1. Introduction • The Navier-Stokes equations present a challenging problem, but there are some exact solutions. • We consider special cases: simple geometry, simple flows. • Sometimes the simple flow solution is observed in experiments. • But sometimes other, possibly unsteady/turbulent, flows are observed instead:
• The concept of flow stability was introduced to explain these scenarios. • Add a small disturbance to a flow, determine whether the disturbance grows (⇒ unstable flow) or does not (⇒ stable flow). The basic idea of stability theory
O The frictionless simple pendulum: a mass m attached to a light rod of θ length l that pivots freely about O. θ¨ Newton’s laws tell us l θ¨ + ω2 sin θ = 0 (1)
mg where ω2 = g/l.
We assume there is a steady solution θ0, and we want to find its stability. Let θ = θ0 + θ1(t) (2) where 1. Linearization
Substitute (2) into (1):
2 0 = θ¨1 + ω sin(θ0 + θ1) 2 2 = θ¨1 + ω sin θ0 cos(θ1) + ω cos θ0 sin(θ1). (3)
Substitute the Taylor expansions
(θ )2 (θ )3 cos(θ ) ∼ 1 − 1 + ..., sin(θ ) ∼ θ − 1 + ... 1 2 1 1 6 into (3) and equate coefficients of O(1) and O():
sin θ0 = 0 (4) 2 θ¨1 + (ω cos θ0)θ1 = 0 (5) Stability of steady pendulum solutions
• The nonlinear equation for steady states (4) has solutions θ0 = 0 and θ0 = π. • The disturbance equation (5) is linear and its coefficients depend on the steady solution θ0.
• Substitute θ0 = 0 into (5):
2 θ¨1 + ω θ1 = 0 ⇒ θ1 = A cos ωt + B sin ωt, (6)
θ1 remains bounded as t → ∞, therefore θ0 = 0 is stable.
• Substitute θ0 = π into (5):
2 ωt −ωt θ¨1 − ω θ1 = 0 ⇒ θ1 = A e + B e , (7)
θ1 → ∞ as t → ∞, therefore θ0 = π is unstable. Hydrodynamic stability theory...... is the application of this approach to the governing equations for fluid motion. Applying Newton’s laws of motion to fluid elements leads to the Navier–Stokes equations:
∇ · u = 0 (8) ∂u 1 + (u · ∇)u = −∇p + ∇2u (9) ∂t Re (incompressible, isothermal, no free boundaries).
The Reynolds number, Re, is based on convenient velocity and length scales, U0, L, and the kinematic viscosity, ν, U L Re = 0 . (10) ν Basic flow plus disturbance
Substitute
u = u0(x, y, z) + u1(x, y, z, t) (11)
p = p0(x, y, z) + p1(x, y, z, t) (12) into (8) and (9), then equating terms of O(1) gives the nonlinear equations for the steady solution:
∇ · u0 = 0 (13) 1 (u · ∇)u = −∇p + ∇2u . (14) 0 0 0 Re 0 Linearized Navier–Stokes equations
Equating terms of O() gives
∇ · u1 = 0 (15) ∂u 1 1 + (u · ∇)u + (u · ∇)u = −∇p + ∇2u (16) ∂t 1 0 0 1 1 Re 1 to be solved for u1 after u0 has been found from (13), (14). • Although (15), (16) are linear, their solution, in general, would require a numerical approach. • Nonetheless, analytic progress can be made in certain special cases. • We hope to understand more of the general problem by studying special cases in detail. Summary of basic steps of linear theory
• Find steady solutions. • Expand solution as a sum of a steady flow and small unsteady disturbance. • Substitute this sum into the full nonlinear equations and linearize in disturbance quantities. • Solve the resulting linear disturbance equation. • If all disturbances decay, or remain bounded, then the steady flow is stable. • If any disturbance grows then the steady flow is unstable. Beyond basic linear theory
• When linear theory predicts stability: • Sufficiently small disturbances decay, or remain bounded. • But how small do they have to be to decay? • Famous examples of stable flows become turbulent, like pipe flow. • Disturbances can still grow transiently. • When linear theory predicts instability: • Sufficiently small disturbances grow exponentially. • As they grow, they eventually become large enough to invalidate the linear theory. Then what happens? • Or the flow could remain laminar in the region of interest if disturbances are swept downstream quickly enough. • Weakly nonlinear theories predict the next stages of disturbance evolution, if growth rates are weak. • Absolute/convective instability theory addresses the issue of downstream propagation. Shear layers
There are many flows that have regions of parallel, or nearly parallel, streamlines where significant shear takes place: • Channel flows, e.g.
Plane Poiseuille flow Plane Couette flow Poiseuille-Couette • Boundary layers, e.g. U00 = 0
∂P/∂x < 0 ∂P/∂x = 0 ∂P/∂x > 0 Tw > Tf • Unbounded shear layers, e.g.
jets wakes mixing layers • Of these, the channel flows are exactly parallel, the others become ‘more parallel’ as the Reynolds number increases (but needs WKB theory, see later). • For a parallel flow, or under the parallel flow approximation, the basic flow is described by a single velocity profile:
u0 = U(y)i. (17)
• There are also axisymmetric versions of some of these flows. Derivation of disturbance equations
We shall consider two-dimensional disturbances, and work in cartesian coordinates: ∂u˜ ∂v˜ + = 0 (18) ∂x ∂y ∂u˜ ∂u˜ ∂u˜ ∂p˜ 1 ∂2u˜ ∂2u˜ +u ˜ +v ˜ = − + + (19) ∂t ∂x ∂y ∂x Re ∂x2 ∂y 2 ∂v˜ ∂v˜ ∂v˜ ∂p˜ 1 ∂2v˜ ∂2v˜ +u ˜ +v ˜ = − + + (20) ∂t ∂x ∂y ∂y Re ∂x2 ∂y 2 Linearized disturbance equations The stability of velocity profile U(y) is found by substituting
u˜ = U(y) + 0uˆ(x, y, t) (21)
v˜ = 0vˆ(x, y, t) (22)
p˜ = P(x) + 0pˆ(x, y, t), (23) where 0 1, into (18) – (20), then equating coefficients of O(0): ∂uˆ ∂vˆ + = 0 (24) ∂x ∂y ∂uˆ ∂uˆ ∂pˆ 1 ∂2uˆ ∂2uˆ + U + U0vˆ = − + + (25) ∂t ∂x ∂x Re ∂x2 ∂y 2 ∂vˆ ∂vˆ ∂pˆ 1 ∂2vˆ ∂2vˆ + U = − + + (26) ∂t ∂x ∂y Re ∂x2 ∂y 2 Normal modes
• Our solution of (24) – (26) relies crucially on the streamlines being parallel, i.e. U is independent of x. • This makes the solution separable:u ˆ = X (x)u(y)T (t) etc. • Attempting a solution of this type, one deduces that X and T are exponential functions of x and t respectively. • We therefore adopt the normal mode form
(ˆu, vˆ, pˆ) = (u(y), v(y), p(y))ei(αx−ωt) (27)
i.e. travelling waves with wavenumber α, frequency ω. • The linearity of (24) – (26) allows a general solution to be constructed from a superposition of (27) with different α and ω (e.g. by taking Fourier transforms). Orr-Sommerfeld equation
• Substituting (27) into (24) – (26) gives the ODEs
iαu + v 0 = 0 (28) 1 − iωu + iαUu + U0v = −iαp + u00 − α2u (29) Re 1 − iωv + iαUv = −p0 + v 00 − α2v , (30) Re where 0 = d/dy. • Eliminating u and p gives the Orr-Sommerfeld equation: 1 (U−c)(v 00−α2v)−U00v = v 0000 − 2α2v 00 + α4v , (31) iαRe where c = ω/α. • This reduction to an ODE is only possible for a parallel flow, otherwise PDEs of the form (24) – (26) must be solved. Boundary conditions • No flow through a solid boundary: v = 0. • No slip at a solid boundary: u = 0 ⇒ v 0 = 0 from (28). • If the flow is unbounded as y → ∞ then v → 0 and v 0 → 0 as y → ∞. • The four boundary conditions may be summarised as
0 0 v(y1) = 0, v (y1) = 0, v(y2) = 0, v (y2) = 0, (32)
where y1 and/or y2 could be finite or infinite depending on whether we are considering channel flow, a boundary layer or an unbounded shear layer. • Nontrivial solutions satisfying these homogeneous boundary conditions are only possible for certain α and ω. • These values satisfy a relation of the form ∆(α, ω) = 0 called the dispersion relation. • Roots of ∆(α, ω) = 0 are called eigenvalues. Instability criteria • Suppose that a Fourier mode with a real α has been chosen giving a complex eigenvalue, ω = ωr + iωi :
ei(αx−ωt) = ei(αx−ωr t−iωi t) = eωi t ei(αx−ωr t). (33)
• Therefore, if for any real α
ωi > 0 ⇒ exponential growth in time ⇒ instability. (34)
• If for all real α
ωi < 0 ⇒ exponential decay in time ⇒ stability. (35)
• Obtaining growth/decay in time is called temporal stability theory. • Spatial stability theory (real ω, complex α) and spatio-temporal stability theory (complex ω, complex α), i.e. convective/absolute instabilities will be discussed later. Rayleigh equation • In the inviscid limit the Orr-Sommerfeld equation (31) reduces to the Rayleigh equation
(U − c)(v 00 − α2v) − U00v = 0. (36)
• The non-slip boundary conditions are dropped for inviscid flow, leaving v(y1) = 0, v(y2) = 0 (37) (the Rayleigh equation is only 2nd order, while the Orr-Sommerfeld equation is 4th order). • Therefore, when Re 1, there will be thin viscous layers near solid boundaries where the non-slip condition is retained. • Sometimes these viscous layers are important to the stability properties (see later). • But we start by ignoring them, and just solve the inviscid problem...