VINCA – Institute of Nuclear Sciences Laboratory for Thermal Engineering and Energy 114 TURBULENCE...

VINCA – Institute of Nuclear SciencesLaboratory for Thermal Engineering and Energy

111414

TURBULENCE HEAT-FLUX MODELLING OF TURBULENCE HEAT-FLUX MODELLING OF NATURAL CONVECTION IN TWO-DIMENSIONAL NATURAL CONVECTION IN TWO-DIMENSIONAL

THIN-ENCLOSURETHIN-ENCLOSURE

Zarko M. StevanovicZarko M. Stevanovic

VINCA Institute for Nuclear SciencesVINCA Institute for Nuclear SciencesLaboratory for Thermal Engineering and Energy Laboratory for Thermal Engineering and Energy

11001 Belgrade, P.O. Box 522, Yugoslavia11001 Belgrade, P.O. Box 522, [email protected]@rt270.vin.bg.ac.yu


221414

C O N T E N TC O N T E N T

Objective of WorkObjective of Work 33Turbulence ModellingTurbulence Modelling 55Differential - to - Algebraic TruncationDifferential - to - Algebraic Truncation 66Computational DetailsComputational Details 88Computational ResultsComputational Results 99Conclusions Conclusions 1313


331414

O b j e c t i v e o f W o r kO b j e c t i v e o f W o r k

• Application of the second-moment closure, even in its simplest form, to the solution of Application of the second-moment closure, even in its simplest form, to the solution of thermal buoyancy-driven flows represent still a formidable task. thermal buoyancy-driven flows represent still a formidable task.

• This is not so much because of a large of differential equations involved, but more This is not so much because of a large of differential equations involved, but more because of still too high uncertainties in modeling various terms in the equations that because of still too high uncertainties in modeling various terms in the equations that obscure real physics. obscure real physics.

• It should be kept in mind that in buoyant convection, irrespective of how large the bulk It should be kept in mind that in buoyant convection, irrespective of how large the bulk Rayleigh number may be, the molecular effects will remain important in a significant Rayleigh number may be, the molecular effects will remain important in a significant portion of the flow domain, requiring the implementation of low-Reynolds number portion of the flow domain, requiring the implementation of low-Reynolds number modifications. modifications.

• This makes the use of standard wall function inapplicable and requires the integration This makes the use of standard wall function inapplicable and requires the integration straight up to the wall, which in turns calls for a very fine numerical mesh. straight up to the wall, which in turns calls for a very fine numerical mesh.

• There is, therefore, much to be gained if the differential model can be truncated to an There is, therefore, much to be gained if the differential model can be truncated to an algebraic form in which transport equations will be solved only for major scalar algebraic form in which transport equations will be solved only for major scalar quantities, primarily quantities, primarily k,k, , ,

• Therefore, the application of simpler models (at most at the algebraic level) should be Therefore, the application of simpler models (at most at the algebraic level) should be served as a convenient alternative for handling complex industrial situations. served as a convenient alternative for handling complex industrial situations.

2


441414

• These prospects have motivated present work, aimed at verifying These prospects have motivated present work, aimed at verifying the current practice in modeling of the transport equations for the the current practice in modeling of the transport equations for the turbulent heat-flux and temperature-variance and at exploring the turbulent heat-flux and temperature-variance and at exploring the limits of simpler turbulence models at the algebraic level. limits of simpler turbulence models at the algebraic level.

• Results of direct numerical simulation and experimental data of Results of direct numerical simulation and experimental data of turbulent natural convection between two differentially heated turbulent natural convection between two differentially heated vertical plates for vertical plates for RaRa = 8.227x10 = 8.227x1055 and for and for RaRa = 5.4x10 = 5.4x1055 have been have been used to validate the algebraic heat flux model.used to validate the algebraic heat flux model.


551414

T u r b u l e n c e M o d e l l i n gT u r b u l e n c e M o d e l l i n g

D( )

D

u u

tP G

i j

ij ij ij ij ij D

D( )

D

k

tP Gk D

D( )

D

t k

C f P C G C f S D 1 1 1 2 2

D( )

D

u

tu u

T

xu

U

xg C

kui

i i jj

ji

ji i i D 2

1

D( )

D

2

2 2t

P D

D( )

D

tC P C P

kC C f

k D 1 2 3 4

2

2 5

ij i j ij ij ij ij ijCk

u u k Ck

P P C G G 123 2

23 3

23

P u uU

xu u

U

xij i k

j

kj k

i

k

G g u g uij i j j i P u u

U

xi ji

j

G g ui i P uT

xjj

D

x xC u u

xj ji j

i

Sk

max . ,0 83 1 0

2 2

Transport Equations:Transport Equations:

Modelled Terms:Modelled Terms:

Specification of CoefficientsSpecification of Coefficients

1.00.090.090.090.070.090.220.60.61.8

0.82.20.721.31.441.921.440.60.65 C1 C2 C3 C

1 C

3 C

4 C

5

C1 C3 Cs Ck C C C C

C1

f C2

i i k k in

a u u n nx

12

k 3 2/

C xn


661414

f

D i f f e r e n t i a l - t o - A l g e b r a i c T r u n c a t i o nD i f f e r e n t i a l - t o - A l g e b r a i c T r u n c a t i o n

uT

xit

i

1. The simple gradient transport model (Boussinesq)1. The simple gradient transport model (Boussinesq)( isotropic eddy viscosity / diffusivity formulation ) ( isotropic eddy viscosity / diffusivity formulation )

( proportionality between transport of turbulent heat flux and ( proportionality between transport of turbulent heat flux and transports of temperature variance and turbulent kinetic energy )transports of temperature variance and turbulent kinetic energy )

3. The more refined formulation ( )3. The more refined formulation ( ) D( / ) / D u k ti2 0

u

u uT

xu

U

xg

Ck

uT

x ku u

U

xg u

i

i kk

ki

ki

kk

i ki

ki i

2

1 2

1

22

1

2

( neglecting the transport entirely )( neglecting the transport entirely ) 2. The reduced algebraic expression ( )2. The reduced algebraic expression ( ) D / D u ti i D 0

uC

ku u

T

xC u

U

xC gi i k

kk

i

ki

11 1

12 3

2


771414

Reduced Model to the Three-Equation Model: Reduced Model to the Three-Equation Model: kk - - - - 2θ

- Lam-Bremhorst Low-Re - Lam-Bremhorst Low-Re k k - - model model

- Additional Transport Equation for- Additional Transport Equation for 2θ

- Reduced Algebraic Expression for- Reduced Algebraic Expression for iθu


881414

C o m p u t a t i o n a l D e t a i l sC o m p u t a t i o n a l D e t a i l s

Th = 1 U = 0 W = 0

Tc = 0 U = 0 W = 0

T /x =0; W = 0

Ra = 5.4 x 105 Ra = 8.227 x 105

Pr = 0.71

2h = 1

z

x

T /x = 0; W = 0

Test Problem ConfigurationTest Problem Configuration Basic References:Basic References:

• Betts, P.L. and Boghari, I.H., (1996), Betts, P.L. and Boghari, I.H., (1996), “Experiments on turbulent natural “Experiments on turbulent natural convection of air in tall cavity”,convection of air in tall cavity”, Proc. of Proc. of the 5th ERCOFTAC workshop on the 5th ERCOFTAC workshop on Refined Flow Modeling, Chatou, France.Refined Flow Modeling, Chatou, France.

• Versteegh, T.A.M., (1998), Versteegh, T.A.M., (1998), “Numerical “Numerical simulation of natural convection in a simulation of natural convection in a differentially heated, vertical channel”,differentially heated, vertical channel”, Ph.D. Thesis, Printed by Ponsen & Ph.D. Thesis, Printed by Ponsen & Looijen, The Netherlands.Looijen, The Netherlands.


991414

C o m p u t a t i o n a l R e s u l t sC o m p u t a t i o n a l R e s u l t s

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

W / u*

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

uu / ( u* ) 2

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

k / ( u* ) 2

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

7

8

vv / ( u* ) 2

x / (2h)

u*

.

Dynamic field profiles for Dynamic field profiles for Ra = Ra = 8.227 x 108.227 x 1055 , scaled by friction velocity , scaled by friction velocity ( ( - Betts and Bokhari, [1]; - Betts and Bokhari, [1]; - Versteegh, [18]); - Versteegh, [18]); AHF Model )AHF Model )

21

wall

* dxdWνu


10101414

0.0 0.1 0.2 0.3 0.4 0.5-0.5

0.0

0.5

1.0

1.5

2.0

2.5

uw / ( u* ) 2

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

14

16

ww / ( u* ) 2

x / (2h)

Dynamic field profiles for Dynamic field profiles for Ra = Ra = 8.227 x 108.227 x 1055 , scaled by friction velocity , scaled by friction velocity ( ( - Betts and Bokhari, [1]; - Betts and Bokhari, [1]; - Versteegh, [18]); - Versteegh, [18]); AHF Model )AHF Model )

21

wall

* dxdWνu


11111414

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

T / *

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0

u / ( *u* )

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

2 / ( * ) 2

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.5-1

0

1

2

3

4

5

w / ( *u* )

x / (2h)

u*

*

.

Thermal field profiles for Thermal field profiles for Ra =Ra = 8.227 x 10 8.227 x 1055 , scaled by friction velocity , scaled by friction velocity and temperature and temperature ( ( - Betts and Bokhari, [1]; - Betts and Bokhari, [1]; - Versteegh, [18]; - Versteegh, [18]; AHF Model )AHF Model )

wall

dxdTuaθ


12121414

0.0 0.1 0.2 0.3 0.4 0.50

2

4

6

8

10

12

T / *

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50.0

0.5

1.0

1.5

2.0

u / ( *u* )

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

2 / ( * ) 2

x / (2h)

0.0 0.1 0.2 0.3 0.4 0.5-1

0

1

2

3

4

5

w / (*u* )

x / (2h)

u*

*

.

Thermal field profiles for Thermal field profiles for Ra =Ra = 5.4 x 10 5.4 x 1055 , scaled by friction velocity , scaled by friction velocity and temperatureand temperature

( ( - Betts and Bokhari, [1]; - Betts and Bokhari, [1]; - Versteegh, [18]; - Versteegh, [18]; AHF Model )AHF Model )

walldxdTuaθ


13131414

2

C o n c l u s i o n sC o n c l u s i o n s

1.1. Results of direct numerical simulation (DNS) of the natural-convection flow between Results of direct numerical simulation (DNS) of the natural-convection flow between differentially heated infinite vertical plates of Versteegh, and as well as experimental differentially heated infinite vertical plates of Versteegh, and as well as experimental data of Betts and Bokhari, were used for validation of the algebraic heat-flux (AHF) data of Betts and Bokhari, were used for validation of the algebraic heat-flux (AHF) model for computing turbulent thermal convection in two-dimensional thin-enclosure. model for computing turbulent thermal convection in two-dimensional thin-enclosure.

2.2. Despite of the large differences in the turbulence parameter-by-parameter comparison, Despite of the large differences in the turbulence parameter-by-parameter comparison, the complete second-moment closure should be replaced by low-the complete second-moment closure should be replaced by low-Re- k Re- k - - - turbulence - turbulence model with the algebraic heat-flux terms.model with the algebraic heat-flux terms.

3.3. AHF model reproduces well the mean flow (average velocity and temperature) and major AHF model reproduces well the mean flow (average velocity and temperature) and major turbulence property (turbulence kinetic energy) in the considered thin-cavity case. turbulence property (turbulence kinetic energy) in the considered thin-cavity case.

4.4. No gain is achieved by employing the refined, computationally more inconvenient, No gain is achieved by employing the refined, computationally more inconvenient, algebraic refined expression, as compared with the reduced expression. algebraic refined expression, as compared with the reduced expression.

5.5. However, even the reduced expressions show important advantages as compared with However, even the reduced expressions show important advantages as compared with simple isotropic eddy-diffusivity model, because the heat-flux includes all major simple isotropic eddy-diffusivity model, because the heat-flux includes all major generation terms (mean temperature, and velocity gradients, and thermal turbulence generation terms (mean temperature, and velocity gradients, and thermal turbulence self-amplification through the temperature variance).self-amplification through the temperature variance).

2


14141414

Finally, we can conclude that one of the minimum level of modeling Finally, we can conclude that one of the minimum level of modeling which can reproduce the major flow features in more complex which can reproduce the major flow features in more complex geometry is the algebraic model which accounts for all major geometry is the algebraic model which accounts for all major sources of turbulent heat flux and thus far expresses the turbulent sources of turbulent heat flux and thus far expresses the turbulent heat flux vector in terms of the mean temperature gradient, mean heat flux vector in terms of the mean temperature gradient, mean flow deformation and temperature variance interacting with the flow deformation and temperature variance interacting with the gravitational vector.gravitational vector.

Also, in the complex geometry of natural convection fluid flow Also, in the complex geometry of natural convection fluid flow problems, one of the most convenience turbulence model is LVEL problems, one of the most convenience turbulence model is LVEL model, proposed by professor D.B. Spalding.model, proposed by professor D.B. Spalding.

VINCA – Institute of Nuclear Sciences Laboratory for Thermal Engineering and Energy 114 TURBULENCE...

Documents

Transcript of VINCA – Institute of Nuclear Sciences Laboratory for Thermal Engineering and Energy 114 TURBULENCE...