Appendix II
Test calculations
II.1. Validation of linear terms
To validate the Galerkin projections of the linear terms of equations (1)-(5) we consider the
problem of the stability of the quiescent state in a vertical cylinder that rotates with an angular
velocity  and is heated from below(Rayleigh-Benard instability in a rotating cylinder). For
constant temperatures on the top and the bottom of the cylinder, the initial no-flow state is a linear
distribution of temperature along the axis and a solid-body rotation of the whole system:
  1  z,
V
R2
e  ,

U W  0
(II.1)


where the temperature is scaled as   *  cold / hot  cold  . The linearized stability
problem for infinitely small perturbations of the velocity, the temperature, and the pressure, in the
coordinate system rotating with the angular velocity  is defined by:
v
 Tae  v  p  v  Rae z
t
Pr
(II.2)

 w  
t
(II.3)
 v  0
(II.4)
where Ra  g b hot  cold R 3   is the Rayleigh number and Ta  2R 2  is the Taylor
number. Since equations (II.2)-( II.4) are linear, the solution of this problem is a good test for the
validation of the Galerkin approximations of the linear terms of equations (1)-(5).
We consider two sets of the boundary conditions for the system (II.2)-( II.4). The first one
(no-slip and no perturbation of temperature at all boundaries) corresponds to the calculations of
Hardin et al. (1990) for Ta = 0:
u  v  w  0,
0
at
z  0,  and
at
r 1
(II.5)
The second one (stress-free top and bottom with no perturbation of temperature, no-slip sidewall
with no perturbation of heat flux) corresponds to the analytical solution of Jones & Moore (1979)
(see also Goldstein et al. 1993) for Ta  0:
u v

 w  0,
z z
0
at
z  0,  ;
u  v  w  0,
34

0
r
at
r 1
(II.6)
It is known that in the case of a stationary cylinder heated from below the most unstable
azimuthal modes are k = 0 for  = 1, and k = 1 for  = 0.5 and 2 . All bifurcations are steady. The
critical Rayleigh numbers for the problem with boundary conditions (II.5), aspect ratios  = 0.5, 1,
and 2, and different azimuthal wavenumbers 0  k  10 were computed by Gelfgat & Tanasawa
(1993), using different number of basis functions in the series (29) and (36), NrNz = 44  1010.
The results corresponding to the most unstable modes were compared with the numerical result of
Hardin et al., 1990. The results of Gelfgat & Tanasawa (1993) showed that 1010 basis functions
suffice to make the values of the critical Rayleigh number converge for all values of the azimuthal
number in the interval 0  k  10 and the converged values of the critical azimuthal modes good
agree up to the third digit with the numerical result of Hardin et al., 1990.
Gelfgat & Tanasawa (1993) also considered the case of the rotating cylinder with the
boundary conditions (II.6). In this case the problem has an analytical solution (Jones & Moore,
1979; Goldstein et al., 1993), which shows that the instability is oscillatory and the critical
azimuthal wavenumber k grows with the increase of the Taylor number. The results of Gelfgat &
Tanasawa (1993) for   = 1 and Pr = 6.7 and Ta = 20, 40, 100, and 400 show that as in the
previous case, the values of the critical Raleigh number converge already with the use of 1010
basis functions. The converged values are in agreement (up to 6 decimal digits) with the analytical
solution reported in Goldstein et al., 1993.
These two test cases allow us to conclude that the linear terms of equations (1)-(5) are
correctly approximated. The spectrum of Galerkin projections of the linear terms rapidly converges
to the spectrum of the corresponding differential operators (6 digits of the critical Rayleigh number
coincide with the analytical solution already for 1010 basis functions).
II.2. Validation of bilinear terms
To validate the Galerkin projections of bilinear terms of equations (1)-(5) we use the relation
(  is the flow region,  is its boundary).
v  v, v

  2

 2
v
v
 v  v  vd     2   v    v  vd    2   vd 








  v2  v2

v2
     v  
  v d   v  n d  0
2
2
2





(II.7)
which holds for incompressible flow (v = 0) with no-throughflow (vn = 0) on the boundaries.
The corresponding relation for the energy equation is
35
v  , 



2
 


 2
2



 v    d   v   2  d      v 2   2





2
  vd   v  n d  0 (II.8)
2


Similar relations exist for each pair of basis functions of velocity or a pair of basis functions of
velocity and temperature. The Galerkin projection of the integrals (II.7) and (II.8) can be extracted
from the dynamical system (39) and written as
Nˆ ijl Z i Z j Z l  0
(II.9)
for each vector Z (summation over repeated indices is assumed). Here N̂ ijk is a matrix containing
projections of all bilinear terms of (1)-(5). A particular case when Z i =1 for all i leads to the
relation

Nˆ ijl  0
(II.10)
ijk
which can be used as a test for calculated values of N̂ ijk . This relation was used in Gelfgat et al.,
1996 for the validation of bilinear terms of the axisymmetric part of the problem.
The relation (II.10) cannot be used directly in the present case when only axisymmetrybreaking bifurcations are the object of the study. There are two reasons for this. First, we do not
need the whole matrix N̂ ijk , but only its part corresponding to the linearized system of equations
(20)-(24). The second reason is the complex form of the basis functions (31), (32) and (36) which
leads to the following definition of the scalar product:

u, v  u  v d 

2 1
   u  v*rdrdzd
(II.11)
000
where the overbar means complex conjugation. The presence of a complex conjugate in (II.11) does
not allow a direct use of (II.7) and (II.8) for the validation of bilinear terms.
To obtain relations for the validation of the bilinear terms, analogous to (II.9), we denote the
basic axisymmetric flow by V (which is a real function) and the three-dimensional perturbation by
v = (u,v,w) (which is a complex function), and vˆ  v  v (which is a real function by definition).
Then we rewrite (II.7) as
0
V  vˆ  V  vˆ , V  vˆ 
 V   V, V  vˆ   vˆ , vˆ 
 V   V, vˆ  vˆ   vˆ , V 
 V   vˆ , V  V   vˆ , vˆ 
 vˆ   V, V  vˆ   V, vˆ
36
(II.12)
According to (II.7)
V  V, V
 v̂  v̂ , v̂  0
(II.13)
The orthogonality of a constant ( k = 0 ) and a function with complex exponents exp(ik) for k  0
on the interval [0,2] gives:
V  vˆ , V
 V   V, vˆ  vˆ   V, V  0
(II.14)
Using (II.13) and (II.14) we can rewrite (II.12) as
V  v̂ V  v̂, V  v̂
 v̂  v̂ , V  V  v̂ , v̂  v̂  V, v̂  0
(II.15)
Now, recalling that a part of the matrix Nijk in (39), corresponding to the Navier-Stokes equation, is
defined as




Nijl  V j   v i , v l  v i   V j , v l
(II.16)
(here and in the following the vectors v with indices i, j and l correspond to any of the vector basis
functions Vijk, Wijk in (35) and (36) with the fixed index k and varying indices i and j; vectors V with
indices i, j and l correspond to any of the basis functions U ijr , z  in (4) ), we can rewrite (II.15) as
 
2 Re al Nijl  V  v̂ , v̂  v̂  V, v̂   v̂  v̂ , V  2Real v̂  v̂ , V
ijl
~
If we define a new matrix N ijl as


~
N ijl  v i   v j ,Vk
(II.17)
(II.18)
then the test relation for the real part of Nijk is the following:
 Real N
ijk

ijl  
 Real N~
ijl

(II.19)
ijk
A similar relation may be obtained for that part of the matrix Nijl in (II.9), which corresponds to the
energy equation. Denoting by  the axisymmetric solution,  the three-dimensional perturbation,
and ̂     , we obtain:
0
V  vˆ    ˆ ,   ˆ 
 V   ,   vˆ   ˆ , ˆ 
 V   , ˆ  vˆ   ˆ ,  
 V   ˆ ,   V   ˆ , ˆ 
 vˆ   ,   vˆ   , ˆ
37
(II.20)
According to (II.8):
V  ,
 vˆ  ˆ , ˆ  0
(II.21)
and due to orthogonality, as in (II.14) :
V  ˆ ,  V  ,ˆ  vˆ  ,  0
(II.22)
Two parts of the matrix Nijl in (II.9), which correspond to the energy equation, are defined as



(1)
N ijl
 V j   i , l ,
We can rewrite (II.20) as
Real N

ijk
2

(II.23)
 vˆ  , ˆ 
(II.24)
(2 )
N ijl
 v i    j ,l

(1)
( 2)
ijl  Nijl 
V  ˆ ,ˆ
  vˆ  ˆ ,   2Re al v  , 
( 3)
If we define a new matrix Nijl
as
( 3)
N ijl
 vi   j , l
(II.25)
(2)
(1)
then the test relation for the real parts of N ijl
and N ijl
is
 Real Nijl(1)  Nijk(2)    Real Nijl(3) 
ijk
(II.26)
ijl
The imaginary part of the matrix Nijl in (II.9) is not zero if the azimuthal component of the
basic axisymmetric flow is not zero. To validate the imaginary part of Nijl
we consider the
following integral relation (V is the azimuthal component of the axisymmetric basic state V):


I  Ve   v, v  v  Ve , v  v  v,Ve 

1 u
1
V
1 v
1
V
V
, u  2 Vv, u  u
,v  V
, v  Vu, v  w
,v 
r 
r
r
r 
r
z

1 w
v
1
v
v
V
,w  u
,V  uv,V  v
,V  w
,V
r 
r
r

z
(II.27)
Taking into account that v/ = ikv, we can rewrite (II.27) as:
I  ik
1
1
V
1
1
V
Vu, u  2 Vv , u  u
, v  ik Vv , v  Vu, v  w
,v 
r
r
r
r
r
z
ik
 ik 



1
v
1
1
v
Vw, w  u
,V  uv ,V  ik vv ,V  w
,V 
r
r
r
r
z
V 2
V
1
1
V
v V
v
2
u  w d   uv
 2 Vuv  Vuv + wv
 Vu
 uv- Vw d
 r
r
r
r
z
r r
z 

38
(II.28)
Applying integration by parts, we can obtain:
 uv

1
1
1 

V
V 

d    ruv
dr ddz    rVuv dr   V
ruv
dr
  ddz 
r
r 
r
r
0


0

0
1


1

r 1

u
v
  rVuv 
  V ruv dr ddz    V  uv + rv
 ru  dr ddz 
r 0
r
r
r  


0 
0

u
u
v
   V  v + v
 u  d
 r
r
r 
(II.29)

and





V

V





rdrd 
vw
d


vw
dz
rdrd


Vvw
dz

V
vw
dz




 z
  z 
  z
 z


o
o

o






z

w
v  




  Vvw
 V
vwdz rdrd    V  v z + w z  dz rdrd 
z  0  z


o

o




w
v
   V  v
+ w  d
 z
z 
(II.30)

Using (II.29) and (II.30) we can rewrite (II.28) as
I  ik 



V 2
2
u  w d 
r
u 1
v
1
1
w
v
v V
v
  Vv
 Vvu  Vu
 2 Vuv  Vuv- Vv
 Vw
 Vu
 uv- Vw d 
r r
r
r
r
z
z
r r
z 


 ik 



V 2
u u w
v
v
v
v 1
1 

2
u  w d   V v  +
 - u
u
w - w
 uv  3 uvd


r
r

r

z

r

r

z

z
r
r 


(II.31)
Now, using equation (18),
I  ik 


ik
u u w
v   +
  and
 r r z 
r

V 2
v
v
v
v 1
1
2
2
u  v  w d   V - u
u
w - w
 uv  3 uvd
 r
r
r
z
z r
r 
(II.32)

it follows from (29)-(32),(37),(38) that
uv ,V   uv,V ,
u
v
v
,V   u
,V ,
r
r
w
v
v
,V   w
,V
z
z
(II.33)
and
I  ik 



V 2
1
2
2
u  v  w d  4  Vuvd
 r

r

39
(II.34)
Finally, for the imaginary part of Nijl, which is defined as
  



iIm N ijl  Vi e    v j , v l  v j   Vi e  , v l
we obtain the following relation:
 Im N ijl    Im N ijl 
~
ijl
(II.35)
(II.36)
ijl
where
 


 1

~
Im N ijl  Im k
v j  v l , Vi  4 u j vl , Vi 
 r

(II.37)
~
(3)
Relations similar to (I.29)-(I.73) are obtained for the calculation of N ijl and N ijl
. For the
validation of the bilinear terms we require that the relations (II.19), (II.26) and (II.36) be satisfied
analytically for any number of the basis functions NrNz and for each azimuthal wavenumber k . For
example, if NrNz = 10, the relations (II.19), (II.26) and (II.36) request an analytical equality of
sums of 106 numbers, which is a rather strong requirement. Note that the relations (II.19), (II.26)
and (II.36) can be used for validation of the conservative properties of other numerical methods as
well.
Convergence of the axisymmetric steady states was already studied in Gelfgat et al. (1996)
and results were compared with the solutions obtained by the finite volume method. It was found
that truncation up to 3030 basis functions in r- and z-directions, respectively, provides a
considerably good accuracy (e.g., 3 correct decimal digits for maximal value of the stream function
and coordinates of the maximum). Figure 7 illustrates convergence of the Galerkin series,
corresponding to steady base states. The coefficients Aij versus their indices i and j (see eq.(19)) are
plotted there for the steady states, calculated at the critical Re, for  = 1 and 3.5.
40