Computational methodologies for forced convection

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The convection problems that can be solved analytically – some of which were discussed in the preceding sections – are limited to the cases for which the boundary layer theory is valid. Even for cases where the boundary layer theory can be used to simplify the governing equations, analytical solutions can be obtained only if the similarity solution exists or if integral approximate solutions can be obtained. The analytical solution can be very complicated for cases with variable wall temperature or heat flux. When the boundary layer theory cannot be applied, or a simple analytical solution based on boundary layer theory cannot be obtained, a numerical solution becomes the desirable approach. Convection problems are complicated by the presence of advection terms in the governing equations. For incompressible flow, the problem is further complicated by the fact that there is not a governing equation for pressure. This section will cover (1) the numerical solution of the convection-diffusion equation with a known flow field, and (2) the algorithm to determine the flow field.

The governing equation for a convection problem in the Cartesian coordinate system can be expressed into the following generalized form:

  & \frac{\partial (\rho \varphi )}{\partial t}+\frac{\partial (\rho u\varphi )}{\partial x}+\frac{\partial (\rho v\varphi )}{\partial y}+\frac{\partial (\rho w\varphi )}{\partial z} \\ 
 & =\frac{\partial }{\partial x}\left( \Gamma \frac{\partial \varphi }{\partial x} \right)+\frac{\partial }{\partial y}\left( \Gamma \frac{\partial \varphi }{\partial y} \right)+\frac{\partial }{\partial z}\left( \Gamma \frac{\partial \varphi }{\partial z} \right)+S \\ 


where \varphi is a general variable that can represent the directional components of velocity (u, v, or w), temperature, or mass concentration. The general diffusivity Γ can be viscosity, thermal conductivity, or mass diffusivity. S is the volumetric source term. Compared with the governing equation for heat conduction, the convection terms seem to be the only new terms introduced into eq. (4.200). As will become evident later, the contribution of the convection term on the overall heat transfer depends on the relative scale of convection over diffusion. Therefore, these two effects are always handled as one unit in the derivation of the discretization scheme (Patankar, 1980; Patankar, 1991). While analytical solutions of convection problems presented in the preceding sections based on boundary layer theory are few, the numerical solutions of convection based on boundary layer theory have been abundant (Tao, 2001; Minkowycz et al., 2006). With significant advancement of computational capability, the numerical solution of convection problems can now be performed by solving the full Navier-Stokes equation. The algorithms that will be discussed in this section are therefore based on the solution of the full Navier-Stokes equation and the energy equation, not on the boundary layer equations.

One-Dimensional Steady-State Convection and Diffusion
Multidimensional Convection and Diffusion Problems
Numerical Solution of Flow Field
Numerical Simulation of Interfaces and Free Surfaces