Hydrodynamic Stability: Absolute and Convective Instability 

Welcome back. In our hydrodynamic stability series so far, we have explored methods for determining whether a flow is linearly stable or unstable. Today, we will augment these discussions by investigating two different types of instability: absolute and convective. If a flow is absolutely unstable, then a small perturbation at a particular location will grow both upstream and downstream as time passes. In contrast, a disturbance introduced in a convectively unstable flow will grow downstream of its initial location, but the flow upstream of the disturbance will remain unperturbed. Making the distinction between absolute and convective instability is important because it has implications on the degree to which a flow can be controlled, as well as how turbulence can sustain itself (more on this later). Therefore, we will spend the rest of this blog setting up the mathematical framework to discuss absolute and convective instabilities. We will then conclude with a brief summary of typical flows that are absolutely or convectively unstable. 

To begin, let’s suppose we have linearized the Navier-Stokes equations around some base flown and wish to explore the dynamics of a small perturbation 𝜙(x,t), that varies in the streamwise direction x and time t. In this general formulation, 𝜙 could be a velocity, vorticity, stream function, or velocity potential disturbance. Also, 𝜙 might vary in cross-stream directions (y or z), but this dependence on other spatial variables is suppressed for notational convenience. The linearized Navier-Stokes equations can be rearranged to form the following evolution equation for the disturbance:

D(i∂_t, -i∂_x; R) 𝜙(x,t) = s(x,t), (1)

where D is a spatio-temporal linear differential operator, R is the Reynolds number, and s(x,t) is a source term (or forcing). If Eq. (1) is Fourier transformed in space and time, then the operator D becomes an algebraic expression in terms of a complex streamwise wavenumber k and complex temporal frequency 𝜔. The equation 

D(𝜔, k; R) = 0 (2)

turns out to be very useful, and is referred to as the dispersion relation. In this derivation, the forward and reverse Fourier transforms in x are defined as

𝛷(k, t) = F_x[𝜙(x,t)] = ∫_-∞^∞ 𝜙(x,t) e^-ikx dx

𝜙(x,t) = F_x^-1[𝛷(k, t)] = 1/(2𝜋) ∫_F 𝛷(k, t) e^ikx dk,

where F is a contour in the complex k plane. Meanwhile, the forward and reverse Fourier transforms in t are

𝜙ˆ(x, 𝜔) = F_t[𝜙(x,t)] = ∫_-∞^∞ 𝜙(x,t) e^i𝜔t dt

𝜙(x,t) = F_t^-1[𝜙ˆ(x, 𝜔)] = 1/(2𝜋) ∫_L 𝜙ˆ(x, 𝜔) e^-i𝜔t d𝜔,

where L is a contour in the complex 𝜔 plane. After Fourier transforming Eq. (1) in space and time, the disturbance 𝜙(x,t) can be written in terms of Fourier integrals:

𝜙(x,t) = 1/(2𝜋)² ∫_F dk ∫_L d𝜔 S(k, 𝜔) / D(𝜔, k; R) e^i(kx-𝜔t). (3)

For positive times t > 0, this integral depends on residues from the dispersion relation (Eq. 2). Furthermore, if we assume that there is only one temporal mode (with 𝜔 depending on k), then D can be expanded into a power series (where we recall that D = 0):

D(𝜔, k; R) = ∂D/∂𝜔|_𝜔(k) [𝜔 – 𝜔(k)] + … .

After the residue theorem is applied to the 𝜔 integral in Eq. (3), the formula for the disturbance becomes 

𝜙(x,t) = 1/(2𝜋)² ∫_F dk {-2𝜋i S(k, 𝜔(k)) / [∂D/∂𝜔|_{(𝜔(k), k; R)}] e^{i[kx-𝜔(k)t]}}

= -i/(2𝜋) ∫_F dk S(k, 𝜔(k)) / [∂D/∂𝜔|_{(𝜔(k), k; R)}] e^{i[kx-𝜔(k)t]}. 

Now, we can evaluate 𝜙(x,t) for large times t → ∞ by the method of steepest descents. To do this, we note that the integral for 𝜙(x,t) can be expressed as 

𝜙(x,t) = ∫_F f(k) e^{t h(k, x/t)} dk,

where 

f(k) = S(k, 𝜔(k)) / [∂D/∂𝜔|_{(𝜔(k), k; R)}],

and 

 h(k, x/t) = i [k x/t – 𝜔(k)] 

is the phase function. To obtain 𝜙(x,t) asymptotically as t → ∞, we can deform the path F in the complex k plane along a path of steepest descent through the saddle point of h(k, x/t). We note that a saddle point k = k^* occurs when 

dh/dk(k^*, x/t) = 0, 

which is to say that 

dh/dk = i (x/t – d𝜔/dk) = 0,

where d𝜔/dk is the complex group velocity. Therefore, at a saddle point, 

d𝜔/dk(k^*) = x/t. 

The goal now is to choose the complex value of k = k^* such that the group velocity d𝜔/dk is real and equal to x/t, which is a ray in physical space. Near the saddle point, k = k^*, h(k, x/t) can be expanded in terms of a power series in k:

h(k) = h(k^*) + ½ d²h/dk²|_k* (k – k^*)² + … .

Now, we can decompose h into its real part and imaginary part: 

h(k) = h_r(k) + i h_i(k). 

We can also let k – k^* = r e^i𝜃 and d²h/dk²|_k* = 𝜌 e^i𝛼 (this is expressing these complex numbers in polar form). Then, 

h(k) – h(k^*) ≈ 𝜌r²/2 e^{i(𝛼 + 2𝜃)} = 𝜌r²/2 {cos(𝛼 + 2𝜃) + i sin(𝛼 + 2𝜃)}. 

Note that lines of constant phase passing through the saddle point k^* have:

Im[h(k) – h(k^*)] = 0.

These correspond to the lines sin(𝛼 + 2𝜃) = 0. The lines with Re[h(k)-h(k^*)] = 𝜌r²/2 are directions of steepest ascent, while lines with Re[h(k)-h(k^*)] = -𝜌r²/2 are directions of steepest descent. By identifying these directions, we can deform the path F such that the integrand does not oscillate rapidly (Im (h-h^*) = 0) and the real part Re(h-h^*) decays rapidly away from the saddle point. In this fashion, it is possible to obtain the dominant contribution to the integral from the immediate neighborhood of the saddle point. After deforming the path, we can approximate the disturbance as 

𝜙(x,t) ~ f(k^*) e^{h(k*) t} ∫_{D+ – D-} exp[½ d²h/dk²|_k* (k – k^*)² t] dk, 

where D₊ and D_– are segments in the complex k plane aligned with the “valley” near the saddle point k^* (please see a textbook for more details). After a change of variables, the integral can be evaluated to yield 

𝜙(x,t) ~ – e^i𝜋/4/sqrt(2𝜋) S(k^*) e^{i [k* x – 𝜔(k*) t]} / {∂D/∂𝜔|_k* [d²𝜔/dk²|_k* t]^½}, (4)

where d𝜔/dk|_k* = x/t. Along a particular ray selected by the real group velocity d𝜔/dk|_k* = x/t, the disturbance 𝜙(x,t) behaves as 𝜙(x,t) ~ e^𝜎t, where the growth rate is 

𝜎 = 𝜔_i(k^*) – k_i^* d𝜔/dk|_k*. (5)

The sign of 𝜎 in Eq. (5) and the asymptotic behavior of 𝜙 determines the stability of the flow.

1. The flow is unstable if there exist any rays along which 𝜎 > 0. 
2. The flow is absolutely unstable if 𝜙(x,t) → ∞ as t → ∞ for all fixed x.
3. The flow is convectively unstable if 𝜙(x,t) → 0 as t → ∞ for all fixed x with rays along which 𝜎 > 0.

Now let’s consider the wavenumber, k₀, for which the group velocity vanishes. This ray will allow us to distinguish between absolute and convective instability. By construction, we know that d𝜔/dk(k₀) = 0 and we can set 𝜔₀ = 𝜔(k₀). 

This absolute frequency 𝜔₀ is an algebraic branch point of k(𝜔), where, in the vicinity of 𝜔₀, 

𝜔 ≈ 𝜔₀ + d²𝜔/dk²|_k0 (k – k₀)².

It follows that 

(k – k₀) ≈ ± [2 / d²𝜔/dk²|_k0]^½ (𝜔 – 𝜔₀)^½. (6)

This equation relates the absolute wavenumber k₀ with the absolute frequency 𝜔₀.

If 𝜔₀ is in the lower half 𝜔 plane, then the flow is convectively unstable. If 𝜔₀ is in the upper half 𝜔 plane, then the flow is absolutely unstable. Therefore, if we want to determine whether a flow is absolutely or convectively unstable, we can find roots of the function k(𝜔) in the 𝜔 plane. Then, the position of the branch point 𝜔₀ determines the type of instability. 

To conclude, we can mention that typical absolutely unstable flows include Rayleigh-Benard convection, the temporal mixing layer, near wakes, and Taylor-Couette flow. Typical convectively unstable flows include spatially developing mixing layers, far wakes, and boundary layers. Furthermore, absolutely unstable flows are insensitive to weak forcing, intrinsic flow dynamics are dominant, and temporal instability theory is relevant. Convectively unstable flows are sensitive to weak forcing, extrinsic flow dynamics are dominant, and spatial instability theory is relevant. 

Next week, we will explore more fluid dynamics topics. Until then, please take care. 

Acknowledgements 

This blog has adapted lecture notes prepared for the graduate hydrodynamic stability course, AME 621, at USC. The author is deeply indebted to this material and the faculty at USC.

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