One-Dimensional Steady-State Convection and Diffusion

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{{Comp Method for Forced Convection Category}}
{{Comp Method for Forced Convection Category}}
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Exact Solution
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==Exact Solution==
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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
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The objective of this article 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
{| class="wikitable" border="0"
{| class="wikitable" border="0"
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</center>
</center>
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|{{EquationRef|(1)}}
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|{{EquationRef|(11)}}
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|}
The right-hand side of eq. (1) can be obtained by assuming the distribution of <math>\varphi </math> between any two neighboring grid points is piecewise linear, i.e.,
The right-hand side of eq. (1) can be obtained by assuming the distribution of <math>\varphi </math> between any two neighboring grid points is piecewise linear, i.e.,
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</center>
</center>
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|{{EquationRef|(2)}}
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|{{EquationRef|(12)}}
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Defining the mass flux and diffusive conductance
Defining the mass flux and diffusive conductance
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<math>F=\rho u,\text{  }D=\frac{\Gamma }{\delta x}</math>
<math>F=\rho u,\text{  }D=\frac{\Gamma }{\delta x}</math>
</center>
</center>
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|{{EquationRef|(3)}}
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|{{EquationRef|(13)}}
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</center>
</center>
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|{{EquationRef|(4)}}
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|{{EquationRef|(14)}}
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where  
where  
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<math>a_{E}=D_{e}-\frac{1}{2}F_{e}</math>
<math>a_{E}=D_{e}-\frac{1}{2}F_{e}</math>
</center>
</center>
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|{{EquationRef|(5)}}
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|{{EquationRef|(15)}}
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<math>a_{W}=D_{w}+\frac{1}{2}F_{w}</math>
<math>a_{W}=D_{w}+\frac{1}{2}F_{w}</math>
</center>
</center>
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|{{EquationRef|(6)}}
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|{{EquationRef|(16)}}
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<math>\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})</math>
<math>\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})</math>
</center>
</center>
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|{{EquationRef|(7)}}
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|{{EquationRef|(17)}}
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This scheme is termed the central difference scheme because the values of <math>\varphi </math> at the faces of the control volume are taken as the averaged value between two grid points. The continuity equation requires that <math>F_{e}=F_{w}</math> and therefore, eq. (7) reduces to
This scheme is termed the central difference scheme because the values of <math>\varphi </math> at the faces of the control volume are taken as the averaged value between two grid points. The continuity equation requires that <math>F_{e}=F_{w}</math> and therefore, eq. (7) reduces to
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<math>\varphi _{P}=\frac{1}{2}\left[ \left( 1-\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{E}+\left( 1+\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{W} \right]</math>
<math>\varphi _{P}=\frac{1}{2}\left[ \left( 1-\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{E}+\left( 1+\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{W} \right]</math>
</center>
</center>
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|{{EquationRef|(8)}}
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|{{EquationRef|(18)}}
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where
where
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<math>\text{Pe}_{\Delta }=\frac{\rho u\delta x}{\Gamma }=\frac{F}{D}</math>
<math>\text{Pe}_{\Delta }=\frac{\rho u\delta x}{\Gamma }=\frac{F}{D}</math>
</center>
</center>
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|{{EquationRef|(9)}}
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|{{EquationRef|(19)}}
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is the Peclet number using grid size as the characteristic length, which is referred to as the grid Peclet number. The grid Pe is a ratio of the strength of convection over diffusion. To ensure stability of the discretization scheme, the value of <math>\varphi _{P}</math> should always fall between <math>\varphi _{E}</math> and <math>\varphi _{W}</math>, which requires that the coefficients, <math>\varphi _{E}</math> and <math>\varphi _{W}</math>, are positive, i.e.,  
is the Peclet number using grid size as the characteristic length, which is referred to as the grid Peclet number. The grid Pe is a ratio of the strength of convection over diffusion. To ensure stability of the discretization scheme, the value of <math>\varphi _{P}</math> should always fall between <math>\varphi _{E}</math> and <math>\varphi _{W}</math>, which requires that the coefficients, <math>\varphi _{E}</math> and <math>\varphi _{W}</math>, are positive, i.e.,  
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<math>\left| \text{Pe}_{\Delta } \right|\le 2</math>
<math>\left| \text{Pe}_{\Delta } \right|\le 2</math>
</center>
</center>
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|{{EquationRef|(10)}}
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|{{EquationRef|(20)}}
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</center>
</center>
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|{{EquationRef|(1)}}
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|{{EquationRef|(21)}}
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where  
where  
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<math>a_{E}=D_{e}+\left[\!\left[ -F_{e},0 \right]\!\right]</math>
<math>a_{E}=D_{e}+\left[\!\left[ -F_{e},0 \right]\!\right]</math>
</center>
</center>
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|{{EquationRef|(2)}}
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|{{EquationRef|(22)}}
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<math>a_{W}=D_{w}+\left[\!\left[ F_{w},0 \right]\!\right]</math>
<math>a_{W}=D_{w}+\left[\!\left[ F_{w},0 \right]\!\right]</math>
</center>
</center>
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|{{EquationRef|(3)}}
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|{{EquationRef|(23)}}
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<math>\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})</math>
<math>\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})</math>
</center>
</center>
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|{{EquationRef|(4)}}
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|{{EquationRef|(24)}}
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The above scheme is referred to as the upwind scheme because the value of <math>\varphi </math> at the grid point on the upwind side was used as the value of <math>\varphi </math> at the face of the control volume to discretize the convection term. The upwind scheme ensures that the coefficients in eq. (4.221) are always positive so that a physically unrealistic solution can be avoided.
The above scheme is referred to as the upwind scheme because the value of <math>\varphi </math> at the grid point on the upwind side was used as the value of <math>\varphi </math> at the face of the control volume to discretize the convection term. The upwind scheme ensures that the coefficients in eq. (4.221) are always positive so that a physically unrealistic solution can be avoided.
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<math>a_{E}/D_{e}=1-\frac{1}{2}\text{Pe}_{\Delta e},\text{  Central difference scheme}</math>
<math>a_{E}/D_{e}=1-\frac{1}{2}\text{Pe}_{\Delta e},\text{  Central difference scheme}</math>
</center>
</center>
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|{{EquationRef|(1)}}
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|{{EquationRef|(25)}}
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<math>a_{E}/D_{e}=1+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right],\text{  Upwind scheme}</math>
<math>a_{E}/D_{e}=1+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right],\text{  Upwind scheme}</math>
</center>
</center>
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|{{EquationRef|(2)}}
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|{{EquationRef|(26)}}
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The hybrid scheme can then be expressed as
The hybrid scheme can then be expressed as
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<math>a_{E}/D_{e}=\left[\!\left[ -\text{Pe}_{\Delta e},1-\frac{1}{2}\text{Pe}_{\Delta e},0 \right]\!\right]</math>
<math>a_{E}/D_{e}=\left[\!\left[ -\text{Pe}_{\Delta e},1-\frac{1}{2}\text{Pe}_{\Delta e},0 \right]\!\right]</math>
</center>
</center>
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|{{EquationRef|(3)}}
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|{{EquationRef|(27)}}
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The coefficient for the west neighbor grid point can be obtained using a similar approach.
The coefficient for the west neighbor grid point can be obtained using a similar approach.
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<math>a_{W}/D_{w}=\left[\!\left[ \text{Pe}_{\Delta w},1+\frac{1}{2}\text{Pe}_{\Delta w},0 \right]\!\right]</math>
<math>a_{W}/D_{w}=\left[\!\left[ \text{Pe}_{\Delta w},1+\frac{1}{2}\text{Pe}_{\Delta w},0 \right]\!\right]</math>
</center>
</center>
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|{{EquationRef|(4)}}
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|{{EquationRef|(28)}}
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The above hybrid scheme combines the advantages of the central difference and upwind schemes to yield better results for cases where <math>\left| \text{Pe}_{\Delta } \right|\to \infty </math> or <math>\left| \text{Pe}_{\Delta } \right|\sim 0</math>. However, there is still room for improvement of the solution when <math>\left| \text{Pe}_{\Delta } \right|</math> is near 2 (see Problem 4.23).
The above hybrid scheme combines the advantages of the central difference and upwind schemes to yield better results for cases where <math>\left| \text{Pe}_{\Delta } \right|\to \infty </math> or <math>\left| \text{Pe}_{\Delta } \right|\sim 0</math>. However, there is still room for improvement of the solution when <math>\left| \text{Pe}_{\Delta } \right|</math> is near 2 (see Problem 4.23).
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<math>\frac{d}{dx}\left( \rho u\varphi -\Gamma \frac{d\varphi }{dx} \right)=0</math>
<math>\frac{d}{dx}\left( \rho u\varphi -\Gamma \frac{d\varphi }{dx} \right)=0</math>
</center>
</center>
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|{{EquationRef|(1)}}
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|{{EquationRef|(29)}}
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Defining the total flux of <math>\varphi </math> due to convection and diffusion  
Defining the total flux of <math>\varphi </math> due to convection and diffusion  
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<math>J=\rho u\varphi -\Gamma \frac{d\varphi }{dx}</math>
<math>J=\rho u\varphi -\Gamma \frac{d\varphi }{dx}</math>
</center>
</center>
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|{{EquationRef|(2)}}
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|{{EquationRef|(30)}}
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eq. (4.229) becomes
eq. (4.229) becomes
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<math>\frac{dJ}{dx}=0</math>
<math>\frac{dJ}{dx}=0</math>
</center>
</center>
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|{{EquationRef|(3)}}
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|{{EquationRef|(31)}}
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Integrating eq. (4.231) over the control volume P (shaded area in Fig. 4.17), yields
Integrating eq. (4.231) over the control volume P (shaded area in Fig. 4.17), yields
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<math>\begin{matrix}{}\\\end{matrix}J_{e}=J_{w}</math>
<math>\begin{matrix}{}\\\end{matrix}J_{e}=J_{w}</math>
</center>
</center>
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|{{EquationRef|(4)}}
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|{{EquationRef|(32)}}
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Instead of assuming piecewise linear distribution of <math>\varphi </math> as with central difference scheme or assuming <math>\varphi </math> at the face of the control volume is equal to the value of <math>\varphi </math> at the grid point on the upwind side in the upwind scheme, the distribution of <math>\varphi </math> between grid points can be taken as that obtained from the exact solution, eq.  (4.210). Applying eq. (4.210) between grid points E and P, we have  
Instead of assuming piecewise linear distribution of <math>\varphi </math> as with central difference scheme or assuming <math>\varphi </math> at the face of the control volume is equal to the value of <math>\varphi </math> at the grid point on the upwind side in the upwind scheme, the distribution of <math>\varphi </math> between grid points can be taken as that obtained from the exact solution, eq.  (4.210). Applying eq. (4.210) between grid points E and P, we have  
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<math>\frac{\varphi (x)-\varphi _{P}}{\varphi _{E}-\varphi _{P}}=\frac{\exp [\text{Pe}_{\Delta \text{e}}(x-x_{P})/(\delta x)_{e}]-1}{\exp (\text{Pe}_{\Delta \text{e}})-1}</math>
<math>\frac{\varphi (x)-\varphi _{P}}{\varphi _{E}-\varphi _{P}}=\frac{\exp [\text{Pe}_{\Delta \text{e}}(x-x_{P})/(\delta x)_{e}]-1}{\exp (\text{Pe}_{\Delta \text{e}})-1}</math>
</center>
</center>
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|{{EquationRef|(5)}}
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|{{EquationRef|(33)}}
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Substituting eq. (4.233) into eq. (4.230) and evaluating the result at <math>x=x_{e}</math>, the total flux of <math>\varphi </math> at the face of control volume becomes
Substituting eq. (4.233) into eq. (4.230) and evaluating the result at <math>x=x_{e}</math>, the total flux of <math>\varphi </math> at the face of control volume becomes
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<math>J_{e}=F_{e}\left[ \varphi _{P}+\frac{\varphi _{P}-\varphi _{E}}{\exp (\text{Pe}_{\Delta e})-1} \right]</math>
<math>J_{e}=F_{e}\left[ \varphi _{P}+\frac{\varphi _{P}-\varphi _{E}}{\exp (\text{Pe}_{\Delta e})-1} \right]</math>
</center>
</center>
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|{{EquationRef|(6)}}
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|{{EquationRef|(34)}}
|}  
|}  
Similarly, the total flux at the west face of the control volume is
Similarly, the total flux at the west face of the control volume is
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<math>J_{w}=F_{w}\left[ \varphi _{W}+\frac{\varphi _{W}-\varphi _{P}}{\exp (\text{Pe}_{\Delta w})-1} \right]</math>
<math>J_{w}=F_{w}\left[ \varphi _{W}+\frac{\varphi _{W}-\varphi _{P}}{\exp (\text{Pe}_{\Delta w})-1} \right]</math>
</center>
</center>
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|{{EquationRef|(7)}}
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|{{EquationRef|(35)}}
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Substituting eqs. (4.234) and (4.235) into eq. (4.232) and rearranging the resulting equation yields
Substituting eqs. (4.234) and (4.235) into eq. (4.232) and rearranging the resulting equation yields
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</center>
</center>
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|{{EquationRef|(8)}}
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|{{EquationRef|(36)}}
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where  
where  
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<math>a_{E}=\frac{F_{e}}{\exp (\text{Pe}_{\Delta e})-1}</math>
<math>a_{E}=\frac{F_{e}}{\exp (\text{Pe}_{\Delta e})-1}</math>
</center>
</center>
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|{{EquationRef|(9)}}
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|{{EquationRef|(37)}}
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<math>a_{W}=\frac{F_{w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}</math>
<math>a_{W}=\frac{F_{w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}</math>
</center>
</center>
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|{{EquationRef|(10)}}
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|{{EquationRef|(38)}}
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<math>\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})</math>
<math>\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})</math>
</center>
</center>
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|{{EquationRef|(11)}}
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|{{EquationRef|(39)}}
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Equations (4.237) and (4.238) can be rewritten in a format similar to that of eqs. (4.225) – (4.228), i.e.,
Equations (4.237) and (4.238) can be rewritten in a format similar to that of eqs. (4.225) – (4.228), i.e.,
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<math>a_{E}/D_{e}=\frac{\text{Pe}_{\Delta e}}{\exp (\text{Pe}_{\Delta e})-1}</math>
<math>a_{E}/D_{e}=\frac{\text{Pe}_{\Delta e}}{\exp (\text{Pe}_{\Delta e})-1}</math>
</center>
</center>
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|{{EquationRef|(12)}}
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|{{EquationRef|(40)}}
|}
|}
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<math>a_{W}/D_{w}=\frac{\text{Pe}_{\Delta w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}</math>
<math>a_{W}/D_{w}=\frac{\text{Pe}_{\Delta w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}</math>
</center>
</center>
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|{{EquationRef|(13)}}
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|{{EquationRef|(41)}}
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The comparison of <math>a_{E}/D_{e}</math> for different schemes is shown in Fig. 1. It can be seen that the hybrid scheme can be viewed as an envelope of the exponential scheme. The hybrid scheme is a good approximation if the absolute value of the grid Peclet number is either very large or near zero.   
The comparison of <math>a_{E}/D_{e}</math> for different schemes is shown in Fig. 1. It can be seen that the hybrid scheme can be viewed as an envelope of the exponential scheme. The hybrid scheme is a good approximation if the absolute value of the grid Peclet number is either very large or near zero.   
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<math>a_{E}/D_{e}=\left[\!\left[ 0,\left( 1-0.1\left| \text{Pe}_{\Delta e} \right| \right)^{5} \right]\!\right]+\left[\!\left[ 0,-\text{Pe}_{\Delta e} \right]\!\right]</math>
<math>a_{E}/D_{e}=\left[\!\left[ 0,\left( 1-0.1\left| \text{Pe}_{\Delta e} \right| \right)^{5} \right]\!\right]+\left[\!\left[ 0,-\text{Pe}_{\Delta e} \right]\!\right]</math>
</center>
</center>
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|{{EquationRef|(14)}}
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|{{EquationRef|(42)}}
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<math>J^{*}=\frac{J}{\Gamma /\delta x}=\text{Pe}_{\Delta }\varphi -\frac{d\varphi }{d(x/\delta x)}</math>
<math>J^{*}=\frac{J}{\Gamma /\delta x}=\text{Pe}_{\Delta }\varphi -\frac{d\varphi }{d(x/\delta x)}</math>
</center>
</center>
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|{{EquationRef|(1)}}
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|{{EquationRef|(43)}}
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which relates to the values of <math>\varphi </math> at grid points i and i+1 (see Fig. 4.19). The first term on the right side of eq. (4.243) will be related to some weighted average of  <math>\varphi _{i}</math> and <math>\varphi _{i+1}</math>, and the second term will be related to the difference between  <math>\varphi _{i}</math> and <math>\varphi _{i+1}</math>. Thus, one can express  
which relates to the values of <math>\varphi </math> at grid points i and i+1 (see Fig. 4.19). The first term on the right side of eq. (4.243) will be related to some weighted average of  <math>\varphi _{i}</math> and <math>\varphi _{i+1}</math>, and the second term will be related to the difference between  <math>\varphi _{i}</math> and <math>\varphi _{i+1}</math>. Thus, one can express  
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<math>J^{*}=B(\text{Pe}_{\Delta })\varphi _{i}-A(\text{Pe}_{\Delta })\varphi _{i+1}</math>
<math>J^{*}=B(\text{Pe}_{\Delta })\varphi _{i}-A(\text{Pe}_{\Delta })\varphi _{i+1}</math>
</center>
</center>
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|{{EquationRef|(2)}}
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|{{EquationRef|(44)}}
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|}
where A and B are dimensionless coefficients that are functions of the grid Peclet number. If the field of <math>\varphi </math> is uniform, we will have <math>d\varphi /dx=0</math> and  eq. (4.243) becomes  
where A and B are dimensionless coefficients that are functions of the grid Peclet number. If the field of <math>\varphi </math> is uniform, we will have <math>d\varphi /dx=0</math> and  eq. (4.243) becomes  
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<math>J^{*}=\text{Pe}_{\Delta }\varphi _{i}=\text{Pe}_{\Delta }\varphi _{i+1}</math>
<math>J^{*}=\text{Pe}_{\Delta }\varphi _{i}=\text{Pe}_{\Delta }\varphi _{i+1}</math>
</center>
</center>
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|{{EquationRef|(3)}}
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|{{EquationRef|(45)}}
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Comparing eqs. (4.244) and (4.245) yields
Comparing eqs. (4.244) and (4.245) yields
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<math>\begin{matrix}{}\\\end{matrix}B(\text{Pe}_{\Delta })-A(\text{Pe}_{\Delta })=\text{Pe}_{\Delta }</math>
<math>\begin{matrix}{}\\\end{matrix}B(\text{Pe}_{\Delta })-A(\text{Pe}_{\Delta })=\text{Pe}_{\Delta }</math>
</center>
</center>
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|{{EquationRef|(4)}}
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|{{EquationRef|(46)}}
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<math>J^{*}=B(-\text{Pe}_{\Delta })\varphi _{i+1}-A(-\text{Pe}_{\Delta })\varphi _{i}</math>
<math>J^{*}=B(-\text{Pe}_{\Delta })\varphi _{i+1}-A(-\text{Pe}_{\Delta })\varphi _{i}</math>
</center>
</center>
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|{{EquationRef|(5)}}
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|{{EquationRef|(47)}}
|}
|}
The symmetric properties of A and B can be obtained by comparing eqs. (4.244) and (4.247), i.e.,
The symmetric properties of A and B can be obtained by comparing eqs. (4.244) and (4.247), i.e.,
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<math>\begin{matrix}{}\\\end{matrix}A(-\text{Pe}_{\Delta })=B(\text{Pe}_{\Delta })</math>
<math>\begin{matrix}{}\\\end{matrix}A(-\text{Pe}_{\Delta })=B(\text{Pe}_{\Delta })</math>
</center>
</center>
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|{{EquationRef|(6)}}
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|{{EquationRef|(48)}}
|}
|}
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<math>\begin{matrix}{}\\\end{matrix}B(-\text{Pe}_{\Delta })=A(\text{Pe}_{\Delta })</math>
<math>\begin{matrix}{}\\\end{matrix}B(-\text{Pe}_{\Delta })=A(\text{Pe}_{\Delta })</math>
</center>
</center>
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|{{EquationRef|(7)}}
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|{{EquationRef|(49)}}
|}
|}
For the exponential schemes discussed above, one can obtain <math>J^{*}</math> from eq(4.234)or (4.235), i.e.,  
For the exponential schemes discussed above, one can obtain <math>J^{*}</math> from eq(4.234)or (4.235), i.e.,  
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<math>A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ -\text{Pe}_{\Delta },0 \right]\!\right]</math>
<math>A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ -\text{Pe}_{\Delta },0 \right]\!\right]</math>
</center>
</center>
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|{{EquationRef|(8)}}
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|{{EquationRef|(50)}}
|}
|}
Similarly, the expression of B for any grid Peclet number can be expressed as (see Problem 4.24).  
Similarly, the expression of B for any grid Peclet number can be expressed as (see Problem 4.24).  
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<math>B(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ \text{Pe}_{\Delta },0 \right]\!\right]</math>
<math>B(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ \text{Pe}_{\Delta },0 \right]\!\right]</math>
</center>
</center>
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|{{EquationRef|(9)}}
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|{{EquationRef|(51)}}
|}
|}
Therefore, different discretization schemes for the convection-diffusion terms can be characterized by different  
Therefore, different discretization schemes for the convection-diffusion terms can be characterized by different  
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<math>J_{e}^{*}\text{D}_{e}=J_{w}^{*}\text{D}_{w}</math>
<math>J_{e}^{*}\text{D}_{e}=J_{w}^{*}\text{D}_{w}</math>
</center>
</center>
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|{{EquationRef|(10)}}
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|{{EquationRef|(52)}}
|}
|}
The total fluxes at the faces of the control volumes can be obtained from eq. (4.244), i.e.,   
The total fluxes at the faces of the control volumes can be obtained from eq. (4.244), i.e.,   
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<math>J_{e}^{*}=B(\text{Pe}_{\Delta e})\varphi _{P}-A(\text{Pe}_{\Delta e})\varphi _{E}</math>
<math>J_{e}^{*}=B(\text{Pe}_{\Delta e})\varphi _{P}-A(\text{Pe}_{\Delta e})\varphi _{E}</math>
</center>
</center>
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|{{EquationRef|(11)}}
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|{{EquationRef|(53)}}
|}
|}
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<math>J_{w}^{*}=B(\text{Pe}_{\Delta w})\varphi _{W}-A(\text{Pe}_{\Delta w})\varphi _{P}</math>
<math>J_{w}^{*}=B(\text{Pe}_{\Delta w})\varphi _{W}-A(\text{Pe}_{\Delta w})\varphi _{P}</math>
</center>
</center>
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|{{EquationRef|(12)}}
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|{{EquationRef|(54)}}
|}
|}
Substituting the above expressions into eq. (4.252) and rearranging the resulting equation yields
Substituting the above expressions into eq. (4.252) and rearranging the resulting equation yields
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<math>a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}</math>
<math>a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}</math>
</center>
</center>
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|{{EquationRef|(13)}}
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|{{EquationRef|(55)}}
|}
|}
where
where
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<math>a_{E}=D_{e}A(\text{Pe}_{\Delta e})=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}</math>
<math>a_{E}=D_{e}A(\text{Pe}_{\Delta e})=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}</math>
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<math>a_{W}=D_{w}B(\text{Pe}_{\Delta w})=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}</math>
<math>a_{W}=D_{w}B(\text{Pe}_{\Delta w})=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}</math>
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<math>a_{P}=a_{E}+a_{W}+(F_{e}-F_{w})</math>
<math>a_{P}=a_{E}+a_{W}+(F_{e}-F_{w})</math>
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Revision as of 05:44, 21 July 2010

 Related Topics Catalog
Computational methodologies for forced convection
  1. One-Dimensional Steady-State Convection and Diffusion
    1. Central Difference Scheme
    2. Upwind Scheme
    3. Hybrid Scheme
    4. Exponential and Power Law Schemes
    5. A Generalized Expression of Discretization Schemes
  2. Multidimensional Convection and Diffusion Problems
  3. Numerical Solution of Flow Field
    1. Special Difficulties
    2. Staggered grid
    3. Pressure Correction Equation
    4. The SIMPLE Algorithm
  4. Numerical Simulation of Interfaces and Free Surfaces
  5. Application of Computational Methods

Exact Solution

The objective of this article 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 (1) 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)

\begin{matrix}{}\\\end{matrix}\Phi =0,\text{  X}=0

(7)

\begin{matrix}{}\\\end{matrix}\Phi =1,\text{  }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. (6) - (8) 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
Integrating the governing equation over the control volume P (shaded area in the figure to the right), one obtains

Control volume for one-dimensional problem
Control volume for one-dimensional problem.

(\rho u\varphi )_{e}-(\rho u\varphi )_{w}=\left( \Gamma \frac{d\varphi }{dx} \right)_{e}-\left( \Gamma \frac{d\varphi }{dx} \right)_{w}

(11)

The right-hand side of eq. (1) can be obtained by assuming the distribution of \varphi between any two neighboring grid points is piecewise linear, i.e.,

\left( \Gamma \frac{d\varphi }{dx} \right)_{e}=\Gamma _{e}\frac{\varphi _{E}-\varphi _{P}}{(\delta x)_{e}}


\left( \Gamma \frac{d\varphi }{dx} \right)_{w}=\Gamma _{w}\frac{\varphi _{P}-\varphi _{W}}{(\delta x)_{w}}

where Γe and Γw are the diffusivities at the faces of the control volume. To ensure that the flux of \varphi across the faces of the control volume is continuous, the harmonic mean diffusivity at the faces should be used. To evaluate the left hand side of eq. (1), it is necessary to know the values of \varphi at the faces of the control volume. If the piecewise linear profile of \varphi is chosen, it follows that

\varphi _{e}=\frac{\varphi _{E}+\varphi _{P}}{2}


\varphi _{w}=\frac{\varphi _{P}+\varphi _{W}}{2}

Therefore, eq. (1) becomes

(\rho u)_{e}\frac{\varphi _{E}+\varphi _{P}}{2}-(\rho u)_{w}\frac{\varphi _{P}+\varphi _{W}}{2}=\Gamma _{e}\frac{\varphi _{E}-\varphi _{P}}{(\delta x)_{e}}-\Gamma _{w}\frac{\varphi _{P}-\varphi _{W}}{(\delta x)_{w}}

(12)

Defining the mass flux and diffusive conductance

F=\rho u,\text{  }D=\frac{\Gamma }{\delta x}

(13)


eq. (2) can be rearranged as

a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}

(14)

where

a_{E}=D_{e}-\frac{1}{2}F_{e}

(15)

a_{W}=D_{w}+\frac{1}{2}F_{w}

(16)

\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})

(17)

This scheme is termed the central difference scheme because the values of \varphi at the faces of the control volume are taken as the averaged value between two grid points. The continuity equation requires that Fe = Fw and therefore, eq. (7) reduces to

aP = aW + aE

To evaluate the performance of the central difference scheme, let us consider the case of a uniform grid, i.e., x)e = (δx)w = δx, for which case eq. (2) can be rearranged as

\varphi _{P}=\frac{1}{2}\left[ \left( 1-\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{E}+\left( 1+\frac{\text{Pe}_{\Delta }}{2} \right)\varphi _{W} \right]

(18)

where

\text{Pe}_{\Delta }=\frac{\rho u\delta x}{\Gamma }=\frac{F}{D}

(19)

is the Peclet number using grid size as the characteristic length, which is referred to as the grid Peclet number. The grid Pe is a ratio of the strength of convection over diffusion. To ensure stability of the discretization scheme, the value of \varphi _{P} should always fall between \varphi _{E} and \varphi _{W}, which requires that the coefficients, \varphi _{E} and \varphi _{W}, are positive, i.e.,

\left| \text{Pe}_{\Delta } \right|\le 2

(20)

This is the criterion for stability of the central difference scheme. It can be demonstrated that the central difference becomes unstable if eq. (10) is violated. The fact that the central difference scheme is stable under small grid Peclet number indicates that the central difference scheme is accurate only if the convection is not very significant.

Upwind Scheme
The central difference scheme assumes that the effects of the values of \varphi at two neighboring grid points on the value of \varphi at the face of the control volume are equal. This assumption is valid only if the effect of diffusion is dominant. If, on the other hand, the convection is dominant, one can expect that the effect of the grid point upwind is more significant than that of the point downwind. If we can assume that the value of \varphi at the face of the control volume is dominated by the value of \varphi at the grid point at the upwind side and that the effect of the value of \varphi at the downwind side can be neglected, the two terms on the left hand side of eq. (4.211) can be expressed as

(\rho u\varphi )_{e}=\left\{ \begin{matrix}
   F_{e}\varphi _{P},\text{  }F_{e}>0  \\
   F_{e}\varphi _{E},\text{  }F_{e}<0  \\
\end{matrix} \right.


(\rho u\varphi )_{w}=\left\{ \begin{matrix}
   F_{w}\varphi _{W},\text{  }F_{w}>0  \\
   F_{w}\varphi _{P},\text{  }F_{w}<0  \\
\end{matrix} \right.

The above two equations can be expressed in the following compact form:

(\rho u\varphi )_{e}=\varphi _{P}\left[\!\left[ F_{e},0 \right]\!\right]-\varphi _{E}\left[\!\left[ -F_{e},0 \right]\!\right]


(\rho u\varphi )_{w}=\varphi _{W}\left[\!\left[ F_{w},0 \right]\!\right]-\varphi _{P}\left[\!\left[ -F_{w},0 \right]\!\right]

where the operator \left[\!\left[ A,B \right]\!\right] denotes the greater of A and B (Patankar, 1980). Substituting the above expression into the left hand side of eq. (4.211) and using central difference for the right hand side of eq. (4.211), the discretized equation becomes

a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}

(21)

where

a_{E}=D_{e}+\left[\!\left[ -F_{e},0 \right]\!\right]

(22)

a_{W}=D_{w}+\left[\!\left[ F_{w},0 \right]\!\right]

(23)

\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})

(24)

The above scheme is referred to as the upwind scheme because the value of \varphi at the grid point on the upwind side was used as the value of \varphi at the face of the control volume to discretize the convection term. The upwind scheme ensures that the coefficients in eq. (4.221) are always positive so that a physically unrealistic solution can be avoided.

Hybrid Scheme
The upwind scheme uses the value of \varphi from the grid point at the upwind side as the value of \varphi at the face of the control volume regardless of the grid Peclet number. While this treatment can yield accurate results for cases with high Peclet number, the result will not be accurate for cases where the grid Peclet number is near zero; for which cases the central difference scheme can produce better results. Spalding (1972) proposed a hybrid scheme that uses the central difference scheme when \left| \text{Pe}_{\Delta } \right|\le 2 and the upwind scheme when \left| \text{Pe}_{\Delta } \right|>2.

To observe the difference between the central difference and upwind schemes, the coefficient for the east neighboring grid point, eqs. (4.215) and (4.222), can be rewritten as

a_{E}/D_{e}=1-\frac{1}{2}\text{Pe}_{\Delta e},\text{  Central difference scheme}

(25)

a_{E}/D_{e}=1+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right],\text{  Upwind scheme}

(26)

The hybrid scheme can then be expressed as

a_{E}/D_{e}=\left\{ \begin{matrix}
   -\text{Pe}_{\Delta e}\text{                 Pe}_{\Delta e}<-2  \\
   1-\frac{1}{2}\text{Pe}_{\Delta e}\text{    }-\text{2}\le \text{ Pe}_{\Delta e}\le 2  \\
   0\text{                       Pe}_{\Delta e}>2  \\
\end{matrix} \right.

which can be rewritten in the following compact form

a_{E}/D_{e}=\left[\!\left[ -\text{Pe}_{\Delta e},1-\frac{1}{2}\text{Pe}_{\Delta e},0 \right]\!\right]

(27)

The coefficient for the west neighbor grid point can be obtained using a similar approach.

a_{W}/D_{w}=\left[\!\left[ \text{Pe}_{\Delta w},1+\frac{1}{2}\text{Pe}_{\Delta w},0 \right]\!\right]

(28)

The above hybrid scheme combines the advantages of the central difference and upwind schemes to yield better results for cases where \left| \text{Pe}_{\Delta } \right|\to \infty or \left| \text{Pe}_{\Delta } \right|\sim 0. However, there is still room for improvement of the solution when \left| \text{Pe}_{\Delta } \right| is near 2 (see Problem 4.23).

Exponential and Power Law Schemes
Since the exact solution of eq. (4.201) exists, one can reasonably expect that an accurate scheme can be derived if the result of the exact solution, eq. (4.210), is utilized. Equation (4.201) can be rewritten as

\frac{d}{dx}\left( \rho u\varphi -\Gamma \frac{d\varphi }{dx} \right)=0

(29)

Defining the total flux of \varphi due to convection and diffusion

J=\rho u\varphi -\Gamma \frac{d\varphi }{dx}

(30)

eq. (4.229) becomes

\frac{dJ}{dx}=0

(31)

Integrating eq. (4.231) over the control volume P (shaded area in Fig. 4.17), yields

\begin{matrix}{}\\\end{matrix}J_{e}=J_{w}

(32)

Instead of assuming piecewise linear distribution of \varphi as with central difference scheme or assuming \varphi at the face of the control volume is equal to the value of \varphi at the grid point on the upwind side in the upwind scheme, the distribution of \varphi between grid points can be taken as that obtained from the exact solution, eq. (4.210). Applying eq. (4.210) between grid points E and P, we have

\frac{\varphi (x)-\varphi _{P}}{\varphi _{E}-\varphi _{P}}=\frac{\exp [\text{Pe}_{\Delta \text{e}}(x-x_{P})/(\delta x)_{e}]-1}{\exp (\text{Pe}_{\Delta \text{e}})-1}

(33)

Substituting eq. (4.233) into eq. (4.230) and evaluating the result at x = xe, the total flux of \varphi at the face of control volume becomes

J_{e}=F_{e}\left[ \varphi _{P}+\frac{\varphi _{P}-\varphi _{E}}{\exp (\text{Pe}_{\Delta e})-1} \right]

(34)

Similarly, the total flux at the west face of the control volume is

J_{w}=F_{w}\left[ \varphi _{W}+\frac{\varphi _{W}-\varphi _{P}}{\exp (\text{Pe}_{\Delta w})-1} \right]

(35)

Substituting eqs. (4.234) and (4.235) into eq. (4.232) and rearranging the resulting equation yields

a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}

(36)

where

a_{E}=\frac{F_{e}}{\exp (\text{Pe}_{\Delta e})-1}

(37)

a_{W}=\frac{F_{w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}

(38)

\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})

(39)

Equations (4.237) and (4.238) can be rewritten in a format similar to that of eqs. (4.225) – (4.228), i.e.,

a_{E}/D_{e}=\frac{\text{Pe}_{\Delta e}}{\exp (\text{Pe}_{\Delta e})-1}

(40)

a_{W}/D_{w}=\frac{\text{Pe}_{\Delta w}\exp (\text{Pe}_{\Delta w})}{\exp (\text{Pe}_{\Delta w})-1}

(41)

The comparison of aE / De for different schemes is shown in Fig. 1. It can be seen that the hybrid scheme can be viewed as an envelope of the exponential scheme. The hybrid scheme is a good approximation if the absolute value of the grid Peclet number is either very large or near zero.

Comparison among different schemes
Figure 1: Comparison among different schemes.

While the exponential scheme is accurate, the computational time is much longer than for the central difference, upwind or hybrid schemes. Patankar (1981) proposed a power law scheme that has almost the same accuracy as the exponential scheme but a substantially shorter computational time. The coefficient of the neighbor grid point on the east side can be obtained by


a_{E}/D_{e}=\left\{ \begin{matrix}
   -\text{Pe}_{\Delta e}\text{                        Pe}_{\Delta e}<-10  \\
   (1+0.1\text{Pe}_{\Delta e})^{5}-\text{Pe}_{\Delta e}\text{  }-1\text{0}\le \text{Pe}_{\Delta e}<0  \\
   (1-0.1\text{Pe}_{\Delta e})^{5}\text{             0}\le \text{Pe}_{\Delta e}\le 10  \\
   0\text{                            Pe}_{\Delta e}>10  \\
\end{matrix} \right.

which can be rewritten in the following compact form

a_{E}/D_{e}=\left[\!\left[ 0,\left( 1-0.1\left| \text{Pe}_{\Delta e} \right| \right)^{5} \right]\!\right]+\left[\!\left[ 0,-\text{Pe}_{\Delta e} \right]\!\right]

(42)

A Generalized Expression of Discretization Schemes
The above discretization schemes can be expressed in a single generalized form. The total flux J at the interface between two grid points that were defined in eq. (4.230) can be used to define:

J^{*}=\frac{J}{\Gamma /\delta x}=\text{Pe}_{\Delta }\varphi -\frac{d\varphi }{d(x/\delta x)}

(43)

which relates to the values of \varphi at grid points i and i+1 (see Fig. 4.19). The first term on the right side of eq. (4.243) will be related to some weighted average of \varphi _{i} and \varphi _{i+1}, and the second term will be related to the difference between \varphi _{i} and \varphi _{i+1}. Thus, one can express J * the total flux as (Patankar, 1980)

J^{*}=B(\text{Pe}_{\Delta })\varphi _{i}-A(\text{Pe}_{\Delta })\varphi _{i+1}

(44)

where A and B are dimensionless coefficients that are functions of the grid Peclet number. If the field of \varphi is uniform, we will have d\varphi /dx=0 and eq. (4.243) becomes

J^{*}=\text{Pe}_{\Delta }\varphi _{i}=\text{Pe}_{\Delta }\varphi _{i+1}

(45)

Comparing eqs. (4.244) and (4.245) yields

\begin{matrix}{}\\\end{matrix}B(\text{Pe}_{\Delta })-A(\text{Pe}_{\Delta })=\text{Pe}_{\Delta }

(46)
Total flux at the interface between grid points i and i+1
Figure 1: Total flux at the interface between grid points i and i+1 .

For the grid system shown in Fig. 1, if we reconsider the problem in a reversed coordinate system x' (x' = − x), the grid Peclet number will become − PeΔ and J * becomes

J^{*}=B(-\text{Pe}_{\Delta })\varphi _{i+1}-A(-\text{Pe}_{\Delta })\varphi _{i}

(47)

The symmetric properties of A and B can be obtained by comparing eqs. (4.244) and (4.247), i.e.,

\begin{matrix}{}\\\end{matrix}A(-\text{Pe}_{\Delta })=B(\text{Pe}_{\Delta })

(48)

\begin{matrix}{}\\\end{matrix}B(-\text{Pe}_{\Delta })=A(\text{Pe}_{\Delta })

(49)

For the exponential schemes discussed above, one can obtain J * from eq(4.234)or (4.235), i.e.,

\begin{align}
  & J^{*}=\text{Pe}_{\Delta }\left[ \varphi _{i}+\frac{\varphi _{i}-\varphi _{i+1}}{\exp (\text{Pe}_{\Delta })-1} \right] \\ 
 & =\frac{\exp (\text{Pe}_{\Delta })\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1}\varphi _{i}-\frac{\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1}\varphi _{i+1} \\ 
\end{align}

Comparing the above expression with eq. (4.244), one obtains

B=\frac{\exp (\text{Pe}_{\Delta })\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1},\text{  }A=\frac{\text{Pe}_{\Delta }}{\exp (\text{Pe}_{\Delta })-1}

It can be verified that the above A and B satisfy eqs. (4.246), and (4.248) – (4.249). The implication of the above properties of A and B is that if the function A(PeΔ) for the case that PeΔ > 0 is known, the expressions of A and B for all PeΔ can be obtained. For example, if PeΔ < 0, eq. (4.246) can be used to obtain

A(PeΔ) = B(PeΔ) − PeΔ

Substituting eq. (4.248) into the above equation yields

A(PeΔ) = A( − PeΔ) − PeΔ

Considering -\text{Pe}_{\Delta }=\left| \text{Pe}_{\Delta } \right| for the case that PeΔ < 0, the above expression can be rewritten as

A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left| \text{Pe}_{\Delta } \right|\text{ for Pe}_{\Delta }<0

Since A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)\text{ for Pe}_{\Delta }>0 the following expression for A under any grid Peclet number can be expressed as

A(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ -\text{Pe}_{\Delta },0 \right]\!\right]

(50)

Similarly, the expression of B for any grid Peclet number can be expressed as (see Problem 4.24).

B(\text{Pe}_{\Delta })=A(\left| \text{Pe}_{\Delta } \right|)+\left[\!\left[ \text{Pe}_{\Delta },0 \right]\!\right]

(51)

Therefore, different discretization schemes for the convection-diffusion terms can be characterized by different A(\left| \text{Pe}_{\Delta } \right|) . To derive the generalized formula for different discretization schemes, let us begin from eq. (4.232), i.e.,

J_{e}^{*}\text{D}_{e}=J_{w}^{*}\text{D}_{w}

(52)

The total fluxes at the faces of the control volumes can be obtained from eq. (4.244), i.e.,

J_{e}^{*}=B(\text{Pe}_{\Delta e})\varphi _{P}-A(\text{Pe}_{\Delta e})\varphi _{E}

(53)

J_{w}^{*}=B(\text{Pe}_{\Delta w})\varphi _{W}-A(\text{Pe}_{\Delta w})\varphi _{P}

(54)

Substituting the above expressions into eq. (4.252) and rearranging the resulting equation yields

\left[ D_{e}B(\text{Pe}_{\Delta e})+D_{w}A(\text{Pe}_{\Delta w}) \right]\varphi _{P}=D_{e}A(\text{Pe}_{\Delta e})\varphi _{E}+D_{w}B(\text{Pe}_{\Delta w})\varphi _{W}

which can be rearranged as

a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}

(55)

where

a_{E}=D_{e}A(\text{Pe}_{\Delta e})=D_{e}\left\{ A(\left| \text{Pe}_{\Delta e} \right|)+\left[\!\left[ -\text{Pe}_{\Delta e},0 \right]\!\right] \right\}

(56)

a_{W}=D_{w}B(\text{Pe}_{\Delta w})=D_{w}\left\{ A(\left| \text{Pe}_{\Delta w} \right|)+\left[\!\left[ \text{Pe}_{\Delta w},0 \right]\!\right] \right\}

(57)


aP = aE + aW + (FeFw)

(58)
Comparison of A(
) for different schemes.

In arriving at eqs. (4.256) and (4.257), A and B were obtained from eqs. (4.250) and (4.251). At this point, it is apparent that different discretization schemes can be characterized by different expressions for A(|PeΔ|). By comparing eqs. (4.256) and (4.257) with different expressions of aE and aW for different schemes, the corresponding A(|PeΔ|) for different schemes can be summarized in Table 1 and plotted in Fig. 2. It should be noted that the difference between the power law and exponential scheme is exaggerated for clear presentation. The generalized formula represented by eqs. (4.255) – (4.258) will be very helpful to develop a generalized computer code for all schemes. A special module or subroutine can be written for different schemes.

Table 1 Summary of A(|PeΔ|) for different schemes

Scheme A(|PeΔ|)
Central difference 1-0.5\left| \text{Pe}_{\Delta } \right|
Upwind 1
Hybrid \left[\!\left[ 0,1-0.5\left| \text{Pe}_{\Delta } \right| \right]\!\right]
Exponential \left| \text{Pe}_{\Delta } \right|/[\exp (\left| \text{Pe}_{\Delta } \right|)-1]
Power Law \left[\!\left[ 0,(1-0.1\left| \text{Pe}_{\Delta } \right|)^{5} \right]\!\right]