OneDimensional SteadyState Convection and Diffusion
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(Created page with 'Exact Solution The objective of this subsection is to introduce various discretization schemes of the convectiondiffusion terms through discussion of the onedimensional steady …') 

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<math>\frac{d(\rho u)}{dx}=0</math>  <math>\frac{d(\rho u)}{dx}=0</math>  
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Equation (4.201) is subject to the following boundary conditions:  Equation (4.201) is subject to the following boundary conditions:  
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<math>\varphi =\varphi _{0},\text{ }x=0</math>  <math>\varphi =\varphi _{0},\text{ }x=0</math>  
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<math>\varphi =\varphi _{L},\text{ }x=L</math>  <math>\varphi =\varphi _{L},\text{ }x=L</math>  
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By introducing the following dimensionless variables  By introducing the following dimensionless variables  
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<math>\Phi =\frac{\varphi \varphi _{0}}{\varphi _{L}\varphi _{0}},\text{ }X=\frac{x}{L}</math>  <math>\Phi =\frac{\varphi \varphi _{0}}{\varphi _{L}\varphi _{0}},\text{ }X=\frac{x}{L}</math>  
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the onedimensional steadystate convection and diffusion problem can be nondimensionalized as  the onedimensional steadystate convection and diffusion problem can be nondimensionalized as  
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<math>\text{Pe}\frac{d\Phi }{dX}=\frac{d^{2}\Phi }{dX^{2}}</math>  <math>\text{Pe}\frac{d\Phi }{dX}=\frac{d^{2}\Phi }{dX^{2}}</math>  
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<math>\Phi =0,\text{ X}=0</math>  <math>\Phi =0,\text{ X}=0</math>  
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<math>\Phi =1,\text{ }X=1</math>  <math>\Phi =1,\text{ }X=1</math>  
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where  where  
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<math>\text{Pe}=\frac{\rho uL}{\Gamma }</math>  <math>\text{Pe}=\frac{\rho uL}{\Gamma }</math>  
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is the Peclet number that reflects the relative level of convection and diffusion. Pe becomes zero for the case of pure diffusion and becomes infinite for the case of pure advection. The exact solution of eqs. (4.206)  (4.208) can be obtained as  is the Peclet number that reflects the relative level of convection and diffusion. Pe becomes zero for the case of pure diffusion and becomes infinite for the case of pure advection. The exact solution of eqs. (4.206)  (4.208) can be obtained as  
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<math>\Phi =\frac{\varphi \varphi _{0}}{\varphi _{L}\varphi _{0}}=\frac{\exp (\text{Pe}X)1}{\exp (\text{Pe})1}</math>  <math>\Phi =\frac{\varphi \varphi _{0}}{\varphi _{L}\varphi _{0}}=\frac{\exp (\text{Pe}X)1}{\exp (\text{Pe})1}</math>  
</center>  </center>  
  {{EquationRef(  +  {{EquationRef(10)}} 
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which will be used as a criterion to check the accuracy of various discretization schemes.  which will be used as a criterion to check the accuracy of various discretization schemes.  
+  
+  [[Central Difference Scheme]]<br>  
+  [[Upwind Scheme]]<br>  
+  [[Hybrid Scheme]]<br>  
+  [[Exponential and Power Law Schemes]]<br>  
+  [[A Generalized Expression of Discretization Schemes]]<br> 
Revision as of 05:35, 1 April 2010
Exact Solution The objective of this subsection is to introduce various discretization schemes of the convectiondiffusion terms through discussion of the onedimensional steady state convection and diffusion problem. For a onedimensional steadystate convection and diffusion problem, the governing equation is

where the velocity, u, is assumed to be known. Both density, ρ, and diffusivity, Г, are assumed to be constants. The continuity equation for this onedimensional problem is

Equation (4.201) is subject to the following boundary conditions:


By introducing the following dimensionless variables

the onedimensional steadystate convection and diffusion problem can be nondimensionalized as

Φ = 0, X = 0 
Φ = 1, X = 1 
where

is the Peclet number that reflects the relative level of convection and diffusion. Pe becomes zero for the case of pure diffusion and becomes infinite for the case of pure advection. The exact solution of eqs. (4.206)  (4.208) can be obtained as

which will be used as a criterion to check the accuracy of various discretization schemes.
Central Difference Scheme
Upwind Scheme
Hybrid Scheme
Exponential and Power Law Schemes
A Generalized Expression of Discretization Schemes