Natural convection in a vertical slotAccurate solution of the linear stability equations

G. D. McBain and S. W. Armfield

geordie.mcbain@aeromech.usyd.edu.au

School of Aerospace, Mechanical, & Mechatronic Engineering

The University of Sydney, AUSTRALIA

CTAC 2003 – p.1/??

Slot convection: base solution

Exact solution of the Oberbeckequations

Waldmann (1938);Jones & Furry (1946);Ostroumov (1952);Gershuni (1953);Batchelor (1954).

linear temperature, cubic velocity

V (x) = (x3 − x)/3

Θ(x) = −x

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Linear stability equations

Derived by Gershuni (1953) and Plapp (1957).

Extends Orr–Sommerfeld equation for convection.

Eigenvalue problem for coupled 4th (vorticity–buoyancy)and 2nd (temperature) order ODEs.

[

iα Gr

64

{

(V − 16c)

(

α2

4− D2

)

+ V ′′

}

+

(

α2

4− D2

)2]

ψ

+ 2Dθ = 0[

(V − 16c) +64

iα Gr Pr

(

α2

4− D2

)]

θ − Θ′ψ = 0

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Previous linear stability studies

All previous methods failed for large Pr.Gershuni (1953), Galerkin methodKorpela et al. (1973), Galerkin methodRuth (1979), method of momentsChen & Perlstein (1989), Galerkin method

Most accurate results by Ruth, but only Pr < 10.

Large Pr behaviour discrepancy

Grcrit ∼ 9400/Pr−1/2 (Gill & Kirkham 1970)

Grcrit ∼ 7520/Pr−1/2 (Birikh et al. 1972)

Present work uses INTERIOR COLLOCATION to obtainvery accurate results across the entire range0 ≤ Pr <∞, including Pr → ∞.

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Discretization: interior collocation

Here, the linear stability equations were discretized byinterior ordinate-based Chebyshev collocation.

In general, to interpolate some function f usingCARDINAL BASIS FUNCTIONS φj(x):

f̂(x) =n

∑

j=1

φj(x)f(xj).

where φj(xi) = δij

so that at a collocation point,

f̂(xi) =n

∑

j=1

φj(xi)f(xj) = f(xi).

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Chebyshev interior collocation

The n collocation points are chosen as the extrema ofthe Chebyshev polynomial of degree n+ 1.

T ′n+1(xi) = 0, i = 1, . . . , n.

Since Tn(x) = cos(n cos−1 x), xi = cos iπn+1 .

Therefore the polynomial cardinal basis functions (orfundamental Lagrangian interpolation polynomials) are

φj(x) =T ′

n+1(x)

(x− xj)T ′′n+1(xj)

.

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Cardinal basis functions (n = 4)

-1

0

1

-1 1x1x2x3x4

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Boundary conditions

In interior collocation methods, all basis functionssatisfy the homogeneous boundary conditions.

Incorporate by multiplying by a coercion function

φ̃j(x) =ω(x)

ω(xj)φj(x).

Interpolation property φj(xi) = δij retained.

For Dirichlet conditions, factor (x− x0)k puts a zero of

order k at x = x0.

ω(x) = (1 + x)(1 − x) = (1 − x2) forces f(±1) = 0.

ω(x) = (1 − x2)2 forces f(±1) = f ′(±1) = 0.

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Dirichlet boundary conditions

0

1

-1 1x1x2x3x4

ω(x)

ω(xj)φj(x) =

1 − x2

1 − x2j

T ′n+1(x)

(x− xj)T ′′n+1(xj)

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‘Clamped’ boundary conditions

0

1

-1 1x1x2x3x4

ω(x)

ω(xj)φj(x) =

(1 − x2)2

(1 − x2j)

2

T ′n+1(x)

(x− xj)T ′′n+1(xj)

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Differentiation matrices

Matrices representing differentiation for vectors offunction values at the collocation points are obtained bydifferentiating the interpolation relation.

f̂ ′(x) =∑

j

φ′j(x)f(xj) or f̂ ′i =∑

j

Dijfj .

Higher orders:f̂ (k)i =

∑

j Dkijfj , where Dk

ij ≡ φ(k)j (xi).

References:Frazer, Duncan, & Collar (1938)Villadsen & Stewart (1967)Welfert (1997): Fortran 77 codeWeideman & Reddy (2000): MATLAB code

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The algebraic problem

After discretization, problem is of form:

Lq = cMq .

Convert to a standard eigenvalue problem by either:1. cancel ‘mass matrix’

M−1

Lq = cq ; or

2. ‘shift-and-invert’Eq = µq

whereE = (L − σM)−1

M

c = σ + 1/µ.

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Convergence

10-12

10-10

10-8

10-6

10-4

10-2

100

102

10 100

RE

LAT

IVE

ER

RO

R IN

GR

OW

TH

RA

TE

NUMBER OF INTERIOR COLLOCATION POINTS, n

Pr=100

Pr=103

Pr=106

(at Gr = 2Grcrit and α = αcrit)

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Results

For each Grashof number, Gr, Prandtl number, Pr, andwavenumber, α, a spectrum of complex eigenvaluewavespeeds c is obtained.

These correspond to modes proportional to

eiα(y−ct) = eα=cteiα(y−<ct)

Exponential amplification rate: α=c.Linear stability criterion: =c < 0.

i.e. spectrum confined to lower-half complex plane.

For a given fluid (Pr), we’re interested in the lowest Grfor which the spectrum crosses into the upper-halfcomplex plane for some α.

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Marginal stability curve: Pr = 0

0

1

2

3

4

5000 10000 15000 20000 25000 30000

WA

VE

NU

MB

ER

, α

GRASHOF NUMBER, Gr

STABLE

Grc = 7930.0551, αc = 2.6883

UNSTABLE

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Determination of critical Gr

Trace marginal stability curves by SKIRTING ALGORITHM

(McBain, this conference).

Having approximately located a minimum marginal Gr,use GOLDEN SECTION SEARCH in α around BISECTIONin Gr.

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The hydrodynamic mode (Pr = 0)

+ =

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Air (Pr = 0.7): neutral curve

0

1

2

3

4

5000 10000 15000 20000 25000 30000

WA

VE

NU

MB

ER

, α

GRASHOF NUMBER, Gr

STABLE

Grc = 8041.4222, αc = 2.8098

UNSTABLE

Very similar to pure hydrodynamic limit Pr → 0.

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Air (Pr = 0.7): critical mode

+ =

+ =CTAC 2003 – p.19/??

Water (Pr = 7): neutral curve

0

1

2

3

4

5000 10000 15000 20000 25000 30000

WA

VE

NU

MB

ER

, α

GRASHOF NUMBER, Gr

STABLE

Grc = 7868.4264, αc= 2.7671

UNSTABLE

Still very similar to pure hydrodynamic limit Pr → 0.

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Water (Pr = 7): critical mode

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Oscillatory mode: Pr = 11.7

0

1

2

3

4

0 5000 10000 15000 20000 25000 30000

WA

VE

NU

MB

ER

, α

GRASHOF NUMBER, Gr

STABLE

Grc = 7872, αc= 2.767

UNSTABLE

Pr→∞

Oscillatory mode appears from αGr ∼ 5.7 × 103, α→ 0.

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Mode crossover: Pr = 12.454

0

1

2

3

4

0 5000 10000 15000 20000 25000 30000

WA

VE

NU

MB

ER

, α

GRASHOF NUMBER, Gr

STABLE

Grc = 7872.9012, αc= 2.7662

UNSTABLE

Pr→∞

Equal minima on monotonic & oscillatory lobes.

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The oscillatory mode (Pr = 12.454)

+ =

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Cusping: Pr = 80

0

1

2

3

4

0 5000 10000 15000 20000 25000 30000

WA

VE

NU

MB

ER

, α

GRASHOF NUMBER, Gr

STABLE

UNSTABLE

Pr→∞

Near Pr = 80, lobes intersect forming a cusp.

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Large Prandtl number: Pr = 1000

0

1

2

3

4

0 5000 10000 15000 20000 25000 30000

WA

VE

NU

MB

ER

, α

GRASHOF NUMBER, Gr

STABLE

UNSTABLE

Pr→0

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Critical Gr: low Pr

2000

3000

4000

5000

6000

7000

8000

9000

10000

10-5 10-4 10-3 10-2 10-1 100 101 102

CR

ITIC

AL

GR

AS

HO

F N

UM

BE

R, G

r

PRANDTL NUMBER, Pr

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Critical Gr: high Pr

103

104

105

100 101 102 103 104 105 106

CR

ITIC

AL

PA

RA

ME

TE

R, S

= G

r Pr

1/2

PRANDTL NUMBER, Pr

Gr ~ 7930.0598

(Birikh et al. 1972) Gr Pr1/2 ~ 7520

(present) Gr Pr1/2 ~ 9435.3767

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Conclusions

numerics

Chebyshev interior collocation method providesaccurate solutions to natural convection linearstability problem.

physicsfeatures of stability margin as Pr increases:

Grcrit ∼ 7930.0598, (Pr → 0)Pr → 0 limit approximates monotonic lobe ∀Pr; .Pr < 11.57: oscillatory lobe appearsPr

.= 12.454: monotonic–oscillatory transition

Pr ≈ 80: lobes cross, form cuspPr → ∞ limit approximates oscillatory lobe ∀Pr

Grcrit ∼ 9435.3767/√

Pr, (Pr → ∞).

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