St.-Petersburg State Polytechnic University Department of Aerodynamics, St.-Petersburg, Russia

A. ABRAMOV, N. IVANOV & E. SMIRNOV

Numerical analysis of turbulentRayleigh-Bénard convection in confined enclosures

using a hybrid RANS/LES approach

E-mail: aerofmf@citadel.stu.neva.ru

“FLOMANIA”, DES WORKSHOP, 2/3 July 2003, St.-Petersburg

“FLOMANIA”, DES WORKSHOP, 2/3 July 2003, St.-Petersburg

Introduction

Problem description

Mathematical model

Computational aspects

Structure of turbulent convection

Heat transfer predictions

Conclusions

OUTLINE

Abramov et al. SPTU, Russia

1. Full Direct Numerical Simulation (DNS): no turbulence model

2. Under-resolved (coarse-grid) DNS: no turbulence model

3. Unsteady Reynolds-Averaged Navier-Stokes (RANS): modeling of all-scales background turbulence

4. Large Eddy Simulation (LES): modeling of subgrid-scale turbulence

5. RANS/LES hybridization, in particular, non-standard DES

3D Unsteady formulations: modeling levels

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Problem descriptionHigh-Ra Rayleigh-Bénard mercury and water

convection in confined enclosures

H

z

H

D = H

z

r

g Cold walls, Tc

Hot walls, Th

Adiabaticwalls

Mercury,Pr = 0.025

Water,Pr = 7Ra > 108

THgVb - buoyancy velocity

ch TTT

Scales:

a

3THgRa

aPr

H

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Mathematical model Navier-Stokes equations averaged/filtered for a RANS/LES model; Boussinesq’s approximation for gravity buoyancy

zeff TeDiv*pt

SVVV 2Div

TTtT

eff aV

0 V

,ppp h* where ,RaPr/ teff

tteff Pr/PrRa/ 1a

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Turbulence Modelling:RANS / LES one-equation turbulence model

(Abramov & Smirnov, 2002)

ykRe),(RefF,yCl,Fl

kyy

RANS 432

3

2

3kCLES 3

zyx

klfC tt

C,Flminlt

151 .A,A

ReexpF y

0101 .A,A

Reexpf y

LESRANS.

tjk

t

j

,maxSxk

xdtdk

22

Modified Wolfshtein model for a RANS zone:

750.C

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Abramov et al. SPTU, Russia

3D Navier-Stokes code SINF Features of the last version (June 2003):

steady and unsteady computation options, stationary and rotatingreferences

conjugate heat transfer in solid-liquid subdomains multi-component mass transfer a number of RANS turbulence models implemented LES and RANS/LES possibilities for domains of complex geometry

block-structured matching and non-matching body-fitted grids second-order physical time stepping second-order finite-volume spatial discretization various upwind schemes, implicit solvers, parallel version

Computational aspects

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Computational program

Grids of about 160000 cells

Water convection:Ra = 5108; 5109

Pr = 7

Conditions of experiments:Zocchi et al. (Physica A.,1990)

Cioni et al. (J. Fluid Mech.,1997)Qiu et al. (Phys. Rev. E., 1998) etc.

Mercury convection:108 < Ra < 5109

Pr = 0.025

Conditions of experiments:Takeshita et al. (Phys. Rev. Lett.,1996)

Cioni et al. (J. Fluid Mech.,1997)Glazier et al. (Nature, 1999)

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Structure of turbulent convection

0.5

0.5

0.70.9

0.50.30.2

0.6

0.4

0.7

0.4Vb

Mercury convection:Ra = 108, Pr = 0.025

Vb

Velocity vector patterns

Temperature isolines

Vertical velocity at middle

horizontal planew

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Structure of turbulent convectionMercury convection:Ra = 108, Pr = 0.025

Equiscalar surfaces of vertical velocity

w = 0.25 (gray) andw = -0.25 (black)

Temperature and velocity vector fields

Vertical velocity distributions(time-averaging over the interval

of 10 time units)

w

0 1 0 0 2 0 0 3 0 0-0 .5 0

-0 .2 5

0 .0 0

0 .2 5

0 .5 0W

Abramov et al. SPTU, Russia

Structure of turbulent convection

B

A

A

B

A

B

Velocity vector and temperature fields

Water convection:Ra = 5109, Pr = 7

Equiscalar surfaces of vertical velocity

w = 0.05 (black) andw = -0.05 (gray)

A

B

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Characteristics of the global circulation

0 .0 0 .2 0 .4 0 .6 0 .80 .0

0 .2

0 .4

0 .6

0 .8

1 .0

108

5108

8.7108

z

Th

0 .0 0 .4 0 .8 1 .20 .0

0 .2

0 .4

0 .6

0 .8

1 .0

108

5108

8.7108

wm

z

wmc=2Vg

0 .0 0 .2 0 .4 0 .60 .0

0 .2

0 .4

0 .6

0 .8

1 .0

5108

5109

z

Th

0 .0 0 .2 0 .4 0 .60 .0

0 .2

0 .4

0 .6

0 .8

1 .0

wm

z

5108

5109

Profiles of maximum horizontal temperature difference and vertical velocity difference

Water Mercury

Reynolds number Reg = VgH /, versus Rayleigh number. Mercury

Abramov et al. SPTU, Russia

Temperature isosurface T = 0.9 colored by vertical velocity

1 0 0 1 5 0 2 0 00 .2

0 .4

0 .6

0 .8

1 .0

Temperature fluctuations near the bottom wall (z = 0.03, r = 0)

Thermal plumes in high-Ra convection

0 5 0 1 0 0 1 5 0 2 0 00 .4 0

0 .4 5

0 .5 0

0 .5 5

Temperature isosurfaces T = 0.45 and T = 0.55

Temperature fluctuations near the top wall (z = 0.96)

TT

Ra = 5109Ra = 5108

w

plumes

plumes

Abramov et al. SPTU, Russia

0 1 0 0 2 0 0 3 0 00 .3

0 .4

0 .5

0 .6

0 .7T

0 .0 1 0 .1 1 f1 E -8

1 E -7

1 E -6

1 E -5

1 E -4

1 E -3

1 E -2

0 .1

1E T -5/3

-4

Mercury, Ra = 5108

z = 0.5

0 1 0 0 2 0 0 3 0 0-0 .5 0

-0 .2 5

0 .0 0

0 .2 5

0 .5 0W

0 .0 1 0 .1 1 f1 E -8

1 E -7

1 E -6

1 E -5

1 E -4

1 E -3

0 .0 1

0 .1

1

1 0E W

-5/3

-4

0 5 0 1 0 0 1 5 0 2 0 00 .4 0

0 .4 5

0 .5 0

0 .5 5

0 4 0 8 0 1 2 0 1 6 0 2 0 0-0 .1 0

-0 .0 5

0 .0 0

0 .0 5

0 .1 0

Turbulent vertical velocityand temperature fluctuations

z = 0.75

T

W

Water, Ra = 5109

z

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Boundary layers near the isothermal walls

0 .0 0 0 .0 5 0 .1 0 0 .1 50 .0

0 .2

0 .4

0 .6

0 .8

1 .0

5108

5109

z

uh

Thermal and viscous boundary layer thicknesses as functions of Ra

Temperature profile near the top wall (mercury)

Mean horizontal velocity profile (water)

Mercury:V < T

Water:V > T

T

0 .0 0 .2 0 .4 0 .60 .9 0

0 .9 2

0 .9 4

0 .9 6

0 .9 8

1 .0 0

z

T

108

5108

8.7108

1 E + 8 1 E + 9

1 E -3

0 .0 1

0 .1

- RANS/LES in water- RANS/LES in mercury- DNS Verzicco et al., 99- Experiment Takeshita et al., 96

Ra

T

1 E + 8 1 E + 9

1 E -3

0 .0 1

0 .1 - DNS Verzicco et al., 99- Experiment Takeshita et al., 96

- RANS/LES in water- RANS/LES in mercury

Ra

V

Ra

Ra

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Heat transfer predictions

0 1 0 0 2 0 0 3 0 0 4 0 02 0

3 0

4 0

5 0Ra = 5108

t

Nu

0 .0 1 0 .1 11 E -7

1 E -6

1 E -5

1 E -4

1 E -3

0 .0 1

0 .1

1

-5/3

-4

fq

f

Eq

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0

8 8

9 2

9 6

1 0 0Nu

Nusselt number fluctuations in mercury and water convection

Ra = 5109

1 E + 8 1 E + 9 1 E + 1 0

1 0

1 0 0

- RANS/LES in water- RANS/LES in mercury- Exp. Cioni et al., 96- Exp. Goldstein, 80- Exp. Glazier, 99

Nu

Ra

t

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Numerical simulations of high-Ra R-B convection was performed with a non-standard DES approach based on the one-equationk-model of unresolved turbulence

The specific patterns of fully developed turbulent convection were analyzed, especially the formation of a large-scale circulation cell and thermal plumes for both the configurations

In mercury the global circulation, velocity and temperature fluctuations are considerably more intensive than in water

Relation between the thicknesses of the viscous layer and the thermal boundary layer was established

Numerically predicted Nusselt numbers were in quantitative agreement with registered experimental laws

CONCLUSIONS

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