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Proceedings of ICHIT- 06

26 February - 5 March 2006, Moscow, Russia




DIRECT NUMERICAL SIMULATION OF TURBULENT
RAYLEIGH-BENARD CONVECTION


IGOR B. PALYMSKIY

Modern Academy for Humanities, Novosibirsk Branch , Novosibirsk, Russia,
630064
palymsky@hnet.ru



Abstract
Turbulent convectional flow of water in horizontal layer with free and
rigid horizontal boundaries, arising by heating from below, is numerically
simulated by spectral method using the Boussinesq model without any
semiempirical relationships (DNS) in 2-D case. The results of the both
numerical simulations compare with experimental data. We studied the time
and space spectrums of temperature pulsations and kinetic energy in both
free and rigid simulations. The Kolmogorov (k-5/3), Obukhov-Bolgiano (k-7/5
for temperature pulsations and

k-11/5 for kinetic energy of pulsations) spectrums have been derived in
numerical simulations. These spectrums were observed earlier in
experimental investigations of turbulent convection of gaseous He. It is
surprising that ranges of k-5/3 and k-7/5 spectrums are partially
coincident.


introduction

At last time many workers have studied thermal Rayleigh-Benard convection
using numerical simulation. As rule, they used spectral methods with
periodic boundary conditions. In numerical simulations were derived
stationary, periodic, quasiperiodic and stochastic regimes [1]. Some
authors performed 2-D and 3-D simulations for high supercriticality with
free [2,3] and rigid [4,5] boundary conditions on the horizontal planes.
The results of correct performed 3-D numerical simulations with rigid
boundary conditions in air, as rule, have good agreement with experimental
data ([6] and [7], for instance). But we have a big troubles with deriving
of time-dependent solutions for 2-D convection in air and gaseous He, up to
large Rayleigh number all solutions are steady state [4].
On the other hand, it is revealed recently that the time-dependent
solutions of 2-D convection with free boundary conditions (stress free) at
Prandtl number is equal to 10 have a good agreement with experimental data
on turbulent convection in air and gaseous He [8]. It is very significant
and practical, as using of free boundary conditions very simplifies the DNS
of turbulent convection, simple and efficient numerical algorithms have
been generated using the formulas of linear stability theory [9,10].
For instance, in table 1 we compare the calculating Nusselt numbers. Here
and below

r = Ra/Racr is supercriticality, when Ra and Racr are Rayleigh number and
the critical value of Rayleigh number, respectively.

Table 1
Comparing of Nusselt Number at r = 33000

|[4], 2-D, rigid, water |24.8 |
|[11], experiment, |25.2 |
|gaseous He |27.5 |
|[12], experiment, air |33.0 |
|[2], 3-D, free, air |28.7 |
|Present, 2-D, free, | |
|water | |

The same situation you can see in [8] for

r = 9800.It shows that for simulations with free boundary conditions the
value of Prandtl number must be higher because of decreasing of effective
Prandtl number by free boundary conditions.
The aim of this work is more detailed comparing of results of 2-D
simulations with free and rigid boundary conditions on horizontal planes
with experimental data on turbulent convection.
PROBLEM FORMULATION

Turbulent convectional flow of water in a horizontal layer numerically is
simulated by heating from below. The fluid is viscous and incompressible.
The flow is time-dependent and two-dimensional. Boundaries of a layer are
isothermal and free (stress-free) or rigid. The model Boussinesq is used
without semiempirical relationships. The dimensionless system of equations
in terms of deviations from an equilibrium solution, representation of
problem solution in the form of sum of eigenfunctions of linear stability
theory, the boundary conditions, the special numerical method, testing and
the results of linear and non linear analysis (on model non linear system)
for free boundary conditions are described in works [9, 10]. In our
simulations with free boundary conditions we used up to 257*63 harmonics at
supercriticality up to r = 34000. In the test simulations we used up to
513*127 of harmonics with free boundary conditions.
For simulations with rigid boundary conditions we used the spectral
representation in x-direction and finite differences in y-direction with
uniform mesh. We used up to 257 harmonics in x-direction and 65 points in y-
direction at supercriticality up to 7000. In the test simulations we used
up to 513*65 (or 257*129) of harmonics with rigid boundary conditions.
We simulated the convection flows for the Prandtl number Pr is equal to 10,
for all simulations the dimensionless periodicity interval is equal to 2?,
the dimensionless distance between the planes is equal

to 1.
So, we are solving the system of equations

[pic]

where ? is a stream function, ? is the vortex, Q is the temperature
deviation from equilibrium profile (the total temperature being T = 1 - y
+ Q), ?f = fxx +fyy is the Laplace operator, Ra = g?Н3dQ/?? is the Rayleigh
number, Pr = ?/? is the Prandtl number, g is the gravitational
acceleration, ?, ?, ? are the coefficients of thermal expansion, kinematics
viscosity and thermal conductivity, respectively, H is the layer height and
dQ is the temperature difference on the horizontal boundaries.


RESULTS AND DISCUSSION

Fig.1 represents the average temperature profile. At figs. 1 - 4 below y
denotes transverse coordinate. At fig.1 symbol ? denotes experimental
results [7]

(r = 5900, air), dash line - experimental results [13]

(r = 5500, water), solid line - results of present work with free boundary
conditions (r = 6000, Pr = 10).

[pic] Figure 1
Average temperature profile at r = 6000

Fig.2 represents the rms of vertical velocity pulsations. Here symbol ?
denotes the experimental results [7] (r = 5900, air), symbol ” -
experimental result [12] (r = 5900, air), solid line - results of present
work with free boundary conditions

(r = 5500, Pr = 10).

[pic]
Figure 2
Rms of vertical velocity pulsations at r = 6000

Fig.3 represents the rms of temperature pulsations at moderate
supercriticality r = 1250. Here symbol ? denotes the experimental results
[7]

(r = 1470, air), symbol ? - experimental result [12]

(r = 1400, air), symbol ? - experimental result of Somerscales, 1965 (r =
1170, data is from work [7]), symbol ? - experimental result [14] (r =
1250, gaseous He), solid line - results of present work with free boundary
conditions (r = 1250, Pr = 10). The experimental data has a big scatter and
derived numerical results have a reasonably good agreement with
experimental data, some waviness is coupled possibly with Gibbs effect for
spectral representations.

[pic]

Figure 3
Rms of temperature pulsations at r = 1250

Figs.1-3, table 1 and [8] demonstrate that results of numerical 2-D
simulation with free boundary conditions on the horizontal planes are
consistent with experimental data in air and gaseous He.
Fig.4 represents the profile of rms temperature pulsations at r = 6000,
here black solid line is result of present simulation (rigid), red and blue
solid lines are theoretical laws [15].

[pic]

Figure 4
Rms of temperature pulsations at r = 6000

Fig.5 represents the profile of rms vertical velocity pulsations at r =
6000, here black solid line is result of present simulation (rigid),
symbols ? and ? - experimental result [16] at r = 7300 and r = 18900,
respectively (water, Pr = 6.1, aspect ratio is equal to 4.5 and Racr — 1820
[17], result is recalculated using

v' ~ r0.44ћPr0.333 for scale [16]), magenta line is theoretical law [15].

[pic]

Figure 5
Rms of vertical velocity pulsations at r = 6000

Fig.6 represents the values of rms vertical velocity pulsations in centre
between the planes divided by Pr1/3, here green solid line - experimental
result of [16] (water, Pr = 6.1), symbol ? - present numerical simulations
(rigid, Pr = 10), symbol ? - experimental result of [7] (air, Pr = 0.71)
and symbol ? - experimental result of [18] (water,

Pr = 6.1).

[pic]

Figure 6
Rms of vertical velocity pulsations in centre

Table 2
Comparing of Nusselt Number at r = 4000
|Work |Nu |Deviati|
| | |on |
| | |in % |
|Present, rigid |16.9|0 |
|O'Toole&Silveston, |15.3|-9.5 |
|1961 [20] | | |


In table 2 we compare the calculating Nusselt number and experimental data
on turbulent convection in water. The agreement is good, but our numerical
result is slightly higher.
Figs.4-6 and table 2 demonstrate that results of numerical 2-D simulation
with rigid boundary conditions on the horizontal planes are consistent with
experimental data in water and theoretical laws.
For free boundary conditions on horizontal plates, the values of Nusselt
number at r > 700 describe by formula:
[pic]

This law practically coincides with experimental laws from [19] (Nu =
1.222•r0.3) and O'Toole and Silveston, 1961 [20] (Nu = 1.222•r0.305) and
close to experimental law [12] (Nu = 1.211•r0.3). The same power law has
been derived also in numerical simulation [3] (Nu ~ r0.301, infinite
Prandtl number model, 2-D, free).
For rigid boundary conditions on horizontal plates, the values of Nusselt
number at r > 300 describe by formula:
[pic]

In recent experimental work [21] was found that

Nu ~ Ra0.309, in some experimental and numerical works the other laws were
found - close to Nu ~ r2/7 [5,11,13,14,16] and close to Nu ~ r1/3 [22]. The
detail review of experimental Nu-Ra laws may be found in work [22] (see
also [20]).


Time and space spectrums

Fig.7 represents the time spectrum of temperature pulsations in center of
cell, here solid line is result of present simulation (free, r = 6500),
blue points are experimental data [23] (gaseous He, Ra = 1.1ћ108,

r — 6400, Racr — 17000 at aspect ratio is equal to 0.5 [17]).
Normalizations are same. Frequency f is in unit of ?/H2.
The green line represents the experimentally defined boundary of two
regimes:

[pic]

above the frequency f0 a power law has slope
-1.4 (Obukhov-Bolgiano spectrum), and below f0 the spectrum is flat.
[pic]
Figure 7
Time spectrum of temperature pulsations (free)

Figs.8 and 9 represent the one-dimensional space spectrums of temperature
pulsations:

[pic]

Here black points are result of present simulation (free, r = 26000).

[pic]
Figure 8
E1(k) space spectrum of temperature (free)

Figure 9
E2(m) space spectrum of temperature (free)
We can see the Kolmogorov (k-5/3), Obukhov-Bolgiano (k-7/5) and k-2.4
spectrums earlier observed in experimental investigations of turbulent
convection of gaseous He [21,23]. Fig.9 shows the slightly distorted
spectrums of Kolmogorov and Obukhov-Bolgiano. Part k-1 is range of
enstrophy transfer inherent to 2-D turbulent flows. It is surprising that
ranges of k-5/3 and k-7/5 spectrums are partially coincident.
Fig.10 represents the one-dimensional space spectrum of temperature
pulsations E2(m) for problem with rigid boundary conditions, here points
are result of present simulation (r = 6000).

[pic]
Figure 10
E2(m) space spectrum of temperature (rigid)

We can see also the Kolmogorov (k-5/3), Obukhov-Bolgiano (k-7/5) spectrums.

We calculated also the one-dimensional spectrum of kinetic energy EK2(m) by
analogous formula. Fig. 11 shows the one-dimensional spectrums of kinetic
energy EK2(m) for rigid (r = 6000) boundary conditions.

[pic]
Figure 11
EK2(m) space spectrum of kinetic energy (rigid)

We can see the Obukhov-Bolgiano spectrum k-2.2 for kinetic energy.

CONCLUSION

We compare the results of our 2-D simulations with free and rigid boundary
conditions on the horizontal planes and experimental data on turbulent
convection. Prandtl number is equal to 10 in a both simulations.
It is revealed that results of simulations with free boundary conditions
have a good agreement with experimental data on turbulent convection in air
and gaseous He. The profiles of mean temperature, rms of temperature and
vertical velocity pulsations are close at enough high supercriticality. We
observe also a good agreement with experimental data in time spectrum of
temperature pulsations in centre of cell. The Nusselt numbers are close
too.
The results of simulations with rigid boundary conditions have a reasonable
agreement with experimental data on turbulent convection in water. The
profiles of rms of temperature and vertical velocity pulsations are close
to experimental data and theoretical laws. The Nusselt numbers at rigid
boundary conditions are slightly higher, but exponent of the Nu-Ra power
law is same for free and rigid simulations.
We studied the time and space spectrums of temperature pulsations and
kinetic energy in both free and rigid simulations. The Kolmogorov (k-5/3),
Obukhov-Bolgiano (k-7/5 for temperature pulsations and k-11/5 for kinetic
energy of pulsations) and k-2.4 spectrums have been derived in our
simulations. These spectrums were observed earlier in experimental
investigations of turbulent convection of gaseous He. It is surprising that
ranges of k-5/3 and k-7/5 spectrums are partially coincident.


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Igor Palymskiy is Professor of Modern University for Humanities,
Novosibirsk Branch, Mathematics Department. His main scientific interests
are Direct Numerical Simulation of Turbulent Flows and Flows with
Hydrodynamical Instabilities.