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Linear instability in the near wake of channels cascade

Abderrahim Achiq

a,*

, Jaafar Khalid Naciri

b

aU.F.R. de Mecanique, Facultedes Sciences et Techniques, BP 146, Mohammadia, Morocco

bU.F.R. de Mecanique, Facultedes Sciences Aõn Choc, Casablanca, Morocco Received 6 August 1999; accepted 6 August 1999

(Communicated by E.S. SßUHUBI)_

Abstract

The growth of linear disturbances in the high-Reynolds number laminar wake of a channels cascade is investigated. Disturbances in the near wake respond according to the Rayleigh equation which is solved numerically for a spatial stability problem. The unperturbed ¯ow is studied using a double deck analysis.

The results of This analysis allow for the basic ¯ow pro®les at di€erent streamwise positions to be nu- merically obtained as solutions of the wake boundary-layer equations. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

The ¯ow in channels or ¯at plates cascade was extensively studied. Recently, Graham [1]

studied the e€ect of a two-dimensional cascade of thin streamwise plates on the homogeneous turbulence.

The present study deals with the laminar wake of a developed ¯ow between ¯at plates. Inside each channel, the basic ¯ow is given by the Poiseuille distribution. This problem is di€erent from that of a classical jet, where only one channel is considered. The jet usually penetrates in a constant pressure environment which leads to a decrease of the jet section [2]. The in®nite number of jets resulting from the wake of channels cascade implies no change in the jet section which alloys an increase of the pressure ®eld inside the wake.

For the analytical study of the near wake of channels cascade, we introduce a double deck analysis in a similar manner to Goldstein [3] who considered the near wake of a single ®nite plate

www.elsevier.com/locate/ijengsci

*Corresponding author. Tel.: +212-3-31-49-05; fax: 212-3-13-53-53.

E-mail address:achiq@uh2m.ac.ma (A. Achiq).

0020-7225/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S0020-7225(99)00129-9

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with a zero pressure gradient assumption. However, this assumption leads to the existence of a singularity in the neighbourhood trailing edge of the plate [4]. Later on a triple deck analysis was used to solve this singularity problem [5].

In a recently paper, Saintlos and Mauss [6] present a systematic asymptotic analysis of the perturbed ¯ow in a channel with accident. They show that for a symmetrical perturbation the ¯ew in the singularity region can be studied using a double deck structure. This analysis can be ex- tended to a ¯ow in the wake of channels cascade due to the ``double symmetry'' imposed in the near wake of every channel [7].

The present work is constituted of two main parts, a ®rst (Section 2) dedicated to the study of the steady basic ¯ow and allows us to get the evolution of the velocity distribution in the near wake of the cascade of channels. Using the results of this part, we treats in the second part (Section 3) the linear spatial instability from a quasi-parallel theory.

2. The study of the basic problem

2.1. Problem formulation

We consider a laminar steady two-dimensional ¯ow of an incompressible Newtonian ¯uid in the wake of channels cascade at high Reynolds number. These channels are identical and arranged in a parallel manner to each other. In Cartesian co-ordinates …X

; Y

† the plates are placed at positions (X

< 0; Y ˆ kL

), where k 2 Z and L

is the width of channels (Fig. 1). For X

> 0, the double symmetry of the problem in Y

ˆ …2k ‡ 1†L

=2 and Y

ˆ kL

reduces the domain of study to 0

6

Y

6

L

=2 for the Y

coordinate.

The streamwise and transverse velocity components are respectively noted U

and V

, and P

the ¯uid pressure. We de®ne dimensionless quantities:

X ˆ X

=L

; Y ˆ Y

=L

; U ˆ U

=U

0

; V ˆ V

=U

0

and P ˆ P

L

=g

U

0

;

Fig. 1. Schematic of channels cascade and the Poiseuille distribution within the channels.

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where U

0

is equal to six times the middle velocity in the channel, and q

and g

are respectively density and the viscosity of the ¯uid.

The dimensionless equations of the steady problem can be written:

Re… ~ V ~ r† ~ V ˆ ÿ rP ~ ‡ D ~ V and ~ r ~ V ˆ 0; …1†

where ~ r…o=oX ;

o=oY

†; D ˆ

o2

=oX

2

‡

o2

=oY

2

; ~ V ˆ …U ; V † and Re ˆ q

U

0

L

=g

the Reynolds number supposed to be large (Re 1).

Inside of channel (X < 0), the ¯ow is given by the Poiseuille distribution.

U…X ; Y † ˆ U

0

…Y † ˆ Y ÿ Y

2

; V …X ; Y † ˆ 0 and dP

dX …X ; Y † ˆ ÿ2:

In the wake (X > 0), it should be noted that the convective acceleration term in Eq. (1), which is equal to zero for the Poiseuille ¯ow, can no longer be negligible because of the great magnitude of Re. Then, noting that the far wake have a perfect nature, it is necessary to introduce an inter- mediate region where convective terms balance viscous terms. In this zone, we have the following scales: X ˆ O…Re†, U ˆ O…1†, V ˆ O…Re

ÿ1

† and P ˆ O…Re† [8]. Therefore, we introduce x ˆ X =Re, y ˆ Y , u ˆ U, v ˆ V Re and p ˆ P =Re, then the ¯uid ¯ow is governed by the classical boundary layer equations:

u

ou ox

‡ v

ou

oy

ˆ ÿ dp dx ‡

o2

u

oy2

;

ou ox

‡

ov

oy

ˆ 0: …2†

In the wake …x > 0† the dimensionless symmetry conditions at y ˆ 0 and y ˆ

12

are given by:

v…y ˆ 0† ˆ

ou

oy

…y ˆ 0† ˆ 0; …a†

u y

ˆ 1 2

ˆ v y

ˆ 1 2

ˆ 0; …b†

…3†

and the initial conditions at x ˆ 0

‡

are set by the Poiseuille distribution:

u…x ˆ 0

‡

; y† ˆ U

0

…y† ˆ y ÿ y

2

; v…x ˆ 0

‡

; y† ˆ 0: …4†

2.2. Double deck analysis

The nature of the singularity at x ˆ 0 leads us to adopt for the near wake …0 < x 1† an analogous analysis to the one used by Goldstein [3]. The stream function w de®ned by: u ˆ

ow=oy

and v ˆ ÿow=ox allows the system (2, 3, 4) can be written as:

ow oy

o2

w

oxoy

ÿ

ow

ox o2

w

oy2

ˆ ÿ dp dx ‡

o3

w

oy3

…5†

with the boundary conditions

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w…x; y ˆ 0† ˆ 0; w x; y

ˆ 1 2

ˆ 1

12 ; …a†

w

00

…x; y ˆ 0† ˆ w

00

x; y

ˆ 1 2

ˆ 0: …b†

…6†

The Poiseuille distribution allows in ®rst approximation: w

0

…y† ˆ …y

2

=2† ÿ …y

3

=3†, which does not verify the symmetry condition at y ˆ 0. Therefore we consider a boundary layer for y 1.

2.2.1. Internal layer (y 1)

We introduce transformations n ˆ x

1=3

…x 1†; g ˆ y=3n …g ˆ O…1††, and w…x; y† ˆ n

2

f …n; g†.

In the case where n is small, we consider the following developments of f …n; g† and p…x†:

f …n; g† ˆ f

0

…g† ‡ nf

1

…g† ‡ n

2

f

2

…g† ‡ p…x† ˆ p

0

‡ np

1

‡ n

2

p

2

‡ ;

where p

n

and f

n

…g† are respectively constants and functions to be determined. Eq. (5) and boundary conditions (6) permit to get p

1

ˆ 0 and the system:

f

0000

‡ 2f

0

f

000

ÿ f

002

ˆ 18p

2

; …a†

f

0

…0† ˆ f

000

…0† ˆ 0; …b†

f

00

…g†

9g ! 1 when g ! 1: …c†

…7†

The condition (7c) is not complete [9]. This completeness can be achieved by considering the external layer …y ˆ 0…1††.

2.2.2. External layer (y ˆ 0(1))

While considering the following development w…x; y†:

w…x; y† ˆ w

0

…y† ‡ nw

1

…y† ‡ n

2

w

2

…y† ‡ ;

where w

n

…y† are functions to be determined, Eq. (5) can be written:

w

00

w

01

‡ w

000

w

1

ˆ ÿp

1

; …a†

2…w

00

w

02

‡ w

000

w

2

† ‡ w

021

‡ w

1

w

001

ˆ ÿ2p

2

;

. . .

…b† …8†

For n

P

1, we have the following boundary conditions:

w

n

…y ˆ 0† ˆ w

n

y

ˆ 1 2

ˆ w

00n

…y ˆ 0† ˆ w

00n

y

ˆ 1 2

ˆ 0; …9†

p

1

is zero, the resolution of (8a) gives w

1

ˆ dw

00

. d is zero because of the symmetry at y ˆ

12

. Using

this fact, the resolution of (8b) permits to get:

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w…x; y† ˆ y

2

2

ÿ y

2

3

ÿ p

2

x

2=3

2U

0

…y†Ln y 1 ÿ y

ÿ U

00

…y†

‡ 0…x†: …10†

This result insure the asymptotic matching between the internal and external layers, and to get f …g† ! …9g

2

=2† ‡ p

2

when g ! 1, and thereafter the condition:

f

00

…g† ! 9g when g ! 1: …11†

2.3. Numerical resolution

The numerical resolution of the system (7) completed by the condition (11) permits to get the values p

2

0:6 and f

00

…0† 2:7, (Fig. 2) and to deduce of the pressure gradient and the velocity at y ˆ

12

:

dp dx ˆ 2

3 p

2

x

ÿ1=3

‡ 0:4x

ÿ1=3

‡ ; …a†

u x; y

ˆ 1 2

ˆ U

0

1

2 ÿ 4p

2

x

2=3

‡ 1

4 ÿ 2:4x

2=3

‡ …b† …12†

The system (2, 3, 4) completed by the conditions (12) is solved numerically. Some of the calcu- lations of the streamwise component of the ¯ow U…X ; Y † ˆ u…y†, at increasing downstream sta- tions, are shown in Fig. 3. The streamwise coordinate indicated is n and is given by x ˆ X

1=3

Re

1=3

, where X is the non-dimensional distance from the trailing edge. The smallest value of n at which

Fig. 2. The solution curves of (7, 11) forf0,f00 andf000versusg.

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eigenvalues were computed is 0.02. This corresponds to a distance X ˆ 8 10

ÿ6

Re from the trailing edge.

The transition from Poiseuille ¯ow to a far wake ¯ow is evident. So we have an acceleration of the ¯ow near y 0, caused by the slipping of the ¯uid after the trailing edge of the plate. At the center of the channel y

12

, the ¯ow is decelerated in order to maintain a constant ¯ow rate across the section …0

6

y

612

†.

3. The linear stability problem

3.1. Formulation

The purpose of this part is to study the linear stability of the near wake. Formally, small perturbations in the ¯ow quantities are introduced in the wake beyond the trailing edge. From the previous analysis, we can deduce the following scales: O…V † ˆ O…oU =oX ˆ O…Re

ÿ2=3

†. At high Reynolds numbers, the basic ¯ow at a ®xed X-station can be written: U…X ; Y † ˆ U …Y † ˆ u…Y † and V …X ; Y † ˆ 0. To study the stability of the ¯ow we let:

U…X ; Y ; t† ˆ U …Y † ‡ U

e

…X ; Y ; t†; V …X ; Y ; t† ˆ V

e

…X ; Y ; t†

and

P …X ; Y ; t† ˆ Re p…X † ‡ P

e

…X ; Y ; t†;

Fig. 3. Unperturbed velocity pro®lesu…y†at di€erent positionsxˆn3 in the near wake.

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where ( U

e

; V

e

) is the velocity disturbance and P

~

is the pressure disturbance. By substituting these expressions in Eqs. (1) and neglecting the quadratic term in the velocity disturbance [10], we obtain the linearized and inviscid equations of motion

o

U

e

ot

‡ U

o

U

e

oX

‡ V

o

U

e

oY

ˆ ÿ

oe

P

oX

;

o

V

e

ot

‡ U

o

V

e

oX

ˆ ÿ

o

P

e

oY

;

o

U

e oX

‡

o

V

e

oY

ˆ 0: …13†

The Rayleigh equation is derived by the introduction of the stream function and normal-mode analysis. More speci®cally, the stream function is taken to be

e

w…X ; y; t† ˆ u…y† exp‰i…aX ÿ xt†Š, where a and x are respectively wave number and frequency and they are complex i.e., a ˆ a

r

‡ ia

i

and x ˆ x

r

‡ ix

i

. Substitution into the system (1) gives

a

e

P ˆ aU

0

u ÿ …aU ÿ x†u

0

…14†

and the classical Rayleigh stability equation:

…aU ÿ x†…/

00

ÿ a

2

u† ˆ aU

00

u …15†

which, together with the boundary conditions

u…0†u 1

2 ˆ 0 …16†

de®nes the basic eigenvalue problem for this inviscid quasi-parallel shear ¯ows. In fact Rayleigh's stability equation is the vorticity equation of the disturbance.

3.2. Numerical resolution and discussion

For the numerical resolution we use the method described by Papageorgiou and Smith [11]. The range ‰0;

12

Š is divided into J equal subintervals of length h. The velocity pro®le is approximated by a piecewise linear distribution in each of the intervals, a fact that considerably simpli®es the Rayleigh equation in each interval since the U

00

term is zero. Hence if the discretization points are y

i:

i ˆ 1; 2;

. . .

; J ‡ 1 with y

1

ˆ 0 and y

J‡1

ˆ

12

, then the stream function in the interval ‰y

i

; y

i‡1

Š, denoted by u

i

, satis®es. …/

00i

ÿ a

2

u

i

† ˆ 0. The solutions are

u

i

ˆ A

i

exp ‰a…y ÿ y

i

†Š ‡ B

i

exp ‰ÿa…y ÿ y

i

†Š; …17†

where A

i

, and B

i

are constants to be determined. These are two equations that connect the values of A

i

and B

i

in neighbouring intervals. These come from continuity of mass ¯ux and pressure (14) across points of discontinuity (U

0

is discontinuous at y ˆ y

i

), and are

A

i

‡ B

i

ˆ A

iÿ1

exp …ah† ‡ B

iÿ1

exp …ÿah†; …a†

…aU

i

ÿ x†…A

i

ÿ B

i

† ÿ U

i0

…A

i

ÿ B

i

†

ˆ …aU

i

ÿ x†‰A

iÿ1

exp …ah† ‡ B

iÿ1

exp …ÿah†Š ÿ U

iÿ10

‰A

iÿ1

exp …ÿah† ‡ B

iÿ1

exp …ÿah†Š; …b†

…18†

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where U

i

ˆ U …y ˆ y

i

† and U

i0

, is the velocity in the interval ‰y

i

; y

i‡1

Š. The boundary conditions at y ˆ y

1

ˆ 0 and y ˆ y

J‡l

ˆ

12

are:

A

1

‡ B

1

ˆ 0 and A

J

exp …ah† ‡ B

J

exp …ÿah† ˆ 0: …19†

The system (18, 19) sets the eigenvalue problem for a and x and it is solved by matrix inversion.

Generally, the object of stability analysis is to solve the Rayleigh Eq. (15) together with the boundary conditions (16) for general values of wave number or frequency and a prescribed un- disturbed velocity pro®te U…y†. This corresponds to an eigenvalue problem for a and x. Three cases are possible: in the ®rst case the eigenvalue problem is obtained for real values of a which produce complex eigenvalues x in general; this is the temporal stability of a prescribed basic ¯ow pro®le. Second, for x kept real the corresponding eigenvalues for a is complex in general, this corresponds to disturbances growing or decaying with downstream distance and is the spatial stability problem. In the third case, a combined stability analysis that involves complex fre- quencies and wave numbers is possible [12]. However, for the stability of the wake, it has been observed both experimentally and numerically [13] that the spatial growth rates describe the stability characteristics very well. So we choose to focus on spatial stability problem …x

r

ˆ 0†. In this case the disturbances are characterized by a real frequency x, a spatial growth rate a

i

and a wavelength k ˆ 2p=a

r

.

In Fig. 4 we show a collection of curves at various streamwise locations. These curves depict the variation of wavelength k as a function of frequency x for the neutral stability …a

i

ˆ 0†. We note that the unstable band broadens with streamwise distance from the trailing edge …n ˆ x

1=3

Re

1=3

).

Fig. 4. Spatial stability of proper wake pro®les at various streamwise positions. Curves of neutral stability in…k;x†, from left to right, atnˆ0:02, 0.04, 0.06, 0.08 and 0.1.

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For n ! 0, the unstable band tend to disappear. This is coherent considering the fact that the stability of the de®nite Poiseuille pro®le in X ˆ 0

‡

. At a streamwise location in the near wake, the disturbances that provoke the instability have small values of frequency and wavelength.

References

[1] J.M.R. Graham, J. Fluid Mech. 356 (1998) 125.

[2] J.P.K. Tillet, J. Fluid Mech. 32 (1968) 273.

[3] S. Goldstein, Proc. Camb. Phil. Soc. 26 (1930) 1.

[4] J. Mauss, A. Achiq, S. Saintlos, C. R. Acad. Sci. Paris 315 (2) (1992) 1611.

[5] K. Stewartson, Mathematika 16 (1969) 106.

[6] S. Saintlos, J. Mauss, int. J. Eng. Sci. 34 (2) (1996) 201.

[7] A. Achiq, These de l'U P. S, Toulouse- France, 1992.

[8] A. Achiq, C. R. Acad Sci. Paris, 327 (IIb), (1999), 1005.

[9] A. Achiq, J. Mauss, 3e Congres de Mecanique Tetouan-Maroc, 2 (1997) 1.

[10] P.G. Drazin, W. Reid, Hydrodynamic Stability, Cambridge University Press, Cambridge, 1985.

[11] D.T. Papageorgiou, F.T. Smith, J. Fluid Mech. 208 (1989) 67.

[12] B.M. Woodley, N. Peake, J. Fluid Mech. 339 (1997) 239.

[13] G.E. Mattingly, W.O. Criminale, J. Fluid Mech. 51 (1972) 233.

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