3Transient growth
III Hydrodynamic Stability
3.1 Motivation
So far, our model of stability is quite simple. We linearize our theory, look at
the individual perturbation modes, and say the system is unstable if there is
exponential growth. In certain circumstances, it works quite well. In others,
they are just completely wrong.
There are six billion kilometers of pipes in the United States alone, where the
flow is turbulent. A lot of energy is spent pumping fluids through these pipes,
and turbulence is not helping. So we might think we should try to understand
flow in a pipe mathematically, and see if it gives ways to improve the situation.
Unfortunately, we can prove that flow in a pipe is linearly stable for all
Reynolds numbers. The analysis is not wrong. The flow is indeed linearly stable.
The real problem is that linear stability is not the right thing to consider.
We get similar issues with plane Poiseuille flow, i.e. a pressure driven flow
between horizontal plates. As we know from IB Fluids, the flow profile is a
parabola:
One can analyze this and prove that it is linearly unstable when
Re =
U
c
d
ν
> 5772.
However, it is observed to be unstable at much lower Re.
We have an even more extreme issue for plane Couette flow. This is flow
between two plates at
z
=
±
1 (after rescaling) driven at speeds of
±
1 (after
scaling). Thus, the base flow is given by
¯
U = z, |z| ≤ 1.
Assuming the fluid is inviscid, the Rayleigh equation then tells us perturbations
obey
(
¯
U − c)
d
2
dz
2
− k
2
−
d
2
dz
2
¯
U
ˆw = 0.
Since
¯
U = z, the second derivative term drops out and this becomes
(
¯
U − c)
d
2
dz
2
− k
2
ˆw = 0.
If we want our solution to be smooth, or even just continuously differentiable,
then we need
d
2
dz
2
− k
2
ˆw = 0. So the solution is of the form
ˆw = A sinh k(z + 1) + B sinh k(z − 1).
However, to satisfy the boundary conditions
ˆw
(
±
1) = 0, then we must have
A = B = 0, i.e. ˆw = 0.
Of course, it is not true that there can be no perturbations. Instead, we have
to relax the requirement that the eigenfunction is smooth. We shall allow it
to be non-differentiable at certain points, but still require that it is continuous
(alternatively, we relax differentiability to weak differentiability).
The fundamental assumption that the eigenfunction is smooth must be
relaxed. Let’s instead consider a solution of the form
ˆw
+
= A
+
sinh k(z −1) z > z
c
ˆw
−
= A
−
sinh k(z + 1) z < z
c
.
If we require the vertical velocity to be continuous at the critical layer, then we
must have the matching condition
A
+
sinh k(z −z
c
) = A
−
sinh k(z + z
c
).
This still satisfies the Rayleigh equation if
¯
U
=
c
at the critical layer. Note that
u is discontinuous at the critical layer, because incompressibility requires
∂w
∂z
= −
∂u
∂x
= −iku.
c
So for all
|c|
=
|ω/k| <
1, there is a (marginally stable) mode. The spectrum
is continuous. There is no discrete spectrum. This is quite weird, compared to
what we have previously seen.
But still, we have only found modes with a real
c
, since
¯
U
is real! Thus, we
conclude that inviscid plane Couette flow is stable! While viscosity regularizes
the flow, but it turns out that does not linearly destabilize the flow at any
Reynolds number (Romanov, 1973).
Experimentally, and numerically, plane Couette flow is known to exhibit a
rich array of dynamics.
– Up to Re ≈ 280, we have laminar flow.
– Up to Re ≈ 325, we have transient spots.
– Up to Re ≈ 415, we have sustained spots and stripes.
– For Re > 415, we have fully-developed turbulence.
In this chapter, we wish to understand transient growth. This is the case when
small perturbations can grow up to some visible, significant size, and then die
off.