wikiwand.com

Upwind differencing scheme for convection - Wikiwand

By taking into account the direction of the flow, the upwind differencing scheme overcomes that inability of the central differencing scheme. This scheme is developed for strong convective flows with suppressed diffusion effects. Also known as ‘Donor Cell’ Differencing Scheme, the convected value of property $\phi$ at the cell face is adopted from the upstream node.

It can be described by Steady convection-diffusion partial Differential Equation:^[1]^: 103^[2]^{[circular reference]} ${\frac {\partial }{\partial t}}(\rho \phi )+\nabla \cdot (\rho \mathbf {u} \phi )\,=\nabla \cdot (\Gamma \nabla \phi )+S_{\phi }$

Continuity equation: $\left(\rho uA\right)_{e}-\left(\rho uA\right)_{w}=0\,$ ^[1]^: 104^[3]^{[circular reference]}

where $\rho$ is density, $\Gamma$ is the diffusion coefficient, $\mathbf {u}$ is the velocity vector, $\phi$ is the property to be computed, $S_{\phi }$ is the source term, and the subscripts $e$ and $w$ refer to the "east" and "west" faces of the cell (see Fig. 1 below).

After discretization, applying continuity equation, and taking source term equals to zero we get^[4]^{[circular reference]}

Central difference discretized equation^[1]^: 105

$F_{e}\phi _{e}-F_{w}\phi _{w}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})$

$F_{e}-F_{w}\,=0$

Lower case denotes the face and upper case denotes node; $E$ , $W$ , and $P$ refer to the "East," "West," and "Central" cell. (again, see Fig. 1 below).

Defining variable F as convection mass flux and variable D as diffusion conductance $F\,=\rho uA$ and $D\,={\frac {\Gamma A}{\delta x}}$

Peclet number (Pe) is a non-dimensional parameter determining the comparative strengths of convection and diffusion

Peclet number: $Pe\,={\frac {F}{D}}\,={\frac {\rho u}{\Gamma /\delta x}}$

For a Peclet number of lower value (|Pe| < 2), diffusion is dominant and for this the central difference scheme is used. For other values of the Peclet number, the upwind scheme is used for convection-dominated flows with Peclet number (|Pe| > 2).

For positive flow direction

${\begin{aligned}u_{w}>0\\u_{e}>0\end{aligned}}$ Corresponding upwind scheme equation:^[1]^: 115

$F_{e}\phi _{P}-F_{w}\phi _{W}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})$

Due to strong convection and suppressed diffusion^[1]^: 115 ${\begin{aligned}\phi _{e}\,=\phi _{P}\\\phi _{w}\,=\phi _{W}\end{aligned}}$

Rearranging equation (3) gives $[(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]\phi _{P}\,=(D_{w}+F_{w})\phi _{W}+D_{e}\phi _{E})$

Identifying coefficients, ${\begin{aligned}a_{P}&=[(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]\\a_{W}&=(D_{w}+F_{w})\\a_{E}&=D_{e}\end{aligned}}$

For negative flow direction ${\begin{aligned}u_{w}<0\\u_{e}<0\end{aligned}}$

Corresponding upwind scheme equation:^[1]^: 115

$F_{e}\phi _{E}-F_{w}\phi _{P}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})$

${\begin{aligned}\phi _{w}=\phi _{P}\\\phi _{e}=\phi _{E}\end{aligned}}$

Rearranging equation (4) gives $[(D_{e}-F_{e})+D_{w}+(F_{e}-F_{w})]\phi _{P}=D_{w}\phi _{W}+(D_{e}-F_{e})\phi _{E}$

Identifying coefficients, ${\begin{aligned}a_{W}&=D_{w}\\a_{E}&=D_{e}-F_{e}\end{aligned}}$

We can generalize coefficients as^[1]^: 116 ${\begin{aligned}a_{W}&=D_{w}+\max(F_{w},0)\\a_{E}&=D_{e}+\max(0,-F_{e})\end{aligned}}$