Page 244 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
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Several details have been neglected in order to keep the discussion brief. For instance,
higher-order elements have not been discussed, and few solution procedures have been
touched upon. Special topics such as adaptive meshing for heat transfer applications is
not mentioned (Lewis et al. 1991; Nithiarasu 2002; Nithiarasu and Zienkiewicz 2000).
However, the CBS scheme for convection heat transfer has been discussed in detail for
linear triangular elements. A complete knowledge of such a single scheme will provide
the reader with a strong starting point for understanding other relevant fluid dynamics and
convection heat transfer solution procedures.

7.15 Exercise

Example 7.15.1 Derive a convection–diffusion equation using a differential control volume
approach.

Example 7.15.2 Derive the CG method for a convection–diffusion equation with the source
term Q.

Example 7.15.3 Derive Navier–Stokes equations in cylindrical and spherical coordinates.

Example 7.15.4 Reduce the incompressible Navier–Stokes equations to solve a one-
dimensional convection heat transfer problem.
Example 7.15.5 For natural convection problems, if α is replaced by ν in the non-
dimensional scaling, derive the new non-dimensional form.

Example 7.15.6 Calculate laminar flow and heat transfer from a hot cylinder at Re = 40
using the CBS flow code. Assume the buoyancy effect is negligible.

Example 7.15.7 Compute the transient vortex shedding phenomenon behind a circular
cylinder at Re = 100 using CBS flow. Assume that the flow is isothermal.

Example 7.15.8 Write a program in any standard scientific language to calculate stream
functions from a computed velocity field.

Bibliography

Aung W and Worku G 1986a Developing flow and flow reversal in a vertical channel with asymmetric
wall temperatures, ASME Journal of Heat Transfer, 108, 299–304.
Aung W and Worku G 1986b Theory of fully developed, combined convection including flow reversal,
ASME J. Heat Transfer, 108, 485–488.
Brooks AN and Hughes TJR 1982 Streamline upwind/Petrov-Galerkin formulation for convection
dominated flows with particular emphasis on incompressible Navier–Stokes equation, Computer
Methods in Applied Mechanics and Engineering, 32, 199–259.
Cheung TJ 2002 Computational Fluid Dynamics, Cambridge University Press, Cambridge, UK.