turbulence heat-flux modeling of natural convection in two-dimensional thin-enclosure

Seminar za ReologijuSeminar za ReologijuBeograd, 22. Februar 2005Beograd, 22. Februar 2005http://www.cm2004.matf.bg.ac.yuhttp://www.cm2004.matf.bg.ac.yu

TURBULENCE HEAT-FLUX MODELING OF NATURAL TURBULENCE HEAT-FLUX MODELING OF NATURAL CONVECTION IN TWO-DIMENSIONAL THIN-ENCLOSURECONVECTION IN TWO-DIMENSIONAL THIN-ENCLOSURE

Laboratorija za termotehniku i energetikuLaboratorija za termotehniku i energetikuInstitut za nuklearne nauke - VinInstitut za nuklearne nauke - Vinčača

ŽŽarkoarko M. Stevanović M. Stevanović

Seminar za Reologiju Beograd, 22. februar 2005http://www.cm2004.matf.bg.ac.yu

2

Introduction ( 1 )Introduction ( 1 )

This This lecturelecture deals with the thermal buoyancy-driven flow in two- deals with the thermal buoyancy-driven flow in two-dimensional thin enclosure with an aspect ratio of 28.6:1, that dimensional thin enclosure with an aspect ratio of 28.6:1, that can be regarded as thermal buoyancy-driven flow between two can be regarded as thermal buoyancy-driven flow between two heated infinite vertical plates. heated infinite vertical plates.

This type of turbulent flow is interesting from a fundamental This type of turbulent flow is interesting from a fundamental point of view, because the production of turbulence by shear point of view, because the production of turbulence by shear and thermal buoyancy takes place in the same direction. and thermal buoyancy takes place in the same direction.

This flow is also interesting from an applied point of view This flow is also interesting from an applied point of view because it is representative for many practical heat transfer because it is representative for many practical heat transfer applications.applications.


3

Introduction ( Introduction ( 22 ) )

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

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

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


4


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

There is, therefore, much to be gained if the differential model There is, therefore, much to be gained if the differential model can be truncated to an algebraic form in which transport can be truncated to an algebraic form in which transport equations will be solved only for major scalar quantities, equations will be solved only for major scalar quantities, primarily primarily k,k, ,, . Therefore, the application of simpler models (at . Therefore, the application of simpler models (at most at the algebraic level) should be served as a convenient most at the algebraic level) should be served as a convenient alternative for handling complex industrial situations. alternative for handling complex industrial situations.

2


5


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


6

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

Dynamic FieldDynamic Field

k

ikj

k

jkiij x

Uuu

x

UuuP

j

iji x

UuuP

ii ugG

The full differential stress/flux second-moment closure for buoyancy-driven flows can be summarizedThe full differential stress/flux second-moment closure for buoyancy-driven flows can be summarized as following:as following:

D( )

D

u u

tP G

i j

ij ij ij ij ij D

D( )

D

t k

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

D( )

D

k

tP Gk D

ijjiij ugugG

ij i j ij ij ij ij ijCk

u u k Ck

P P C G G 123 2

23 3

23

D

x xC u u

xj ji j

i

Sk

max . ,0 83 1 0

2 2

k 3 2/

C xn


7

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

Thermal FieldThermal Field

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

x xC u u

xj ji j

i

P uT

xjj

Specification of CoefficientsSpecification of Coefficients

i i k k in

a u u n nx

12

D( )

D

tC P C P

kC C f

k D 1 2 3 4

2

2 5

There are 21 calibrated empirical coefficients !There are 21 calibrated empirical coefficients !


8

D i f e r 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 e r 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

Presented full differential stress-flux model for three-dimensional flows consists of 17 transport Presented full differential stress-flux model for three-dimensional flows consists of 17 transport equations (for velocity components, pressure, components of Reynolds stresses, temperature, equations (for velocity components, pressure, components of Reynolds stresses, temperature, dissipation of turbulent kinetic energy, components of turbulent heat fluxes, temperature dissipation of turbulent kinetic energy, components of turbulent heat fluxes, temperature variance and its dissipation rate). variance and its dissipation rate).

Furthermore, in the buoyancy-driven flows major interactions occur very close to the wall where Furthermore, in the buoyancy-driven flows major interactions occur very close to the wall where molecular effects are important and resolving the flow details and heat transfer in near-wall molecular effects are important and resolving the flow details and heat transfer in near-wall regions requires the low-Re number modification as well as integration up to the wall for which a regions requires the low-Re number modification as well as integration up to the wall for which a very fine numerical grid is needed. very fine numerical grid is needed.

Much can be gained if the full differential model can be truncated to a simpler form in which Much can be gained if the full differential model can be truncated to a simpler form in which differential transport equations will be solved only for major scalar quantities, e.g. for the differential transport equations will be solved only for major scalar quantities, e.g. for the turbulence kinetic energy, temperature variance and their dissipation rates. turbulence kinetic energy, temperature variance and their dissipation rates.

This implies the elimination of the convective and diffusive transport terms in the equations for This implies the elimination of the convective and diffusive transport terms in the equations for turbulent stresses and turbulent heat fluxes, which are the only terms containing the time and turbulent stresses and turbulent heat fluxes, which are the only terms containing the time and spatial derivatives of spatial derivatives of andand . . u ui j ui


9


The first stepThe first step in the analysis of possible simpler mathematical form that can describe sufficiently in the analysis of possible simpler mathematical form that can describe sufficiently accurately defined phenomena is to recognize the dominant mechanisms that determine the accurately defined phenomena is to recognize the dominant mechanisms that determine the character of the complete problem under consideration. character of the complete problem under consideration.

In the present case it is obvious that the effects of external force in the form of a buoyancy term In the present case it is obvious that the effects of external force in the form of a buoyancy term must be included in the basic equations for the fluid flow and the heat transfer. must be included in the basic equations for the fluid flow and the heat transfer.

Besides this direct influence, the effect of the gravitational field must be taken into account in the Besides this direct influence, the effect of the gravitational field must be taken into account in the analysis of pressure fluctuations. analysis of pressure fluctuations.

The effect of the gravitational field on pressure fluctuations will be expressed in pressure-The effect of the gravitational field on pressure fluctuations will be expressed in pressure-velocity gradient and pressure-temperature gradient correlations via additional terms. velocity gradient and pressure-temperature gradient correlations via additional terms.

These two features of buoyancy-driven flows must be taken into account in any kind of proposed These two features of buoyancy-driven flows must be taken into account in any kind of proposed mathematical form. mathematical form.

The simplest form of turbulence differential model can be derived using the Boussinesq isotropic The simplest form of turbulence differential model can be derived using the Boussinesq isotropic eddy viscosity/diffusivity formulation for Reynolds stresses and turbulent flux components, eddy viscosity/diffusivity formulation for Reynolds stresses and turbulent flux components, knowing as the gradient transport models (SGDH):knowing as the gradient transport models (SGDH):

u u kU

x

U

xi j ij ti

j

j

i

2

3

uT

xit

i

t C f

k

3 2/


10


The next stepThe next step towards the improvement of SGDH is obviously to replace the scalar towards the improvement of SGDH is obviously to replace the scalar eddy diffusivity/conductivity formulations with more general form. eddy diffusivity/conductivity formulations with more general form.

The direction that we followed is to truncate the transport differential equation for The direction that we followed is to truncate the transport differential equation for turbulent heat fluxes in the appropriate algebraic form. turbulent heat fluxes in the appropriate algebraic form.

The development of algebraic models for turbulent heat flux components usually The development of algebraic models for turbulent heat flux components usually followed the approach outlined by Rodi, for development of an algebraic model for followed the approach outlined by Rodi, for development of an algebraic model for Reynolds stresses. Reynolds stresses.

Two different levels in proposed modeling strategy can be distinguished - Two different levels in proposed modeling strategy can be distinguished - reduced reduced and more and more refinedrefined formulation. Assuming that we can neglect the transport entirely formulation. Assuming that we can neglect the transport entirely the reduced algebraic expressionthe reduced algebraic expression of of can be can be

written in the following form:written in the following form: D / D u ti i D 0 ui

uC

ku u

T

xC u

U

xC gi i k

kk

i

ki

11 1

12 3

2


11


The refined formulationThe refined formulation can be derived assuming proportionality between transport of turbulent can be derived assuming proportionality between transport of turbulent heat flux and transports of temperature variance and turbulent kinetic energy:heat flux and transports of temperature variance and turbulent kinetic energy:

D

D

D

D

D

D

u

tu

t k

k

ti

i i k

D D D1

2

1 12

2

This corresponds also to the assumption of weak equilibrium of the thermal field, defined by the This corresponds also to the assumption of weak equilibrium of the thermal field, defined by the condition: condition: D( / ) / D u k ti

2 0 In the case of a fully developed flow between the two infinite parallel plates, the convection is In the case of a fully developed flow between the two infinite parallel plates, the convection is

zero, so that the hypothesis relates only the diffusive transport, zero, so that the hypothesis relates only the diffusive transport, D D D i i ku k 12

2( / / )

Replacing the transport of by the modeled right-hand side of the equation for turbulent heat flux Replacing the transport of by the modeled right-hand side of the equation for turbulent heat flux and the transport of and the transport of kk and and by their sources, yields the algebraic expression for the turbulent by their sources, yields the algebraic expression for the turbulent heat-flux vector:heat-flux vector:

2

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

( )1 2C

( )1 3C


12


The closure of the algebraic expressions, irrespective of the modeling level (reduced The closure of the algebraic expressions, irrespective of the modeling level (reduced or more complete), requires that the basic scalar variables,or more complete), requires that the basic scalar variables, k,k, , , and and be supplied be supplied from separate modeled transport equations from separate modeled transport equations (four-equation model).(four-equation model).

The modeling of the equation is very difficult. The modeling of the equation is very difficult. A further simplificationA further simplification can be achieved by expressing the dissipative ratio of can be achieved by expressing the dissipative ratio of

temperature variancetemperature variance , in terms of , in terms of , , k,k, and and , from the assumed ratio of the thermal , from the assumed ratio of the thermal to mechanical turbulence time scales, to mechanical turbulence time scales, , with either its constant value, or , with either its constant value, or expressed by an algebraic function in terms of available variables. expressed by an algebraic function in terms of available variables.

This will reduce the model to a three-equations model,This will reduce the model to a three-equations model, e.g. e.g. kk -- -- . The constant . The constant time-scale ratio assumption is valid only when the turbulence is in local equilibrium.time-scale ratio assumption is valid only when the turbulence is in local equilibrium.

Fortunately, the numerical simulation for buoyancy-driven flows are not so sensitive Fortunately, the numerical simulation for buoyancy-driven flows are not so sensitive to the choice of calculation , so the thermal to mechanical turbulence time scales ratio to the choice of calculation , so the thermal to mechanical turbulence time scales ratio can be taken constant (can be taken constant (RR=0.5=0.5).).

2

2

R m /

2


13


In buoyancy-driven flows, both laminar and turbulent regions may exist in the same In buoyancy-driven flows, both laminar and turbulent regions may exist in the same flow domain. flow domain.

The model is required to reproduce a gradual transition from one to another regime, The model is required to reproduce a gradual transition from one to another regime, not only across the molecular wall sub-layer, but also at the edge of turbulence zones not only across the molecular wall sub-layer, but also at the edge of turbulence zones away from the wall. away from the wall.

In the latter case, the turbulent fluctuations decay freely under the action of molecular In the latter case, the turbulent fluctuations decay freely under the action of molecular forces. forces.

This process is physically different from the turbulence damping in the near-wall This process is physically different from the turbulence damping in the near-wall region, where the molecular effects are mixed with the eddy splitting due to the wall region, where the molecular effects are mixed with the eddy splitting due to the wall blockage and pressure reflection. blockage and pressure reflection.

A consistent approach would require separate modeling of these effects, as practiced A consistent approach would require separate modeling of these effects, as practiced in some recent second-moment closure models. in some recent second-moment closure models.

In the present model, we have followed the low-Re-number modifications of the In the present model, we have followed the low-Re-number modifications of the kk and and equations appeared to suffice and no direct modifications of the equation were equations appeared to suffice and no direct modifications of the equation were found necessary. We applied Lam-Bremhorst extension of found necessary. We applied Lam-Bremhorst extension of kk-- model, with the model, with the advantage that the model requires no additional source terms. advantage that the model requires no additional source terms.


14

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

In order to test the accuracy of the algebraic heat-flux (AHF) model, we performed a simulation of In order to test the accuracy of the algebraic heat-flux (AHF) model, we performed a simulation of a two-dimensional square thin-enclosure. a two-dimensional square thin-enclosure.

The left wall is heated and the right wall is cooled. The configuration is shown on Fig. 1. The left wall is heated and the right wall is cooled. The configuration is shown on Fig. 1. Since we where only be comparing normalized quantities, the dimensions and fluid property Since we where only be comparing normalized quantities, the dimensions and fluid property

values can be freely chosen. We took values can be freely chosen. We took h = h = 0.50.5 and and T = TT = Thh - T - Tcc = = 1,1, then suitable values should then suitable values should

be assigned to, be assigned to, g,g, and and to give the required Rayleigh numbers. to give the required Rayleigh numbers. The results are normalized using the friction velocity and temperature, defined by:The results are normalized using the friction velocity and temperature, defined by:

u*

dW

dxwall

1 2

a dT

dx wallu

T h = 1U = 0W = 0

T c = 0U = 0W = 0

T / x = 0 ; W = 0

R a = 5 . 4 x 1 0 5

R a = 8 . 2 2 7 x 1 0 5

P r = 0 . 7 1

2 h = 1

zx

T / x = 0 ; W = 0

Ra g T h va ( ) /32


15

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

Figure 2. Figure 2. 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 - Betts and BokhariBokhari (exp.) (exp.);; - Versteegh - Versteegh (DNS) (DNS);; Present Present AHF Model )AHF Model )

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)


16


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)

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)

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 0.5-0.5

0.0

0.5

1.0

1.5

2.0

2.5

uw / ( u* ) 2

x / (2h)


17


Figure 3.Thermal field profiles for Figure 3.Thermal field profiles for Ra =Ra = 8.227 x 10 8.227 x 1055 , scaled by friction velocity and temperature, , scaled by friction velocity and temperature,

( ( - Betts and Bokhari - Betts and Bokhari (exp.) (exp.); ; - Versteegh - Versteegh (DNS) (DNS);; Present Present AHF Model )AHF Model )

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 0.5

0.0

0.5

1.0

1.5

2.0

u / ( *u* )

x / (2h)


18


0.0 0.1 0.2 0.3 0.4 0.5-1

0

1

2

3

4

5

w / ( *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)


19



20



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For the average profiles the main discrepancy of DNS and experimental data is found in the For the average profiles the main discrepancy of DNS and experimental data is found in the average velocity which lies about 25% higher in the measurements. average velocity which lies about 25% higher in the measurements.

Also the average temperature is somewhat higher in the measurements than in the DNS data. Also the average temperature is somewhat higher in the measurements than in the DNS data. The discrepancies in the average velocity and temperature profiles of DNS and measured data The discrepancies in the average velocity and temperature profiles of DNS and measured data

may have been caused by the fact that the experimental enclosure is finite. Due to the side and may have been caused by the fact that the experimental enclosure is finite. Due to the side and top walls, the average flow may have been influenced by three-dimensional effects. Given the top walls, the average flow may have been influenced by three-dimensional effects. Given the higher temperature gradient in the center region of the experimental tank, the mixing properties higher temperature gradient in the center region of the experimental tank, the mixing properties of turbulence may not be as much developed as in an infinite channel. As a result, the buoyant of turbulence may not be as much developed as in an infinite channel. As a result, the buoyant force is larger and the momentum exchange is less in the thank. Thus allowing a higher average force is larger and the momentum exchange is less in the thank. Thus allowing a higher average velocity in the tank. velocity in the tank.

In the central region, the velocity variances lie for the measurements somewhat higher than for In the central region, the velocity variances lie for the measurements somewhat higher than for the numerical data. the numerical data.

Nevertheless, the turbulence intensities relative to the average velocities are nearly the same. Nevertheless, the turbulence intensities relative to the average velocities are nearly the same. The experimental temperature variances, are much lower than in the DNS as well as present The experimental temperature variances, are much lower than in the DNS as well as present

calculations. Furthermore, a near-wall peak that is always found in horizontal and vertical calculations. Furthermore, a near-wall peak that is always found in horizontal and vertical turbulent thermal boundary layers is only visible in the numerical data. Such a near wall peak in turbulent thermal boundary layers is only visible in the numerical data. Such a near wall peak in the measurements of Betts an Bokhari was not found. The reason for that may be that response the measurements of Betts an Bokhari was not found. The reason for that may be that response time of the thermocouple used in the measurements was lower than required. time of the thermocouple used in the measurements was lower than required.

So, it was reasonable that the DNS data of Versteegh have been used for comparing, because So, it was reasonable that the DNS data of Versteegh have been used for comparing, because they seem to be the most accurate.they seem to be the most accurate.


22


The The presentpresent AHF results and the results of Versteegh and Boudjemadi agree AHF results and the results of Versteegh and Boudjemadi agree quite well for the average velocity, average temperature and turbulent kinetic quite well for the average velocity, average temperature and turbulent kinetic energy profiles, respectively. energy profiles, respectively.

Discrepancies in Reynolds-stress components have been expected due to Discrepancies in Reynolds-stress components have been expected due to the eddy-viscosity concept of the eddy-viscosity concept of kk-- turbulence model. turbulence model.

The largest shortcoming of the AHF model is the large deviation of the The largest shortcoming of the AHF model is the large deviation of the vertical heat-flux component from the DNS data. vertical heat-flux component from the DNS data.

This is probably caused by the turbulent-diffusion model in the modeled This is probably caused by the turbulent-diffusion model in the modeled equations and the reduced temperature flux model, which are too dissipative.equations and the reduced temperature flux model, which are too dissipative.

Predictions of Predictions of temperature variansetemperature varianse by AHF model show the same deviation by AHF model show the same deviation in shape, which is caused manly by the turbulent-diffusion model. in shape, which is caused manly by the turbulent-diffusion model.

The wall-normal heat-flux component is predicted quite wellThe wall-normal heat-flux component is predicted quite well..


23

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

Results of direct numerical simulation (DNS) of the natural-convection flow between differentially Results of direct numerical simulation (DNS) of the natural-convection flow between differentially heated infinite vertical plates of Versteegh, and as well as experimental data of Betts and 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 model for computing turbulent thermal Bokhari were used for validation of the algebraic heat-flux model for computing turbulent thermal convection in two-dimensional thin-enclosure. Despite the large differences in the turbulence convection in two-dimensional thin-enclosure. Despite the large differences in the turbulence parameter-by-parameter comparison, the complete second-moment closure should be replaced parameter-by-parameter comparison, the complete second-moment closure should be replaced by low-by low-Re-kRe-k-- turbulence model with the turbulence model with the presented presented algebraic heat-flux terms. algebraic heat-flux terms.

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. No gain is turbulence property (turbulence kinetic energy) in the considered thin-cavity case. No gain is achieved by employing the refined, computationally more inconvenient, algebraic expression, as achieved by employing the refined, computationally more inconvenient, algebraic expression, as compared with the reduced expression. compared with the reduced expression.

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

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


24

Thank you.Thank you.

turbulence heat-flux modeling of natural convection in two-dimensional thin-enclosure

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