Multidimensional Convection and Diffusion Problems
From ThermalFluidsPedia
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Defining the following integrated total flux  Defining the following integrated total flux  
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+   width="100%" <center>  
<math>J_{e}^{*}=\frac{J_{e}}{D_{e}},\text{ }J_{w}^{*}=\frac{J_{w}}{D_{w}},\text{ }J_{n}^{*}=\frac{J_{n}}{D_{n}},\text{ }J_{s}^{*}=\frac{J_{s}}{D_{s}}</math>  <math>J_{e}^{*}=\frac{J_{e}}{D_{e}},\text{ }J_{w}^{*}=\frac{J_{w}}{D_{w}},\text{ }J_{n}^{*}=\frac{J_{n}}{D_{n}},\text{ }J_{s}^{*}=\frac{J_{s}}{D_{s}}</math>  
  +  </center>  
+    
+  }  
where  where  
<math>D_{e}=\frac{\Gamma _{e}}{(\delta x)_{e}}\Delta y,\text{ }D_{w}=\frac{\Gamma _{w}}{(\delta x)_{w}}\Delta y,\text{ }D_{n}=\frac{\Gamma _{n}}{(\delta y)_{n}}\Delta x,\text{ }D_{s}=\frac{\Gamma _{s}}{(\delta y)_{s}}\Delta x</math>  <math>D_{e}=\frac{\Gamma _{e}}{(\delta x)_{e}}\Delta y,\text{ }D_{w}=\frac{\Gamma _{w}}{(\delta x)_{w}}\Delta y,\text{ }D_{n}=\frac{\Gamma _{n}}{(\delta y)_{n}}\Delta x,\text{ }D_{s}=\frac{\Gamma _{s}}{(\delta y)_{s}}\Delta x</math>  
  the integrated fluxes at the east and west faces of the control volume can be evaluated  +  the integrated fluxes at the east and west faces of the control volume can be evaluated: 
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<math>J_{e}=D_{e}J_{e}^{*}=D_{e}[B(\text{Pe}_{\Delta e})\varphi _{P}A(\text{Pe}_{\Delta e})\varphi _{E}]</math>  <math>J_{e}=D_{e}J_{e}^{*}=D_{e}[B(\text{Pe}_{\Delta e})\varphi _{P}A(\text{Pe}_{\Delta e})\varphi _{E}]</math>  
  +  </center>  
  +    
+  }  
+  { class="wikitable" border="0"  
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+   width="100%" <center>  
<math>J_{w}=D_{w}J_{w}^{*}=D_{w}[B(\text{Pe}_{\Delta w})\varphi _{W}A(\text{Pe}_{\Delta w})\varphi _{P}]</math>  <math>J_{w}=D_{w}J_{w}^{*}=D_{w}[B(\text{Pe}_{\Delta w})\varphi _{W}A(\text{Pe}_{\Delta w})\varphi _{P}]</math>  
  +  </center>  
  Substituting  +   
  +  }  
+  Substituting <math>B(\text{P}{{\text{e}}_{\Delta }})A(\text{P}{{\text{e}}_{\Delta }})=\text{P}{{\text{e}}_{\Delta }}</math> into the two equations above yields  
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<math>J_{e}=D_{e}J_{e}^{*}=D_{e}[A(\text{Pe}_{\Delta e})\varphi _{P}+\text{Pe}_{\Delta e}\varphi _{P}A(\text{Pe}_{\Delta e})\varphi _{E}]</math>  <math>J_{e}=D_{e}J_{e}^{*}=D_{e}[A(\text{Pe}_{\Delta e})\varphi _{P}+\text{Pe}_{\Delta e}\varphi _{P}A(\text{Pe}_{\Delta e})\varphi _{E}]</math>  
  +  </center>  
  +    
+  }  
+  { class="wikitable" border="0"  
+    
+   width="100%" <center>  
<math>J_{w}=D_{w}J_{w}^{*}=D_{w}[B(\text{Pe}_{\Delta w})\varphi _{W}B(\text{Pe}_{\Delta w})\varphi _{P}+\text{Pe}_{\Delta w}\varphi _{P}]</math>  <math>J_{w}=D_{w}J_{w}^{*}=D_{w}[B(\text{Pe}_{\Delta w})\varphi _{W}B(\text{Pe}_{\Delta w})\varphi _{P}+\text{Pe}_{\Delta w}\varphi _{P}]</math>  
  +  </center>  
+    
+  }  
+  
Similarly, the integrated total flux at the north and south faces of the control volume can be expressed as  Similarly, the integrated total flux at the north and south faces of the control volume can be expressed as  
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<math>J_{n}=D_{n}J_{n}^{*}=D_{n}[A(\text{Pe}_{\Delta n})\varphi _{P}+\text{Pe}_{\Delta n}\varphi _{P}A(\text{Pe}_{\Delta n})\varphi _{N}]</math>  <math>J_{n}=D_{n}J_{n}^{*}=D_{n}[A(\text{Pe}_{\Delta n})\varphi _{P}+\text{Pe}_{\Delta n}\varphi _{P}A(\text{Pe}_{\Delta n})\varphi _{N}]</math>  
  +  </center>  
  +    
+  }  
+  { class="wikitable" border="0"  
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+   width="100%" <center>  
<math>J_{s}=D_{s}J_{s}^{*}=D_{s}[B(\text{Pe}_{\Delta s})\varphi _{S}B(\text{Pe}_{\Delta s})\varphi _{P}+\text{Pe}_{\Delta s}\varphi _{P}]</math>  <math>J_{s}=D_{s}J_{s}^{*}=D_{s}[B(\text{Pe}_{\Delta s})\varphi _{S}B(\text{Pe}_{\Delta s})\varphi _{P}+\text{Pe}_{\Delta s}\varphi _{P}]</math>  
+  </center>  
+    
+  }  
Substituting the above four integrated total fluxes into eq. (4.263), we have  Substituting the above four integrated total fluxes into eq. (4.263), we have 
Revision as of 06:22, 14 July 2010
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. (4.263), we have

where
are the mass flow rates at the four faces of the control volume. Equation (4.264) can be rearranged to obtain the final form of the following discretized equation:

where







If the continuity equation is satisfied, eq. (4.271) can be simplified as (see Problem 4.4)

Similar to the case of onedimensional convectiondiffusion, different discretization schemes for the discretized equations (4.265) – (4.272) can be obtained by using different expressions for A(PeΔ) from Table 4.3.
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 (Patankar, 1980)

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 Table 4.3. In addition to the six first order discretization schemes described above, some researchers have used higher order schemes such as second order upwind (Leonard et al., 1981) and QUICK (Quadratic Upwind Interpolation of Convective Kinetics; Leonard, 1979) 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 (Patankar, 1980). 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 (Tao, 2001). 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.