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¨ + !2 sin θ = 0 (1) mg where !2 = 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 + ! sin(θ0 + θ1) 2 2 = θ¨1 + ! sin θ0 cos(θ1) + ! 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 + (! 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 + ! θ1 = 0 ) θ1 = A cos !t + B sin !t; (6) θ1 remains bounded as t ! 1, therefore θ0 = 0 is stable. • Substitute θ0 = π into (5): 2 !t −!t θ¨1 − ! θ1 = 0 ) θ1 = A e + B e ; (7) θ1 ! 1 as t ! 1, 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: r · u = 0 (8) @u 1 + (u · r)u = −∇p + r2u (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: r · u0 = 0 (13) 1 (u · r)u = −∇p + r2u : (14) 0 0 0 Re 0 Linearized Navier{Stokes equations Equating terms of O() gives r · u1 = 0 (15) @u 1 1 + (u · r)u + (u · r)u = −∇p + r2u (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) + 0u^(x; y; t) (21) v~ = 0v^(x; y; t) (22) p~ = P(x) + 0p^(x; y; t); (23) where 0 1, into (18) { (20), then equating coefficients of O(0): @u^ @v^ + = 0 (24) @x @y @u^ @u^ @p^ 1 @2u^ @2u^ + U + U0v^ = − + + (25) @t @x @x Re @x2 @y 2 @v^ @v^ @p^ 1 @2v^ @2v^ + 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 ^ = 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 (^u; v^; p^) = (u(y); v(y); p(y))ei(αx−!t) (27) i.e. travelling waves with wavenumber α, frequency !. • The linearity of (24) { (26) allows a general solution to be constructed from a superposition of (27) with different α and ! (e.g. by taking Fourier transforms). Orr-Sommerfeld equation • Substituting (27) into (24) { (26) gives the ODEs iαu + v 0 = 0 (28) 1 − i!u + iαUu + U0v = −iαp + u00 − α2u (29) Re 1 − i!v + 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 = !/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 ! 1 then v ! 0 and v 0 ! 0 as y ! 1. • 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 !. • These values satisfy a relation of the form ∆(α; !) = 0 called the dispersion relation. • Roots of ∆(α; !) = 0 are called eigenvalues. Instability criteria • Suppose that a Fourier mode with a real α has been chosen giving a complex eigenvalue, ! = !r + i!i : ei(αx−!t) = ei(αx−!r t−i!i t) = e!i t ei(αx−!r t): (33) • Therefore, if for any real α !i > 0 ) exponential growth in time ) instability: (34) • If for all real α !i < 0 ) exponential decay in time ) stability: (35) • Obtaining growth/decay in time is called temporal stability theory. • Spatial stability theory (real !, complex α) and spatio-temporal stability theory (complex !, 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....