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Upwind Differencing Scheme

Introduction
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
The Upwind Differencing Scheme by taking into account the direction of the flow 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 ф at the cell face is adopted from the upstream node.

Steady convection-diffusion partial Differential Equation is as follows $$\frac{\partial}{\partial t}(\rho\phi)+\nabla (\rho \mathbf{u} \phi)\,= \nabla (\Gamma \cdot \text{grad} \phi)+S_{\phi} $$$$\;$$  ($$) Continuity equation
 * $$\left(\rho u A \right)_{r}$$ - $$\left(\rho u A \right)_{l}$$ = 0 $$\;$$  ($$)

Where $$\rho$$ is density, $$\Gamma$$ is diffusion coefficient, $$\mathbf{u}$$ is the velocity vector, $$\phi$$ is the property to be computed and $$S_{\phi}$$ is the source term.

After discretization, applying continuity equation, and taking source equals to zero we get

Central Difference Discretized Equation  -
 * $$ F_{e} \phi_{e}-F_{w} \phi_{w}\,= D_{e}(\phi_{E}-\phi_{P})-D_{w}(\phi_{P}-\phi_{W})$$ $$\;$$  ($$)  .      .....(2)

$$ F_{r}-F_{l}\,=0$$ $$\;$$       ($$)       ...(3) Lower case denotes the face and upper case denotes node.

Defining variable F as convection mass flux and variable D as diffusion conductance
 * $$F\,= \rho u A$$$$\;$$ 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) = F/D is taken positive above

For positive Flow Direction Uw > 0 and Ue > 0 Corresponding Upwind scheme equation  - Fe фP – Fw фW = De (фE - фP) - Dw (фP - фW)    ……(2) Because of strong convection Фe became фP    and     фw became фW

Rearranging equation (2) gives [(Dw + Fw) + De + (Fe - Fw)] фP = (Dw + Fw) фW + DeфE Identifying Coefficients aP  =   [(Dw + Fw) + De + (Fe - Fw)], aW = (Dw + Fw), aE  =  DE

For Negative Flow Direction: Uw < 0 and Ue < 0

Corresponding Upwind scheme equation  - Fe фE – Fw фP = De (фE - фP) - Dw (фP - фW) After rearranging we get [(De - Fe) + Dw + (Fe - Fw)] фP = Dw фW + (De – Fe)фE Identifying Coefficients aW = Dw , aE  =  De - Fe

General form of coefficients

aW	                      aE             Dw + max(Fw,0)	                De + max(0,-Fe)

Usage:

Solution in the central difference scheme fails to converge for peclet no. greater than 2 which upwind scheme overcame giving reasonable result. Therefore UDS is applicable for Pe > 2 for positive flow and Pe < -2 for negative flow. For other values of Pe, this scheme doesn’t give effective solution.

Assessment:

Conservativeness: The upwind differencing scheme formulation is conservative.

Boundedness: As the coefficients of the discretised equation are always positive hence satisfying the requirements for boundedness and also the coefficient matrix is diagonally dominant therefore no irregularities occur in the solution.

Transportiveness: Transportiveness is built into the formulation as the scheme already accounts for the flow direction.

Accuracy: Based on the backward differencing formula, the accuracy is only first order on the basis of the Taylor series truncation error. It gives error when flow is not aligned with grid lines. Distribution of transported properties become marked giving diffusion-like appearance, called as the False Diffusion. Refinement of grid serves in overcoming the issue of false diffusion. With decrease in the grid size, false diffusion decrease thus increasing the accuracy.