Upwind differencing scheme for convection

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 }$$\;$ (1) Continuity equation

$\left(\rho uA\right)_{r}$ - $\left(\rho uA\right)_{l}$ = 0 $\;$ (2)

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 -

$F_{e}\phi _{e}-F_{w}\phi _{w}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})$ $\;$ (3) . .....(2)

$F_{r}-F_{l}\,=0$ $\;$ (4) ...(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 uA$$\;$ and $\;$$D\,={\frac {\Gamma A}{\delta x}}$ $\;$ (6)

Peclet number (Pe) is a non-dimensional parameter determining the comparative strengths of convection and diffusion, Peclet number :$Pe\,={\frac {F}{D}}\,={\frac {\rho u}{\Gamma {\text{/}}\delta x}}$ $\;$ (9)

For Peclet no. of lower value (|Pe|<2, diffusion is dominant and for this we use Central Difference Scheme. for other value of and Upwind Scheme is used for Convection dominating flows with Peclet no. (|Pe|>2

For positive Flow Direction

$U_{w}>0andU_{e}>0$

Corresponding Upwind scheme equation -

$F_{e}\phi _{P}-F_{w}\phi _{W}\,=D_{e}(\phi _{E}-\phi _{P})-D_{w}(\phi _{P}-\phi _{W})$ $\;$ .....(3)

Because of strong convection :Failed to parse (syntax error): {\displaystyle Ф_e became ф_P     and     ф_w became ф_W }

 Rearranging equation (2) gives

$[(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]\phi _{P}=(D_{w}+F_{w})\phi _{W}+D_{e}\phi _{E})IdentifyingCoefficients:<math>a_{P}=[(D_{w}+F_{w})+D_{e}+(F_{e}-F_{w})]$

 :$a_{W}=(D_{w}+F_{w})/math>:<math>a_{E}=D_{E}/math>ForNegativeFlowDirection::<math>U_{w}<0andU_{e}<0$

Corresponding Upwind scheme equation

:Failed to parse (syntax error): {\displaystyle F_{e} ф_{E} – F_{w} ф_{P} = D_{e} (ф_{E} - ф_{P}) - D_{w} (ф_{P} - ф_{W})}

After rearranging we get

Failed to parse (syntax error): {\displaystyle [(D_{e} - F_{e}) + D_{w} + (F_{e} - F_{w})] ф_{P} = D_{w} ф_{W} + (D_{e} – F_{e})ф_{E} }

Identifying Coefficients

$a_{W}=D_{w}$

$a_{E}=D_{e}-F_{e}$

General form of coefficients

                   :$a_{W}=D_{w}+max(F_{w},0)$	                      
                   :$a_{E}=D_{e}+max(0,-F_{e})$

Usage:

Solution in the central difference scheme fails to converge for peclet no. greater than 2 which upwind scheme overcame giving reasonable result. Therefore Upwind Differencing Scheme 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.