Global instability of flow over a rotating disc · 2017. 10. 9. · Rotating disc ﬂow A new...

Local and global stability theoryRotating disc flow

A new global frequency selection mechanism

Global instability of flow over a

rotating disc

Jonathan Healey

Keele University

6th January 2011

Jonathan Healey Brighton, 6.1.2011



Outline

1 Local and global stability theory




Outline


2 Rotating disc flow




Outline


2 Rotating disc flow

3 A new global frequency selection mechanism




Stability of shear layers

There are many shear layers that are either parallel, orapproximately parallel, e.g.






channel flows,






channel flows,

boundary layers,

∂P/∂x < 0 ∂P/∂x = 0 ∂P/∂x > 0 Tw > Tf

U ′′ = 0




unbounded shear layers,

jets wakes mixing layers






Channel flows are exactly parallel.






Channel flows are exactly parallel.

The others may become ‘more parallel’ as the Reynoldsnumber increases.




Local and global stability theory

In ‘local’ theory, the basic flow is assumed parallel:

u = U(y) + ǫu(y) exp i(αx − ωt)

v = ǫv(y) exp i(αx − ωt)

p = P(x) + ǫp(y) exp i(αx − ωt)

giving an ODE for v , and an eigenrelation between α and ω.










In ‘global’ theory, the basic flow is nonparallel:

u = U(x , y) + ǫu(x , y) exp(−iωG t)

v = V (x , y) + ǫv(x , y) exp(−iωG t)

p = P(x , y) + ǫp(x , y) exp(−iωG t)

giving a PDE for v , and an eigenrelation for ωG .










In ‘global’ theory, the basic flow is nonparallel:

u = U(x , y) + ǫu(x , y) exp(−iωG t)

v = V (x , y) + ǫv(x , y) exp(−iωG t)

p = P(x , y) + ǫp(x , y) exp(−iωG t)

giving a PDE for v , and an eigenrelation for ωG .

How are local and global theories related when the basic flowvaries slowly in the x direction?




Global frequency selection criteria

Is ωG the most unstable local frequency, ie ω when Im(ω) ismaximized over x and α?






No — this wave may propagate out of region of interest.







Consider propagation of a wavepacket produced impulsively:t t

x xconvective instability absolute instability









Let ω = ω0, where dω/dα = x/t = 0, be the local absolutefrequency.









Let ω = ω0, where dω/dα = x/t = 0, be the local absolutefrequency.

How do the rays curve when the flow varies with x?




Sketch of rotating-disc flow

Ω∗




A rotating-disc experiment




Laminar-turbulent transition on the rotating disc

year Ret

Gregory, Stuart & Walker (1955) 530 Visual, China-clayFederov et al. (1976) 515 Visual, napthaleneClarkson, Chin & Shacter (1980) 562 Visual, dyeCobb & Saunders (1956) 490 Heat transferChin & Litt (1972) 510 Mass transferGregory & Walker (1960) 505 Pressure probeSmith (1946) 557 Hot-wire

Kobayashi et al. (1980) 500 Hot-wireMalik, Wilkinson & Orszag (1981) 520 Hot-wireWilkinson & Malik (1985) 550 Hot-wireLingwood (1996) 508 Hot-wireOthman & Corke (2006) 539 Hot-wire




Transition related to onset of absolute instability?

Lingwood (1995) found, using local theory, that rotating discflow is absolutely unstable for Re > 507.






She argued that this created a global instability destroying thelaminar flow.







But Davies & Carpenter (2003) found, using DNS ofnonparallel linearized equations, that wavepackets couldpropagate through the locally absolutely unstable region.








This implies that rotating disc flow is globally stable.








This implies that rotating disc flow is globally stable.

Othman & Corke (2006) observed, in an experiment,wavepackets propagating through a laminar flow withRe > 507, confirming global stability.




We introduce a new global frequency selection mechanism.





It is relevant to rotating disc flow.






It predicts global instability.






It predicts global instability.

It may explain the observed variation in Ret in experiments.




An amplitude equation approach

In some circumstances, the envelope, A, of a wavepacket

A(X ,T ) exp i(αx − ωt)

satisfies the linearized complex Ginzburg-Landau eqn:

∂A

∂T+ U

∂A

∂X= µA + γ

∂2A

∂X 2.








∂A

∂T+ U

∂A

∂X= µA + γ

∂2A

∂X 2.

U is the group velocity of the packet;








∂A

∂T+ U

∂A

∂X= µA + γ

∂2A

∂X 2.


Re(µ) is the growth rate;








∂A

∂T+ U

∂A

∂X= µA + γ

∂2A

∂X 2.



Im(µ) is the detuning between A and the carrier wave;








∂A

∂T+ U

∂A

∂X= µA + γ

∂2A

∂X 2.



Im(µ) is the detuning between A and the carrier wave;

Re(γ) > 0 is the rate of spreading (dispersion/diffusion).




A model flow

Taking the coefficients U = U(X ), µ = µ(X ) and γ = γ(X )models the propagation of a wavepacket through a spatiallyvarying flow.




A model flow


Considerµ = (1 + iδ)ǫX , U = γ = 1,

where ǫ≪ 1 implies a slowly varying flow.




A model flow




This models a flow that




A model flow





becomes progressively more unstable as X increases,




A model flow





becomes progressively more unstable as X increases,has detuning when δ 6= 0.




A model flow





becomes progressively more unstable as X increases,has detuning when δ 6= 0.

Hunt & Crighton (1991) give an exact impulsive solution forthese coefficients.




Local results

Treat the coefficients U, µ and γ as constants.




Local results


Substituting A = A0 exp i(αX − ωT ) into G-L eqn gives thelocal dispersion relation

ω = Uα+ iµ− iγα2.




Local results




The most unstable local wave has α = 0, giving ω = iµ.




Local results





Our model is therefore locally unstable for X > 0.




Local results






The local absolute frequency is

ω0 = i

(

µ−U2

4γ

)




Local results







ω0 = i

(

µ−U2

4γ

)

Our model is therefore locally absolutely unstable forX > (4ǫ)−1.




Local results







ω0 = i

(

µ−U2

4γ

)

Our model is therefore locally absolutely unstable forX > (4ǫ)−1.

Local instability is independent of δ.




Global results

Take Hunt & Crighton’s exact solution for our variablecoefficients:




Global results


As T → ∞ for any fixed finite X ,

A ∼ exp[ǫ2(1 − δ2 + 2iδ)T 3/12].




Global results



A ∼ exp[ǫ2(1 − δ2 + 2iδ)T 3/12].

Global instability for δ2 < 1.




Global results



A ∼ exp[ǫ2(1 − δ2 + 2iδ)T 3/12].

Global instability for δ2 < 1.

Global decay for δ2 > 1.




An example

For ǫ = 0.01, there is local convective instability for X > 0,and local absolute instability for X > 25.




An example


Contours of |A|:

0 20 40 60 80 100

25

50

75

100

125

150

175

200

0 20 40 60 80 100

25

50

75

100

125

150

175

200T T

X X

δ = 0: global instability δ = 2: global decay




An example


Contours of |A|:

0 20 40 60 80 100

25

50

75

100

125

150

175

200

0 20 40 60 80 100

25

50

75

100

125

150

175

200T T

X X

δ = 0: global instability δ = 2: global decay

Davies, Thomas & Carpenter (2007) argued that rotating discflow has strong enough detuning (large enough δ) to make itglobally stable, despite existence of local absolute instability.




Creation of global instability

Remember that discs in experiments have finite radius!






Hunt & Crighton’s solution has b.c.s A → 0 as X → ±∞.







Keep A → 0 as X → −∞, but let A = 0 at X = h (e.g. atedge of disc).








This creates a discrete spectrum of global modesA = ψ(X ) exp(−iωGT ) where

ψ′′ − ψ′ + [iωG + (1 + iδ)ǫX ]ψ = 0.









ψ′′ − ψ′ + [iωG + (1 + iδ)ǫX ]ψ = 0.

The eigenvalues are

ωG = i

[

(1 + iδ)ǫh − 1/4 + ǫ2/3(1 + iδ)2/3bn

]

= ω0(h)+O(ǫ2/3)

where Ai(bn) = 0, i.e. b1 ≈ −2.34, b2 ≈ −4.09, etc.









ψ′′ − ψ′ + [iωG + (1 + iδ)ǫX ]ψ = 0.

The eigenvalues are

ωG = i

[

(1 + iδ)ǫh − 1/4 + ǫ2/3(1 + iδ)2/3bn

]

= ω0(h)+O(ǫ2/3)

where Ai(bn) = 0, i.e. b1 ≈ −2.34, b2 ≈ −4.09, etc.

Local absolute instability at disc edge ⇒ global instability.




An example

Let ǫ = 0.01, δ = 2 and A → 0 as X → −∞.

0 20 40 60 80 100

25

50

75

100

125

150

175

200

0 20 40 60 80 100

25

50

75

100

125

150

175

200T T

X X

Global decay when Global instability whenA → 0 as X → ∞ . A = 0 at X = 100 .




Stabilizing nonlinearity

Consider

∂A

∂T+∂A

∂X= 0.01(1 + iδ)XA +

∂2A

∂X 2− |A|2A.

0 20 40 60 80 100

25

50

75

100

125

150

175

200

0 20 40 60 80 100

25

50

75

100

125

150

175

200T T

X X

δ = 0 δ = 2




Qualitative behaviour of front

X X = h

globalinstability

XfXC/A

convective instability

hc h

There is no front for h < hc , where hc > XC/A.




Qualitative behaviour of front

X X = h

globalinstability

XfXC/A


hc h

There is no front for h < hc , where hc > XC/A.

As h passes through hc the front appears and moves inwards,approaching the convective-absolute transition location forlarge h.




Rotating disc transition experiments

500 600 700 800 900

450

500

550

600

650

700

750

800

Retrans

discedge

globalinstability


Reedge




Conclusions

The out-flow boundary condition has a global effect on theflow when there is local absolute instability at the out-flow.




Conclusions


The global frequency can then be driven by the out-flow localabsolute instability.




Conclusions



This provides a possible mechanism for global instability inrotating disc flow.




Conclusions




Ret for the disc is correlated to Reedge.




Conclusions




Ret for the disc is correlated to Reedge.

Ret follows the same qualitative dependence as the front inthe nonlinear global mode theory.


Global instability of flow over a rotating disc · 2017. 10. 9. · Rotating disc ﬂow A new...

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