6. INSTABILITY AND TRANSITION SPRING 2009 6.1 Introduction 6.2 The Orr-Sommerfeld equation 6.3 Inviscid instability – Rayleigh’s equation 6.4 Viscous instability 6.5 Instability of particular flows 6.6 From instability to transition 6.7 Effect of free-stream disturbances 6.8 Engineering prediction of transition 6.9 Transition control 6.10 References Examples 6.1 Introduction “Laminar flows have a fatal weakness : poor resistance to high Reynolds numbers ” (White, 1991). The process of change from a smooth laminar flow to a chaotic turbulent one is termed transition . The amplification (or otherwise) of small disturbances may be addressed by linearised instability theory . Note, however, that instability does not lead inexorably to transition and the two may be separated by a significant streamwise distance. In certain circumstances (e.g. a very favourable pressure gradient) a boundary layer may actually relaminarise (“ reverse transition ”). 6.2 The Orr-Sommerfeld Equation This is the fundamental differential equation governing stability, derived independently by Orr (1907) and Sommerfeld (1908). You will meet this equation again in the optional module on Hydrodynamic Stability Theory. Consider a small 2-d perturbation to a plane-parallel base flow U(y). Total velocity: U + u = (U + u,v )0, Pressure: P + p (1) The total momentum and continuity equations are: ∂ 1 momentum : (U + u) + (U + u) • ∇(U + u) = − ∇(P + p) + ∇ 2 (U + u) ∂t continuity : ∇ • (U + u) = 0 (2) Subtract off the base flow (which is assumed to satisfy the equations of motion) and linearise: ∂u 1 + U• ∇u + u • ∇U = − ∇p + ∇ 2u , ∇ •u = 0 (3) ∂t If U = (U(y),0,0) the momentum perturbation equation reduces to: Turbulent Boundary Layers 6 - 1 David Apsley ∂u ∂u dU 1 +U + v = − ∇p + ∇ 2u (4) ∂t ∂x dy In addition u satisfies the non-slip condition at walls and is assumed to vanish at large distances. Component-wise this gives ∂u ∂u dU 1 ∂p x-momentum: +U + v = − + ∇ 2u (5) ∂t ∂x dy ∂x ∂v ∂v 1 ∂p y-momentum: +U = − + ∇ 2v (6) ∂t ∂x ∂y ∂ ∂ Take (5) – (6) to eliminate pressure (this is equivalent to taking the curl and producing ∂y ∂x an equation for the z component of vorticity): ∂ ∂u ∂v ∂ ∂u ∂v dU ∂u ∂v dU d 2U ∂u ∂v ( − ) +U ( − ) + + + v = ∇ 2 ( − ) ∂t ∂y ∂x ∂x ∂y ∂x dy ∂x ∂y dy dy 2 ∂y ∂x But ∂u ∂v + = 0 (incompressible) ∂x ∂y and ∂ ∂ ∂u ∂v u = , v = − ( a streamfunction) ⇒ − = ∇ 2 ∂y ∂x ∂y ∂x Hence, ∂ ∂ d 2U ∂ +U ∇ 2 − = ∇ 2 (∇ 2 ) (7) ∂t ∂x dy 2 ∂x Since the problem is linear, we can investigate the time evolution of individual harmonic components: = f (y)ei(kx − t) (8) Then, in operator terms, ∂ ∂ → ik , → −i ∂x ∂t and hence (7) becomes d 2 d 2U d 2 d 2 (−i + ikU )( −k 2 + ) f − ikf = (−k 2 + )( −k 2 + ) f dy 2 dy 2 dy 2 dy 2 or, dividing by ik and writing c = (9) k for the phase speed , one has the Orr-Sommerfeld Equation d 2 f d 2U i d 4 f d 2 f (U − c)( − k 2 f ) − f = − ( − 2k 2 + k 4 f ) (10) dy 2 dy 2 k dy 4 dy 2 Turbulent Boundary Layers 6 - 2 David Apsley This is more often written in non-dimensional form: d 2 f d 2U i d 4 f d 2 f (U − c)( − k 2 f ) − f = − ( − 2k 2 + k 4 f ) (11) dy 2 dy 2 k Re dy 4 dy 2 where lengths are non-dimensionalised with respect to a characteristic cross-stream length, e.g. , and velocities with respect to, e.g., Ue. Notes . (1) Boundary conditions . Since u and v vanish at walls and the disturbance is assumed to be initially localised, df f = = 0 at walls and at ± ∞ dy (2) Since both equation and boundary conditions are homogeneous this constitutes an eigenvalue problem . We may consider either: temporal instability : k real; = r + i i instability ⇔ i > 0 spatial instability : real; k = kr + ik i instability ⇔ ki < 0 In both cases, instability ⇔ ci > 0, where c = /k is the phase speed. (3) It is found that, for example in temporal instability, distinct regions of k-Re space give unstable eigenvalues ; these are delimited by neutral curves (see later) with i = 0 and there is a minimum value of Re necessary for instability. (4) Squire ’s Theorem (Squire, 1933): for a two-dimensional parallel base flow U(y), the minimum unstable Reynolds number occurs for a two-dimensional disturbance propagating in the same direction. Thus, in so far as we are searching for the minimum Reynolds number at which instability occurs, it is appropriate to use the 2-d analysis. (Note, however, it can be shown that the maximum rate of amplification of disturbances actually occurs for waves propagating at an angle to the base flow.) (5) The terms on the LHS of (10) come from the inertia terms; those on the RHS come from the viscous terms. If the viscous terms are neglected it becomes the Rayleigh equation d 2 f d 2U (U − c)( − k 2 f ) − f = 0 (12) dy 2 dy 2 (6) The linearised 2-d sinusoidal disturbances which satisfy the full viscous Orr-Sommerfeld equation are called Tollmien-Schlichting waves . They are the first (infinitessimal) indications of laminar-flow instability. Turbulent Boundary Layers 6 - 3 David Apsley 6.3 Inviscid Instability – Rayleigh’s Equation If the viscosity = 0 the linearised perturbation equation becomes: Rayleigh’s equation 2 ′′ d f 2 U − k + f = 0 (13) dy 2 U − c 6.3.1 Point-of-Inflexion Criterion For Instability U(y) A point where d 2U/d y 2 = 0 is called a point of inflexion and is crucial in determining the stability of the base flow. The curvature of the velocity profile therefore has a fundamental effect on stability. P.I. Rayleigh (1880) A necessar y [but not sufficient] condition for instability is that the velocity profile has a point of inflexion . Fjørtoft (1950) If a point of inflexion (PI) exists it is further necessary that: dU (1) has a local maximum at the PI; dy d 2U (2) (U −U ) < 0 somewhere on the profile. dy 2 PI y U(y) y y y P.I. P.I. (a) stable (b) stable (c) stable (d) possibly unstable Thus, for inviscid flow , of the velocity profiles sketched above, (a) and (b) are stable (no point of inflexion); (c) is stable (dU/d y has a minimum at the P.I.); (d) may be unstable. The occurrence of a point of inflexion in a boundary layer is inextricably linked to the pressure gradient. This is readily seen from the momentum equation at the boundary: ∂ 2U dP = e (14) ∂ 2 dx y w 2 2 2 2 If d Pe/d x > 0 then ∂ U/∂y > 0 at the wall. But ∂ U/∂y < 0 in the outer part of the boundary layer and hence there must be an intermediate point where ∂2U/∂y2 = 0. For this reason, adverse-pressure-gradient boundary layers are more prone to instability. Turbulent Boundary Layers 6 - 4 David Apsley 6.3.2 Sketch Proof of Rayleigh’s Point-of-Inflexion Criterion. Consider a single harmonic mode: ik (x−ct ) = f (y)e f is complex. There is amplification if ci ≡ Im( c) > 0. Write Rayleigh's equation and its complex conjugate (denoted by *) d 2 f U ′′ −[k 2 + ] f = 0 dy 2 U − c d 2 f * U ′′ −[k 2 + ] f * = 0 dy 2 U − c* Form the (pure imaginary) quantity df * df P = f − f * dy dy Then dP d 2 f * d 2 f U ′′ U ′′ = f − f * = f [k 2 + ] f * − f *[k 2 + ] f dy dy 2 dy 2 U − c* U − c 1 1 2 = U ′′( − ) f U − c* U − c − 2ic U ′′ f 2 = i U − c 2 But P = 0 at boundaries (either wall or at ± ∞). Hence, if ci 0 then P is not identically zero. But since P = 0 at boundaries then there must be a point with d P/d y = 0 (and P 0). At this point one must have U ′′ = 0 ; i.e. a point of inflexion. 6.3.3 Interpretation in Terms of Vorticity. Suppose the converse. WLOG assume U ′′ < 0 everywhere. y y Ω =dU U dy B' B' B B B' B' A' A B C B C' Turbulent Boundary Layers 6 - 5 David Apsley Any upward motion B to B' carries its vorticity with it and therefore intensifies vorticity there. This extra vorticity at B' induces: • a downward velocity taking C to C'; • an upward velocity taking A to A'. These carry their own vorticity with them; in both cases the resultant vorticity induces a velocity tending to bring B' back to B. Thus, if U ′′ < 0 everywhere (and similarly for U ′′ > 0 ) there is always a restoring force, and hence the situation is stable. 6.3.4 Critical Layers Rayleigh (1880) proved that the phase velocity cr of any amplified disturbance must lie between the minimum and maximum values of U(y). Hence, where instability is possible, there must exist a critical layer within the flow where U – c = 0 for neutral disturbances (ci = 0). The critical layer yc is a singular point of Rayleigh’s inviscid stability equation unless, simultaneously, U ′′ = 0 there. Otherwise the perturbation velocity u tends to infinity like U ′′(y ) u = c ln( y − y ) ′ c U (yc ) In this region of large velocity gradients viscous effects must become important.