Linear Stability of Vertical Natural...
Transcript of Linear Stability of Vertical Natural...
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Linear Stability of Vertical NaturalConvection
G. D. McBain
School of Aerospace, Mechanical, & Mechatronic Engineering
The University of Sydney, AUSTRALIA
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Vertical natural convection
We consider here vertical natural convection flows dueto horizontal hearing:
slot single heated wall7ANCW 2003 – p.2/43
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Outline
convection in a vertical slotlinear stability equationsdiscretization by Chebyshev collocationresults for vertical slot
other configurationsthe importance of stratificationsingle vertical wall (mountain & valley winds)
Laguerre collocationthe case for heat flux boundary conditions
nonnormal evolution
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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 σ
(
α2
4− D2
)]
θ − Θ′ψ = 0
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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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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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Convergence
10-12
10-10
10-8
10-6
10-4
10-2
100
102
10 20 30 40 50 60 70 80 90 100
RE
LAT
IVE
ER
RO
R IN
GR
OW
TH
RA
TE
NUMBER OF INTERIOR COLLOCATION POINTS, n
Pr=106
Pr=103
Pr=100
Conducted at twice the critical Grashof number
N.B.: exponential convergence: ε ∝ e−rx, r > 0.7ANCW 2003 – p.9/43
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Results
For each Grashof number, Gr, Prandtl number, σ, 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 (σ), we’re interested in the lowest Gr forwhich the spectrum crosses into the upper-half complexplane for some α.
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Marginal stability curve: σ = 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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The hydrodynamic mode (σ = 0)
+ =
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Air (σ = 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 σ → 0.
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Air (σ = 0.7): critical mode
+ =
+ =7ANCW 2003 – p.14/43
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Water (σ = 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 σ → 0.
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Water (σ = 7): critical mode
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Oscillatory mode: σ = 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: σ = 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 (σ = 12.454)
+ =
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Cusping: σ = 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 σ = 80, lobes intersect forming a cusp.
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Large Prandtl number: σ = 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 σ
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 σ
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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Slot conclusions
numerics
Chebyshev interior collocation method providesaccurate solutions to natural convection linearstability problem.
physicsfeatures of stability margin as σ increases:
Grcrit ∼ 7930.0598, (σ → 0)σ → 0 limit approximates monotonic lobe ∀σ; .σ < 11.57: oscillatory lobe appearsσ.= 12.454: monotonic–oscillatory transition
σ ≈ 80: lobes cross, form cuspσ → ∞ limit approximates oscillatory lobe ∀σGrcrit ∼ 9435.3767/
√σ, (σ → ∞).
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Stratification
‘. . . an initial stratification is usual, and is in fact generallyunavoidable’ (J. C. Patterson, this workshop)
It’s a natural consequence ofhorizontal heating.
Consider a horizontally heatedsphere (McBain & Stephens 2000,7AHMTC)
σ = 0.7,Ra = 5000
T ∼ −x− Grσ
44800(1 − 4r2)(9 − 20r2)y + . . .
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Stratified base solutions
Rewrite Oberbeck equations in terms of temperatureexcess over linear stratification: T− = 2m4y/σR
(
D
Dt− 1
R∇2
)
u = −∇P +2
RT ̂
(
σD
Dt− 1
R∇2
)
T = −2m4
Ru · ̂
∇ · u = 0
And separate vertical-means: T (x, y) = Θ(x) + θ(x, y).
Then vertical-means satisfy
V ′′(x) = −2Θ(x)
Θ′′(x) = 2m4V (x)
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Stratified slot (Ostroumov 1952)
m = 1 m = 2 m = 5V = {sinhm(1−x) sinm(1+x)−sinhm(1+x) sinm(1−x)}/m2dΘ = {coshm(1−x) cosm(1+x)− coshm(1+x) cosm(1−x)}/dd = cosh 2m− cos 2m
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Stratified half-space (Prandtl 1952)
‘Mountain and valleywinds in stratified air’
a.k.a. ‘anabatic/katabatic’winds
V (x) = e−x sin x
Θ(x) = e−x cosx
T (x, y) = Θ(x) + 2y/σR
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Linear stability of anabatic wind
comprehensive study by Gill & Davey (1969, JFM)
repeated here using interior Laguerre collocation
The Chebyshev polynomials are orthogonal on[−1, 1] with respect to weight function
1√1 − x2
=1
√
(1 − x)(1 + x).
The generalized Laguerre functions are orthogonalon [0,∞) with respect to weight function xαe−x.The Chebyshev weight improves performance forboundary value problems.
α = −12 gives a similar effect on [0,∞)
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Linear stability margins
0.4
0.8
100 200 300 400
WA
VE
NU
MB
ER
, α
REYNOLDS NUMBER, R
σ=0.77.0
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
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
‘Thermal’ modes much more important than inunstratified slot at same low Prandtl numbers.
cf. σ = 7 (top-left) and σ = 1000 (top-right) for slot.7ANCW 2003 – p.30/43
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Thermal boundary conditions
A problem with these stratified slot and half-spacesolutions is the boundary conditions on the temperatureat the wall:
Θ = 1
T = Θ + 2m4y/σR
= 1 + 2m4y/σR
. . . linearly increasing with height
Unlikely to occur in nature or industry
Difficult to impose in an experiment
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Thermal boundary conditions II
HOWEVER: Since the (excess) temperature Θ isindependent of height y, the normal temperaturegradient at the wall is too:
T (x, y) = Θ(x) + 2m4y/σR
∂
∂xT (x, y) = Θ′(x) + 0 .
Thus, solutions also match a uniform heat flux b.c.
useful idealization of many boundary conditions innature (e.g. sunshine) and industry (e.g. electricheating)
relatively easy to impose experimentally
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Thermal boundary conditions III
steady–vertical base solutions unchanged
stability ODEs unchanged
[(αR)−1(α2 −D2)2 − {V (α2 −D2) + V ′′}]ψ+ 2i(αR)−1Dθ = −c(α2 −D2)ψ
[iσΘ′+2m4(αR)−1D]ψ
+ [iσV + (αR)−1(α2 −D2)]θ = iσcθ
disturbance stream-function b.c.s unchanged ψ = ψ′ = 0
but disturbance temperature b.c. changes from Dirichletθ = 0 to Neumann θ′ = 0
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Stratified half-space, flux b.c.
0.4
0.8
100 200 300 400
WA
VE
NU
MB
ER
, α
REYNOLDS NUMBER, R
σ=0.77.0
Thermal modes more important again: critical even for low
Prandtl numbers (0.7: air).
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Fastest growing mode at R = 10, σ = 7
+ =
cf. critical R.= 8.58 at α = 0.462
α = 0.454 (wavelength ≈ 10 boundary layer thicknesses)
phase speed: cr = ωr/alpha ≈ 1.18Vmax
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Transient amplification
1000 2000 3000
0.827
0.828
0.829
DNS (Armfield)
0
x(t)
TIME, t
3 d.o.f. model system
1. initial approach to steady state solution
2. transient amplification
3. transient decay back to steady state solution
4. asymptotic instability (as predicted by linear theory)7ANCW 2003 – p.36/43
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Nonnormal evolution
Consider the general dynamical system:
d
dty = Ay.
If the system is autonomous (i.e. ddtA = 0)
and A admits the spectral decomposition
A = ZΛZ−1,Λ = diagλ
columns of Z are A’s eigenvectorsdiagonal elements of Λ are A’s eigenvalues
Then the solution to the initial value problem is
y(t) = ZeΛtZ−1y(0)
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Classification of matrices
Normal matricescommute with adjoint: AA
∗ = A∗A
⇒ eigenvectors are orthogonal: Z−1 = Z
∗
includes Hermitian matrices A∗ = A
Nonnormal matrices
don’t commute with adjoint AA∗ 6= A
∗A
don’t have orthogonal eigenvectors: Z−1 6= Z
∗
eigenvalues ill-conditionedpeculiar evolutionary propertiesOrr–Sommerfeld matrix is nonnormal!
ref.: L. N. Trefethen (1997) ‘Pseudospectra of linearoperators’, SIAM Review 39:383–406.
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Nonnormal v. normal evolution
Consider a simple example (3 degrees of freedom)
d
dt
x
y
z
=
−1 1 1
0 −λ 0
0 0 −µ
x
y
z
Spectral decomposition A = ZΛZ−1 =
1 1 1
1 − λ 0 0
0 1 − µ 0
−λ 0 0
0 −µ 0
0 0 −1
0 11−λ
0
0 0 11−µ
1 1λ−1
1µ−1
And form B = UΛU∗, where U = orth Z; e.g.
[
0 0 11 0 00 1 0
]
.
B is normal, but has identical spectrum to A.7ANCW 2003 – p.39/43
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Nonnormal v. normal evolution
0
x(t)
TIME, t
Normal:
y(t) = UeΛtU∗y(0)
0
x(t)
TIME, t
Nonnormal:
y(t) = ZeΛtZ−1y(0)
same long-time behaviour
transient growth of ‘stable’ modes in nonnormal case
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Conclusions on nonnormality
Eigenvalues control long-time evolution and stability.
Nonnormality can cause large transient growth, even inasymptotically stable systems.
Linear stability analysis only applies to infinitesimalperturbations.
Can nonnormal transient growth generate amplitudeslarge enough to excite nonlinear effects?
Is this an alternative scenario for the breakdown ofsteady laminar flow to the classical one of Landau?
ref.: P. J. Schmid & D. S. Henningson (2001) Stabilityand Transition in Shear Flows, Springer.
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Conclusions
Collocation excels for linear stability analysis.extensively tested against published resultsagrees and typically extends accuracyvery fast results on a typical desktop PC
heat flux b.c. more realistic for stratified cases
heat flux b.c. anabatic wind is a good configuration
Thermal modes increasingly important for higherPrandtl numbers, stratified fluids, and heat flux b.c.
Unusual transient growth and decay in DNS resultsexplained by theory of nonnormal evolution.
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