Upwind differencing scheme for convection

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 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 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: Continuity equation: ( ρ u A ) e − ( ρ u A ) w = 0 {displaystyle left( ho uA ight)_{e}-left( ho uA ight)_{w}=0,} where ρ {displaystyle ho } is density, Γ {displaystyle Gamma } is 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