Upwind Scheme

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Computational methodologies for forced convection
  1. One-Dimensional Steady-State Convection and Diffusion
    1. Central Difference Scheme
    2. Upwind Scheme
    3. Hybrid Scheme
    4. Exponential and Power Law Schemes
    5. A Generalized Expression of Discretization Schemes
  2. Multidimensional Convection and Diffusion Problems
  3. Numerical Solution of Flow Field
    1. Special Difficulties
    2. Staggered grid
    3. Pressure Correction Equation
    4. The SIMPLE Algorithm
  4. Numerical Simulation of Interfaces and Free Surfaces
  5. Application of Computational Methods

The central difference scheme assumes that the effects of the values of \varphi at two neighboring grid points on the value of \varphi at the face of the control volume are equal. This assumption is valid only if the effect of diffusion is dominant. If, on the other hand, the convection is dominant, one can expect that the effect of the grid point upwind is more significant than that of the point downwind. If we can assume that the value of \varphi at the face of the control volume is dominated by the value of \varphi at the grid point at the upwind side and that the effect of the value of \varphi at the downwind side can be neglected, the two terms on the left hand side of eq. (4.211) can be expressed as

(\rho u\varphi )_{e}=\left\{ \begin{matrix}
   F_{e}\varphi _{P},\text{  }F_{e}>0  \\
   F_{e}\varphi _{E},\text{  }F_{e}<0  \\
\end{matrix} \right.


(\rho u\varphi )_{w}=\left\{ \begin{matrix}
   F_{w}\varphi _{W},\text{  }F_{w}>0  \\
   F_{w}\varphi _{P},\text{  }F_{w}<0  \\
\end{matrix} \right.

The above two equations can be expressed in the following compact form:

(\rho u\varphi )_{e}=\varphi _{P}\left[\!\left[ F_{e},0 \right]\!\right]-\varphi _{E}\left[\!\left[ -F_{e},0 \right]\!\right]


(\rho u\varphi )_{w}=\varphi _{W}\left[\!\left[ F_{w},0 \right]\!\right]-\varphi _{P}\left[\!\left[ -F_{w},0 \right]\!\right]

where the operator \left[\!\left[ A,B \right]\!\right] denotes the greater of A and B (Patankar, 1980). Substituting the above expression into the left hand side of eq. (4.211) and using central difference for the right hand side of eq. (4.211), the discretized equation becomes

a_{P}\varphi _{P}=a_{E}\varphi _{E}+a_{W}\varphi _{W}

(1)

where

a_{E}=D_{e}+\left[\!\left[ -F_{e},0 \right]\!\right]

(2)

a_{W}=D_{w}+\left[\!\left[ F_{w},0 \right]\!\right]

(3)

\begin{matrix}{}\\\end{matrix}a_{P}=a_{W}+a_{E}+(F_{e}-F_{w})

(4)

The above scheme is referred to as the upwind scheme because the value of \varphi at the grid point on the upwind side was used as the value of \varphi at the face of the control volume to discretize the convection term. The upwind scheme ensures that the coefficients in eq. (4.221) are always positive so that a physically unrealistic solution can be avoided.