Drazin and Reid 1981 and references therein). If streaky structures play the alleged central role in the bypass transition process, a comparison of streak growth between an ordinary boundary layer (Blasius) and a suction boundary layer would be of interest.

Such a study has recently been done by Fransson and Corbett (2003), who con- sidered the optimal energy growth of linear disturbances in the two flows. The most amplified structure was found to have a spanwise wave number β = 0.53 and the spatial distribution correspond reasonably well to the experimental re- sults of Fransson and Alfredsson (2003). An interesting result was also that the two flows show little difference in algebraic growth whereas it is known that suction effectively dampens the Tollmien-Schlichting wave growth.

Most recent work on transient growth has been concerned with the opti- mal energy growth, where the initial perturbation giving maximum energy is determined, but limited information is obtained of how particular perturba- tions behave. Earlier attempts in this direction are found in the work on plane Poiseuille flow (Henningson et al., 1993) and on the Blasius boundary layer (Lasseigne et al., 1999), but no systematic evaluation seems to have been made.

Also, from a practical point of view, optimal perturbations may not always be

accessible or possible to obtain so it would therefore be of interest to study

more elementary types of perturbations. In the present paper the approach

taken is to study how a localized perturbation (δ-function) applied at a cer-

tain location in the wall-normal direction is developing in both the asymptotic

suction boundary layer and the Blasius boundary layer. With this method, it

is possible to gain more detailed insights of how perturbations behave and by

superposition more complex perturbations can be studied. A further benefit is

that analytical techniques can be used to a large extent and serve as basis both

for numerical evaluations and validation of other numerical results. As the fun-

damental equations of the problem are well-known and the solution techniques

of standard character we give only marginal attention to derivations; an outline

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u v

w

V

0

x y

z

V

0

U(y) U

∞

porous plate

(a)

0 0.5 1

0 1 2 3 4 5 6 7

(b)

y δ

^∗

Figure 1: (a) The flow geometry with coordinate system and direction of velocity components. (b) Velocity profiles (black lines) and their derivative (grey lines) for the flow cases BBL (solid lines) and ASBL (dashed lines).

of the methods and details of some critical steps are given in the appendices.

The paper is organized in the following way: The geometry, the velocity profiles for the two boundary layers, governing equations and boundary condi- tions are presented in section 2, as well as the initial conditions representing a localized perturbation normal to the wall. In section 3 we present the solution for the wall-normal and streamwise velocity components resulting from this per- turbation. Details of the mathematical procedures used are given in appendices A-C. In section 4 a parameter study is made of the influence of position and spanwise scale of the initial perturbation on the temporal development. Here, some specific features such as maximum amplitudes are studied in detail. The results are summarized and discussed in section 5.

2 Basic equations

In this paper we investigate the evolution of perturbations in a 2D, zero pres- sure gradient steady boundary layer flow, above a flat porous plate. When a pressure gradient is applied through such a plate, a wall-normal suction veloc- ity V

0

is induced, resulting in a boundary layer with constant properties in the streamwise direction. This boundary layer is denoted the Asymptotic Suction Boundary Layer (ASBL). With no suction the Blasius Boundary Layer (BBL) flow is obtained, which is the reference case in the following investigation. The flow situation with coordinate system is described in figure 1(a).

2.1 The mean velocity profile

In Drazin and Reid (1981) it is stated that one of the main reasons for the

improved stability characteristics of the ASBL versus the BBL flow is the change

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in shape of the mean velocity profile that the suction provides.

For the BBL the velocity profile is obtained numerically by solving the Bla- sius equation f

⁰⁰⁰

+1/2 f f

⁰⁰

= 0, with the appropriate boundary conditions (here f = U

⁰

).

An analytical expression for the ASBL mean velocity profile is obtained by assuming that the mean-flow only depends on the wall-normal coordinate, namely

U (y) = h

U

_∞

(1 − e

^{−y V}⁰^/ν

), V

0

, 0 i

. (1)

Here U

_∞

, V

0

and ν are the free-stream velocity, suction velocity and kinematic viscosity, respectively. The shape of the two velocity profiles are given in figure 1(b).

The displacement thickness obtained from the ASBL profile (1) is δ

^∗

= ν/V

0

, and the Reynolds number based on δ

^∗

becomes

R = U

_∞

V

0

. (2)

2.2 Evolution equations for small perturbations

The stability equations for the BBL and ASBL are derived in an identical man- ner, following a standard procedure; see for instance Schmid and Henningson (2001). The flow is decomposed as

u

i

= U

i

+ u

⁰_i

, p = P + p

⁰

, (3) where capital letters denote the steady mean flow and prime a perturbation.

Furthermore, the parallel flow assumption is used in the BBL. Thus, for the following analysis to be valid, the length scale of the streak (l) must be smaller than some characteristic length scale of the boundary layer (L), say l/L 1.

Hultgren and Gustavsson (1981) considered this relation and pointed out that it corresponds to the inequality 1 l/δ

^∗

R. Having inserted flow state (3) into the Navier-Stokes equations, the evolution equations are obtained by eliminating the pressure terms. The disturbances are now fully described by two equations, here made non-dimensional by scaling with δ

^∗

and U

_∞

and presented for the ASBL:

( ∂

∂t + U ∂

∂x − 1 R

∂

∂y )∇

²

− ∂

²

U

∂y

²

∂

∂x − 1 R ∇

⁴

v = 0 (4)

∂

∂t + U ∂

∂x − 1 R

∂

∂y − 1 R ∇

²

η = − dU dy

∂v

∂z . (5) The primes denoting fluctuating quantities are dropped in the above equations.

Equation (4) governs the wall-normal velocity and (5) the normal vorticity, η = ∂u/∂z − ∂w/∂x, from which the velocity components u and w are obtained together with the continuity equation.

Corresponding evolution equations for the BBL are obtained by removing

the terms marked with bold text in (4) and (5).

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2.3 Boundary conditions

For the BBL flow, both perturbation and mean velocities are zero at the wall. In the ASBL, however, the wall condition for the wall-normal perturbation velocity is not obvious. An analysis for this must also take into account the flow through the porous wall. Gustavsson (2000) showed by means of Darcy’s law that the relation

d

³

dy

³

− d

²

dy

²

+ iαRU

⁰

+ (α

²

+ β

²

)R G

v = 0 (6)

needs to be fulfilled at the wall. Here α and β are transform variables in stream- wise and spanwise direction, respectively, and since G is a parameter propor- tional to the permeability of the plate, statement (6) enforces v|

y→ 0

= 0 for small values of the permeability. We also consider perturbations initially local- ized in the wall-normal direction decaying far away from the wall.

Thus, the following boundary conditions will be used both in the ASBL and the BBL:

v = ∂v

∂y = η = 0, y = 0 v, η → 0, y → ∞ .

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2.4 Initial conditions

To solve (4), the initial value for v must be specified. By choosing this to be localized in the y-direction, fundamental solutions are obtained which can be superposed to represent an arbitrary perturbation. Thus,

ˆ

v

0

= A(β) δ(y − y

⁰

), (8)

where the Fourier-transform version has been used, and δ denotes Dirac´s func- tion. For ˆ η, we will assume

ˆ

η

0

= 0 (9)

to highlight the specific response given by the forcing term. Physically, ˆ η

0

= 0 means a perturbation axisymmetric in the x-z-plane as discussed by Gustavsson (1991). From a general point of view, a general disturbance must also consist of η

0

6= 0.

3 Analytical solution for streamwise elongated structures

We consider (4) and (5) for longitudinal structures, ∂/∂x = 0. The solution

method are to first apply Laplace and Fourier transforms in t and z (trans-

form variable: β), respectively. Then the (ordinary) differential equations are

solved with variation of the parameters and the Laplace transform is inverted

using residual calculus. Procedures and critical solution steps are presented in

appendices A and B for the ASBL and BBL, respectively.

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3.1 ASBL

3.1.1 The wall-normal perturbation velocity

With the initial perturbation given by (8) inserted in (4), inversion of the Laplace transform gives

ˆ

v/A(β) = 1

√ πT e

^−(β²^{+1/4) T}

· h 1

2 e

^−(y−y⁰^)/2

(e

^−(y−y⁰⁾²^/4T

+e

^−(y+y⁰⁾²^/4T

)

− e

^y⁰^/2

e

^−βy−y²⁰^/4T

i

+ (β −

¹₂

) e

^−β(y+y⁰⁾

e

^y⁰^−βT

× h erfc

(β −

¹₂

) √

T −

₂^√^y⁰_T

−erfc

(β −

¹₂

) √

T −

^y+y₂^√_T⁰

i .

(10) where T = t/R and the bold parts mark terms that are unique to ASBL.

3.1.2 The streamwise perturbation velocity

With ˆ v known, ˆ η is solved via (25) and then ˆ u = −

βⁱ

η since α = 0. The result ˆ obtained by similar operations as for the wall-normal velocity is

ˆ

η = 1

4 i β R A(β) (

−

1 + β − 1

β · (e

^−y+y⁰

− e

^y⁰

)

e

^−β²^T

erfc 1 2

√ T + y + y

0

2 √ T

+ e

^−y

e

^−β²^T

erfc 1

2 √ T − y − y

0

2 √ T

− erfc 1 2

√ T − y + y

0

2 √ T

+ 2e

^βT

e

^−y

e

^−β(y+y⁰⁾

"

erfc

(β + 1

2 ) √

T − y + y

0

2 √ T

− erfc

(β + 1

2 ) √

T − y

0

2 √ T

#

+ e

^−y

e

^−β²^T

erfc 1 2

√ T + y − y

0

2 √ T

+ 1 − 2β

β e

^−βT

e

^−y+y⁰

e

^−β(y+y⁰⁾

"

erfc

(β − 1

2 ) √

T − y + y

0

2 √ T

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− erfc

(β − 1

2 ) √

T − y

0

2 √ T

#

+ 1

β e

^{βT +y}⁰

e

^β(y+y⁰⁾

erfc

(β + 1

2 ) √

T + y + y

0

2 √ T

− 1

β e

^{βT +y}⁰

e

^−y

e

^−β(y−y⁰⁾

erfc

(β + 1

2 ) √

T + y

0

2 √ T

) .

One can note that factors like e

^−β²^T

and e

^βT

show up and that β thus influence

the temporal behaviour in a quite complex manner. Despite the complexity of

this solution it is quite useful to evaluate the temporal behavior of ˆ η.

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3.2 The BBL

For the wall-normal velocity the procedures and initial conditions are identical to the ones used for the ASBL. For the streamwise velocity, however, numerical integration is required since the mean velocity profile is not known analytically.

3.2.1 The wall-normal velocity

The differences between the ASBL and BBL in the stability equation (4) are few, and lead to differences in the solutions that were marked with bold face terms in (10). The solution for the BBL, obtained after invoking the initial assumption (8) once again, is

ˆ

v/A(β) = 1

√ πT e

^−β²^T

· h 1

2 (e

^−(y−y⁰⁾²^/4T

+ e

^−(y+y⁰⁾²^/4T

) − e

^−βy

e

^−y²⁰^/4T

i + β e

^−β(y+y⁰⁾

· erfc

β √

T − y

0

2 √ T

− erfc

β √

T − y + y

0

2 √ T

, (12) with T = t/R.

3.2.2 The streamwise velocity

The initial conditions (8) and (9) are used also for the BBL. The streamwise velocity is obtained by ˆ u = ˆ u(y, t, β) = −

_βⁱ

η which gives ˆ

ˆ

u = 1

iπ e

^−β²^T

Z

_∞

0

( sin(σy)

Z

_∞

0

e

^iσy⁰

˜ˆv

₋

(y

⁰

) − e

^−iσy⁰

˜ˆv

+

(y

⁰

) dU

dy

⁰

dy

⁰

(13) +

Z

y 0

sin(σ(y

⁰

− y)) ˜ˆv

₋

(y

⁰

) − ˜ˆv

⁺

(y

⁰

) dU dy

⁰

dy

⁰

)

e

^−σ²^T

dσ ,

where ˜ˆ v

_±

= ˜ˆ v|

r=±iσ

is retrieved from (31) (with r

3

= r). The integration variable σ comes from integration along the branch-cut defined in appendix A.1. Since the y

⁰

-integrals contain dU /dy

⁰

, ˆ u for the BBL flow must be evaluated numerically.

To verify that the integrations in (13) are carried out correctly, a second independent solution method has been invoked. In Hultgren and Gustavsson (1981) an equation for u was derived, which has the Fourier transform solution

ˆ u =

Z

∞ 0

Z

t 0

U

⁰

v(y ˆ

⁰

, t

⁰

, β) e

^−β²^T^∗

2 √

πT

^∗

e

^−(y⁰^+y)²^/4T^∗

− e

^−(y−y⁰⁾²^/4T^∗

dt

⁰

dy

⁰

, (14)

where T

^∗

= T −T

⁰

= (t−t

⁰

)/ R and ˆ v is given by (12). For this solution numeri-

cal instabilities arise for certain parameter combinations when using moderately

small integration steps. Thus the analysis has been taken one step further in

appendix C, and the evaluation of the inner integral is carried out through

time-integrals in (40) and (41).

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Table 1 shows the obtained disturbance amplitudes for some different sets of parameters for the two different methods. With our integration step the differences is of order 10

⁻²

−10

⁻³

percent, therefore the agreement is considered to be sufficient.

Table 1: Values of streamwise perturbation velocity ˆ u calculated by two meth- ods: ˆ u

A

by (13) and ˆ u

B

by (14), (40) and (41).

y T y

0

β u ˆ

A

u ˆ

B

Difference (%)

2.0 1 8 0.25 6.755·10

⁻⁷

6.754·10

⁻⁷

1.0·10

⁻²

2.0 2 8 0.25 1.837·10

⁻³

1.837·10

⁻³

1.1·10

⁻²

2.0 3 8 0.25 2.584·10

⁻³

2.583·10

⁻³

1.2·10

⁻²

0.5 0.01 0.5 2 1.523·10

⁻²

1.523·10

⁻²

1.5·10

⁻²

0.5 0.1 0.5 2 2.144·10

⁻²

2.144·10

⁻²

4.4·10

⁻³

0.5 0.2 0.5 2 1.128·10

⁻²

1.128·10

⁻²

4.7·10

⁻³

Since the numerical integration of (14), using (40) and (41), is faster this solution has been used for all forthcoming calculations of the streamwise distur- bance development in the BBL.

4 Results or Parameter investigation

The perturbation velocity for both flow cases is described by the solutions for ˆ v and ˆ u. In this section further analysis about the characteristics and differences of the two flows will be done by means of graphical representation of numerical data. There are two parameters in the solution; y

0

and β representing the position of the initial perturbation and the spanwise scale, respectively.

We study perturbations initiated inside as well as outside the boundary layer.

For the wall-normal velocity we only illustrate the behaviour for two cases, but for the streamwise velocity a more thorough study is conducted.

4.1 The wall-normal velocity

The behaviour of the wall-normal perturbation velocity is interesting since it affects the streamwise perturbation as a factor in the forcing term for the normal vorticity stability equations. The characteristics and differences of solutions (10) and (12) for the ASBL and BBL flows are studied below, although not as detailed as for the streamwise perturbation.

The appearance of ˆ v as time increases is given in figure 2 for the free-stream

disturbance with y

0

= 8 and β = 1. Observe that the scale on the ˆ v-axis

decreases as time increases. In accordance with initial condition (8) ˆ v starts

with an amplitude peak localized at y

0

. As time increases the peaks spread and

damp out, with the ASBL peak decaying marginally faster.

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0 0.2 0.4 0.6 0.8

T=0.1

0 0.05 0.1

vˆ

T=1

0 2 4 6 8

x 10⁻³ T=3

0 2 4 6 8 10 12

0 2 4 6 8

x 10⁻⁴ T=5

y

Figure 2: ˆ v for y

0

= 8 and β = 1. Solid line: BBL, dashed line: ASBL. Note

change of vertical scale.

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0 1 2 3 4 5 6 7 8 0

1 2 3 4 5 6 7 8 9

T y

Figure 3: The position of the maxima of ˆ v for y

0

= 8 and β = 2. Solid line:

BBL, dashed line: ASBL.

The prominent effect seen in the figure is that the ASBL peak moves towards the wall as time increases, whereas the BBL peak stays at y ≈ y

0

. This effect is further shown in figure 3, where the position of the peak is plotted versus time. The velocity of the ASBL peak is initially ∆y/∆T = −1. Rewriting this in dimensional quantities gives ∆y

^∗

/∆t

^∗

= −V

0

, i.e. the ASBL peak moves towards the wall with the suction velocity, a not too surprising result. Due to the boundary conditions this process changes character and slows down in the vicinity of the boundary layer. The peak movement stops slightly below y = 3 (' δ

⁹⁵

).

With the perturbation placed inside the boundary layer, an undershoot can be observed as the initial peaks decay. As an example, the appearance of ˆ v is plotted in figure 4. Note that the first frame has a different vertical scale. Here the parameters y

0

= 1 and β = 0.1 are chosen since they give a clear picture of the course of events. The trace of the initial condition can still be observed at T = 0.1, but as time increases the outer part of the maxima overdamp and develop minima, which compared to the initial peak decay rather slowly for both flow cases.

The undershoot is mainly observed for perturbations starting inside the boundary layer, but with small enough wavenumbers it can also be found for perturbations originating at the edge of the boundary layer. This feature has consequences also for the streamwise disturbances, as will be seen in section 4.2.

The behaviour of perturbations placed even further inside the boundary layer is similar to figure 4.

4.2 The streamwise velocity

For the streamwise velocity component we have focused on the following po-

sitions of the initial perturbation: y

0

= 8 (in the free-stream), y

0

= 2.5 (at

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−0.4

−0.2 0 0.2 0.4 0.6

T=0.1

−0.4

−0.2 0

vˆ

T=0.3

−0.4

−0.2 0

T=0.5

0 1 2 3 4 5 6 7 8

−0.4

−0.2 0

T=2

y

Figure 4: The undershoot phenomenon for ˆ v, here with y

0

= 1 and β = 0.1.

Solid line: BBL, dashed line: ASBL. Note change of vertical scale.

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−3

−2

−1

0x 10⁻⁴ T=1

−3

−2

−1

0x 10⁻⁴ T=2

−3

−2

−1 0x 10⁻⁴

u ˆ R __

T=3

−3

−2

−1

0x 10⁻⁴ T=4

0 2 4 6 8 10 12

−3

−2

−1

0x 10⁻⁴ T=5

y

Figure 5: Streamwise velocity for y

0

= 8 and β = 1. Solid line: BBL, dashed line: ASBL.

the edge of the boundary layer) and y

0

= 0.5 (inside the boundary layer).

Comparisons are here done by evaluating solution (11) for the ASBL and the numerically integrated solution (14) for the BBL. The initial perturbations are placed as mentioned above, and for each position (y

0

) the spanwise wavenumber is varied. For each value of β, the evolution of ˆ u is calculated and its largest (positive and negative) amplitudes are recorded together with their positions and times. These amplitudes are denoted the absolute minima (ˆ u

min

, i.e. the largest value of negative velocity) and the absolute maxima (ˆ u

max

, the largest value of positive velocity).

y

0

= 8: The typical behaviour of ˆ u for the free-stream disturbances is seen in figure 5, where β = 1. An initial minimum develops at y

0

and its subsequent development is seen to be different for the two cases. Whereas the BBL pertur- bation is mainly governed by diffusion, the ASBL perturbation is also advected against the wall. Thus, the action of the term U

⁰

∂v/∂z in (5) will appear earlier and lead to much larger amplitudes for the ASBL-case.

The temporal evolution of the amplitude peaks is shown in figure 6. The ASBL minimum follows a slightly curved line with an initial slope corresponding to a velocity (cf. section 4.1) of about 2V

0

. Thus, the minimum in u moves faster than the suction velocity V

0

. This may seem surprising but since the initial development of u is essentially given by u ∼ R

t

0

U

⁰

v dt

⁰

(Hultgren and

(19)

0 0.5 1 1.5 2 2.5 3 3.5 4 1

2 3 4 5 6 7 8

T ymin

Figure 6: Position of the minima for y

0

= 8 and β = 1. Solid line: BBL, dotted line: ASBL.

Gustavsson, 1981) the connection between u and v is not transparent. However, the relation indicates that those parts of v that are subject to large shear will contribute more to the development of u than the peak region, cf. figure 1(b).

The absolute minima is plotted in figure 7 together with the position and time it appears. As shown in figure 7(a) there is an optimum obtained for β w 0.13 and β w 0.08 in the ASBL and BBL, respectively. Also, the ASBL amplitudes are larger. Figure 7(b) shows that the absolute minima in the ASBL are reached faster, as discussed above. In figure 7(c) one can note that the positions where the absolute minima are reached are fairly independent of β (for β . 1.35 in the ASBL case), and lie inside the boundary layer. At β & 1.35 in the ASBL, the y

min

values make a sudden jump which indicates a two- peak phenomenon in the determination of y

min

. The ˆ u

min

amplitudes are here very small and since a similar phenomenon appears for perturbations inside the boundary layer, we present details in that context.

y

0

= 2.5: Data for the absolute minima for perturbations initiated at the edge of the boundary layers are found in figure 8. As seen in figure 8(a) the largest amplitudes are reached for β w 0.21 in the ASBL and β w 0.15 in the BBL. The results here are different to what was found for free-stream distur- bances in several aspects: Much larger amplitudes are attained, and at much earlier time (figure 8b). Also, ASBL gives smaller amplitudes than BBL.

The position of the absolute minima, however, are still found inside the boundary layer and varies weakly with β; cf. figure 8(c). It seems that external disturbances have a certain penetration depth which they reach independent of spanwise scale. However, the time at which this occurs varies with β so the spanwise coherence is distorted during the process.

y

0

= 0.5: For perturbations initiated within the boundary layer, the tem-

poral behaviour of the streamwise velocity is quite different from the previous

cases. The development is characterized by an overshoot following the initial

(20)

0 0.5 1 1.5 2 0

0.002 0.004 0.006 0.008 0.01

β

|uˆmin| / R

(a)

0 0.5 1 1.5 2

0 2 4 6 8 10

β Tmin

(b)

0 0.5 1 1.5 2

0 1 2 3 4 5 6 7 8

β ymin

(c)

Figure 7: Amplitude (a), time (b) and position (c) of the absolute minima for

y

0

= 8. (·): ASBL, (X): BBL.

(21)

0 0.5 1 1.5 2 0

0.01 0.02 0.03 0.04

β

|uˆmin| / R

(a)

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1 1.2

β Tmin

(b)

0 0.5 1 1.5 2

0 1 2 3

β ymin

(c)

Figure 8: Amplitude (a), time (b) and position (c) of the absolute minima for

y

0

= 2.5. (·): ASBL, (X): BBL.

(22)

−0.02 0 0.02 0.04

T=0.005

−0.02 0 0.02 0.04

T=0.05

−0.02 0 0.02 0.04

u ˆ R __

T=0.2

−0.02 0 0.02 0.04

T=1

0 1 2 3 4 5 6

−0.02 0 0.02 0.04

T=2.5

y

Figure 9: Streamwise velocity for y

0

= 0.5 and β = 0.25. Solid line: BBL, dashed line: ASBL.

minimum, as shown in figure 9, where β = 0.25. The overshoot is associated with momentum transport away from the wall, and its timescale is large com- pared to the minima.

Data of the absolute minima are found in figure 10. Figure 10(a) shows that the amplitudes vary only marginally with β (with a broad peak at about β = 0.8). These minima are of secondary interest, however, since the absolute amplitude for the overshoot is larger and dominates the course of events. Ac- cording to figure 10(b, c) the absolute minima are reached very quickly and close to the original perturbation location. Also, here a small effect of suction is noticed.

For larger times, the perturbation development is dominated by the over- shoot phenomenon, for which data are presented in figure 11. The amplitude obtains its absolute maxima for both ASBL and BBL as β → 0, as seen in figure 11(a), and it is at least twice as large as the optimal absolute minima. For these wavenumbers also the time T

max

is largest, as shown in figure 11(b). It may be noted that the time to reach this maximum is comparable with the time to reach the minimum for a perturbation at the edge of the boundary layer (y

0

= 2.5;

cf. figure 8b). Also, the position of the maximum is comparable with those in

figure 8(c). For relatively large values of β, the maximum contains a double

structure. It shows up as two peaks in the temporal development, illustrated in

(23)

0 0.5 1 1.5 2 0

0.005 0.01 0.015 0.02 0.025 0.03

β

|uˆmin| / R

(a)

0 0.5 1 1.5 2

0 0.01 0.02 0.03 0.04 0.05 0.06

β Tmin

(b)

0 0.5 1 1.5 2

0 0.1 0.2 0.3 0.4 0.5 0.6

β ymin

(c)

Figure 10: Amplitude (a), time (b) and position (c) of the absolute minima for

y

0

= 0.5. (·): ASBL, (X): BBL.

(24)

0 0.5 1 1.5 2 0

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08

β

|uˆmax| / R

(a)

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1

β Tmax

(b)

0 0.5 1 1.5 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

β ymax

(c)

0 0.2 0.4 0.6 0.8 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

x 10⁻³

T

|uˆmax| / R

β=1.35 β=1.40

(d)

Figure 11: Amplitude (a), time (b) and position (c) of the absolute maxima for

y

0

= 0.5. (·): ASBL, (X): BBL. (d) Temporal development of the maxima in

the ASBL.

(25)

figure 11(d) for the ASBL. Since the largest obtained amplitude is shifted from the second to the first peak as β becomes larger than 1.35, our procedure for recording the absolute maximum leads to a discontinuity in T

max

, as shown in figure 11(b). Consequently, the motion of the peak gives a corresponding jump in the y-position seen in figure 11(c). A similar behaviour is observed for BBL at β ' 1.15 although not presented here. The relevance of these discontinuities could however be discussed, since the actual amplitudes at these wavenumbers are two orders of magnitude smaller than for β → 0.

As in all previous cases the y-position where the maximum is obtained is rather insensitive to the spanwise wavenumber. However, the time when the maximum is reached varies considerably with spanwise wavenumber so, again, the spanwise coherence for a given perturbation will be distorted. The largest value is found for about the same wavenumber as the maximum amplitude, which again agree reasonably well with previous cases.

5 Conclusions and discussion

The linear evolution of streamwise elongated structures triggered by a localized disturbance has been investigated. Two flows are compared throughout the investigation, namely the asymptotic suction boundary layer (ASBL) and the Blasius boundary layer (BBL).

Some interesting observations can be made of the results presented. For the wall-normal velocity the two cases differ little in the formal solutions but the difference is sufficient to move the peak of the perturbation for the ASBL- case towards the wall with a velocity which, not surprisingly, initially equals the suction velocity. For the streamwise velocity the solutions show that the perturbation amplitude scale with the Reynolds number at both flows.

The dramatic development occurs when the perturbation experiences shear which shows up in the streamwise component. The migration of v towards the wall for ASBL starts the shear-induced process at an earlier time than for BBL and it also results in a larger response. The slow rate for BBL is due to the fact that v in that case is diffusing towards the wall and thus the transient growth starts later. This process gives a simple explanation for how free-stream perturbations penetrate the boundary layer. It may be of some interest to note that the y-position where the response is largest seems to be independent of the spanwise scale. This y-position agrees reasonably well with the position of the global optima as obtained by Fransson and Corbett (2003) and Andersson et al.

(1999) for the ASBL and the BBL, respectively.

For a perturbation originating at the edge of the boundary layer (y

0

= 2.5) the shear induced process starts earlier, giving much larger amplitudes but now with the BBL having the largest response. Also, the position of the maximum amplitude varies slightly with β and at a slightly lower value than for y

0

= 8.

For a disturbance originating well inside the boundary layer the character

of the response changes drastically. Here, the process of overshoot dominates

the amplitude development and gives amplitudes outside the original position

(26)

well above those obtained in the previous cases. This fact opens possibilities for control of external perturbations, since the maxima are reached for times similar and below that of the minima of free-stream disturbances.

It is also of interest to note that ˆ u

min

for perturbations inside the boundary layer is insensitive to the spanwise scale, whereas the overshoot is not.

It may be of significance that the time scale for the perturbation development shortens drastically for a disturbance initiated within the boundary layer. The time scale is locally given by 1/U

⁰

as identified from (5). Thus, an analysis accounting for both inner and outer phenomena needs (at least) a two-timing formulation.

For all parameters the chosen initial perturbation shows the effect of tran- sient growth before the viscous decay sets in, with the maximum response ob- tained for β = 0.21 (ASBL) and β = 0.15 (BBL). The calculations of optimal growth show corresponding values of β = 0.53 for the ASBL (Fransson and Cor- bett, 2003), and for the BBL Butler and Farrell (1992) found a value of β = 0.65 (temporal calculations), while Andersson et al. (1999) reports β = 0.77 (spatial calculations, in the large-Reynolds-number limit). The difference is a conse- quence of our choice of initial condition diverging from the streamwise-oriented vortices given by the optimal linear growth theory.

Overall, the obtained differences between the two studied cases are surpris- ingly small. The globally largest amplitudes are of the same order of magnitude, although smaller in the ASBL. A similar relationship is also found for optimal disturbances by Fransson and Corbett (2003). Experimentally, Fransson and Alfredsson (2003) showed that suction gives an elimination of downstream en- ergy growth (for a boundary layer subjected to free stream turbulence). Thus, it is obvious that an explanation for these experimental results must include aspects of the transient growth process outside the scope studied here. This could involve e.g. non-linear interactions of oblique waves. It may also show the weakness of studying only α = 0 perturbations. In fact, the transient growth operates also for α 6= 0 and the extent of the growth areas (in the αβ-plane) may be as important for the non-linear interactions as the peak value at α = 0.

For optimal disturbances, Fransson and Corbett (2003) show that the ASBL has a much narrower growth area than the BBL, a fact that should definitely motivate further studies.

Acknowledgements

This work has been financed through the program of Energy Related Fluid

Mechanics operated by the Swedish Energy Agency.

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A Solutions for the ASBL

A.1 The wall-normal velocity

Solution is obtained by applying Fourier transformation in the spanwise direc- tion and Laplace transform in time, defined as:

ˆ

v(y, β, t) = Z

∞

−∞

e

^−iβz

v(y, z, t) dz (15)

˜ˆv(y, β, s) = Z

∞ 0

e

^−st

ˆ v(y, β, t) dt. (16) Applying these transformations to the governing equation (4) together with basic rules for derivatives and integrals of Fourier and Laplace transforms gives

(sR − D) − (D

²

− β

²

) (D

²

− β

²

)˜ˆ v = R (D

²

− β

²

)ˆ v

0

, (17) where D = ∂/∂y and ˆ v|

^t=0

= ˆ v

0

denotes the initial disturbance. The roots of the characteristic (homogeneous) equation of (17) are

( r

1,2

= ± β r

3,4

= − 1

2 ± r 1

4 + sR + β

²

≡ − 1

2 ± a. (18)

The method of variation of parameters uses these roots to superpose the eigen- functions, and applying the boundary conditions one obtains

˜ˆv = R 2a

2a β + r

4

e

^−βy

+ e

^r³^y

− β + r

3

β + r

4

e

^r⁴^y

Z

_∞

0

ˆ

v

0

e

^−r³^y⁰

dy

⁰

− R

2a e

^r³^y

Z

y

0

ˆ

v

0

e

^−r³^y⁰

dy

⁰

+ R 2a e

^r⁴^y

Z

y 0

ˆ

v

0

e

^−r⁴^y⁰

dy

⁰

.

(19)

The last step to retrieve a solution is to apply the inverse Laplace transforma- tion, which needs to be done with some care. It is defined as

ˆ

v(y, β, t) = lim

b→∞

1 2πi

Z

c+ib c−ib

e

^st

˜ˆv(y, β, s) ds , (20) i.e. integration is done along the vertical infinite line in the complex s-plane shown in fig. A.1. As the inversion contour is closed to the left, contributions to the inverse comes from poles and branch cuts. However, the terms containing the pole r

4

= −β cancel, as also the terms containing a = 0. Left is the branch cut starting at a = 0. Thus inversion of (20) turns into integration along this branch cut (where s = iσ and 0 < σ < ∞), as

i π ˆ v = Z

_∞

0

e

^−(1/4+β²^+σ²^)t/R

˜ˆv σ R dσ

a=−iσ

− Z

_∞

0

e

^−(1/4+β²^+σ²^)t/R

˜ˆv σ R dσ

_a=iσ

.

(21)

(28)

Im{s}

Re{s}

(c, i b)

(c, −i b) a = 0

β+1/4 R

Figure 12: The integration contour in the complex s-plane.

By specifying the initial perturbation as ˆ

v

0

= A(β) δ(y − y

0

), (22)

the integrals in (19) can be evaluated and different terms contribute depending on the relative values of y and y

0

. However, independent of the values of y vs.

y

0

, the same solution is obtained after inversion. The result is:

ˆ

v/A(β) = 1

√ πT e

^−(β²^{+1/4) T}

· h 1

2 e

^−(y−y⁰^)/2

(e

^−(y−y⁰⁾²^/4T

+e

^−(y+y⁰⁾²^/4T

)

− e

^y⁰^/2

e

^−βy−y⁰²^/4T

i

+ (β −

¹₂

) e

^−β(y+y⁰⁾

e

^y⁰^−βT

×

erfc

(β − 1

2 ) √ T − y

0

2 √ T

−erfc

(β − 1

2 ) √

T − y + y

0

2 √ T

.

(23)

A.2 The streamwise perturbation velocity

Transformations according to (15) and (16) are applied to the stability equation (5) for the normal vorticity η. Thus an equation with the characteristic roots

r

1, 2

= − 1

2 ± a = − 1 2 ± r 1

4 + sR + β

²

, (24)

is obtained. Note that these roots are the same as r

3, 4

for the wall-normal velocity, given in (18). Solving by using variation of parameters together with the boundary conditions give

˜ˆη = 1

a e

⁻¹²^y

sinh (ay) Z

∞

0

e

^−r¹^y⁰

I dy

⁰

(29)

+ 1 a e

⁻¹²^y

Z

y 0

e

¹²^y⁰

sinh (a(y

⁰

− y)) I dy

⁰

, (25) where I = R(ˆ η

0

− ∂˜ˆv/∂z · dU/dy) = R(ˆη

⁰

− iβ˜ˆv dU/dy) originates from the inhomogeneous part of the stability equation. The initial value ˆ η

0

is now set to zero, while ˆ v

0

is left unspecified at this moment. Inserting the known wall- normal velocity (23) and using integration by parts gives the result:

˜ˆη = iβR

²

2a e

^−y/2

(

− 1

r

2

sinh ay Z

∞

0

ˆ

v

0

e

^r²^y⁰

dy

⁰

+ 1 2r

2

r

1

+ β

r

2

+ β e

^−(a+1)y

− 1

2r

1

e

^(−1+a)y

+ 2a

(r

²₂

− β

²

)(r

1

− β) e

^−(β+1/2)y

− a(β

²

− r

2

β + r

1

) r

1

r

2

(r

1

− β)(r

²

− β) e

^−ay

! Z

∞ 0

ˆ

v

0

e

^−r¹^y⁰

dy

⁰

+ 1

2r

1

e

^(−1+a)y

Z

y

0

ˆ

v

0

e

^−r¹^y⁰

dy

⁰

− 1 2r

2

e

^−(1+a)y

Z

y

0

ˆ

v

0

e

^−r²^y⁰

dy

⁰

− 1

2r

1

e

^−ay

Z

y

0

ˆ

v

0

e

^r¹^y⁰

dy

⁰

+ 1 2r

2

e

^ay

Z

y

0

ˆ

v

0

e

^r²^y⁰

dy

⁰

)

. (26)

The Laplace inversion formula reads ˆ

η (y, β, t) = lim

b→∞

1 2πi

Z

c+ib c−ib

e

^st

˜ˆη(y, β, s) ds . (27) Residual theory gives the same branch-cut as for the wall-normal perturbation velocity, i.e. integration according to fig. A.1 applies again. Thus the inversion (27) turns into

i π ˆ η = Z

∞

0

e

^−(1/4+β²^+σ²⁾^R^t

˜ˆη σ R dσ

a=−iσ

− Z

∞

0

e

^−(1/4+β²^+σ²⁾^R^t

˜ˆη σ R dσ

a=iσ

. (28) An explicit expression for ˆ η is obtained by inserting the retrieved solution (26) and the initial condition 22 into the above inversion. The result becomes

ˆ

η = 1

4 i β R A(β) (

−

1 + β − 1

β · (e

^−y+y⁰

− e

^y⁰

)

e

^−β²^T

erfc 1 2

√ T + y + y

0

2 √ T

+ e

^−y

e

^−β²^T

erfc 1

2 √ T − y − y

0

2 √ T

− erfc 1 2

√ T − y + y

0

2 √ T

+ 2e

^βT

e

^−y

e

^−β(y+y⁰⁾

"

erfc

(β + 1

2 ) √

T − y + y

0

2 √ T

− erfc

(β + 1

2 ) √

T − y

0

2 √ T

#

+ e

^−y

e

^−β²^T

erfc 1 2

√ T + y − y

⁰

2 √

T

(30)

+ 1 − 2β

β e

^−βT

e

^−y+y⁰

e

^−β(y+y⁰⁾

"

erfc

(β − 1

2 ) √

T − y + y

0

2 √ T

(29)

− erfc

(β − 1

2 ) √

T − y

0

2 √ T

#

+ 1

β e

^{βT +y}⁰

e

^β(y+y⁰⁾

erfc

(β + 1

2 ) √

T + y + y

0

2 √ T

− 1

β e

^{βT +y}⁰

e

^−y

e

^−β(y−y⁰⁾

erfc

(β + 1

2 ) √

T + y

0

2 √ T

) .

B Solutions for the BBL

B.1 The wall-normal velocity

Fourier and Laplace transformations (15) and (16) are applied to the stabil- ity equation governing the BBL flow. An ODE is obtained with roots to the characteristic equation given by

( r

1,2

= ±β

r

3,4

= ± psR + β

²

. (30)

Solving the ODE using variation of parameters and boundary conditions gives the result

2 s ˜ˆ v = r

₃²

− β

²

r

3

e

^r³^y

− 2(r

3

+ β) e

^−βy

+ (r

3

+ β)

²

r

3

e

^−r³^y

Z

_∞

0

ˆ

v

0

e

^−r³^y⁰

dy

⁰

− r

₃²

− β

²

r

3

e

^r³^y

Z

y

0

ˆ

v

0

e

^−r³^y⁰

dy

⁰

+ r

²₃

− β

²

r

3

e

^−r³^y

Z

y

0

ˆ

v

0

e

^r³^y⁰

dy

⁰

. (31) The Laplace inversion (20) applies again and the integral is replaced by a similar branch cut as shown in fig. A.1,

i π ˆ v = Z

∞

0

e

^−(β²^+σ²^{) t/R}

˜ˆv σ R dσ

r3=iσ

− Z

∞

0

e

^−(β²^+σ²^{) tR}

˜ˆv σ R dσ

r3=−iσ

. (32)

Inserting the wall-normal velocity retrieved in (31) and the initial perturbation (22) the final result becomes

ˆ

v/A(β) = 1

√ πT e

^−β²^T

· h 1

2 (e

^−(y−y⁰⁾²^/4T

+ e

^−(y+y⁰⁾²^/4T

) − e

^−βy

e

^−y²⁰^/4T

i + β e

^−β(y+y⁰⁾

· erfc

β √

T − y

0

2 √ T

− erfc

β √

T − y + y

0

2 √ T

,

(33)

again independent of whether y is less or larger than y

0

.

(31)

B.2 The streamwise velocity

Fourier and Laplace transformations according to (15) and (16) are applied to the normal vorticity stability equation (5). The obtained ODE has the charac- teristic roots

r

1, 2

= ± pβ

²

+ sR. (34)

Note that these roots are common with r

3, 4

for the wall-normal velocity (30).

Variation of parameters and use of boundary conditions gives

˜ˆη = 1

r

1

sinh(r

1

y) Z

∞

0

e

^−r¹^y⁰

I dy

⁰

+ 1

r

1

Z

y 0

sinh (r

1

(y

⁰

− y)) I dy

⁰

, (35)

where I = R (ˆ η

0

−i β ˜ˆv dU/dy). The Laplace inversion (27) is for this flow turned into

i π ˆ η = Z

∞

0

e

^−(β²^+σ²^{) t/R}

˜ˆη σ R dσ

r1=−iσ

− Z

∞

0

e

^−(β²^+σ²^{) t/R}

˜ˆη σ R dσ

r1=iσ

(36)

by residual calculus, i.e. the integration contour follows a similar branch-cut as given by fig. A.1. Now (35) is inserted and the initial condition ˆ η

0

= 0 is used in accordance with the ASBL calculations, and the result is given by

ˆ

u = 1

iπ e

^−β²^T

Z

_∞

0

( sin(σy)

Z

_∞

0

e

^iσy⁰

˜ˆv

₋

(y

⁰

) − e

^−iσy⁰

˜ˆv

+

(y

⁰

) dU

dy

⁰

dy

⁰

(37) +

Z

y 0

sin(σ(y

⁰

− y)) ˜ˆv

₋

(y

⁰

) − ˜ˆv

⁺

(y

⁰

) dU dy

⁰

dy

⁰

)

e

^−σ²^T

dσ ,

where ˜ˆ v

_±

= ˜ˆ v|

^r³=±iσ

is taken from (31).

C Time integrals

In solving (14) the time integration Z

t

0

ˆ

v(y

⁰

, t

⁰

, k) 1 2 √

πT

^?

e

^−k²^T^∗

e

^−(y⁰^+y)²^/4T^∗

− e

^−(y−y⁰⁾²^/4T^∗

dt

⁰

(38) consists of two types of integrals upon inserting ˆ v, namely

I

1

= Z

T

0

√ 1

T

⁰

T

^∗

e

^−α²^/T^∗^−β²^/T⁰

dT

⁰

I

2

=

Z

T 0

√ 1

T ∗ e

^−(k^√^T^∗^+α/^√^T^∗⁾²

erfc k √

T

⁰

− β/ √ T

⁰

dT

⁰

.

(39)

(32)

Here α and β are constants, T

^∗

= T − T

⁰

and T = t/R is time scaled by the Reynolds number. Trigonometric substitution and parametric differentiation with respect to α, leads to the formula

I

1

= π

1 − erfc | α| + | β| √ T

. (40)

For I

2

parametric differentiation has been carried out with respect to β, and further analysis show that

I

2

= −

√ π k e

^−2kα

(

e

^2kβ−k²^T

1−erf | α| + | β|

√ T

−e

^−k²^T

1−erf | α|

√ T

+ e

^−2k|α|

erf | α| + | β|

√ T − k √ T

− erf | α|

√ T − k √ T )

+ 2 √ T

Z

1 0

e

^−(k^√^{T x+α/(x}^√^{T ))}²

erf

k pT (1 − x

²

)

dx . (41)

The numeric calculation of (14) has been carried out with the inner integral replaced by terms involving (40) and (41).

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