Stability of open flows: basic ideas
3.3 Convective and absolute instabilities .1 The criterion for absolute instability
For a wave packet centered on a wave number k0and propagating without decay-ing or growdecay-ing (both k andω are real), the envelope of the packet propagates, as discussed above, at the real group velocity
cg(k0) =∂ω
∂k(k0). (3.28)
Let us now see what happens in the neighborhood of an instability threshold (with k andω complex) for a value of the bifurcation parameter R close to the critical value Rc (Huerre and Rossi, 1998, §3). At the threshold R= Rc the curve describ-ing the growth rate ωi(kr) of the temporal normal modes is tangent to the axis kr at real critical wave number kc, and so∂ωi/∂kr is necessarily zero. The group velocity of a wave packet centered on the critical wave number kc,
cg(kc) =∂ω
∂k(kc, Rc) =∂ωr
∂kr =∂ωi
∂ki, (3.29)
is indeed real.4
Above the threshold, if there exists a wave number k0 for which the group velocity cg(k0) is zero and the temporal growth rate (i.e., the imaginary part of ω0= ω(k0)) is positive, then the packet grows in place (Figure 3.3c). This result is obtained by calculating the impulse response G(x,t) along the ray x/t = 0, given by (3.24) with ˆS(k,ω) = 1 inside the integral. The integral is calculated by generalizing the stationary-phase method to the case where the argument of the exponential is not purely imaginary, but contains a real part (Bender and Orszag, 1978, §6.6). The resulting asymptotic response at long times is given by
G(x,t) ∼ 1
√2πω(k0)t
eiπ/4
∂ωD(k0,ω0)ei(k0x−ω0t), (3.30) where k0andω0are defined through
∂ω
∂k(k0) = 0, ω0= ω(k0). (3.31) It follows that the instability criterion is simply that the imaginary part of ω0, referred to as the absolute growth rate, be positive.
From this discussion we can deduce an essential characteristic of convective and absolute instabilities. A convective instability amplifies any unstable perturbation, and advects it downstream. If the perturbations are viewed as “noise,” an unstable convective system then behaves as a noise amplifier. On the other hand, a system which is absolutely unstable responds selectively to perturbations: its response is
4 For a function Z= X +iY of a complex variable z = x +iy to be differentiable at the point z0, it is necessary and sufficient that X(x, y) and Y (x, y) be differentiable at (x0, y0) and that at this point (the Cauchy conditions)
∂ X
∂x =∂Y
∂y, ∂ X
∂y = −∂Y
∂x. The derivative Zis then given by
Re(Z) =∂ X
∂x =∂Y
∂y, Im(Z) = −∂ X
∂y=∂Y
∂x.
dominated by the mode with zero group velocity, which grows in place, while the other modes are “swept away” by the flow. Since this mode is selected by the dispersion relation, which is intrinsic to the system, an absolutely unstable system behaves like an oscillator with its own natural frequency.
3.3.2 The spatial branches of a convective instability
Let us consider the situation shown in Figure3.3b of a perturbation with given frequency advected by a flow. We want to determine the spatial growth rate of this perturbation. This problem makes sense only in the case of a convective instability, because for an absolute instability the perturbation field is dominated by the mode with zero group velocity, which grows in place. The spatial growth rate is obtained by using the dispersion relation D(k,ω)=0 to find the spatial branches, that is, the complex wave number(s) kr+ikicorresponding to a real frequencyω. The spatial growth rate is then−kiand the speed of the wave isω/kr.
3.3.3 Illustrations
As a simple example, let us consider the Ginzburg–Landau equation, for which the dispersion relation is(3.26). The (complex) group velocity of a wave number k is then
cg=∂ω
∂k = V −2ik, (3.32)
which vanishes for wave number k0= V/2i. Inserting this wave number into the dispersion relation(3.26), we find the corresponding frequencyω0= i(R − V2/4).
We conclude that for 0< R < V2/4 the base state u0= 0 is convectively unstable (Figure3.3b), and that for R> V2/4 it is absolutely unstable (Figure3.3c).
When the instability is convective, i.e., R< V2/4, the two spatial branches (or spatial modes) are obtained from (3.26):
k±(ω) =1 2
−iV ±
4(R +iω)− V2
, (3.33)
from which we easily obtain the spatial growth rate−k±,i. We stress the fact that the spatial branches can be determined only when we are sure that the instability is convective.
We briefly mention some hydrodynamical illustrations which will be treated in more detail later on. The mixing layer between two concurrent and parallel flows with different velocities is a typical example of a flow which is convectively unsta-ble. The Poiseuille flow between flat plates or in a pipe, as well as the boundary
layer on a flat plate, are two other important examples. In such flows, any per-turbation which is forced at an upstream location is amplified downstream; the instability does not have its own dynamics. On the other hand, the wake down-stream from a solid obstacle is absolutely unstable beyond a critical Reynolds number of a few tens. For example, Rec=U∞d/ν =48.5 for a cylinder of diameter d in a flow of unperturbed velocity U∞. At this critical Reynolds number the flow bifurcates from a stationary state to an oscillatory state characterized physically by the appearance of a K´arm´an vortex street (Williamson, 1996; Huerre and Rossi, 1998, §6). The oscillation frequency f corresponds to a well-defined value of the Strouhal number f d/U∞, which is quite insensitive to perturbations of the flow (St = 0.19 for the cylinder at moderate Reynolds numbers). In terms of dynamical systems, the appearance of this oscillation corresponds to a Hopf bifurcation.
3.3.4 The Gaster relation
The Gaster relation (1962) is an expression allowing the temporal growth rate to be related to the spatial growth rate in the vicinity of marginal stability. Let us consider a marginal normal mode(kc,ωc) such that ωci=0 and kci=0. Expanding the dispersion relationω(k) in a Taylor series in the neighborhood of the marginal stability, to first order we have
ω −ωc=∂ω
∂k(kc, R)(k −kc), from which, taking the imaginary part, we find
ωi=∂ωi
∂kr(kc, R)(kr−kc)+∂ωr
∂kr(kc, R)ki. (3.34) For a temporal mode, ki is zero and the above relation gives the temporal growth rateω(T)i :
ωi(T)=∂ωi
∂kr(kc, R)(kr−kc).
For a spatial mode,ωiis zero and (3.34) gives the spatial growth rate−ki(S): 0=∂ωi
∂kr(kc, R)(kr−kc)+∂ωr
∂kr(kc, R)ki(S).
Subtracting these two equations from each other and using cg= ∂ωr/∂kr, we find ωi(T)= −cgk(S)i . (3.35)
1000 1400 1800 Reδ1
2k i × 10
Spatial analysis Temporal analysis
2200 2600
–1.2 –0.8 –0.4 0.0 0.4 0.8
Figure 3.6 Spatial growth rate (m−1) of the eigenmode with frequencyων/U∞2 = 0.4 × 10−4 in a boundary layer as a function of the Reynolds number Reδ1 obtained by (—) spatial analysis and (− −) temporal analysis and the Gaster relation(3.35) (calculation by G. Casalis, ONERA).
The above Gaster relation shows that in the limit of weak departures from marginal conditions, the temporal growth rateω(T)i and the spatial growth rate ki(S) are related by the group velocity. In other words, for a spatially advected wave packet which is observed in the reference frame of the group velocity, it is the temporal growth rate which is measured.
The usefulness of the Gaster relation is illustrated in Figure3.6,which displays the spatial growth rate of the instability of the Blasius boundary layer on a flat plate, as a function of the Reynolds number Reδ1 (cf. Chapter 5). Two different calculations are compared: the spatial growth rate obtained from the spatial stabil-ity analysis (real frequency), and that inferred from the temporal stabilstabil-ity analysis (real wave number) and the Gaster relation(3.35). We observe that the curves coin-cide exactly for ki= 0, corresponding to marginal stability, and remain very close to each other even far from marginal stability.
3.4 Exercises
3.4.1 Dispersion of a wave packet Consider a wave packet u(x,t) with Gaussian spectrum
ˆu(k) = ˆu0e−σ2(k−k0)2, k> 0.
1. Show that u(x,t) is given at long times (t 1 with = ω0/2σ2) by
u(x,t) ∼ ˆu0√ π 2σ
√1
t e−
(x−cgt)2
(2σt)2 cos [k0x−ω0t+φ(x,t)], where the phaseφ(x,t) is defined as
φ(x,t) = −θ
2+(x −cgt)2
4σ2t , tanθ = t .
2. Show that the local wave number (the gradient of the phase) varies spatially as
∂xφ(x,t) =x−cgt ω0t . Interpret this variation.
3. For gravity waves in deep water and δk = 0.1k0, at what distance from the amplitude maximum are the wave numbers k0±δk found?
3.4.2 Spatial branches of a convective instability
Determine the spatial growth rate (3.33)of perturbations of the solution u(x,t) = 0 of the Ginzburg–Landau equation (3.14) when the instability is convective (R< V2/4).