Appl. Math. Mech. -Engl. Ed.     2014, Vol. 35 Issue (5) : 541–548     PDF       
http: //dx. doi. org/10.1007/s10483-014-1811-7
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Article Information

Hua -shu DOU, V. GANESH 2014.
Short wave st ability of homogeneous shear flows with variable topography
Appl. Math. Mech. -Engl. Ed., 35 (5) : 541–548
http: //dx. doi. org/10.1007/s10483-014-1811-7

Article History

Received 2013-03-25;
in final form 2013-11-13
Citation
Hua-shu DOU, V. GANESH 2014. Short wave st ability of homogeneous shear flows with variable topography[J]. Appl. Math. Mech. -Engl. Ed., 35 (5) : 541–548.
Short wave st ability of homogeneous shear flows with variable topography
Hua -shu DOU , V. GANESH        
Faculty of Mechanical Engineering & Automation, Zhejiang Sci-Tech University, Hangzhou 310018, P. R. China
Received 2013-03-25; in final form 2013-11-13
Fund program：supported by the Natural Science Foundation of Zhejiang Sci-Tech University (No. 11130032241201)
Corresponding author: Zhe-wei ZHOU, Professor, Ph. D., E-mail: zhwzhou@shu.edu.cn
ABSTRACT：For the stability problem of homogeneous shear flows in sea straits of arbitrary cross section, a sufficient condition for stability is derived under the condition of inviscid flow. It is shown that there is a critical wave number, and if the wave number of a normal mode is greater than this critical wave number, the mode is stable.
Keywords： hydrodynamic stability        shear flow        variable bottom         sea strait       

1 Introduction

The stability of homogeneous and stratified shear flows of an inviscid fluid to infinitesimal normal mode disturbances has been studied extensively[1 −2]. These studies are mostly restricted to rectangular cross sections. However,sea straits have rarely rectangular cross sections. The stability analysis of homogeneous and stratified shear flows in sea straits of arbitrary cross section was initiated in Ref. [3],while a more general theory for transversely uniform,timedependent,and stratified flows in a channel with an arbitrary cross section was studied in Ref. [4].

For the homogeneous shear flows,the mentioned problem reduces to the extended Rayleigh problem of hydrodynamic stability,and a number of general analytical results have been obtained[4 −6].

In this study,a basic flow with velocity U0(z) and variable topography

is considered,where b = b(z) is the width function. It is proved that for a special class of flows with decreasing topography,a sufficient condition for stability is

where

Furthermore ,it is proved that for a class of flows ,ci = 0 when k > kc (ci expresses the degree of damping or amplification of the disturbance),where kc is the critical value of wave number k. That means the short waves are stable. 2 Extended Rayleigh problem

The extended Rayleigh problem[4] is given by the second-order ordinary differential equation

with boundary conditions

For the problem defined by (1) and (2),the equation below is established[7].

3 Results of stability a nalysis

Theorem 1 If ci > 0,then we have the relations as follows:

(i)

(ii)

Proof Multiplying (1) by (bW *),integrating that over [0,D],and using (2),we get

and

Since ci > 0,it follows

From (6),it follows that = 0 at z = zs. In the following sections,the expression U0(zs) = U0s is used.

Theorem 2 If ci > 0,then we have the relation as follows:

Proof Multiplying (6) by (cr − U0s) and adding the obtained result with (4),we get

Theorem 3 If ci > 0,then (U0 − U0s ) < 0 at some point z = zp ≠ zs.

Proof In (7),dropping the first integration term being non-negative,we obtain

This result implies that (U0 − U0s ) < 0 and K(z) > 0 at some point z = zp ≠ zs,where

Theorem 4 If ci > 0,then we have the relations as follows:

and

Proof (1) can be rewritten as

Multiplying (9) by W∗,integrating that over [0,D],and then using (2) and (3),we get

and

Theorem 5 If ci > 0,then we have the relations as follows:

and

Proof Multiplying (1) by ,integrating that over [0,D],and using (2),we get

and

Multiplying (10) by k2 and adding the resultant to (12),we get

From (14) and multiplying (7) by ,we get

Theorem 6 If T'≤ 0,then an estimation for the growth rate is given by

Proof In (15),the terms

are non-negative. Thus,we can get

Theorem 7 A sufficient criterion of instability is that 0 < K(z) ≤ in the flow domain [0,D].

Proof From (16),if K(z) ≤ ,it follows that ci = 0,which means that the flow is stable.

Theorem 8 If ci > 0,then we have the relation as follows:

Proof Adding (12) and (13) which is multiplied by ,we can get

Theorem 9 If ci > 0,then we have the relation as follows:

Proof Multiplying (1) by and integrating that over [0,D],we obtain

From (17) and multiplying (11) by 2k2,we can get

Theorem 10 For an unstable mode with wave number k > 0 and cr = U0s ,it is necessary that

Proof For the homogeneous case,a neutral s-wave is represented by a solution of the eigen value problem defined by (1) and (2) that corresponds to cr = U0s whose existence was shown in Ref. [8].

From (18),it follows that

i.e.,

Remark 1 As λ = 2π/k is the length of the wave,it follows that the wave length of an unstable mode must be sufficiently large as small k corresponds to large λ. Let

Then,it follows that k > kc implies that the normal mode with wave number k is stable,and the provided kc is finite. Similar,the type of result has been proved for this problem but with constant topography in Ref. [6].

In the following,the theorems are used for some examples.

(i) Let the basic velocity be U0(z) = sin(z2) and topography be T = 1/z in 0 ≤ z ≤ 2π. Then,= −4z2 sin(z2),which changes sign at zs = √π. Thus,it satisfies the necessary condition for instability. Then,using (19),we obtain k > kc = (≤2√2π) which implies the stability of the mode.

(ii) Let the basic velocity be U0 = ez sin z and topography T = 2 (b = e2z) in 0 ≤ z ≤ 2π. Then,= −2ez sin z,which changes sign at zs = π. Thus,it satisfies the necessary condition for instability. Then,using (19),we get k > kc = 1,which implies the stability of the mode 4 Conclusions

In this paper,some general analytical results for the stability problem of homogeneous shear flows in sea straits of arbitrary cross section are obtained. The conclusions are summarized as follows:

(i) A sufficient condition for the stability of basic flow is proved as 0 < K(z) ≤ in [0,D],where

(ii) For flows in which the basic velocity U0(z) is second-order continuously differentiable, and T is the topography,it is proved that there is a critical wave number kc satisfying

If the wave number of a normal mode k > kc,then the mode is stable.

References
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