Linear stability of natural convectionon an

evenly heated vertical wall

Geordie McBain & Steve ArmfieldSydney University Fluid Dynamics Group

December 2004

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The problem

• infinite vertical wall

• uniform heat flux out of wall

• into unbounded fluid on one side

• stable linear stratification far from wall

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The general problem: temperature-rise

What we want to know is:

• How hot does the wall get?

• In terms of:

– the wall heat flux;– the stratification;– kinematic viscosity, thermometric conductivity, & expansion coefficients.

• i.e. Nusselt number in terms of Rayleigh and Prandtl numbers (+ geometry)

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The particular problem: laminar or turbulent convection

• Temperature-rise is controlled by radiation and natural convection.

– Here we consider only natural convection.

• The single biggest factor influencing the heat transfer here is whether thenatural convection is laminar or turbulent.

• The present aim is a criterion for the breakdown of laminar convection.

• We use linear stability theory.

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Overview

• Governing equations

• The base solution

• Linearized stability equations

• Discretization & algebraic solution

• Critical modes at various Prandtl numbers

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Governing equations: Oberbeck–Boussinesq

∇ · u = 0

R

(∂

∂t+ u ·∇

)u = −R∇p+ ∇2u + 2T ey

σR

(∂

∂t+ u ·∇

)T = ∇2T

• Dimensionless parameters: Reynolds number R , Prandtl number σ

• At wall (x = 0 ): u = 0 , ∂T∂x = −1

• Far-field (x→∞ ): u ∼ 0 , T ∼ 2y/σR

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Oberbeck–Boussinesq for stratified fluid

. . . but subtract the background stratification from T :

∇ · u = 0

R

(∂

∂t+ u ·∇

)u = −R∇p+ ∇2u + 2T ey

σR

(∂

∂t+ u ·∇

)T = ∇2T+2v

so that the far-field condition becomes T ∼ 0 .

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Base solution

Solutions of the form V (x) , Θ(x) satisfy

0 =∂2V

∂x2+ 2Θ

0 =∂2Θ

∂x2+ 2V

and V (0) = 0 and Θ′(0) = −1 . The solution is:

V (x) = e−x sinx

Θ(x) = e−x cosx

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Prandtl’s (1952) anabatic wind solution

hue: T = e−x cosx

profile: v = e−x sinx

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Comments on the base solution

• Boundary conditions

– Prandtl derived the solution for the condition T0 = T∞ + ∆T– but since the solution is independent of y , T ′(0) is too.

• Realization

– Prandtl’s b.c. implies linearly varying wall temperature.– Uniform heat flux is more natural.– System occurs in side-heated cavity (Kimura & Bejan 1984, JHT )– and electrochemical cells∗ Eklund et al. (1991, Electrochimica Acta); Bark et al. (1992, JFM )

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Linearizing the stability equations

• Now put u = V (x)ey + δu and T = Θ(x) + δT

• subtract base equations

• neglect second-order quantities

• introduce stream-function for velocity perturbation

• Fourier transform in t and y

– since all coefficients are independent of them

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The linearized stability equations

{[(κ2 −D2)2 2D

2D κ2 −D2

]+ iκR

[V (κ2 −D2) + V ′′ 0

σΘ′ σ V

]}[ψθ

]

= iκR c

[κ2 −D2 0

0 σ

] [ψθ

]

• D ≡ d/dx

• a linear non-Hermitian generalized eigenvalue problem

• eigenvalue: c, (complex) wave speed; eigenvector: ψ & θ .

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Comments on the linear stability equations

• Like Orr–Sommerfeld equations, but with

– buoyancy term & θ-equation (Gershuni 1953; Plapp 1957)– stratification term (Gill & Davey 1969)

• Assumes base flow parallel, which it is! (Unlike many other boundary layers.)

• Assumes periodicity in streamwise direction y , which is ultimately unphysical.

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Discretization

• Use orthogonal collocation, based on generalized Laguerre polynomials.

• The L(α)k (x) are orthogonal on (0,∞) with weight xαe−x .

• Multiply basis functions by e−x/2 to enforce far-field decay.

• Also by x or x2 to get f(0) = 0 or f(0) = f ′(0) = 0 .

• Factor for imposing just f ′(0) = 0 (without f(0) = 0 ) is messier to derive,

– though relatively straightforward to implement.

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The algebraic generalized eigenvalue problem

• Discretization gives Lq = cMq an algebraic generalized eigenvalue problem.

• where L and M are matrices, q is an eigenvector, and c is an eigenvalue.

• All are complex, and L is non-Hermitian (L∗ 6= L ).

• When M is non-singular, we can convert to standard form:

M−1Lq = cq

• Then solve by LAPACK SUBROUTINE ZGEEV, via Octave function eig.

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Orthogonal collocation is very accurate

10-12

10-10

10-8

10-6

10-4

10-2

20 40 60 80 100

ER

RO

R

NUMBER OF NODES

Convergence of the marginal R at σ = 7 and κ = 0.4612 .

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Interpretation of the eigenvalues

• Consider the real (< c ) and imaginary (= c ) parts of the wave speed c .

• The eigenvectors are related to the perturbations by:

δΨ = <{ψ(x)eiκ(y−ct)

}=eκt= c {(< ψ) cosκ(y − t< c)− (= ψ) sinκ(y − t< c)}

δT = <{θ(x)eiκ(y−ct)

}=eκt= c {(< θ) cosκ(y − t< c)− (= θ) sinκ(y − t< c)} ,

so the perturbations grow like eκt= c .

• Thus if = c > 0 for any eigenvalue c , the perturbation grows without bound.

• If = c < 0 for the whole spectrum, the base flow is linearly stable.

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The linear stability margin

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

• For a given Prandtl number σ , thelinear stability margin divides thelinearly stable and unstable points inthe Reynolds number–wave number(R–κ) plane.

• The minimum R on the margin is thecritical Reynolds number.

• For R < Rc , all modes decay(according to the linear theory).

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Tracing the stability margin

Sometimes the marginisn’t smooth so tracingit is tricky.Here (σ = 7 ), it’scusped where marginsfor different modescross.

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

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An algorithm for stability margins

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0 50 100 150 200

WA

VE

NU

MB

ER

, α

REYNOLDS NUMBER, R

STABLE

UNSTABLEstable ##1-2

stable #3

unstable #1

unstable ##2-3

Here we use our skirting algorithm (McBain 2004, ANZIAM J.).

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Results: Stability margins

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

σ=0

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

σ=0.1

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

σ=0.2

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

σ=0.2163

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

σ=0.7

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

σ=7

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Critical Reynolds numbers: 0 < σ 6 1000

0

50

100

150

200

10-2 100 102

CR

ITIC

AL

RE

YN

OLD

S N

UM

BE

R, R

c

PRANDTL NUMBER, σ

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Low & high Prandtl numbers

• Clearly σ < 0.2163 and σ > 0.2163 are verydifferent.

• There are two distinct critical modes, differing in:

– speed– wavelength– shape

0.0

0.2

0.4

0.6

0.8

1.0

0 100 200 300 400

κ

R

σ=0.2163

0

50

100

150

200

10-2 100 102

CR

ITIC

AL

RE

YN

OLD

S N

UM

BE

R, R

c

PRANDTL NUMBER, σ

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Critical wave numbers and speeds

0.2

0.3

0.4

0.5

0.6

0.7

10-2 100 102

WA

VE

NU

MB

ER

, κ

PRANDTL NUMBER, σ

0

0.1

0.2

0.3

0.4

0.5

0.6

10-2 100 102

WA

VE

SP

EE

D, c

PRANDTL NUMBER, σ

Vmax=2-1/2e-π/4

Above σ = 0.2163 , modes are longer and faster.

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The low Prandtl number mode

isotherms

σ = 0

σ = 0.1streamlines

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The high Prandtl number mode

isotherms

σ = 0.7

σ = 7streamlines

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Fastest growing mode at R = 10, σ = 7

+ =

• cf. Rc = 8.58 at this Prandtl number

• wavelength ≈ 10 boundary layer thicknesses; phase speed ≈ 1.18maxV (x)

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Comparison of T (0) & T ′(0) boundary conditions

0

50

100

150

200

10-2 100 102

CR

ITIC

AL

RE

YN

OLD

S N

UM

BE

R, R

c

PRANDTL NUMBER, σ

θ’(0)=0 (present work)θ(0)=0 (Gill & Davey 1969)

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Low and high Prandtl number modes

• The low-σ mode has Rc roughly independent of σ .

– It’s essentially a shear instability near the inflexion-point.

• The high-σ mode has Rc roughly independent of thermal boundary condition.

– It’s due to a thermal disturbance in outer boundary layer.

• So the two modes can be characterized as ‘hydrodynamic’ and ‘thermal’.

• Transition (σ = 0.2163 ) is quite low; cf. other vertical convection flows.

• ‘High-σ’ relevant for most fluids, including air & water.

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Conclusions

• Generalized Laguerre collocation works well here.

• Like other vertical natural convection flows, there are two different criticalmodes, depending on the Prandtl number.

• Low mode (σ < 0.2163 ) is slow and short, basically hydrodynamic.

• ‘High’ mode (σ > 0.2163 ) is fast and long, basically thermal.

• Accurate critical data tabulated in paper for 0 6 σ 6 1000 .

• Results at σ = 7 agree with fully nonlinear numerical solutions.

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