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

1. TURBULENCE HEAT-FLUX MODELLING OF NATURAL CONVECTION IN TWO-DIMENSIONAL THIN-ENCLOSURE Zarko M. Stevanovic VINCA Institute for Nuclear Sciences Laboratory for Thermal Engineering and Energy 11001 Belgrade, P.O. Box 522, Yugoslavia zare@rt270.vin.bg.ac.yu

2. C O N T E N T Objective of Work 3 Turbulence Modelling 5 Differential - to - Algebraic Truncation 6 Computational Details 8 Computational Results 9 Conclusions 13

3. Application of the second-moment closure, even in its simplest form, to the solution of thermal buoyancy-driven flows represent still a formidable task. • 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 obscure real physics. • 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 portion of the flow domain, requiring the implementation of low-Reynolds number modifications. • 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. • 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 quantities, primarily k,, • Therefore, the application of simpler models (at most at the algebraic level) should be served as a convenient alternative for handling complex industrial situations. O b j e c t i v e o f W o r k

4. These prospects have motivated present work, aimed at verifying the current practice in modeling of the transport equations for the turbulent heat-flux and temperature-variance and at exploring the limits of simpler turbulence models at the algebraic level. • Results of direct numerical simulation and experimental data of turbulent natural convection between two differentially heated vertical plates for Ra = 8.227x105 and for Ra = 5.4x105 have been used to validate the algebraic heat flux model.

5.   5 0.6 1.3 0.6 1.44 1.92 1.44 0.72 2.2 0.8 0.22 1.0 1.8 0.6 0.6 0.09 0.07 0.09 0.09 0.09 T u r b u l e n c e M o d e l l i n g Transport Equations: Modelled Terms: Specification of Coefficients

6. 1. The simple gradient transport model (Boussinesq) ( isotropic eddy viscosity / diffusivity formulation ) 2. The reduced algebraic expression ( ) ( neglecting the transport entirely ) 3. The more refined formulation ( ) ( proportionality between transport of turbulent heat flux and transports of temperature variance and turbulent kinetic energy ) 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 n

7. Reduced Model to the Three-Equation Model: k-  - - Lam-Bremhorst Low-Re k -  model - Additional Transport Equation for - Reduced Algebraic Expression for

8. Test Problem Configuration C o m p u t a t i o n a l D e t a i l s • Basic References: • Betts, P.L. and Boghari, I.H., (1996), “Experiments on turbulent natural convection of air in tall cavity”, Proc. of the 5th ERCOFTAC workshop on Refined Flow Modeling, Chatou, France. • Versteegh, T.A.M., (1998), “Numerical simulation of natural convection in a differentially heated, vertical channel”, Ph.D. Thesis, Printed by Ponsen & Looijen, The Netherlands.

9. Dynamic field profiles for Ra = 8.227 x 105 , scaled by friction velocity (  - Betts and Bokhari, [1];  - Versteegh, [18]);AHF Model ) C o m p u t a t i o n a l R e s u l t s .

10. Dynamic field profiles for Ra = 8.227 x 105 , scaled by friction velocity (  - Betts and Bokhari, [1];  - Versteegh, [18]);AHF Model )

11. . Thermal field profiles for Ra = 8.227 x 105 , scaled by friction velocity and temperature ( - Betts and Bokhari, [1];  - Versteegh, [18];AHF Model )

12. . Thermal field profiles for Ra = 5.4 x 105 , scaled by friction velocity and temperature (  - Betts and Bokhari, [1];  - Versteegh, [18];AHF Model )

13. C o n c l u s i o n s • 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 Bokhari, were used for validation of the algebraic heat-flux (AHF) model for computing turbulent thermal convection in two-dimensional thin-enclosure. • Despite of the large differences in the turbulence parameter-by-parameter comparison, the complete second-moment closure should be replaced by low-Re- k -  - turbulence model with the algebraic heat-flux terms. • 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 achieved by employing the refined, computationally more inconvenient, algebraic refined expression, as compared with the reduced expression. • However, even the reduced expressions show important advantages as compared with simple isotropic eddy-diffusivity model, because the heat-flux includes all major generation terms (mean temperature, and velocity gradients, and thermal turbulence self-amplification through the temperature variance).

14. Finally, we can conclude that one of 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 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 gravitational vector. Also, in the complex geometry of natural convection fluid flow problems, one of the most convenience turbulence model is LVEL model, proposed by professor D.B. Spalding.