One-Dimensional Steady-State 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 convection-diffusion terms through discussion of the one-dimensional 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 one-dimensional steady-state convection and diffusion problem can be nondimensionalized as
the one-dimensional steady-state 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>
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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 convection-diffusion terms through discussion of the one-dimensional steady state convection and diffusion problem. For a one-dimensional steady-state convection and diffusion problem, the governing equation is

\frac{d(\rho u\varphi )}{dx}=\frac{d}{dx}\left( \Gamma \frac{d\varphi }{dx} \right)

(1)

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 one-dimensional problem is

\frac{d(\rho u)}{dx}=0

(2)

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

\varphi =\varphi _{0},\text{  }x=0

(3)

\varphi =\varphi _{L},\text{  }x=L

(4)

By introducing the following dimensionless variables

\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}},\text{  }X=\frac{x}{L}

(5)

the one-dimensional steady-state convection and diffusion problem can be nondimensionalized as

\text{Pe}\frac{d\Phi }{dX}=\frac{d^{2}\Phi }{dX^{2}}

(6)

Φ = 0, X = 0

(7)

Φ = 1, X = 1

(8)

where

\text{Pe}=\frac{\rho uL}{\Gamma }

(9)

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

\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}}=\frac{\exp (\text{Pe}X)-1}{\exp (\text{Pe})-1}

(10)

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