Channel entrance flow and its linear stability

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Channel entrance flow and its linear stability

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JST A T (2004) P06003

ournal of Statistical Mechanics:

An IOP and SISSA journal

J Theory and Experiment

Channel entrance flow and its linear stability

Ahmed Hifdi

, Mohammed Ouazzani Touhami and Jaˆ afar Khalid Naciri

Laboratoire de M´ ecanique, Facult´ e des Sciences Ain chock, BP 5366 Maˆ arif, Casablanca, Morocco

E-mail: [email protected], [email protected] and [email protected]

Received 7 April 2004 Accepted 4 June 2004 Published 15 June 2004

Online at stacks.iop.org/JSTAT/2004/P06003

doi:10.1088/1742-5468/2004/06/P06003

Abstract. In this work, we present a temporal linear stability analysis of developing channel flow. For the main flow, the considered solution is analytic. It is based on the hypothesis of small disturbances from fully developed flow and it is valid for intermediate Reynolds numbers. The disturbances are separated into symmetric and anti-symmetric eigenmodes of the velocity. We deal subsequently with the linear stability of this main flow, taking into account the nearly parallel flow assumption. The stability problem formulation leads to the Orr–Sommerfeld equation. This equation is then resolved using the Chebyshev spectral collocation method. The stability results depend essentially on the shape and amplitude of the velocity profiles imposed at the channel entry.

Keywords: hydrodynamic instabilities

1 Author to whom any correspondence should be addressed.

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Contents

1. Introduction 2

2. The study of the basic problem 3

3. The linear stability problem 5

4. Results and discussion 9

5. Conclusions 14

References 15

1. Introduction

The study of the pipe entrance flow stability is a relatively classical theme for which extensive analytical investigations have been performed [1]–[6] during the second half of the last century. These works are numeric [1, 2], [4]–[6] or experimental [3] and are carried out in order to assess the influence of the both axisymmetric [2, 3, 5, 6] and non- axisymmetric [1]–[4] disturbance effects on the stability of developing laminar pipe flow.

For the numerical approaches, the entry profile is generally that of Sparrow [1, 2] or that of Hornbeck [4, 5]. The numerical results do not compare well with experimental data in the case of the Sparrow profile. However, the temporal linear stability of channel entrance flow is an unresolved issue even if the channel entrance flow is a classical problem in fluid mechanics that has been a subject of numerous analyses. In spite of the existence of a voluminous literature, the analytic data of the entry channel profile are not available.

Only numerical solutions for the symmetric modes of Poiseuille flow perturbations are given [7, 8] to describe the channel entrance flow. For this reason, we intend in this work to approach the temporal linear stability problem of channel entrance flow by using the basic analytic solution that we have established in a recent paper [9].

The present work is constituted of three main parts. The first (section 2) is dedicated to the analytical study of the steady basic flow based on the hypothesis of small disturbances from fully developed flow which is valid for intermediate Reynolds numbers.

The obtained analytic solutions of disturbances are separated into symmetric and anti- symmetric eigenmodes of the velocity. They allow us to get a large spectrum of eigenmodes and to assess the effects of anti-symmetric eigenmodes on the channel entrance length [9].

For any given velocity profile at the entry channel slightly perturbed from Poiseuille

flow, these solutions are used to obtain the evolution of the velocity distribution in the

whole channel entry region. The second part (section 3) is dedicated to the formulation

of the temporal linear stability problem from nearly parallel flow theory. It leads to

the classical Orr–Sommerfeld equation. This equation is then resolved using Chebyshev

spectral collocation method [10]. To validate our calculation code, we find some classical

results already reported by Orszag [11] and Dongarra et al [12] when the main flow is

supposed to be Poiseuille flow.

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Using our results obtained in the first part for the basic flow, we investigate, in the third part (section 4), the temporal linear stability for some perturbed plane Poiseuille flows. It is found that the stability characteristics are very sensitive to the velocity field in the inlet region.

2. The study of the basic problem

We consider the two-dimensional steady flow in the inlet zone of a long straight channel of width 2h. The fluid is incompressible with kinematic viscosity ν. We take (x

, y

) to be the coordinate system, with x

= 0 lying at some arbitrary location (for example the position of the channel entry), and y = 0 lying along the centreline of the channel, and so the walls are located at y

= ± h. The flow is supposed to be slightly perturbed from plane Poiseuille flow. The dimensional velocity vector has components (U

, V

) and may thus be represented by

U

= U

(y) + u

(x, y), V

= v

(x, y ), P

= P

(x) + p

(x, y) (1) where U

(y) and P

(x) denote the velocity and the pressure of the Poiseuille flow, respectively, and u

(x, y), v

(x, y) and p

(x, y ) denote small perturbations.

The problem is scaled using the average velocity U

of the Poiseuille profile with L and h as the downstream and transverse characteristic length scale of these perturbations respectively. The intermediate Reynolds number is defined as Re = U

h/ν where = h/L. In the case 1, we substitute (1) into the continuity equation and the Navier–Stokes equations. This is followed by subtraction of the Poiseuille flow and neglect of squares of perturbation quantities. They lead to the following dimensionless linearized perturbation equations [13]:

Re

U

∂u

∂x + v ∂U

∂y

= − ∂p

∂x + ∂

u

∂y

, ∂p

∂y = 0. (2a,b)

The relevant boundary conditions have the following form:

u(x, 1) = u(x, − 1) = v (x, 1) = v (x, − 1) = 0 at x ≥ 0, U(0, y) = U

(y) with

−1

U

(y) dy = 4/3, Poiseuille flow at x → ∞ , − 1 ≤ y ≤ 1.

The purpose of this section is to recall how we have determined the analytic solution of system (2a,b) in our recent paper [9]. Following Wilson [14], let us assume that the stream function for the small two-dimensional perturbation to Poiseuille flow can be written as Ψ

(x, y ) = Φ(y)e

where α > 0 is a real number. Substitution of this representation of the flow perturbations into the field equations (2a,b) leads, after some algebra, to the third-order linear ordinary differential equation in the form

Φ

(y) + α Re(1 − y

)Φ

(y) + 2α Re y Φ(y) = e

dp/dx, (3) subject to homogeneous boundary conditions

Φ(1) = Φ( − 1) = Φ

(1) = Φ

( − 1) = 0,

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where Φ

(y) = d

Φ(y)/dy

. The right-hand side of equation (3) is obviously constant (=k) and its solutions can be separated into even and odd eigenfunctions. An eigenfunction is called even or odd depending on the symmetry or the anti-symmetry of Φ

, respectively, which is related to the streamwise velocity component. The constant k, which is related to the pressure gradient, turns out to be zero for odd eigenfunctions. Consequently, the pressure field is uniform in the case of a small anti-symmetric streamwise velocity perturbation.

Equation (3) can be differentiated once to get

Φ

(y) + α Re(1 − y

)Φ

(y) + 2α Re Φ(y) = 0, (4) subject to the same homogeneous boundary conditions as equation (3). Equation (4) is an eigenvalue problem which has been solved numerically in [14]–[16] to calculate only eigenvalues and not eigenfunctions. The even eigenfunctions only have been determined numerically in [7, 8].

The analytic solution of equation (4) is obtained as follows.

(i) We consider the following combination of equations (3) and (4):

[Φ

(y) − yΦ

(y)] + α Re(1 − y

)[Φ

(y) − yΦ

(y) + 2Φ(y)] = − ky. (5) (ii) Let us consider a change of function in the form

Θ(y) = Φ

(y) − yΦ

(y) + 2Φ(y). (6)

Equation (5) becomes

Θ

(y) + α Re(1 − y

)Θ(y) = − ky. (7)

The analytical solution of this equation (7) has the form Θ(y) = kβ

π/2

˜

u(a, y/β )

0

t˜ v(a, t) dt − v(a, y/β) ˜

0

t u(a, t) dt ˜

+ d

v(a, y/β ˜ ) + d

u(a, y/β), ˜ (8)

where ˜ u(a, y/β) and ˜ v(a, y/β) denote the parabolic cylinder functions [17] with a =

−

α Re/4, β = √

2(α Re)

/2. d

and d

denote integration constants.

Calculating the analytical solution of equation (6), the function Φ

(y) has the form Φ

(y) = y

0

tΘ(t) dt − y π/2

0

F (t/ √

2)Θ(t)(t

− 1)e

dt + 2c

y +

y

π/2F (y/ √

2) − e

c

+

0

Θ(t)(t

− 1)e

dt

, (9)

where F (y/ √

2) = 2/π

0

e

dt. c

and c

denote integration constants.

Consequently, the x-velocity component is symmetric for c

= 0 and anti-symmetric

for c

= 0. Table 1 contains the first 12 eigenvalues. They are alternately odd and even

and are numbered in increasing order for each kind. The first five eigenvalues are in good

agreement with those given by Wilson [14] and Kumar et al [7], when the differing form

of U

(y) is taken into account.

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Table 1. The eigenvalues α

Re of equation (3).

i Odd Even

1 21.67960 28.22083 2 73.30818 86.28399 3 156.64606 175.93671 4 271.84257 297.34978 5 418.95346 450.60010 6 598.00650 635.73381 7 809.01696 852.77493 8 1051.99528 1101.74256 9 1326.94734 1382.64954 10 1633.87888 1695.50541 11 1972.79208 2040.31744 12 2343.69121 2417.09136

It can be shown from equation (4) that the obtained anti-symmetric eigenfunctions Φ

(y) (i = 1 to n) and symmetric eigenfunctions Φ

(y) (i = 1 to n), which are associated with eigenvalues α

Re and α

Re, constitute two bases, respectively. These bases can be written in their orthonormalized forms Φ

(y) (i = 1 to n) and Φ

(y) (i = 1 to n), using the Gram–Schmidt procedure [18].

Taking into account the above remarks and the linearity of system (2a,b), we can write the x-velocity component of small perturbations as

u(x, y) =

i=1

γ

Φ

(y)e

+

i=1

β

Φ

(y)e

, (10) where

γ

=

−1

u

(y)Φ

(y) dy

and β

=

−1

u

(y)Φ

(y) dy

. The functions u

(y) and u

(y) are defined respectively by u

(y) = [u(0, y) + u(0, − y)]/2 and u

(y) = [u(0, y) − u(0, − y)]/2. Finally, the complete flow solution in the inlet zone of the channel is

U(x, y) = 1 − y

+ u(x, y). (11)

The two-dimensional basic flow presented above is said to be nearly parallel [19] since V (x, y) U(x, y) and ∂U/∂x 1.

3. The linear stability problem

The purpose of this part is to study the temporal linear stability of the entrance flow in a channel which has been defined in the previous section. A reasonable approximation to this flow stability is to make the ‘parallel flow assumption’. The basic flow at a fixed x- station can be written U (x, y) = U (y) and V (x, y) = 0 (negligible). To study the stability of the flow, we let

U ˜ (x, y, t) = U (y) + u

(x, y, t), V ˜ = v

(x, y, t), P ˜ (x, y, t) = P (x) + p

(x, y, t), (12)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Poiseuille profile (R=10

, λ =1)

Even

Odd

c

=0.66845 S

P A

–1 –0.8 –0.6 –0.4 – 0.2 0 0.2

Figure 1. The spectrum for plane Poiseuille flow.

where (u

, v

) is the velocity disturbance and p

is the pressure disturbance. Then we substitute (12) into the continuity equation and the Navier–Stokes equations. This is followed by subtraction of the basic flow and neglect of squares of disturbance quantities.

They lead to the linearized disturbance equations [19]. Next, the Orr–Sommerfeld equation is derived by the introduction of the stream function and normal-mode analysis.

More specifically, the stream function is taken to be H(x, y, t) = G(y)e

where λ > 0 is a real wavenumber, c = c

+ ic

is the complex wave speed and t is the dimensionless time based on h and U

. Substitution of H(x, y, t) into the disturbance equations leads to the classical Orr–Sommerfeld equation below:

A G = c B G, G( ± 1) = D

G( ± 1) = 0, (13)

where A = R

(D

− λ

)

− iλU (D

− λ

) + iλD

U, B = − iλ(D

− iλ

) and D

= d

/dy

. R denotes the Reynolds number corresponding to disturbances of the basic flow and it is defined similarly to Re in the previous section i.e., R = U

h

/L

ν where L

is the downstream characteristic length of these disturbances and U

the average velocity of Poiseuille profile. The choice of L

is, of course, not unique but we consider only that the two components of the disturbance velocity are of the same order of magnitude [19]. In the present discussion, we take L

= h and it follows that R = U

h/ν.

Equation (13) is an eigenvalue problem for the unknown complex wave speed, c,

and the amplitude of the stream function, G(y). If there is an eigenvalue with positive

imaginary part, then the flow is linearly unstable. For the numerical resolution of this

equation, we use the Chebyshev spectral method based on the most commonly used

collocation points of Gauss–Lobatto. For a complete description of this method, we refer

to Canuto et al [10].

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Figure 2. Evolution of flow 1 in the entry zone. ——: U (x, y). – – –: U

(y).

◦ ◦ ◦ ◦ : U (0, y) imposed.

Figure 3. Evolution of flow 2 in the entry zone when the profile U (0, y) =

− (π/3) cos[π(1 − y/2)] is imposed. ——: U (x, y). – – –: U

(y). ◦ ◦ ◦ ◦ : U (0, y)

imposed.

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Figure 4. Evolution of flow 3 in the entry zone. ——: U (x, y). – – –: U

(y).

◦ ◦ ◦ ◦: U (0, y) imposed.

Figure 5. Evolution of flow 4 in the entry zone when the profile U (0, y) =

−(π/3) cos[π(1 − y/2)] + 0.25 sin[π(1 − y)] is imposed. ——: U (x, y). – – –:

U

(y). ◦ ◦ ◦ ◦ : U (0, y) imposed.

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

x/2Re

R c

Flow 3 Flow 1 Rc=5772.22

Figure 6. Critical Reynolds number distribution for flows 1 and 3.

In order to validate our code of calculation that will be able to compute odd modes and even modes separately, we have determined the critical Reynolds number for instability of plane Poiseuille flow. The critical Reynolds number R

is defined as the smallest value of R for which an unstable eigenmode exists. The mode that becomes unstable at R

is even [11]. We find that the critical Reynolds number is R

= 5772.2223 for λ

= 1.020 55 with the number of collocation points N = 80. We have also computed the eigenvalues for R = 10

, λ = 1 and N = 100. We obtain complete agreement with the list of Orszag [11, table 5]. The spectrum, in the range c

∈ [ − 1, 0.2], c

∈ [0, 1], is given in figure 1. This figure agrees with that reported by Dongarra et al [12] when they used the so-called D

alternative of the Chebyshev tau-QZ algorithm method. However, like these authors, we find an odd extra eigenvalue c = 0.212 725 7823 − 0.199 360 6947i between positions 17 and 18 of Orszag [11, table 5].

4. Results and discussion

To carry out a numerical analysis of the stability characteristics of the hydrodynamically developing region of the channel flow, we consider some sample flows in this region. The choice of these flows will be discussed below. They are shown in figures 2–5 and are identified by flow 1–4. At each axial location specified by x/2Re, the values of U (y) are given by equation (11). Flows 1 and 2 are symmetric and they differ from each other at the inlet. The first has a top-hat shape while the second has a jet shape.

Figures 6 and 7 show the variation of R

at several axial locations for both flows 1

and 2 respectively. In the near entry region of the channel, the first is more stable than

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 1000

2000 3000 4000 5000 6000

x/2Re

R

Flow 4 Flow 2 Rc=5772.22

Figure 7. Critical Reynolds number distribution for flows 2 and 4.

plane Poiseuille flow while the second is less stable than the latter. Consequently, the shape of the considered profile has an influence on the stability of symmetric flows in the entry region.

To assess the influence due to the presence of the anti-symmetric part in the entrance flow on its stability, we superpose, at the inlet, to flows 1 and 2 the same following anti- symmetric correction: 0.25 sin[π(1 − y)]. This defines flows 3 and 4 respectively, which are shown in figures 4 and 5 respectively. Figures 6 and 7 show the variation of R

at several axial locations for both flows. The first becomes less stable than plane Poiseuille flow while the second becomes more unstable than flow 2. Consequently, the anti-symmetric part of entry flow increases its instability.

To characterize this effect, we analyse the eigenvalue spectrum for all cited flows when R = 10

and λ = 1. Thus, we can compare them to those obtained in [11, 12]

corresponding to plane Poiseuille flow. They are shown in figures 8–11 in the range c

∈ [ − 1, 0.2], c

∈ [0, 1.2], at x/2Re = 0.007. The choice of this station has been made since flow 1 becomes unstable while the others are unstable from the inlet as we can note from table 2. The c

values of the first eigenvalue, that belongs to the so-called A branch (as the standard notation in the fluid dynamics literature [19]), for flows 1–4 are reported at several axial locations with the equivalent c

for Poiseuille flow. As indicated in figure 1, the P branch is composed of degenerate pairs of even and odd eigenvalues for Poiseuille flow. The same branch is also composed of some degenerate pairs for flow 1 and of numerous degenerate pairs for flow 2 as shown in figures 8 and 9 respectively.

In contrast, these degenerate pairs are not observed for both flows 3 and 4 as shown in

figures 10 and 11 respectively.

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Even

Odd R=10

, λ =1, x/2Re=0.007

c

=0.66886 S

P A

–1 –0.8 –0.6 –0.4 –0.2 0 0.2

Figure 8. The spectrum for flow 1 at x/2Re = 0.007.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

R=10

, λ =1, x/2Re=0.007

Odd

Even

S P A

cr=0.667908

–0.8 –0.6 –0.4 –0.2 0 0.2

Figure 9. The spectrum for flow 2 at x/2Re = 0.007.

Our stability analysis depends on the velocity profile imposed at the channel entry.

Thus, our main purpose is to generalize this analysis to all profiles slightly perturbed

from Poiseuille flow. It is worth noting that these profiles differ from each other, at

any given axial location x, only by their coefficients γ

e

and β

e

that appear in

equation (10). In addition, the eigenmodes Φ

(y) and Φ

(y) are not related to the

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 cr

c i

R=10⁴ ,λ=1, x/2Re=0.007

Even

Odd

P A

S

cr=0.787152 –1

–0.8 –0.6 –0.4 –0.2 0 0.2

Figure 10. The spectrum for flow 3 at x/2Re = 0.007.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

c

=0.787444

R=10

, λ =1, x/2Re=0.007

Even Odd

A

P

S

–1 –0.8 –0.6 –0.4 –0.2 0 0.2

Figure 11. The spectrum for flow 4 at x/2Re = 0.007.

velocity profiles imposed at the channel entry. Then, for a given x, the solution (11) can be written as

U(y) = 1 − y

+

i=1

a

Φ

(y) +

i=1

a

Φ

(y), (14)

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0 1 2 3 4 5 6 7 8 9 10 11 12

0 1000 2000 3000 4000 5000 6000 7000

8000 a

f=0.001 af=0.005 af=0.01 af=0.03 af=0.05 af=0.1

Figure 12. Stability of the profiles defined by (15) in the case of symmetric eigenmodes when 0.001 ≤ a

≤ 0.1.

0 1 2 3 4 5 6 7 8 9 10 11 12

0 1000 2000 3000 4000 5000 6000 7000 8000

af=0.001 a_f=0.005 a_f=0.01 af=0.03 af=0.05 a_f=0.1

Figure 13. Stability of the profiles defined by (15) in the case of anti-symmetric

eigenmodes when 0.001 ≤ a

≤ 0.1.

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Table 2. The c

values of the first eigenvalue for flows 1–4 when R = 10

and λ = 1 at different stations. (Note: c

= 0.003 739 670.)

x/2Re Flow 1 Flow 2 Flow 3 Flow 4

0.001 −0.002 164 765 0.011 518 739 0.003 807 596 0.019 787 040 0.003 −0.001 328 792 0.006 866 078 0.005 103 872 0.017 150 634 0.005 −0.000 607 159 0.005 468 621 0.005 687 662 0.015 351 375 0.007 +0.000 013 523 0.004 881 740 0.006 023 881 0.013 973 188 0.009 +0.000 545 831 0.004 580 653 0.006 223 773 0.012 854 053 0.011 +0.001 001 405 0.004 404 140 0.006 335 358 0.011 914 588 0.013 +0.001 390 637 0.004 289 672 0.006 385 074 0.011 108 858 0.015 +0.001 722 849 0.004 209 257 0.006 389 679 0.010 407 386 0.017 +0.002 006 233 0.004 148 962 0.006 360 727 0.009 789 867 0.019 +0.002 247 887 0.004 101 347 0.006 306 721 0.009 241 623 0.021 +0.002 454 024 0.004 062 168 0.006 234 048 0.008 751 646 0.023 +0.002 629 948 0.004 028 920 0.006 147 648 0.008 311 448

where a

and a

are constant coefficients which can be interpreted as amplification factors of the eigenmodes. Instead of analysing the stability of the velocity fields obtained from equation (14), it is convenient to do it for the profiles defined by

U(y) = 1 − y

+ a

Φ

(y), (15)

for different values of i and given values of a

. Note that we can also obtain information about the related influence to each eigenmode on the instability process. Results concerning profiles given by equation (15) are shown in figures 12 and 13 for symmetric and anti-symmetric eigenmodes respectively and for some values of a

belonging to the range [0.001, 0.1]. The eigenmodes with i ≥ 2 become more unstable as a

increases. The anti-symmetric eigenmode i = 1 has a particular behaviour; its stability increases with a

and reaches a maximum at a

such that 0.05 < a

< 0.1.

These results allow us to obtain, without further computations, information on the stability of any developing flow once the values of the coefficients a

and a

are known.

Thus, a protocol to control the stability of channel entrance flow is provided.

5. Conclusions

This work reveals the great interest of the analytic solution availability to study easily

the entrance channel flow and its stability. The presented results permit us to raise the

influence of the shape of the considered profiles on the occurrence of instabilities in the

entry region. In particular, we have shown the destabilizing effect of the anti-symmetric

part of these profiles. We are also able to characterize, and thereafter to control, the

contribution of each eigenmode of the Poiseuille flow in the stability process. Finally, we

have to note that in the case of developing pipe flow, all authors have been limited by

Sparrow and Hornbeck profiles, whereas this work allows us to consider a large spectrum

of entry channel profiles slightly perturbed from plane Poiseuille flow and therefore to

study its stability.

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