Skip to main content
SpringerLink
Log in

Computation of a turbulent natural convection in a rectangular cavity with the low-reynolds-number differential stress and flux model

  • Published:
KSME International Journal Aims and scope Submit manuscript

Abstract

A numerical study of a natural convection in a rectangular cavity with the low-Reynoldsnumber differential stress and flux model is presented. The primary emphasis of the study is placed on the investigation of the accuracy and numerical stability of the low-Reynolds-number differential stress and flux model for a natural convection problem. The turbulence model considered in the study is that developed by Peeters and Henkes (1992) and further refined by Dol and Hanjalic (2001), and this model is applied to the prediction of a natural convection in a rectangular cavity together with the two-layer model, the shear stress transport model and the time-scale bound\(\overline {v^2 } - f\) model, all with an algebraic heat flux model. The computed results are compared with the experimental data commonly used for the validation of the turbulence models. It is shown that the low-Reynolds-number differential stress and flux model predicts well the mean velocity and temperature, the vertical velocity fluctuation, the Reynolds shear stress, the horizontal turbulent heat flux, the local Nusselt number and the wall shear stress, but slightly under-predicts the vertical turbulent heat flux. The performance of the\(\overline {v^2 } - f\) model is comparable to that of the low-Reynolds-number differential stress and flux model except for the over-prediction of the horizontal turbulent heat flux. The two-layer model predicts poorly the mean vertical velocity component and under-predicts the wall shear stress and the local Nusselt number. The shear stress transport model predicts well the mean velocity, but the general performance of the shear stress transport model is nearly the same as that of the two-layer model, under-predicting the local Nusselt number and the turbulent quantities.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Abbreviations

gi :

Gravitational acceleration

Gk :

Buoyancy generation term

H:

Height of the cavity

h:

Heat transfer coefficient

k:

Turbulent kinetic energy

kf :

Conductivity of fluid

L:

Width of the cavity

ni :

i-th component of normal vector at the wall

Nu:

Nusselt number

p:

Pressure

Pk :

Generation term of turbulent kinetic energy

Pθ :

Generation term of temperature variance

Pr:

Prandtl number

Prθ :

Turbulent Prandtl number

Ra:

Rayleigh number

RT :

Turbulent Reynolds number\(\left( { = \frac{{\rho k^2 }}{{\mu \varepsilon }}} \right)\)

t:

Time

T:

Time scale

Ui :

Cartesian velocity components

\(\overline {u_i u_j } \) :

Reynolds stresses

Uτ :

Wall friction velocity\(\left( { = \left( {\frac{\mu }{\rho }\left| {\frac{{\partial U_P }}{{\partial x_n }}} \right|_w } \right)^{1/2} } \right)\)

Xi :

Cartesian Coordinates

xn :

Normal distance from the wall

x + x :

Dimensionless normal distance from the\(\left( { = \frac{{\rho U_\tau x_n }}{\mu }} \right)\)

β:

Gas expansion coefficient

Δn :

Normal distance from the wall

ε:

Dissipation rate of turbulent kinetic energy

εθ:

Dissipation rate of temperature fluctuation

γ:

Dynamic viscosity

γM ij :

Pseudo eddy-viscosity for momentum equations

γT j :

Pseudo eddy-viscosity for energy equation

γT :

Turbulent eddy viscosity

υ:

Kinematic viscosity

ρ:

Density

Θ:

Temperature

\(\overline {\theta u_i } \) :

Turbulent heat fluxes

\(\overline {\theta ^2 } \) :

Temperature variance

H:

Pertaining to hot wall

ε:

Pertaining to dissipation rate of turbulent kinetic energy

θ:

Pertaining to temperature

N-1 :

Pertaining to previous iteration level

M:

Pertaining to momentum equation

T:

Pertaining to energy equation

References

  • Ampofo, F. and Karayiannis, T. G., 2003, “Experimental Benchmark Data for Turbulent Natural Convection in an Air Filled Square Cavity,”Int. J. Heat Mass Transfer, Vol. 46, pp. 3551–3572.

    Article  Google Scholar 

  • Betts, P. L. and Bokhari, I. H., 2000, “Experiments on Turbulent Natural Convection in an Enclosed Tall Cavity,”Int. J. Heat Fluid Flow, Vol.21, pp. 675–683.

    Article  Google Scholar 

  • Boudjemadi, R., Maupu, V., Laurence, D., Quere and P. Le, 1997, “Budgets of Turbulent Stresses and Fluxes in a Vertical Slot Natural Convection Flow at Rayleigh Ra=105 and 5.4x 105,”Int. J. Heat Fluid Flow, Vol. 18, pp. 70–79.

    Article  Google Scholar 

  • Cheesewright, R., King, K. J. and Ziai, S., 1986, “Experimental Data for the Validation of Computer Codes for the Prediction of Two-Dimensional Buoyant Cavity Flows,”Proceeding of ASME Meeting, HTD, Vol. 60, pp. 75–81.

    Google Scholar 

  • Chen, H. C. and Patel, V. C., 1988, “Near-Wall Turbulence Models for Complex Flows Including Separation,”AIAA J., Vol. 26, pp. 641–648.

    Article  Google Scholar 

  • Choi, S. K., 2003, “Evaluation of Turbulence Models for Natural Convection,”KAERI Internal Report, FS200-AR-01-R0-03, (in Korean).

  • Choi, S. K., Kim, E. K. and Kim, S. O., 2004, “Computation of Turbulent Natural Convection in a Rectangular Cavity with the\(k - \varepsilon - \overline {v^2 } - f\) 2-f Model,”Numer. Heat Transfer, Part B, Vol. 45, pp. 159–179.

    Google Scholar 

  • Davidson, L. 1990, “Calculation of the Turbulent Buoyancy-Driven Flow in a Rectangular Cavity Using an Efficient Solver and Two Different Low Reynolds Numberk-ε Turbulence Model,”Numer. Heat Transfer, Part A, Vol. 18, pp. 129–147.

    Article  Google Scholar 

  • Dol, H. S. and Hanjalic, K., 2001, “Computational Study of Turbulent Natural Convection in a Side-Heated Near-Cubic enclosure at a High Rayleigh Number,”Int. J. Heat Mass Transfer, Vol. 44, pp. 2323–2344.

    Article  MATH  Google Scholar 

  • Dol, H. S., Hanjalic, K. and Kenjeres, S., 1997, “A Comparative Assessment of the Second-Moment Differential and Algebraic Models in Turbulent Natural Convection,”Int. J. Heat Fluid Flow, Vol. 18, pp. 4–14.

    Article  Google Scholar 

  • Dol, H. S. Opstelten I. J., and Hanjalic, K., 2000, “Turbulent Natural Convection in a Side- Heated Near-Cubic Enclosure: Experiments and Model Computations,”Technical report APTFR/ 00-02, Thermal and Fluids Science Section, Department of Applied Physics, Delft University of Technology, Delft, The Netherlands.

    Google Scholar 

  • Hanjalic, K., 2002, “One-Point Closure Models for Buoyancy-Driven Turbulent Flows,”Annu. Rev. Fluid Mech. Vol. 34, pp. 321–347.

    Article  MathSciNet  Google Scholar 

  • Henkes, R. A. W. M. and Hoodendoorn, C. J., 1995, “Comparison Exercise for Computations of Turbulent Natural Convection in Enclosures,”Numer. Heat Transfer, Part B, Vol. 28, pp. 59–78.

    Article  Google Scholar 

  • Henkes, R. A. W. M., Van Der Vlugt and F. F. Hoodendoorn, C. J., 1991, “Natural-Convection Flow in a Square Cavity Calculated with Low- Reynolds-Number Turbulence Models,”Int. J. Heat Mass Transfer, Vol. 34, pp. 377–388.

    Article  Google Scholar 

  • Hsieh, K. J. and Lien, F. S., 2004, “Numerical Modeling of Buoyancy-Driven Turbulent Flows in Enclosures,”Int. J. Heat Fluid Flow, in press.

  • Ince, N. Z. and Launder, B. E., 1989, “On the Computation of Buoyancy-Driven Turbulent Flows in Rectangular Enclosures,”Int. J. Heat Fluid Flow, Vol. 10, pp. 110–117.

    Article  Google Scholar 

  • Ince, N. Z. and Launder, B. E., 1995, “Three- Dimensional and Heat-Loss Effects on Turbulent Flow in a Nominally Two-Dimensional Cavity,”Int. J. Heat Fluid Flow, Vol. 16, pp. 171–177.

    Article  Google Scholar 

  • Kenjeres, S., 1998, “NumericalModelling of Complex Buoyancy-Driven Flows,” Ph. D Thesis, Delft University of Technology, The Netherlands.

    Google Scholar 

  • Kenjeres, S., and Hanjalic, K., 1995, “Prediction of Turbulent Thermal Convection in Concentric and Eccentric Annuli,”Int. J. Heat Fluid Flow, Vol. 16, pp. 429–439.

    Article  Google Scholar 

  • King, K.J., 1989, “TurbulentNatural Convection in Rectangular Air Cavities,” Ph. D Thesis, Queen Mary College, University of London, U. K.

    Google Scholar 

  • Janssen, R. J. A., 1994, ”Instabilitiesin Natural- Convection flows in Cavities,” Ph. D thesis, Delft University of Technology, Delft, The Netherlands.

    Google Scholar 

  • Lai, Y. G., and So, R. M. C., 1990, “Near-Wall Modeling of Turbulent Heat Fluxes,”Int. J. Heat Mass Transfer, Vol. 33, pp. 1429–1440.

    Article  Google Scholar 

  • Launder, B. E. and Sharma, B. I., 1974, “Application of the Energy Dissipation Model of Turbulence to the Calculation of Flow Near Spinning Disc,”Lett. In Heat and Mass Transfer, Vol. 1, pp. 131–138.

    Article  Google Scholar 

  • Lien, F. S., and Leschziner, M. A., 1991, “Second-Moment Modelling of Recirculating Flow with a Non-orthogonal Collocated Finite-Volume Algorithm,”Turbulent Shear Flows, Vol. 8, pp. 205–222.

    Google Scholar 

  • Medic, G. and Durbin, P. A., 2002, “Toward Improved Prediction of Heat Transfer on Turbine Blades,”J. Turbomachinery, Vol. 124, pp. 187- 192.

    Article  Google Scholar 

  • Menter, F. R., 1994, “Two-Equation Eddy- Viscosity Turbulence Models for Engineering Applications,”AIAA Journal, Vol. 32, No. 8, pp. 1598–1604.

    Article  Google Scholar 

  • Opstelten, I. J., 1994, “ExperimentalStudy on Transition Characteristics of Natural- Convection Flow,” Ph. D thesis, Delft University of Technology, Delft, The Netherlands.

    Google Scholar 

  • Patankar, S. V., 1980, “NumericalHeat Transfer and Fluid Flow” Hemisphere, New York, USA.

  • Peeters, T. W. J. and Henkes, R. A. W. M., 1992, “The Reynolds-Stress Model of Turbulence Applied to the Natural-Convection Boundary Layer along a Heated Vertical Plate,”Int. J. Heat Mass Transfer, Vol. 35, pp. 403–420.

    Article  MATH  Google Scholar 

  • Peng, S. H. and Davidson, L., 2001, “Large Eddy Simulation of Turbulent Buoyant Flow in a Confined Cavity,”Int. J. Heat Fluid Flow, Vol. 22, pp. 323–331.

    Article  Google Scholar 

  • Shikazo, N. and Kasagi, N., 1996, “Second Moment Closure for Turbulent Scalar Transport at Various Prandtl Numbers,”Int. J. Heat Mass Transfer, Vol. 39, pp. 2977–2987.

    Article  Google Scholar 

  • Shin, J. K., An, J. S. and Choi, Y. D. 2004, “Near Wall Modelling of Turbulent Heat Fluxes by Elliptic Equation.”Transactions of the Korean Society of Mechanical Engineers, Part B. Vol. 28, pp. 526–534.

    Google Scholar 

  • Shin, J. K., Choi, Y. D. and Lee, G. H., 1993, “A Low-reynolds Number Second Moment Closure for Turbulent Heat Fluxes,”Transactions of the Korean Society of Mechanical Engineers, Part B. Vol. 17, pp. 3196–3207.

    Google Scholar 

  • Tian, Y. S. and Karayiannis T. G., 2000, “Low Turbulence Natural Convection in an Air Filled Square Cavity part I: The Thermal and Fluid Flow Fields,”Int. J. Heat Mass Transfer, Vol. 43, pp. 849–866.

    Article  MATH  Google Scholar 

  • Tieszen, S., Ooi, P., Durbin, P. A. and Behnia, M., 1998, “Modeling of Natural Convection Heat Transfer,”Proc. Summer Program, Center of Turbulence Research, Stanford University, Stanford, CA, U. S. A., pp. 287–302.

    Google Scholar 

  • Tsuji, T and Nagano, Y., 1987, “Turbulence Measurements in a Natural Convection Boundary Layer along a Vertical Flat Plate,”Int. J. Heat Mass Transfer, Vol. 31, pp. 2101–2111.

    Article  Google Scholar 

  • Van Leer, B., 1974, “Towards the Ultimate Conservative Difference Scheme: Monotonicity and Conservation Combined in a Second Order Scheme,”J. Comput. Phy., Vol. 14, pp. 361–370.

    Article  Google Scholar 

  • Versteegh, T.A.M. and Nieuwstat, F. T. M., 1998, “Turbulence Budgets of Natural Convection in an Infinite, Differentially Heated, Vertical Wall,”Int. J. Heat Fluid Flow, Vol. 19, pp. 135–149.

    Article  Google Scholar 

  • Zhu, J., 1991, “A Low-Diffusive and Oscillation Free Convection Scheme,”Comm. Appl. Numer. Methods, Vol. 7, pp. 225–232.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fluid System Engineering Division, Korea Atomic Energy Research Institute, 150 Deokjin-dong, Yuseong-gu, 305-353, Daejeon, Korea

    Seok-Ki Choi, Eui-Kwang Kim, Myung-Hwan Wi & Seong-O Kim

Authors
  1. Seok-Ki Choi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eui-Kwang Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Myung-Hwan Wi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Seong-O Kim
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Seok-Ki Choi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Choi, SK., Kim, EK., Wi, MH. et al. Computation of a turbulent natural convection in a rectangular cavity with the low-reynolds-number differential stress and flux model. KSME International Journal 18, 1782–1798 (2004). https://doi.org/10.1007/BF02984327

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02984327

Key Words

Search

Navigation