natural convection in a vertical slot: accurate solution of the linear stability equations
TRANSCRIPT
Natural convection in a vertical slotAccurate solution of the linear stability equations
G. D. McBain and S. W. Armfield
School of Aerospace, Mechanical, & Mechatronic Engineering
The University of Sydney, AUSTRALIA
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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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