The upwind differencing scheme is a method used in numerical methods in computational fluid dynamics for convection–diffusion problems. This scheme is specific for Peclet number greater than 2 or less than −2
Description 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 ϕ {\displaystyle \phi } at the cell face is adopted from the upstream node. It can be described by Steady convection-diffusion partial Differential Equation:
∂ ∂ t ( ρ ϕ ) + ∇ ⋅ ( ρ u ϕ ) = ∇ ⋅ ( Γ ∇ ϕ ) + S ϕ {\displaystyle {\frac {\partial }{\partial t}}(\rho \phi )+\nabla \cdot (\rho \mathbf {u} \phi )\,=\nabla \cdot (\Gamma \nabla \phi )+S_{\phi }}
Continuity equation: ( ρ u A ) e − ( ρ u A ) w = 0 {\displaystyle \left(\rho uA\right)_{e}-\left(\rho uA\right)_{w}=0\,}
where ρ {\displaystyle \rho } is density, Γ {\displaystyle \Gamma } is the diffusion coefficient, u {\displaystyle \mathbf {u} } is the velocity vector, ϕ {\displaystyle \phi } is the property to be computed, S ϕ {\displaystyle S_{\phi }} is the source term, and the subscripts e {\displaystyle e} and w {\displaystyle 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 Central difference discretized equation
Lower case denotes the face and upper case denotes node;
E {\displaystyle E} , W {\displaystyle W} , and P {\displaystyle 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 = ρ u A {\displaystyle F\,=\rho uA} and D = Γ A δ x {\displaystyle D\,={\frac {\Gamma A}{\delta x}}}
Peclet number (Pe) is a non-dimensional parameter determining the comparative strengths of convection and diffusion Peclet number:
P e = F D = ρ u Γ / δ x {\displaystyle 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
u w > 0 u e > 0 {\displaystyle {\begin{aligned}u_{w}>0\\u_{e}>0\end{aligned}}}
Corresponding upwind scheme equation:
Due to strong convection and suppressed diffusion
ϕ e = ϕ P ϕ w = ϕ W {\displaystyle {\begin{aligned}\phi _{e}\,=\phi _{P}\\\phi _{w}\,=\phi _{W}\end{aligned}}}
Rearranging equation (3) gives
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