Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic...
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Transcript of Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic...
![Page 1: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/1.jpg)
Lecture 19-20: Natural convection in a plane layer.
Principles of linear theory of hydrodynamic stability
1
z
x
kGrTvpvvtv
TPr
TvtT
1
0div v
Governing equations:
T=0
T=A
h=1
We will distinguish two cases:
A =-1 – layer heated from aboveA =1 – layer heated from below
![Page 2: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/2.jpg)
Quiescent state
2
1. Quiescent state: 00 t
v ,
AzczcTT
zTkTkTkTkGrTp
2100
000000
0
000
![Page 3: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/3.jpg)
Linear stability of a quiescent state
3
ppp
TT
v
0
0
Let us analyse the time evolution of a small perturbation of the quiescence
The linearised equations for a perturbation read
kGrvptv
PrTv
t1
0
0div v
v0 =0, p0 and T0 is the basic state;v, θ and p’ is a small perturbation
![Page 4: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/4.jpg)
4
Grvzp
tv
vxp
tv
z
z
x
x
PrAv
t z
1
0
zv
xv
zx
For 2D flow, we may introduce the stream-function defined as
xv
zv
zx
,
The continuity equation is satisfied automatically 0
22
xzxzzv
xv zx
Taking x- and z-projections of the Navier-Stokes equation gives
![Page 5: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/5.jpg)
5
The Navier-Stokes and heat transfer equation can be re-written as
Grxz
pxt
zxp
zt
PrxA
t1
z-derivative of the first equation minus x-derivative of the second equations gives
xGr
t
2
PrxA
t1
Boundary conditions:
00 zx
vv ,
0For temperature, at the upper and lower plate:
For velocity, at the upper and lower plate(rigid walls):
00
xz
,or, in terms of stream-function
![Page 6: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/6.jpg)
6
iri
tittir
expexpexp Time dependence of a perturbation:
In general,
If λr>0 then the perturbation will exponentially grow (with the rate λr).If λr<0 then the perturbation will exponentially decay (with the rate λr).
If λi ≠ 0 then the growth (or decay) is oscillatory.If λi = 0 then the growth (or decay) is monotonic.
Basic idea of the stability analysis: (i)Seek a solution in the form of the normal modes. (ii)Find λ that satisfy the equations (we may find several discrete values of λ or even continuous spectrum of λ).(iii)If at least one λ has positive real part then the considered basic state is unstable.
We will analyse stability of the quiescent state only in respect to normal modes:
ikxtztzxikxtztzx exp,,,exp,,
![Page 7: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/7.jpg)
7
For this problem it can be shown that perturbations develop monotonically, i.e. λi = 0.
ikGrkkk iv 422 2
21k
PrikA
Substitution of the normal modes gives Boundary conditions:
000 ,,
Next, system (*) together with the above boundary conditions can be solved numerically.
For the case, when the upper and lower plates are both free surfaces (which is not a good assumption as both plates cannot be free surfaces), the solution can be obtained analytically.
(*)
At a free boundary:
000 ,,
000
,,xv
zv
v zx
xzz
In terms of the stream-function, for normal perturbations:
![Page 8: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/8.jpg)
8
z sin0
Functions and satisfy the boundary conditions. Let us substitute these functions into system (*):
z sin0
00
4224
0
22 2 ikGrkkkr
0
22
00
1 kPr
ikAr
or
000
22222 ikGrkkr
01
0
22
0
k
PrikA
r
We obtained the homogeneous system of linear equations,
0
0
022021
012011
aa
aa
This system has a non-trivial (non-zero) solution if
or 00
0
2221
1211
aa
aa
02221
1211
aa
aadet
(**)
![Page 9: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/9.jpg)
9
The last condition written for equations (**) is
01 22
22222
kPr
ikA
ikGrkk
r
r
det
or
01 2222222
AGrkk
Prkk
rr
or
011
122
2
222222
AGr
kk
kPr
kPr rr
This quadratic equation can be written in the following form:
AGrk
kk
Prc
kPr
bcbrr
22
2
222
222
1
01
10
,
![Page 10: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/10.jpg)
10
A basic state is unstable if λr>0. For this, we need
0440 222 ccbbcbbr
This equation will have two solutions defined by the formula:
242 cbb
r
01
22
2
222
AGrk
kk
Pr
or
Finally,
3
2
322
,gL
PrGrRak
kARa
For the layer heated from above, A=1. The instability may occur if
2
322
kk
Ra
But Ra>0, this condition is never satisfied. Hence, the layer heated from above is hydrodynamically stable. Fluid will remain at the quiescent state.
-- Rayleigh number
![Page 11: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/11.jpg)
11
For the layer heated from below, A=-1. The instability may occur if
2
322
kk
Ra
Ra
unstable
stable
neutral curve, λr=0(stability curve)
kkc
Rac
Let us determine Rac. Condition of minimum:
2222
0 .d
d c
kk
Ra
6584
27 4
cc
kRaRa
Quiescence becomes unstable for the layer heated from below if the temperature difference between the plates is high enough for Ra>Rac.
Convective rolls with dimensions of will be observed. c
kh
2
ck2
![Page 12: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/12.jpg)
12
For the case of rigid-rigid boundaries, the stability diagram is very similar but
For the free-rigid boundaries,
1708123 cc
Rak ,.
1101682 cc
Rak ,.
![Page 13: Lecture 19-20: Natural convection in a plane layer. Principles of linear theory of hydrodynamic stability 1 z x Governing equations: T=0T=0 T=AT=A h =1.](https://reader035.fdocuments.in/reader035/viewer/2022062720/56649f155503460f94c2a035/html5/thumbnails/13.jpg)
Cloud streets
13
Horizontal convective rolls producing cloud streets (lower left portion of the image) over the Bering Sea
Simple schematic of the production of cloud streets by horizontal convective rolls
Good pictures:http://www.meteorologynews.com/2009/10/29/cloud-streets-photographed-over-gulf-of-mexico/
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14
Remarks:
(i) The method used to define the instability threshold is universal. This method can be used for finding the thresholds of instability of any solution for any partial differential equations.
(ii) Next, we can take the convective rolls as a basic state; represent all physical quantities as sums of a basic state with small disturbances; linearise the equations; and determine the conditions when the found rolls become unstable.
(iii) At the first instability threshold, the state of quiescence is replaced by convective rolls with a typical horizontal size kc. Passing the next instability threshold, the convective motion will represent the combination of the rolls of two different sizes. And so on. At large Gr~105, convective motion becomes turbulent: superposition of the rolls with the sizes determined by continuous spectrum of k.
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John William Strutt, 3rd Baron Rayleigh (12 November 1842 – 30 June 1919) was an English physicist who, with William Ramsay, discovered the element argon, an achievement for which he earned the Nobel Prize in Physics in 1904. He also discovered the phenomenon now called Rayleigh scattering, explaining why the sky is blue, and others.