Streak stability in the suction boundary layer

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MASTER’S THESIS

2002:112 CIV

MASTER OF SCIENCE PROGRAMME Department of Applied Physics and Mechanical Engineering

Division of Fluid Mechanics

Streak Stability in the Suction Boundary Layer

NIKLAS DAVIDSSON

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Streak Stability in the

Suction Boundary Layer

Niklas Davidsson

Department of Mechanical Engineering Division of Fluid Mechanics

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Abstract

A study of the stability characteristics of streamwise streaky ( ^∂

∂x = 0) structures in parallel boundary layer ﬂow above a porous surface is conducted.

The analytical solution for the perturbed velocities is sought by applying Fourier transformation in the spanwise direction and Laplace transformation in time. Assuming a localized initial perturbation in the y-direction and applying inverse Laplace transform the solution in wave number space is obtained.

The obtained ﬂow shows non-algebraic initial growth. The streamwise velocity component reaches largest amplitudes for small spanwise wavenum- bers and the initial perturbation placed in the boundary layer. For both vertical and streamwise velocities the amplitude peak diﬀuses towards the boundary in time, and a change of the amplitude sign occurs for some values of the parameters.

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Chapter 1 Introduction

Fluid dynamics is present in many industrial applications. Research is done in order to understand the complicated flow that arises and to influence the flow in different ways. The idea of flow control is to obtain the optimum balance of different properties such as mixing effects, lift forces and friction by for example preventing the transition to turbulence. One way of exercising flow control is suction, where fluid is forced through a porous boundary by applying a pressure gradient. This thesis handles the boundary layer of a parallel flow above an infinite large porous plate where suction is applied, called the asymptotic suction boundary layer (ASBL).

Long streamwise structures of low and high speed ﬂuid, so called streaks, have been observed experimentally as a preceding state of turbulent spots in for example the Blasius boundary layer. The photograph on the cover of this report shows a visualization of a typical boundary layer transition, studied by Alfredsson and Matsubara [1].

The streaks have shown to both lengthen and grow wider downstream, become unstable and break down to turbulence. Motivated by these obser- vations, Hultgren and Gustavsson [2] investigated long streamwise structures for Blasius boundary layer ﬂow and found initial algebraic growth.

One property (and problem) of the Blasius flow is the growth of the boundary layer with streamwise distance which will modify the results of a parallel flow assumption. This makes an investigation of the ASBL partic- ularly interesting, since it is a strictly parallel flow allowing the assumption

∂

∂x = 0 for the streaks. Thus, as we shall see, the stability equations of the perturbated velocity ﬁeld is possible to solve analytically.

Experimental work on suction has shown it´s ability to stabilize the boundary layer. For instance Poll [3] found that suction induced robust laminar characteristics in the boundary layer. Fransson [4] conducted ex- periments on both ASBL and Blasius ﬂow and found some properties that

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changed when suction was used, for instance the disturbance amplitude reached a saturation level in the suction case.

With the above reasons in mind it seems natural to investigate streaks in the ASBL.

The main objective of this work is to draw conclusions and information about the ASBL by analytically solving the stability equations of an incom- pressible ﬂuid. The calculations follow closely and serves as a veriﬁcation of the work done by Gustavsson [5]. It also extends it in terms of nonlinear analysis of the stability equations and further analysis of the solutions.

The outline is that the governing stability equations and the boundary conditions are established in chapter two, the methods to solve these equations are presented in chapter three and four (with some procedures moved to the appendices) and analysis of the obtained solutions are found in chapter ﬁve and six. The last chapter consists of concluding remarks.

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Chapter 2 Stability of the Asymptotic Suction Boundary Layer

(ASBL)

In this chapter the Navier-Stokes equations will be manipulated in order to find the governing equations of the perturbed velocity field, which together with the velocity profile, known initial conditions and boundary conditions states the stability problem of the ASBL.

The aim is to study streamwise extended perturbations through diﬀerent calculations for the vertical velocity v and the normal vorticity

η = ∂u

∂z − ∂w

∂x. (2.1)

With these calculations we have a complete description of the flow field since it is straight forward to find u, w with v, η known. Observe for the calcula- tions below that variables with an asterisk as superfix are dimensional, and velocities with capital letters U = (U, V, W ) denotes the mean flow.

2.1 The geometry

The flow situation consists of a parallel flow with free stream velocity U_∞^∗ above an infinite large porous plate. When suction is applied through the plate the geometry of the ASBL is obtained, as shown in fig. (2.1) where V₀^∗ is the suction velocity.

The theory to derive the mean ﬂow is for instance dealt with by Schlicht- ing [6]. By using Navier-Stokes equations, the assumption of constant pressure along the plate and appropriate boundary conditions calculations give

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Figure 2.1: Coordinate system and velocity proﬁle for a porous plate with suction.

u^∗

U_∞^∗ = 1− e^−y∗V0^ν (2.2)

and

V^∗ =−V0. (2.3)

Here ν denotes the kinematic viscosity. Choosing velocity scale U_∞^∗ and length scale ν/V₀^∗ the velocities (2.2) and (2.3) can be used to describe the mean velocity proﬁle in dimensionless form as

U = (1− e^−y, −1

R, 0), (2.4)

where R = ^U_V^∞^∗∗

0 is the Reynolds number. To get an idea of the length scales one can notice that with this mean velocity proﬁle the displacement thickness becomes

δ^∗ = 1 (2.5)

( and 99% of U_∞^∗ is reached for y 5 ).

2.2 Governing stability equations

For a review of the stability equations of the ASBL see Drazin/Reid [7]

and references therein. This investigation will be done in the same manner,

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but with emphasis on 3-D perturbations. It will also consider a nonlinear analysis.

The basic idea is to extract the stability equations for the perturbed vertical velocity and normal vorticity through manipulation of the dimensionless Navier-Stokes equations (2.6) and the continuity equation (2.7):

∂ui

∂t + uj∂ui

∂xj

= −∂p

∂xi

+ 1

R∇²ui (2.6)

∂ui

∂xi = 0. (2.7)

Now introduce a ﬂow state as

ui = Ui+ u_i, p = P + p (2.8) where Ui, P is the steady solution of the above equations, Ui is given by (2.2) and u_i, p is the perturbation.

Inserting the quantities ui,p into (2.6) and (2.7) and subtracting for the mean ﬂow will lead to equations (2.9) and (2.10) below. Here the primes denoting perturbations are dropped, and from now on the perturbion velocity ﬁeld is denoted by (u, v, w).

∂ui

∂t + Uj∂ui

∂xj + uj ∂

∂xj(Ui+ ui) = −∂p

∂xi

+ 1

R∇²ui (2.9)

∂u

∂x + ∂v

∂y +∂w

∂z = 0 (2.10)

The next step is to ﬁnd an expression for the pressure and then to eliminate it. These calculations are performed both for the linear and the non-linear case.

2.2.1 Linear equations

We linearize (2.9) and use the fact that for the mean ﬂow U = U(y). The results for the three directions are:

∂u

∂t + U∂u

∂x + V ∂u

∂y + v∂U

∂y =−∂p

∂x + 1

R∇²u (2.11)

∂v

∂t + U∂v

∂x + V ∂v

∂y =−∂p

∂y + 1

R∇²v (2.12)

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∂w

∂t + U∂w

∂x + V∂w

∂y =−∂p

∂z + 1

R∇²w. (2.13)

If we apply the divergence operator on the equations above to calculate

∂

∂x(2.11) + _∂y^∂(2.12) + _∂z^∂ (2.13) and use continuity, the relation for the pressure is obtained as

∇²p = −2dU dy

∂v

∂x. (2.14)

Implementing this result in (2.12) gives the sought stability equation for v:

(∂

∂t + U ∂

∂x − 1 R

∂

∂y)∇²− ∂²U

∂y²

∂

∂x − 1 R∇⁴

v = 0. (2.15) The normal vorticity η is calculated by the operations _∂z^∂ (2.11) − _∂x^∂(2.13), resulting in

∂

∂t + U ∂

∂x − 1 R

∂

∂y − 1 R∇²

η = −dU dy

∂v

∂z. (2.16)

2.2.2 Non-linear equations

Now we want to derive the non-linear version of the stability equations for v and η. The operations are the same as for the last section except that no linearization is done for eq. (2.9). The resulting stability equations are

(∂

∂t + U ∂

∂x − 1 R

∂

∂y)∇²−∂²U

∂y²

∂

∂x − 1 R∇⁴

v =

∂

∂y

2∂v

∂x

∂u

∂y + 2∂w

∂y

∂v

∂z + 2∂w

∂x

∂u

∂z + (∂u

∂x)²+ (∂v

∂y)²+ (∂w

∂z)²

−∇²

u∂v

∂x + v∂v

∂y + w∂v

∂z

(2.17) and

∂

∂t+ U ∂

∂x − 1 R

∂

∂y − 1 R∇²

η +

∂u

∂x + ∂w

∂z + u ∂

∂x + v ∂

∂y + w ∂

∂z

η =

−dU dy

∂v

∂z − ∂v

∂z

∂u

∂y + ∂v

∂x

∂w

∂y. (2.18)

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This result has been derived by Benney and Gustavsson [8] for the Blasius boundary layer with the only diﬀerence that the term -¹

R(^∂

∂y)∇²v drops out from (2.17) and -_R¹(_∂y^∂)η from (2.18). So it seems that only linear terms separate the stability of the Blasius boundary layer from the ASBL.

2.3 Boundary conditions

For the perturbed velocity field we will show that the boundary conditions at the plate does not differ from the no-suction case. The non-slip condition for a viscous fluid flowing parallel to a solid surface still requires that (u, w)|y=0 = 0. Since those velocities are zero on the plate it must also be true that

∂u

∂z|y=0= ^∂w

∂x|y=0 = 0 and therefore

η|y=0 = 0. (2.19)

We must however consider the normal velocity v closer. This is done by Gustafsson [9], who uses Darcy´s law for porous materials to show that

v|y=0= 0 (2.20)

in the limit of small permeability of the plate. By continuity and the fact that the derivatives of u, w parallel to the plate is zero we can conclude that also

∂v

∂y|y=0 = 0, (2.21)

which means that the boundary conditions on the plate indeed are the same for Blasius and ASBL ﬂow if the suction velocity V₀^∗ is small.

The condition for the far ﬁeld is that the perturbation velocities v and η are assumed to be bounded, which yields

(v, η)|y→∞ = 0. (2.22)

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Chapter 3 The vertical velocity

This chapter is devoted to solving the linearized governing equation (2.15) for v. In the last chapter it was concluded that both this equation and the boundary conditions are homogeneous. We shall see later that with the number of accessible boundary conditions it is possible to retrieve a unique solution if the initial state is known, so for now we assume

v|t=0 = v0(y) (3.1)

and leave the choice of v0till later. We will apply spanwise Fourier-transform, Laplace transform in time and make use of the assumption for streaky structures ^∂

∂x = 0. Then, with derivatives of time and downstream coordinate x dropped out it is possible to solve the problem and look for inverse transformation.

3.1 Solution of the stability equation

Since the streaky structure condition leads to _∂x^∂ = 0, Fourier transformation in the streamwise direction drops out. This condition eliminates the x-dependence. Applying the Fourier transform for the homogeneous coordi- nate z (spanwise) yields

ˆv(y, β, t) =

_∞

−∞e^−iβzv(y, z, t) dz (3.2) and continuing with Laplace transformation in time we get

ˆ˜

v(y, β, s) =

_∞

0 e^−stv(y, β, t) dt.ˆ (3.3) The rules and conventions

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







∂

∂z˜v = −iβˆ˜v

L(f(t)) = sF (s) − f (0)

∇² = D²− β²

together with (2.15) gives the stability equation for ˆ˜v as

(sR − D) − (D²− β²)(D²− β²)ˆv = R(D˜ ²− β²)ˆv0, (3.4) where D = _∂y^∂ . This is a fourth order diﬀerential equation. It will be solved with Lagrange´s method of variation of parameters, described in for example [10]. The roots of the characteristic homogeneous equation from (3.4) are

r1,2 = ±β r3,4 = −1

2 ± a (3.5)

where

a =

1

4 + sR + β². (3.6)

The method uses these roots to build a solution according to ˆ˜

v =

4 i=1

Bi(y)e^rⁱ^y. (3.7)

The appropriate choice of the functions Bi and the boundary conditions yield the solution (3.8), where for simplicity r3 and r4 has been kept. A more detailed description of these calculations is given in appendix A.

ˆ˜

v = R

2a

β + r4 e^−βy+ e^r³^y −β + r3

β + r4 e^r⁴^y

_∞

0 vˆ0e^−r³^ydy

− R

2ae^r³^y

_y

0 vˆ0e^−r³^ydy+ R 2ae^r⁴^y

_y

0 vˆ0e^−r⁴^ydy (3.8)

3.2 The initial perturbation

For the initial vertical perturbation we have so far been able to work with the general assumption (3.1) without problems or restrictions, but deﬁning ˆv0will

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simplify the integrals of (3.8) and the upcoming inverse Laplace transform.

Thus, the initial perturbation is deﬁned as a delta function acting at distance y0 above the plate, together with an arbitrary function of the span-wise wavenumber β:

vˆ0 = A(β)δ(y − y0). (3.9) Since the stability equation is linear it is possible to superpose δ-functions to represent arbitrary initial perturbations.

The properties of the δ-function make the integrands drop out and the result, now depending on the position y, is

ˆ˜

v = A(k)R 2a











2a

β + r4e^−βy−r³^y⁰ − β + r3

β + r4e^r⁴^y−r³^y⁰ + e^r³^(y−y⁰⁾ , y < y0

2a

β + r4e^−βy−r³^y⁰ − β + r3

β + r4e^r⁴^y−r³^y⁰ + e^r⁴^(y−y⁰⁾ , y > y0. (3.10) The last step of solving the stability equation of the vertical perturbation is to apply the inverse Laplace transform.

3.3 Laplace inversion

Inverting the Laplace transform is non-trivial and needs a closer look. It is deﬁned as

v(y, β, t) = limˆ

b→∞

1 2πi

_c+ib

c−ib e^stv(y, β, s)dsˆ˜ (3.11) where the integration is carried out in the complex s-plane. The constant c is positive, real and assigned a value to the right of all singularities of ˆv.˜

As shown in appendix B we can use residual theory to build a closed contour around poles and branches of ˆ˜v. The only contribution to ˆv will be the branch-cut in the contour of ﬁgure (3.1), where the integration variable σ is deﬁned by

s = −1/4 − β²− σ²

R , 0 < σ < ∞. (3.12) The integral (3.11) will then be evaluated as

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Figure 3.1: Contour for the Laplace inversion of ˆv.˜

ˆ˜

v = _πi¹

_∞

0 e^−(1/4+β²^+σ²⁾^R^t ˆ˜vσ Rdσ

a=−iσ

− _πi¹ ^∞

0 e^−(1/4+β²^+σ²⁾^R^t ˆ˜vσ Rdσ

a=iσ.

(3.13)

The approach of the Dirac initial perturbation has given two separate solutions in (3.10), above and below y0. However, these solutions yield the same result after the inversion (3.13), namely

v =ˆ A(k)

π e^−(1/4+β²^)T

e⁻¹²^(y−y⁰⁾

_∞

0 e^−σ²^T cos σ(y − y0) dσ + e⁻¹²^(y−y⁰⁾

_∞

0 e^−σ²^T(σ²− (β − ¹₂)²) cos σ(y + y0) + (1− 2β)σ sin σ(y + y₀)

σ²+ (β − ¹₂)² dσ

+ 2e^−βy+¹²^y⁰

_∞

0 e^−σ²^T(β − ¹₂)σ sin σy0− σ²cos σy0

σ²+ (β − ¹₂)² dσ

(3.14)

where T = _R^t is time scaled with the Reynolds number. The integrals in (3.14) are of standard type (see for instance [11]) and can be expressed in terms of the complementary error function,

erfc(x) = 1 − erf(x) = 1 − 2

√π

_∞

0 e^−t²dt. (3.15) This gives the ﬁnal result:

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v = A(k)ˆ ^√¹_πTe^−(β²⁺¹⁴^)T

·¹₂e⁻¹²^(y−y⁰⁾(e⁻^(y−y0)2^4T + e⁻^(y+y0)2^4T )− e^−βy+¹²^y⁰e⁻^y20^4T

+ A(k)(β − ¹₂)e^−β(y+y0)e^y⁰e^−βT

·erfc(β −¹₂)√

T −₂^y^√⁰_T− erfc(β −¹₂)√

T − ^y+y₂^√_T⁰.

(3.16)

3.4 Blasius solution

The procedure for finding the vertical velocity for the suction-free case, as done by Gustavsson [5], is similar as for the ASBL. The stability equation for the Blasius flow lacks the term −_R¹_∂y^∂v in (2.15), and a slightly different solution is obtained:

v = A(k)ˆ ^√¹_πTe^−β²^T ·¹₂e⁻^(y−y0)2^4T − e^−βye⁻^4T^y20 +¹₂ e⁻^(y+y0)2^4T + A(k)β e^−β(y+y0)·erfcβ√

T −^y+y₂^√_T⁰− erfcβ√

T − ₂^√^y⁰_T.

(3.17)

This solution will be used for comparison of the Blasius boundary layer and the ASBL in chapter ﬁve and six.

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Chapter 4 The normal vorticity

In this chapter the linearized governing equation (2.16) for η shall be solved.

The fact is that knowing v and η yields the remaining parts of the ﬂow ﬁeld u and w in wave number space by

u =ˆ i

(α²+ β²)

α∂ˆv

∂y − βˆη

w =ˆ i

(α²+ β²)

αη + β∂ˆv

∂y

, (4.1)

where α is the Fourier transform variable in the downstream direction. But since we are working with streaks, i.e. α = _∂x^∂ = 0, the previous equation yields

u = −ˆ i βηˆ w =ˆ i

β

∂ˆv

∂y. (4.2)

The governing equation of the normal vorticity (2.16) has an important property that separates it from the vertical velocity: It is inhomogeneous.

The factors in the forcing term consist of the y-derivative of the mean ﬂow and the z-derivative of the vertical perturbation. Therefore one can suspect that the largest eﬀect on η occurs within the boundary layer where ^∂U_∂y = 0.

The solution for η is obtained by the same technique as used for v in chapter 3, i.e. by applying Fourier- and Laplace transformations in space and time respectively, and solving with variation of parameters. Again, we shall see that the accessible boundary conditions will be suﬃcient for obtaining a unique solution when the initial perturbation is considered to be known.

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4.1 Solution of the stability equation

Applying the transformations (3.2) and (3.3) on the stability equation (2.16) together with the streak assumption _∂x^∂ = 0, rules about the derivative of transforms and the convention D = _∂y^∂ yields

D²+ D − (β²+ sR)ˆ˜η = −Rˆη0+ iβRe^−y∂ˆ˜v

∂y. (4.3)

Here

η|ˆ_y=0 = ˆη0 (4.4)

is considered to be a known function but choosing it is left for later. The roots of the (homogeneous) characteristic equation are

r1,2 = −1

2 ± a , (4.5)

where a =¹₄ + sR + β², and we see that these roots are identical to r3 and r4 for ˆv in equation (3.5). By applying the method of variation of parameters,˜ as described in appendix A, and using the boundary conditions from chapter 2.3 to determine the constants, the solution obtained is

ˆ˜η = 1

ae⁻¹²^ysinh ay

_∞

0 e^−r¹^yI dy + 1

ae⁻¹²^y

_y

0 e¹²^ysinh a(y − y)I dy, (4.6) where I is the inhomogeneous part of equation (4.3) with reversed sign, i.e.

I = Rˆη0− iβRe^−y∂ˆ˜v

∂y. (4.7)

By choosing

ηˆ0 = 0, (4.8)

which from (4.2) means that there is no initial downstream disturbance, and using ˆv from (3.8),˜

ˆ˜

v(y) = R 2a

2a

β + r4e^−βy + e^r³^y − β + r3

β + r4e^r⁴^y

_∞

0 vˆ0e^−r³^ydy

− R

2ae^r³^y

_y

0 vˆ0e^−r³^ydy+ R 2ae^r⁴^y

_y

0 vˆ0e^−r⁴^ydy (4.9)

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written with changed dummy indices to ﬁt into equation (4.6), an equation with double integrals is retrieved. The inner integrals are removed with integration by parts, resulting in

ˆ˜η = ^iβR_2a² e⁻¹²^y

− _r¹₂ sinh ay

_∞

0 ˆv0e^r²^ydy+

1 2r2

r1+ β

r2+ βe^−(a+1)y

− 1

2r1e^(−1+a)y+ 2a

(r²₂− β²)(r1− β)e^−(β+¹²^)y

− a(β²− r2β + r1) r1r2(r1− β)(r2− β)e^−ay

_∞

0 vˆ0e^−r¹^ydy

+ 1

2r1e^(−1+a)y

_y

0 vˆ0e^−r¹^ydy − 1

2r2 e^−(1+a)y

_y

0 ˆv0e^−r²^ydy

− 1

2r1e^−ay

_y

0 vˆ0e^r¹^ydy+ 1 2r2 e^ay

_y

0 vˆ0e^r²^ydy

.

(4.10)

4.2 Laplace inversion

Again, the inversion formula reads η(y, β, t) = limˆ

b→∞

1 2πi

_c+ib

c−ib e^stˆ˜η(y, β, s) ds (4.11) and the choice of c is the same as in inversion (3.11). A more careful survey of this inversion is found in appendix B. There are many possible singularities but, as shown in appendix, the integration contour of ﬁgure (3.1) is also valid this time. Therefore the inversion becomes

ˆ˜η = _πi¹ ^∞

0 e^−(1/4+β²^+σ²⁾^R^t ˆ˜ησ Rdσ

a=−iσ

− _πi¹ ^∞

0 e^−(1/4+β²^+σ²⁾^R^t ˆ˜ησ Rdσ

a=iσ.

(4.12)

Choosing ˆv0 as a Dirac function in (4.10) suggests that diﬀerent results are obtained for y < y0 and y > y0. However, inverting the Laplace transfor- mation according to (4.12) leads to the same expression whether y is less or larger than y0.

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η =ˆ A(k)βR

2πi e⁻⁽¹⁴^+β²^)Te⁻¹²^(y−y⁰⁾

_∞

0 e^−σ²^T

1

2e^−y⁰ 1 σ²+ ¹₄

cos σ(y + y0)− cos σ(y − y₀)

−2σ sin σ(y + y0)− 2σ sin σ(y − y0)

+ 1

β(β + 1)

1

σ²+¹₄ − 1 σ²+ (β +¹₂)²

·2σ(β²+ 1 2β − 1

2) sin σ(y + y0)

−2σ²cos σ(y + y0) + 1

2 1

σ²+ ¹₄e^−y [2σ sin σ(y − y0)− cos σ(y − y0)]

+ 1

2βe^−(β+¹²^)y

− 1

σ²+ (β +¹₂)² + 1 σ²+ (β − ¹₂)²

·

·4σ(1

2 − β) sin σy₀+ 4σ²cos σy0

+ 1

β(β − 1)e^−y

1

σ²+¹₄ − 1 σ²+ (β − ¹₂)²

·(2β − 3

2)σ²+ 1

2(β − 1 2)²·

· cos σ(y + y0) +

(2β − β² −3

4)σ + σ³

sin σ(y + y0)

dσ, (4.13)

where T = _R^t. Similar to ˆv, the integrals in (4.13) can be expressed in terms of error functions, resulting in

Streak stability in the suction boundary layer

MASTER’S THESIS

Streak Stability in the Suction Boundary Layer

Streak Stability in the

Suction Boundary Layer

Niklas Davidsson

Contents

Chapter 1 Introduction

Chapter 2

Stability of the Asymptotic Suction Boundary Layer

(ASBL)

Chapter 3

The vertical velocity

Chapter 4

The normal vorticity