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Upwind differencing scheme for convection : ウィキペディア英語版
Upwind differencing scheme for convection

The upwind differencing scheme is a method used in numerical methods in computational fluid dynamics for convectiondiffusion 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.
It can be described by Steady convection-diffusion partial Differential Equation〔H.K Versteeg & W. Malalasekera (1995). An introduction to Computational Fluid Dynamics.Chapter:5, Page103.〕〔Central differencing scheme#Steady-state convection diffusion equation〕 –
: \frac(\rho\phi)+\nabla \cdot (\rho \mathbf \phi)\,= \nabla \cdot (\Gamma\operatorname \phi)+S_
Continuity equation: \left(\rho u A \right)_ - \left(\rho u A \right)_w = 0 \, 〔H. K. Versteeg & W. Malalasekera (1995). ''An introduction to Computational Fluid Dynamics''. Chapter 5, page 104.〕〔Central differencing scheme#Formulation of Steady state convection diffusion equation
where
\rho is density,
\Gamma is diffusion coefficient,
\mathbf is the velocity vector,
\phi is the property to be computed,
S_\phi is the source term,
and the subscripts e and 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 differencing scheme#Formulation of Steady state convection diffusion equation
Central difference discretized equation
: F_ \phi_-F_ \phi_\,= D_(\phi_-\phi_)-D_(\phi_-\phi_) \;.〔H.K Versteeg & W. Malalasekera. An introduction to Computational Fluid Dynamics.Chapter:5. Page 105.〕.....(1)
: F_-F_\,=0 \;〔H.K Versteeg & W. Malalasekera . An introduction to Computational Fluid Dynamics.Chapter:5. Page 105.〕.....(2)
Lower case denotes the face and upper case denotes node;
E, W, and 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\,= \rho u A\; and \;D\,= \frac \;
Peclet number (Pe) is a non-dimensional parameter determining the comparative strengths of convection and diffusion
Peclet number:
: Pe \,= \frac \,= \frac \;
For a Peclet number of lower value (|Pe| < 2, diffusion is dominant and for this we use the central difference scheme. For other values of and upwind scheme is used for convection dominating flows with Peclet number (|Pe| > 2).
For positive flow direction
: u_ > 0
: u_ > 0
Corresponding upwind scheme equation:
: F_ \phi_-F_ \phi_\,= D_(\phi_-\phi_)-D_(\phi_-\phi_)〔H.K Versteeg & W. Malalasekera . An introduction to Computational Fluid Dynamics.Chapter:5.Page 115.〕.....(3)
Due to strong convection and suppressed diffusion
:\phi_ \,= \phi_〔H. K. Versteeg & W. Malalasekera ). ''An Introduction to Computational Fluid Dynamics, Chapter 5, page 115.〕
:\phi_ \,= \phi_
Rearranging equation (3) gives
:(D_+ (F_-F_) )\phi_\, =(D_+F_)\phi_+D_\phi_)\;
Identifying coefficients,
:a_\,= (+ F_) + D_ + (F_ - F_) ) \;
:a_\,=(D_ + F_)
:a_\,= D_
For negative flow direction
:u_ < 0
:u_ < 0
Corresponding upwind scheme equation:
: F_ \phi_-F_ \phi_\,= D_(\phi_-\phi_)-D_(\phi_-\phi_)〔H.K Versteeg & W. Malalasekera. An introduction to Computational Fluid Dynamics.Chapter:5. Page115.〕.....(4)
:\phi_\,= \phi_
:\phi_\,= \phi_
Rearranging equation(4) gives
: (D_e - F_e ) + D_w + ( F_e - F_w ) ) \phi_ = D_w \phi_ + ( D_e - F_e ) \phi_
Identifying coefficients,
:a_\,= D_
:a_\,= D_ - F_
We can generalize coefficients as〔H. K. Versteeg & W. Malalasekera. ''An Introduction to Computational Fluid Dynamics'', Chapter 5, page 116.〕 –
:a_=D_ + \max(F_,0)

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