False diffusion

Fig 2: West face is at 100°C while south wall is at 0°C. Heat is diffused across the diagonal XX

In this case, heat from west and south wall is carried by convection flow towards north and east wall. Heat is also diffused across the diagonal XX from upper to lower triangle. Figure 2 shows the approximate temperature distribution.

In this case heat from west and south wall is convected by flow towards north and east. There will be no diffusion across the diagonal XX but, when the upwind scheme is applied the results are similar to case (i) where actual diffusion is occurring. This error is known as false diffusion.

False diffusion with the upwind scheme is reduced by increasing the mesh density. In the results of figure 3 and 4 the false diffusion error is lowest in figure 4(b) with finer mesh size.

False diffusion error also can be reduced by using schemes such as the power law scheme, QUICK scheme, exponential scheme, and SUCCA, and others.[3][4]

False diffusion with the simple upwind scheme occurs because the scheme does not take into account grid/flow direction inclination. An approximate expression for the false-diffusion term in two dimensions has been given by de Vahl Davis and Mallinson(1972)[5]

${\displaystyle {\tau _{fc}^{\star }={\frac {\rho \ U\,\Delta x\,\Delta y\sin 2\theta }{4(\Delta y\sin ^{3}\theta +\Delta x\cos ^{3}\theta )}}}}$

(1)

where U is the resultant velocity and θ is the angle made by the velocity vector with the x direction. False diffusion is absent when the resultant flow is aligned with either of the sets of grid lines and is greatest when the flow direction is 45˚ to the grid lines.

Using Taylor series for ${\displaystyle \phi _{W}}$ and ${\displaystyle \phi _{P}}$ at the time t + kt are

${\displaystyle \phi _{Wk}=\phi _{wk}-\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}+{\frac {1}{2!}}\left({\frac {\delta x_{i}}{2}}\right)^{2}\left({\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)_{wk}+\cdots .}$

(2a)

${\displaystyle \phi _{Pk}=\phi _{wk}-\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}+{\frac {1}{2!}}\left({\frac {\delta x_{i}}{2}}\right)^{2}\left({\frac {\partial ^{2}\phi }{\partial x^{2}}}\right)_{wk}+\cdots .}$

(2b)

according to the upwind approximation for convection (UAC),${\displaystyle {\phi _{wk}=\phi _{Wk}}}$. Neglecting the higher order in equation (2a), the error of convected flux due to this approximation is ${\displaystyle {-\rho _{w}u_{w}\delta y_{i}\left({\frac {\delta x_{i}}{2}}\right)\left({\frac {\partial \phi }{\partial x}}\right)_{wk}}}$. It has the form of flux of ${\displaystyle {\phi }}$ by false diffusion with a diffusion co-efficient[6]

${\displaystyle \tau _{fc,UAC}^{\star }={\rho _{w}u_{w}\left({\frac {\Delta x_{i}}{2}}\right)}}$

(3)

The subscript fc is a reminder that this is a false diffusion arising from the estimate of the convected flux at the instant ${\displaystyle t+k\,\Delta t}$ using UAC.

Fig 5:SUCCA grid cell cluster

SUCCA takes the local flow direction into account by introducing the influence of upwind corner cells into the discretized conservation equation in the general governing transport equation. In Fig 5, SUCCA is applied within nine cell grid cluster. Considering the SW corner inflow for cell P, the SUCCA equations for the convective transport of the conserved species ${\displaystyle {\phi }}$ are

${\displaystyle C_{P}\phi _{P}=\left({\dot {m}}_{w}-{\frac {({\dot {m}}_{s})^{2}}{{\dot {m}}_{w}}}\right)\phi _{W}+\left({\dot {m}}_{s}+{\frac {({\dot {m}}_{s})^{2}}{{\dot {m}}_{w}}}\right)\phi _{SW}+0.\phi _{S}{\text{ for }}0<\theta \leq 45}$

(4)

i.e.,

${\displaystyle C_{P}\phi _{P}=C_{w}\phi _{W}+C_{s}w\phi _{SW}}$

(5)

${\displaystyle C_{P}\phi _{P}=\left({\dot {m}}_{s}-{\frac {({\dot {m}}_{w})^{2}}{{\dot {m}}_{s}}}\right)\phi _{W}+{\left({\dot {m}}_{w}+{\frac {({\dot {m}}_{w})^{2}}{{\dot {m}}_{s}}}\right)\phi _{SW}}+0.\phi _{W}{\text{ for }}45<\theta <90}$

(6)

i.e.,

${\displaystyle C_{P}\phi _{P}=C_{s}\phi _{S}+C_{s}w\phi _{SW}}$

(7)

This formulation satisfies all the criteria of convergence and stability.[7]

Fig 6: Comparison of different schemes

In Fig. 6, as mesh is refined, the upwind scheme gives more accurate results but SUCCA offers a nearly exact solution and is more useful in avoiding multidimensional false diffusion errors.