# Convection-Diffusion Phenomena and a Navier-Stokes Processor

• C. J. Hoogendoorn
• Th. H. van der Meer
Chapter
Part of the Notes on Numerical Fluid Mechanics book series (volume 17)

## Summary

For heat and mass transport in complex flow situations computational methods are very important. Many technological processes can be simulated by a set of convection-diffusion equations. These equations can numerically be solved using a single algorithm based on the finite domain or control volume method. For turbulent transport a k-ɛ model is often used. This requires that in some cases an experimental validation for a completely new flow situation has to be done.

For two examples the application will be shown. The natural convection in a square cavity both for laminar and turbulent cases will be discussed. For flows and heat transfer in living spaces good predictions including radiative exchange can be given. The second example is the simulation model “Furnace”. The flow, combustion and heat transfer in a glass furnace can be predicted. A full 3-dimensional model has been developed. For fine grids, and for time dependent or 3-D situations the computational effort is large. The elliptic flows and the coupling of a large set of partial differential equations give a slow convergence. CPU time on a main-frame computer may run in many hours. This has led us to the development of a processor to directly solve the convection-diffusion algorithm for the finite control volume method as well as the transport equations. This will be applied in a special purpose dedicated Navier-Stokes computer with the capabilities of a super-computer for this special algorithm. It can be expected that this tool will enhance the application of numerical transport phenomena studies strongly.

## References

1. [1]
Spalding, D.B.: “A unified theory of friction, heat transfer in the turbulent boundary layer”, Int.J. Heat and Mass Transfer, 7 (1964) pp. 743–761.
2. [2]
Patankar, S.V., Spalding, D.B.: “A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows”, Int.J. Heat and Mass Transfer, 15 (1972) pp. 1787–1806.
3. [3]
Schinkel, W.M.M., Hoogendoorn, C.J.: “Core stratification effects in inclined cavities”, Appl.Sc.Res., 12 (1985) pp. 109–130.
4. [4]
Linthorst, S.J.M., Hoogendoorn, C.J.: “Numerical calculations of heat transfer by natural convection in a cubical enclosure”, Proc. 2nd Int. Conf. Num. Methods in Laminar and Turbulent Flow, Venice (1981) pp. 1069–1078.Google Scholar
5. [5]
Linthorst, S.J.M., Hoogendoorn, C.J.: “Natural convective heat transfer in three dimgjisional inclined small aspect ratio enclosures”, Proc. 8 Int. Heat Tr. Conf., San Francisco (1986) pp. 1501–1505.Google Scholar
6. [6]
Bos, W.G., Elsen, T. van den, Hoogendoorn, C.J., Test, F.L.: “Numerical study of a smoke layer in a corridor”, Comb. Sc. and Techn., 38. (1984) pp. 227–243.
7. [7]
Dalhuijsen, A.J., Meer, Th.H. van der, Hoogendoorn, C.J., Hoogvliet, J., Bennekom, W.P. van: “Hydrodynamic properties and mass transfer characteristics of electrochemical flow- through cells of the confined wall-jet type”, J. Electroanal. Chem., 182 (1985) pp. 295–313.
8. [8]
Singhal, A.K.: “A critical look at the progress in numerical heat transfer and some suggestions for improvement”, Num. Heat Transfer, 8 (1985) pp. 505–517.
9. [9]
Patel, V.C., Rodi, W., Scheuerer, G.: “Evaluation of turbulence models for near wall and low-Reynolds number flows”, Proc.3rd Symp. Turbulent Shear Flows, Davis, Cal. (1981) pp. 1–11 8.Google Scholar
10. [10]
Patankar, S.V.: Numerical heat transfer and fluid flow, McGraw-Hill, New York (1980).
11. [11]
Doornmaal, J.P. van, Raithby, G.D.: “Enhancement.of the SIMPLE method for predicting incompressible fluid flows”, Num. heat transfer, 7 (1984) pp. 147–163.
12. [12]
Vahl Davis, G. de, Jones, I.P.: “Natural convection in a square cavity, a comparison excercise”, Int.J. Num. Meth. in Fluids, 3 (1983) pp. 227–248.
13. [13]
Jones, I.P.: “A comparison problem for numerical methods in fluid dynamics, the double glazing problem”, Num. Meth. in Therm. Probl., Pineridge Press, UK (1979) pp. 338–348.Google Scholar
14. [14]
Euser, H., Hoogendoorn, C.J., Ooyen, H. van: “Airflow in a room as induced by natural convection streams”, Energy Cons, in Heating, Cooling, Ventil. Build., Ed. Hoogendoorn, C.J., Afgan, N.H., Hemisphere Publ., USA, 1 (1978) pp. 259–270.Google Scholar
15. [15]
Vermogen van radiatoren bij niet genormeerde opstelling. Publ. nr.l Stichting ISSO, Rotterdam, Holland (1980) pp. 29–31.Google Scholar
16. [16]
Simonis, F., Waal, H. de, Beerkens, R.C.G.: “Influence of furnace design and operation parameters on the residence time distribution of glass tanks, predicted by 3-D computer simulations”, Proc. 14 Int. Conf. on Glass, India (1986).Google Scholar
17. [17]
Peterson, V.L.: “Impact of computers on aerodynamics research and development”, Proc. I.E.E.E. 72 (1984) no.l pp. 68–79.Google Scholar
18. [18]
Hilhorst, H.J., Bakker, A.F., Bruin, C., Compagner, A., Hoogland, A.: “Special purpose computers in physics”, J. of Statistical Physics 34 (1984) pp. 987–1000.
19. [19]
Heller, D.: “Some aspects of the cyclic reduction algorithm for block tri-diagonal linear systems”, SIAM J. Numer. Anal., 13 (1976) no.4 pp 484–496.

© Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1987

## Authors and Affiliations

• C. J. Hoogendoorn
• 1
• Th. H. van der Meer
• 1
1. 1.Dept. of Applied PhysicsTechnical University DelftDelftThe Netherlands