Multidimensional Convection and Diffusion Problems
From ThermalFluidsPedia
Computational methodologies for forced convection

Twodimensional problem
The heat transfer problems discussed in the preceding subsection are steadystate convectiondiffusion problems with the general variable varying in one dimension only. We now turn our attention to the unsteady state twodimensional convectiondiffusion problem which includes a source term S. The problem is described by

which can be rewritten as

where


Integrating eq. (2) with respect to t in the interval of (t, t+Δt) and over the control volume P, we have
where the source term is treated as a linear function of . Assuming the total fluxes are uniform on all faces of the control volume and employing fullyimplicit scheme, the above equation becomes
where the superscript 0 represents the values at the previous time step. Introducing the integrated total fluxes , , and , and dividing the above equation by Δt yields

Defining the following integrated total flux

where
the integrated fluxes at the east and west faces of the control volume can be evaluated:


Substituting B(Pe_{Δ}) − A(Pe_{Δ}) = Pe_{Δ} into the two equations above yields


Similarly, the integrated total flux at the north and south faces of the control volume can be expressed as


Substituting the above four integrated total fluxes into eq. (5), we have

where


are the mass flow rates at the four faces of the control volume. Equation (6) can be rearranged to obtain the final form of the following discretized equation:

where







If the continuity equation is satisfied, eq. (13) can be simplified as

Similar to the case of onedimensional convectiondiffusion, different discretization schemes for the discretized equations (7) – (14) can be obtained by using different expressions for A(PeΔ) from the following table.
Table Summary of A(PeΔ) for different schemes
Scheme  A(PeΔ) 
Central difference  
Upwind  1 
Hybrid  
Exponential  
Power Law 
Threedimensional problem
The discretized equation for a transient threedimensional convectiondiffusion problem can be obtained by integrating the conservation equation with respect to t in the interval of (t, t+Δt) and over the threedimensional control volume P (formed by considering two additional neighbors at top, T, and bottom, B). The final form of the governing equation is ^{[1]}

where









The expressions for conductance at the faces of the control volume are

and the flow rates are:

The Different discretization schemes for the above threedimensional problem can be obtained by using different expressions for A(PeΔ) from the above table. In addition to the six first order discretization schemes described above, some researchers have used higher order schemes such as second order upwind ^{[2]} and QUICK (Quadratic Upwind Interpolation of Convective Kinetics)^{[3]} schemes to overcome the false diffusion problem, which is referred to as error caused by using the discretization scheme with accuracy less than the second order ^{[1]}. The error due to false diffusion could potentially be severe for (1) transient problems, (2) multidimensional steadystate problems, or (3) problems with nonconstant source terms ^{[4]}. While the accuracies of these higher order schemes are better than the first order schemes, their computational time is much greater than that of the first order schemes.^{[5]}
References
 ↑ ^{1.0} ^{1.1} Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC.
 ↑ Leonard 1981, B.P., “A Survey of Finite Differences with Upwinding for Numerical Modeling of the Incompressible Convection Diffusion Equation,” Computational Techniques in Transient and Turbulent Flows, Taylor, C., and Morgan, K., Eds., Pineridge Press, Swansea, pp. 135.
 ↑ Leonard, B.P., 1979, “A Stable and Accurate Convective Modeling Procedure based on Quadratic Upstream Interpolation,” Computer Methods in Applied Mechanics and Engineering, Vol. 29, pp. 5998.
 ↑ Tao 2001, W.Q., Numerical Heat Transfer, 2nd Ed., Xi’an Jiaotong University Press, Xi’an, China (in Chinese).
 ↑ Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.