Turbulent Boundary Layers 6

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: = f (y)eik (x−ct ) 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.

6.4 Viscous Instability

The zero-pressure-gradient flat-plate boundary layer, channel flow and pipe flow have no internal point of inflexion (actually, the boundary-layer velocity profile must have a P.I. on the boundary itself – see equation (14)) and are therefore not subject to inviscid instability. Nevertheless, common experience indicates that all become unstable at some critical Reynolds number and, therefore, viscosity must be responsible for the instability.

O. Tietjens (1922) demonstrated that if the effects of viscosity were confined to a small region near the wall then instability occurred for all Reynolds numbers! Tollmien (1929) resolved this paradox by demonstrating that the effect of viscosity must be included right across the shear layer and, in particular, at the critical layer ( U = c). In this pre-computer era, Tollmien made an amazing (but slightly inaccurate) calculation of the minimum Reynolds number for instability in the flat plate boundary layer (without the effects of non-parallel base flow – see below).

Even with sufficiently powerful modern computers, numerical solution of the Orr-Somerfeld equation is complicated by the form of the equation in which the highest-order derivative (d 4f/ dy4) is multiplied by a tiny factor (1/Re). In the free stream ( y/ → ∞), where U ′′ = 0 , the Orr-Somerfeld equation (in non-dimensional units) becomes d 2 f d 4 f d 2 f ik Re( U − c)( − k 2 f ) = − 2k 2 + k 4 f dy 2 dy 4 dy 2

Turbulent Boundary Layers 6 - 6 David Apsley which has (spatially-decaying) exponential solutions of the form f = Ae −ky + Be − y where 2 = k 2 + ik Re( U − c) The first exponential term is also a solution of the inviscid equation and so is called the inviscid solution . That with exponent involving γ is the viscous solution .

Since Re is large, the arguments of the exponentials are very different and, numerically, the faster-varying part tends to obscure the other. This numerical challenge was finally met by the “purification” scheme of Kaplan which separated slow and fast-varying solutions.

Numerical calculations can now be undertaken routinely and produce curves of neutral stability ( ci = 0) in the form of “thumb” curves of two types as shown. Small disturbances with wavenumbers inside the “thumbs” are unstable.

Type A Type B y y kδ P.I. c i=0 U U

Type A stable unstable

Type B inviscid instability

Re δ Re ins Re ins (Type A) (Type B)

Type A : profile has a point of inflexion; subject to inviscid instability ; a region of instability persists to infinite Re.

Type B : no point of inflexion; subject to viscous instability only over a finite range of Reynolds numbers.

In both cases there is a minimum Reynolds number (Re ins – Schlichting refers to this as the indifference Reynolds number) below which instability does not arise. This tends to be smaller (more prone to instability) for profiles with inflexion points.

Note that since the Reynolds number of the boundary layer grows with downstream distance, a disturbance of a particular wavenumber may pass through a region of amplification and then out of it again – instability does not lead inexorably to transition.

Turbulent Boundary Layers 6 - 7 David Apsley 6.5 Instability of Particular Flows

6.5.1 Flat-Plate Boundary Layer

The base flow is the Blasius solution. Neglecting non-parallel effects (i.e. the streamwise growth of the boundary layer) the critical Reynolds number (based on displacement thickness) for instability is

Re = 520 corresponding to Re = 91000 *,ins x,ins (If non-parallel effects are taken into account the first of these is reduced to 400).

* * The maximum wavenumber for instability is k 0.35, corresponding to min / 18 or min / 0.99 6. Thus, unstable Tollmien-Schlichting waves are long compared to boundary- layer thickness.

6 For a low-disturbance free stream the point of final transition is Re x,tr 3×10 , or more than an order of magnitude further downstream.

6.5.2 Channel Flow

Numerical calculations give Re D,ins for instability as 5767, whereas experiments indicate transition at Re D 1000. Linear instability is clearly not the instigator of transition and the explanation has variously been assigned to non-linear instability and development region effects (see below).

6.5.3 Pipe Flow

This is even more paradoxical in that numerical computations based on linear instability theory predict no instability at any (so-far-computed ) Reynolds number ! Experimentally, transition is usually observed to occur at about ReD = 2300. The favoured explanation is that large disturbances may arise in the developing entrance flow (the growth of the boundary layer on the pipe walls is like a flat-plate boundary layer, which is known to become unstable) and may then be further amplified by non-linear effects. It is known that experiments conducted with exceeding care to eliminate disturbances in the development 5 region can maintain laminar profiles in a very-smooth-walled pipe up to Re D ~ 10 .

Turbulent Boundary Layers 6 - 8 David Apsley 6.6 From Instability to Transition

The stages in the development of a boundary layer depend somewhat on geometry and are not fully understood, but may be roughly summarised as follows (see White, 1991).

0 1 2 3 4 5 6

(0) A laminar region extending from the trailing edge. (1) Initial instability. (2) Unstable 2-dimensional Tollmien-Schlichting (T-S) waves aligned across, and propagating in the direction of, the free stream. (3) 3-dimensional secondary instability of the T-S waves themselves (“ vortices”) (4) Cascading vortex breakdown (loss of the T-S natural frequency). (5) Formation and growth of wedge-shaped “turbulent spots”. (6) The coalescence of turbulent spots to form a fully-turbulent boundary layer.

Some of these stages may be bypassed, particularly in the presence of large free-stream disturbances, the presence of side walls or large roughness.

On curved surfaces the boundary layer may separate while still laminar, the separated shear layer then becoming turbulent and reattaching as a fully-turbulent boundary layer. This occurs, for example, for smooth cylinders over a range of Reynolds numbers.

The neutral curves may also be established experimentally by the technique pioneered by Schubauer and Skramstad (1947). A fine metallic ribbon is stretched across the flow close to the wall. This is vibrated at known frequencies by passing an alternating current through it in the field of a magnet, just the other side of the wall. The resulting disturbance is then traced further downstream.

6.7 Effect of Free-Stream Disturbances

The transition point for a flat-plate boundary layer is notoriously difficult to predict, varying 5 6 from about Re x,tr = 5×10 for a moderately noisy free stream to 3 ×10 for very low free- stream disturbances. White (1991) cites van Driest and Blumer for the following correlation: − + + 2 2/1 = 1 1 132500 T Re x,tr (15) 39 .2T 2 where the free-stream turbulence intensity is here defined as 1 (u 2 + v 2 + w2 ) T = 3 (16) U

Turbulent Boundary Layers 6 - 9 David Apsley 6.8 Engineering Prediction of Transition

White (1991) gives a number of “engineering” methods for locating transition. Simple but effective methods are due to Michel (1952) and Wazzan et al. (1981). These depend, respectively, on knowing the variation of the laminar boundary-layer momentum thickness and shape factor H. For these latter two quantities it is common to use the correlations of Thwaites (1949) (for some insight into the latter laminar-boundary-layer method see the Examples section).

Thwaites’s correlation for the momentum thickness is x 2 = 45.0 ⌠ 5 ′ ′ 6  U e (x ) dx (17) U e ⌡0 A boundary-layer parameter related to pressure gradient is defined as 2 d( U /d x) = e (18) with shape factor H a function of λ. In particular, at separation, = –0.09 and H = 3.55.

For other values of , White (1991) gives the following “desperation polynomial” fit to Thwaites’s table of data for H( ): H ( ) = 0.2 + 14.4 z − 83 5. z 2 + 854 z 3 − 3337 z 4 + 4576 z 5 , (z = 25.0 − ) (19)

Michel (1952) Transition Line (Michel) Calculate the momentum thickness (x) by, e.g., Thwaites’s 1.E+04 method (above), and form ≡ U e (x) (x) 1.E+03 Re (x) (20) θ θ θ θ Re Transition is predicted to occur when Re θ meets the transition Transition Line 1.E+02 Flat Plate line : 4.0 ¡ = Re 9.2 Re x (21) 1.E+01 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 = × 6 Re x This gives Re x,tr 3.2 10 for a flat-plate boundary layer if using Thwaites’ method for (or 2.5 ×10 6 if using the Blasius value).

Transition Line (Wazzan et al.) Wazzan et al. (1981) 1.0E+09

Calculate the shape factor H(x) by, e.g. Thwaites’s method. 1.0E+08 1.0E+07 x Re Transition is predicted to occur when (quoting White, as I have 1.0E+06 no access to the original reference): = − + − 2 + 3 1.0E+05 log 10 (Re x,tr ) 40 .4557 64 .8066 H 26 .7538 H .3 3819 H 1.0E+04 (22) 2 2.2 2.4 2.6 2.8 3 For a flat-plate boundary layer, taking H = 2.59 (Blasius), this H = × 6 method gives Re x,tr 8.4 10 .

Turbulent Boundary Layers 6 - 10 David Apsley

6.9 Transition Control

Control of transition is made to either: • deliberately force transition (e.g. by trip wire or roughened surface): this may be of use in delaying or preventing separation (hence greatly reducing pressure drag) at the expense of a small increase in friction drag; • deliberately delay transition to reduce viscous drag.

6.9.1 “Tripping” a Boundary Layer

For isolated trip wires of diameter k laid across the stream, Schlichting gives k+ = 7 as the maximum height (in wall units) for the wire to have no effect on transition, whilst transition occurs at the wire if k+ exceeds 15 - 20.

White (1991) gives the following criterion for a wire to trip a boundary layer into turbulence: U k e > 850 (23)

6.9.2 Delaying Transition

In accordance with the findings of inviscid stability theory, boundary layers may be made 2 2 more stable (i.e. transition delayed) by making ( ∂ U/∂y )w more negative. This can be achieved by a number of means including: (1) favourable pressure gradient (d Pe/d x < 0 or d Ue/d x > 0); (2) wall suction ( Vw < 0) (3) wall cooling (gases) or heating (liquids).

These may be inferred from the boundary layer equations: ∂U ∂U dP ∂ ∂U (U +V ) = − e + ( ) ∂x ∂y dx ∂y ∂y

At the wall, U = 0 and V = Vw (which is non-zero if suction or blowing is applied) and hence  ∂ 2  dP   ∂    ∂   U  = e + −    U  w  Vw      (24)  ∂ 2  dx ∂y ∂y  y  w    w    w

2 2 The velocity profile is more stable if ( ∂ U/∂y )w < 0, and this is favoured by d Pe/dx < 0 (favourable pressure gradient), Vw < 0 (wall suction) or ∂ /∂y > 0. Since viscosity increases with temperature for gases, this corresponds to wall cooling. The opposite is true for liquids.

Wall suction also acts to reduce the thickness of the boundary layer, in itself enhancing stability. This is readily seen from the integral momentum equation including a wall transpiration velocity: d dU c V + 2( + H ) e = f + w (25) dx U e dx 2 U e

Turbulent Boundary Layers 6 - 11 David Apsley The “passive” technique (1), shaping the body such that the point of maximum velocity moves as far downstream as possible is used to design “laminar aerofoils”, with application to gliders. The “active” techniques (2) and (3) have considerable energy penalties.

There is also some evidence that (4) compliant boundaries may enhance stability and delay transition, but this has yet to find commercial favour.

6.10 References

Fjørtoft, R., 1950, Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex , Geofys. Pub. Oslo , 17, 1-52. Michel, R., 1952, Etude de la transition sur les profils d’aile establissement d’un point de transition et calcul de la trainée de profil en incompressible, ONERA Rept. 1/1578A. Orr, W.M.F., 1907, The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, Part I: A perfect liquid; Part II: A viscous liquid, Proc. Roy. Irish Acad., A , 27, 9-68 and 69-138. Rayleigh, Lord, 1880, On the stability of certain fluid motions, Proc. Math. Soc. London , 11, 57; see Scientific Papers , Vol. 1, 474-487, Dover, New York, 1964. Sommerfeld, A., 1908, Ein Beitrag zur hydrodynamischen Erklärung der turbulenten Flüssigkeitsbewegungen, Proc. 4 th Internat. Cong. Math. , Rome, 3, 116-124. Schubauer, G.B. and Skramstad, 1947, Laminar boundary layer oscillations and transition on a flat plate, J. Res. Nat. Bur. Standards , 38, 251-292. Squire, H.B., 1933, On the stability of three-dimensional distribution of viscous fluid between parallel walls, Proc. Roy. Soc., Series A , 142, 621-628. Thwaites, B., 1949, Approximate calculation of the laminar boundary layer, Aeronaut. Quart. , 1, 245-280. Tollmien, W., 1929, Über die Entstehung der Turbulenz, Nachr . Ges. Wiss. Göttingen Math.- Phys. Kl. II , 21-44, (English translation in NACA Technical Memo. 609). Wazzan, A.R., Gazley, C. and Smith, A.M.O., 1981, H-Rx method for predicting transition, AIAA J., 19, 810-812.

Turbulent Boundary Layers 6 - 12 David Apsley Examples

Question 1. What values does van Driest and Blumer’s correlation (equation (15)) give for the transitional Reynolds number of a flat plate boundary layer when the free-stream turbulence intensity T is (a) 0 (ideal); (b) 1% (slightly noisy wind tunnel)?

Question 2. (Thwaites’s method) (a) By multiplying by 2 Ue / and rearranging, show that the integral momentum equation d dU c + 2( + H ) e = f dx U e dx 2 can be rewritten in the form d 2 U ( ) = [2 S − 2( + H ) ] (26) e dx where 2 d( U /d x) S = w , = e U e

(b) By collecting experimental and analytical results, Thwaites was able to show that the dimensionless shear stress S and shape factor H were reasonably represented as functions of the dimensionless boundary-layer parameter . Call the RHS of equation (26) F( ). If F( ) = a – b , show that (assuming (0) = 0), 2 x = a ⌠ b−1 b  U e dx U e ⌡0

(d) If S( ) = ( + 09.0 ) 62.0 what value of corresponds to separation?

Question 3. Derive an Orr-Somerfeld equation for (the Fourier components of) v by eliminating u and p from the momentum equations.

Question 4. Using the method of Michel find the location of transition for: (a) a zero-pressure-gradient flat-plate boundary layer; = φ φ (b) potential free-stream flow across a circular cylinder, U e 2U 0 sin , (where is the 6 angle measured from the upstream stagnation point) at Re D = 5×10 .

Turbulent Boundary Layers 6 - 13 David Apsley Question 5. Calculate Re x for transition in a flat-plate boundary layer by the method of Wazzan et al., and compare with that of Michel, above.

Question 6. A decelerating flow may be represented by the free-stream-velocity variation = − U e U 0 1( x/L ) with characteristic Reynolds number U 0 L/ .Use Michel’s method with Thwaites’s correlation to estimate the value x/L at which transition occurs for a Reynolds number 6 Re L = 5×10 . (You may assume that transition occurs before separation).

Answers

(1) (a) 2.9 ×10 6 (b) 5.0 ×10 5

(2) (d) –0.09

6 (4) (a) Re x = 2.3 ×10 (b) x/R = 2.02 (or φ = 116º) – I’m not sure that I believe this

6 (5) Re x = 4.8 ×10

(6) 0.109

Turbulent Boundary Layers 6 - 14 David Apsley