(Difference between revisions)
 Revision as of 05:33, 1 April 2010 (view source) (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 …')← Older edit Revision as of 05:35, 1 April 2010 (view source)Newer edit → Line 16: Line 16: $\frac{d(\rho u)}{dx}=0$ $\frac{d(\rho u)}{dx}=0$ - |{{EquationRef|(1)}} + |{{EquationRef|(2)}} |} |} Equation (4.201) is subject to the following boundary conditions: Equation (4.201) is subject to the following boundary conditions: Line 25: Line 25: $\varphi =\varphi _{0},\text{ }x=0$ $\varphi =\varphi _{0},\text{ }x=0$ - |{{EquationRef|(1)}} + |{{EquationRef|(3)}} |} |} Line 33: Line 33: $\varphi =\varphi _{L},\text{ }x=L$ $\varphi =\varphi _{L},\text{ }x=L$ - |{{EquationRef|(1)}} + |{{EquationRef|(4)}} |} |} By introducing the following dimensionless variables By introducing the following dimensionless variables Line 42: Line 42: $\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}},\text{ }X=\frac{x}{L}$ $\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}},\text{ }X=\frac{x}{L}$ - |{{EquationRef|(1)}} + |{{EquationRef|(5)}} |} |} 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 Line 51: Line 51: $\text{Pe}\frac{d\Phi }{dX}=\frac{d^{2}\Phi }{dX^{2}}$ $\text{Pe}\frac{d\Phi }{dX}=\frac{d^{2}\Phi }{dX^{2}}$ - |{{EquationRef|(1)}} + |{{EquationRef|(6)}} |} |} Line 59: Line 59: $\Phi =0,\text{ X}=0$ $\Phi =0,\text{ X}=0$ - |{{EquationRef|(1)}} + |{{EquationRef|(7)}} |} |} Line 67: Line 67: $\Phi =1,\text{ }X=1$ $\Phi =1,\text{ }X=1$ - |{{EquationRef|(1)}} + |{{EquationRef|(8)}} |} |} where where Line 76: Line 76: $\text{Pe}=\frac{\rho uL}{\Gamma }$ $\text{Pe}=\frac{\rho uL}{\Gamma }$ - |{{EquationRef|(1)}} + |{{EquationRef|(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 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 Line 85: Line 85: $\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}}=\frac{\exp (\text{Pe}X)-1}{\exp (\text{Pe})-1}$ $\Phi =\frac{\varphi -\varphi _{0}}{\varphi _{L}-\varphi _{0}}=\frac{\exp (\text{Pe}X)-1}{\exp (\text{Pe})-1}$ - |{{EquationRef|(1)}} + |{{EquationRef|(10)}} |} |} 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]]
+ [[Upwind Scheme]]
+ [[Hybrid Scheme]]
+ [[Exponential and Power Law Schemes]]
+ [[A Generalized Expression of Discretization Schemes]]

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.