# Computational methodologies for forced convection

(Difference between revisions)
 Revision as of 06:09, 14 July 2010 (view source)← Older edit Current revision as of 01:16, 27 July 2010 (view source) (2 intermediate revisions not shown) Line 1: Line 1: {{Comp Method for Forced Convection Category}} {{Comp Method for Forced Convection Category}} - 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 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 topic 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: The governing equation for a convection problem in the Cartesian coordinate system can be expressed into the following generalized form: Line 14: Line 14: |} |} - 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 $\Gamma$ 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. (1). 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, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC. Patankar, S.V., 1991, Computation of Conduction and Duct Flow Heat Transfer, Innovative Research.. + 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 $\Gamma$ 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. (1). 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, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC. Patankar, S.V., 1991, Computation of Conduction and Duct Flow Heat Transfer, Innovative Research.Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.. 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, W.Q., Numerical Heat Transfer, 2nd Ed., Xi’an Jiaotong University Press, Xi’an, China (in Chinese). Minkowycz, M.J., Sparrow, E.M., and Murthy, J.Y., eds., 2006, Handbook of Numerical Heat Transfer, 2nd ed. John Wiley & Sons, Hoboken, NJ.. 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. 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, W.Q., Numerical Heat Transfer, 2nd Ed., Xi’an Jiaotong University Press, Xi’an, China (in Chinese). Minkowycz, M.J., Sparrow, E.M., and Murthy, J.Y., eds., 2006, Handbook of Numerical Heat Transfer, 2nd ed. John Wiley & Sons, Hoboken, NJ.. 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. Line 23: Line 23: [[Numerical Simulation of Interfaces and Free Surfaces]]
[[Numerical Simulation of Interfaces and Free Surfaces]]
[[Application of Computational Methods]] [[Application of Computational Methods]] + + ==References== + {{Reflist}}

## Current revision as of 01:16, 27 July 2010

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 topic 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:

 \begin{align} & \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 \\ \end{align} (1)

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. (1). 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 [1][2][3].

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 [4][5]. 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.

## References

1. Patankar, S.V., 1980, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC.
2. Patankar, S.V., 1991, Computation of Conduction and Duct Flow Heat Transfer, Innovative Research.
3. Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.
4. Tao 2001, W.Q., Numerical Heat Transfer, 2nd Ed., Xi’an Jiaotong University Press, Xi’an, China (in Chinese).
5. Minkowycz, M.J., Sparrow, E.M., and Murthy, J.Y., eds., 2006, Handbook of Numerical Heat Transfer, 2nd ed. John Wiley & Sons, Hoboken, NJ.