Cohen I.M., Kundu P.K. Fluid Mechanics

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502 Instability

The continuity equation can be satisﬁed by deﬁning a streamfunction through

u =

∂ψ

∂z

,w=−

∂ψ

∂x

Equations (12.52) and (12.53) then become

− ψ

+ ψ

U =−

−ψ

− ψ

U =−

gρ

−

+ Uρ

= 0,

(12.54)

where subscripts denote partial derivatives.

As the coefﬁcients of equation (12.54) are independent of x and t, exponential

variations in these variables are allowed. Consequently, we assume normal mode

solutions of the form

[ρ,p,ψ]=[ˆρ(z), ˆp(z),

ψ(z)] e

ik(x−ct)

where quantities denoted by ( ˆ) are complex amplitudes. Because the ﬂow is

unbounded in x, the wavenumber k must be real. The eigenvalue c = c

+ic

can be

complex, and the solution is unstable if there exists a c

> 0. Substituting the normal

modes, equation (12.54) becomes

(U − c)

− U

ψ =−

ˆp, (12.55)

(U − c)

ψ =−

g ˆρ

−

ˆp

, (12.56)

(U − c) ˆρ +

ψ = 0. (12.57)

We want to obtain a single equation in

ψ. The pressure can be eliminated by

taking the z-derivative of equation (12.55) and subtracting equation (12.56). The

density can be eliminated by equation (12.57). This gives

(U − c)



− k



ψ −U

ψ +

U − c

ψ = 0. (12.58)

This is the Taylor–Goldstein equation, which governs the behavior of perturbations

in a stratiﬁed parallel ﬂow. Note that the complex conjugate of the equation is also

a valid equation because we can take the imaginary part of the equation, change the

sign, and add to the real part of the equation. Now because the Taylor–Goldstein

equation does not involve any i, a complex conjugate of the equation shows that if

is an eigenfunction with eigenvalue c for some k, then

∗

is a possible eigenfunction

with eigenvalue c

∗

for the same k. Therefore, to each eigenvalue with a positive c

there

7. Instability of Continuously Stratiﬁed Parallel Flows 503

is a corresponding eigenvalue with a negative c

. In other words, to each growing mode

there is a corresponding decaying mode. A nonzero c

therefore ensures instability.

The boundary conditions are that w = 0 on rigid boundaries at z = 0, d. This

requires ψ

= ik

ψ exp(ikx − ikct) = 0 at the walls, which is possible only if

ψ(0) =

ψ(d) = 0. (12.59)

Richardson Number Criterion

A necessary condition for linear instability of inviscid stratiﬁed parallel ﬂows can be

derived by deﬁning a new variable φ by

φ ≡

√

U − c

ψ = (U − c)

1/2

φ.

Then we obtain the derivatives

= (U − c)

1/2

φU

2(U − c)

1/2

= (U − c)

1/2

+ (1/2)φU

(U − c)

1/2

−

φU

(U − c)

3/2

The Taylor–Goldstein equation then becomes, after some rearrangement,

{(U − c)φ

}−



(U − c) +

(1/4)U

− N

U − c



φ = 0. (12.60)

Now multiply equation (12.60) by φ

∗

(the complex conjugate of φ), integrate from

z = 0toz = d, and use the boundary conditions φ(0) = φ(d) = 0. The ﬁrst term

gives



{(U − c)φ

}φ

∗

dz =





{(U − c)φ

∗

}−(U − c)φ

∗



=−



(U − c)|φ

dz,

where we have used φ = 0 at the boundaries. Integrals of the other terms in equa-

tion (12.60) are also simple to manipulate. We ﬁnally obtain



− (1/4 )U

U − c

|φ|

dz =



(U − c){|φ

+ k

|φ|

} dz



|φ|

dz. (12.61)

The last term in the preceding is real. The imaginary part of the ﬁrst term can be found

by noting that

U − c

∗

|U − c|

U − c

+ ic

|U − c|

504 Instability

Then the imaginary part of equation (12.61) gives



− (1/4 )U

|U − c|

|φ|

dz =−c



{|φ

+ k

|φ|

} dz.

The integral on the right-hand side is positive. If the ﬂow is such that N

everywhere, then the preceding equation states that c

times a positive quantity equals

times a negative quantity; this is impossible and requires that c

= 0 for such a

case. Deﬁning the gradient Richardson number

Ri(z) ≡

, (12.62)

we can say that linear stability is guaranteed if the inequality

Ri >

(stable), (12.63)

is satisﬁed everywhere in the ﬂow.

Note that the criterion does not state that the ﬂow is necessarily unstable if

Ri <

somewhere, or even everywhere, in the ﬂow. Thus Ri <

is a necessary

but not sufﬁcient condition for instability. For example, in a jetlike velocity proﬁle

u ∝ sech

z and an exponential density proﬁle, the ﬂow does not become unstable until

the Richardson number falls below 0.214. A critical Richardson number lower than

is also found in the presence of boundaries, which stabilize the ﬂow. In fact, there is no

unique critical Richardson number that applies to all distributions of U(z) and N(z).

However, several calculations show that in many shear layers (having linear, tanh,

or error function proﬁles for velocity and density) the ﬂow does become unstable to

disturbances of certain wavelengths if the minimum value of Ri in the ﬂow (which is

generally at the center of the shear layer) is less than

. The “most unstable” wave,

deﬁned as the ﬁrst to become unstable as Ri is reduced below

, is found to have a

wavelength λ  7h, where h is the thickness of the shear layer. Laboratory (Scotti

and Corcos, 1972) as well as geophysical observations (Eriksen, 1978) show that the

requirement

min

is a useful guide for the prediction of instability of a stratiﬁed shear layer.

Howard’s Semicircle Theorem

A useful result concerning the behavior of the complex phase speed c in an inviscid

parallel shear ﬂow, valid both with and without stratiﬁcation, was derived by Howard

(1961). To derive this, ﬁrst substitute

F ≡

U − c

7. Instability of Continuously Stratiﬁed Parallel Flows 505

in the Taylor–Goldstein equation (12.58). With the derivatives

= (U − c)F

+ U

= (U − c)F

+ 2U

+ U

Equation (12.58) gives

(U − c)[(U − c)F

+ 2U

− k

(U − c)F ]+N

F = 0,

where terms involving U

have canceled out. This can be rearranged in the form

[(U − c)

]−k

(U − c)

F + N

F = 0.

Multiplying by F

∗

, integrating (by parts if necessary) over the depth of ﬂow, and

using the boundary conditions, we obtain

−



(U − c)

∗

dz − k



(U − c)

|F |

dz +



|F |

dz = 0,

which can be written as



(U − c)

Qdz =



|F |

dz,

where

Q ≡|F

+ k

|F |

is positive. Equating real and imaginary parts, we obtain



[(U − c

)

− c

]Qdz =



|F |

dz, (12.64)



(U − c

)Q dz = 0. (12.65)

For instability c

= 0, for which equation (12.65) shows that (U − c

) must change

sign somewhere in the ﬂow, that is,

min

max

, (12.66)

which states that c

lies in the range of U . Recall that we have assumed solutions of

the form

ik(x−ct)

= e

ik(x−c

which means that c

is the phase velocity in the positive x direction, and kc

is the

growth rate. Equation (12.66) shows that c

is positive if U is everywhere positive,

506 Instability

and is negative if U is everywhere negative. In these cases we can say that unstable

waves propagate in the direction of the background ﬂow.

Limits on the maximum growth rate can also be predicted. Equation (12.64) gives



+ c

− 2Uc

− c

]Qdz > 0,

which, on using equation (12.65), becomes



− c

)Q dz > 0. (12.67)

Now because (U

min

− U) < 0 and (U

max

− U) > 0, it is always true that



min

− U )(U

max

− U)Qdz  0,

which can be recast as



max

min

+ U

− U(U

max

+ U

min

)]Qdz  0.

Using equation (12.67), this gives



max

min

+ c

− U(U

max

+ U

min

)]Qdz  0.

On using equation (12.65), this becomes



max

min

+ c

− c

max

+ U

min

)]Qdz  0.

Because the quantity within []is independent of z, and



Qdz > 0, we must have

[] 0. With some rearrangement, this condition can be written as



−

max

+ U

min

)



+ c





max

− U

min

)



This shows that the complex wave velocity c of any unstable mode of a disturbance

in parallel ﬂows of an inviscid ﬂuid must lie inside the semicircle in the upper half of

the c-plane, which has the range of U as the diameter (Figure 12.20). This is called

the Howard semicircle theorem. It states that the maximum growth rate is limited by

max

− U

min

The theorem is very useful in searching for eigenvalues c(k) in numerical solution of

instability problems.

8. Squire’s Theorem and Orr–Sommerfeld Equation 507

Figure 12.20 The Howard semicircle theorem. In several inviscid parallel ﬂows the complex eigenvalue

c must lie within the semicircle shown.

8. Squire’s Theorem and Orr–Sommerfeld Equation

In our studies of the B´enard and Taylor problems, we encountered two ﬂows in which

viscosity has a stabilizing effect. Curiously, viscous effects can also be destabilizing,

as indicated by several calculations of wall-bounded parallel ﬂows. In this section we

shall derive the equation governing the stability of parallel ﬂows of a homogeneous

viscous ﬂuid. Let the primary ﬂow be directed along the x direction and vary in the

y direction so that U =[U(y),0, 0]. We decompose the total ﬂow as the sum of the

basic ﬂow plus the perturbation:

u =[U + u, v, w],

˜p = P + p.

Both the background and the perturbed ﬂows satisfy the Navier–Stokes equations.

The perturbed ﬂow satisﬁes the x-momentum equation

∂u

∂t

+ (U + u)

∂

∂x

(U + u) + v

∂

∂y

(U + u)

=−

∂

∂x

(P + p) +

∇

(U + u), (12.68)

where the variables have been nondimensionalized by a characteristic length scale L

(say, the width of ﬂow), and a characteristic velocity U

(say, the maximum velocity

of the basic ﬂow); time is scaled by L/U

and the pressure is scaled by ρU

. The

Reynolds number is deﬁned as Re = U

L/ν.

508 Instability

The background ﬂow satisﬁes

0 =−

∂P

∂x

∇

Subtracting from equation (12.68) and neglecting terms nonlinear in the perturbations,

we obtain the x-momentum equation for the perturbations:

∂u

∂t

+ U

∂u

∂x

+ v

∂U

∂y

=−

∂p

∂x

∇

u. (12.69)

Similarly the y-momentum, z-momentum, and continuity equations for the

perturbations are

∂v

∂t

+ U

∂v

∂x

=−

∂p

∂y

∇

∂w

∂t

+ U

∂w

∂x

=−

∂p

∂z

∇

∂u

∂x

∂v

∂y

∂w

∂z

= 0.

(12.70)

The coefﬁcients in the perturbation equations (12.69) and (12.70) depend only on y,

so that the equations admit solutions exponential in x, z, and t. Accordingly, we

assume normal modes of the form

[u,p]=[

u(y), ˆp(y)]e

i(kx+mz−kct)

. (12.71)

As the ﬂow is unbounded in x and z, the wavenumber components k and m must be

real. The wave speed c = c

+ ic

may be complex. Without loss of generality, we

can consider only positive values for k and m; the sense of propagation is then left

open by keeping the sign of c

unspeciﬁed. The normal modes represent waves that

travel obliquely to the basic ﬂow with a wavenumber of magnitude

√

+ m

and

have an amplitude that varies in time as exp(kc

t). Solutions are therefore stable if

< 0 and unstable if c

> 0.

On substitution of the normal modes, the perturbation equations (12.69) and

(12.70) become

ik(U − c) ˆu +ˆvU

=−ik ˆp +

[ˆu

− (k

+ m

) ˆu],

ik(U − c) ˆv =−ˆp

[ˆv

− (k

+ m

) ˆv],

ik(U − c) ˆw =−im ˆp +

[ˆw

− (k

+ m

) ˆw],

ik ˆu +ˆv

+ im ˆw = 0,

(12.72)

where subscripts denote derivatives with respect to y. These are the normal mode

equations for three-dimensional disturbances. Before proceeding further, we shall

ﬁrst show that only two-dimensional disturbances need to be considered.

8. Squire’s Theorem and Orr–Sommerfeld Equation 509

Squire’s Theorem

A very useful simpliﬁcation of the normal mode equations was achieved by Squire in

1933, showing that to each unstable three-dimensional disturbance there corresponds

a more unstable two-dimensional one. To prove this theorem, consider the Squire

transformation

k = (k

+ m

)

1/2

, ¯c = c,

k ¯u = k ˆu + m ˆw, ¯v =ˆv,

¯p

ˆp

Re = k Re.

(12.73)

In substituting these transformations into equation (12.72), the ﬁrst and third of equa-

tion (12.72) are added; the rest are simply transformed. The result is

k(U − c)¯u +¯vU

=−i

k ¯p +

[¯u

−

¯u],

k(U − c)¯v =−¯p

[¯v

−

¯v],

k ¯u +¯v

= 0.

These equations are exactly the same as equation (12.72), but with m =ˆw = 0. Thus,

to each three-dimensional problem corresponds an equivalent two-dimensional one.

Moreover, Squire’s transformation (12.73) shows that the equivalent two-dimensional

problem is associated with a lower Reynolds number as

k>k. It follows that the

critical Reynolds number at which the instability starts is lower for two-dimensional

disturbances. Therefore, we only need to consider a two-dimensional disturbance if

we want to determine the minimum Reynolds number for the onset of instability.

The three-dimensional disturbance (12.71) is a wave propagating obliquely to the

basic ﬂow. If we orient the coordinate system with the new x-axis in this direction, the

equations of motion are such that only the component of basic ﬂow in this direction

affects the disturbance. Thus, the effective Reynolds number is reduced.

An argument without using the Reynolds number is now given because Squire’s

theorem also holds for several other problems that do not involve the Reynolds number.

Equation (12.73) shows that the growth rate for a two-dimensional disturbance is

exp(

k ¯c

t), whereas equation (12.71) shows that the growth rate of a three-dimensional

disturbance is exp(kc

t). The two-dimensional growth rate is therefore larger because

Squire’s transformation requires

k>kand ¯c = c. We can therefore say that the

two-dimensional disturbances are more unstable.

Orr–Sommerfeld Equation

Because of Squire’s theorem, we only need to consider the set (12.72) with

m =ˆw = 0. The two-dimensionality allows the deﬁnition of a streamfunction

ψ(x, y, t) for the perturbation ﬁeld by

u =

∂ψ

∂y

,v=−

∂ψ

∂x

510 Instability

We assume normal modes of the form

[u, v, ψ]=[ˆu, ˆv, φ] e

ik(x−ct)

(To be consistent, we should denote the complex amplitude of ψ by

ψ; we are using

φ instead to follow the standard notation for this variable in the literature.) Then we

must have

ˆu = φ

, ˆv =−ikφ.

A single equation in terms of φ can now be found by eliminating the pressure

from the set (12.72). This gives

(U − c)(φ

− k

φ) − U

φ =

ik Re

[φ

yyyy

− 2k

+ k

φ], (12.74)

where subscripts denote derivatives with respect to y. It is a fourth-order ordinary

differential equation. The boundary conditions at the walls are the no-slip conditions

u = v = 0, which require

φ = φ

= 0aty = y

and y

. (12.75)

Equation (12.74) is the well-known Orr–Sommerfeld equation, which governs

the stability of nearly parallel viscous ﬂows such as those in a straight channel or in

a boundary layer. It is essentially a vorticity equation because the pressure has been

eliminated. Solutions of the Orr–Sommerfeld equations are difﬁcult to obtain, and

only the results of some simple ﬂows will be discussed in the later sections. However,

we shall ﬁrst discuss certain results obtained by ignoring the viscous term in this

equation.

9. Inviscid Stability of Parallel Flows

Useful insights into the viscous stability of parallel ﬂows can be obtained by ﬁrst

assuming that the disturbances obey inviscid dynamics. The governing equation can

be found by letting Re →∞in the Orr–Sommerfeld equation, giving

(U − c)[φ

− k

φ]−U

φ = 0, (12.76)

which is called the Rayleigh equation. If the ﬂow is bounded by walls at y

and y

where v = 0, then the boundary conditions are

φ = 0aty = y

and y

. (12.77)

The set (12.76) and (12.77) deﬁnes an eigenvalue problem, with c(k) as the eigenvalue

and φ as the eigenfunction. As the equations do not involve i, taking the complex

conjugate shows that if φ is an eigenfunction with eigenvalue c for some k, then

∗

is also an eigenfunction with eigenvalue c

∗

for the same k. Therefore, to each

eigenvalue with a positive c

there is a corresponding eigenvalue with a negative c

9. Inviscid Stability of Parallel Flows 511

In other words, to each growing mode there is a corresponding decaying mode. Stable

solutions therefore can have only a real c. Note that this is true of inviscid ﬂows only.

The viscous term in the full Orr–Sommerfeld equation (12.74) involves an i, and the

foregoing conclusion is no longer valid.

We shall now show that certain velocity distributions U(y)are potentially unsta-

ble according to the inviscid Rayleigh equation (12.76). In this discussion it should

be noted that we are only assuming that the disturbances obey inviscid dynamics; the

background ﬂow U(y) may be chosen to be chosen to be any proﬁle, for example,

that of viscous ﬂows such as Poiseuille ﬂow or Blasius ﬂow.

Rayleigh’s Inﬂection Point Criterion

Rayleigh proved that a necessary (but not sufﬁcient) criterion for instability of an

inviscid parallel ﬂow is that the basic velocity proﬁle U(y) has a point of inﬂection.

To prove the theorem, rewrite the Rayleigh equation (12.76) in the form

− k

φ −

U − c

φ = 0,

and consider the unstable mode for which c

> 0, and therefore U −c = 0. Multiply

this equation by φ

∗

, integrate from y

to y

, by parts if necessary, and apply the

boundary condition φ = 0 at the boundaries. The ﬁrst term transforms as follows:



∗

dy =[φ

∗

]

−



∗

dy =−



|φ

dy,

where the limits on the integrals have not been explicitly written. The Rayleigh

equation then gives



[|φ

+ k

|φ|

]dy +



U − c

|φ|

dy = 0. (12.78)

The ﬁrst term is real. The imaginary part of the second term can be found by multi-

plying the numerator and denominator by (U − c

∗

). The imaginary part of equation

(12.78) then gives



|φ|

|U − c|

dy = 0. (12.79)

For the unstable case, for which c

= 0, equation (12.79) can be satisﬁed only if

changes sign at least once in the open interval y

<y<y

. In other words, for

instability the background velocity distribution must have at least one point of inﬂec-

tion (where U

= 0) within the ﬂow. Clearly, the existence of a point of inﬂection

does not guarantee a nonzero c

. The inﬂection point is therefore a necessary but not

sufﬁcient condition for inviscid instability.

Fjortoft’s Theorem

Some seventy years after Rayleigh’s discovery, the Swedish meteorologist Fjortoft in

1950 discovered a stronger necessary condition for the instability of inviscid parallel