False diffusion
False diffusion is a type of error observed when the
Definition
False diffusion is defined as an error having a diffusion-like appearance, obtained when the upwind scheme is used in multidimensional cases to solve for the distribution of transported properties flowing non-orthogonally to one or more of the system's major axes. The error is absent when the flow is orthogonal or parallel to each major axis.
Example
In figure 1, u = 2 and v = 2 m/s everywhere so the
Case (i)
In this case, heat from west and south walls is carried by convection flow towards north and east walls. Heat is also diffused across the diagonal XX from upper to lower triangle. Figure 2 shows the approximate temperature distribution.
Case (ii)
In this case heat from west and south walls 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.
Background
In early approaches,
Reducing errors
Finer mesh
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.
Other schemes
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]
Improving the upwind scheme
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]
-
(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.
Determining the accuracy of approximation for the convection term
Using Taylor series for and at the time t + kt are
-
(2a)
-
(2b)
according to the upwind approximation for convection (UAC),. Neglecting the higher order in equation (2a), the error of convected flux due to this approximation is . It has the form of flux of by false diffusion with a diffusion co-efficient[6]
-
(3)
The subscript fc is a reminder that this is a false diffusion arising from the estimate of the convected flux at the instant using UAC.
Skew upwind corner convection algorithm (SUCCA)
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 are
-
(4)
i.e.,
-
(5)
-
(6)
i.e.,
-
(7)
This formulation satisfies all the criteria of
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.
See also
- Computational fluid dynamics
- Navier–Stokes equations
- Numerical diffusion
- Finite volume method
- Taylor series
References
- .
- ^ Torrance, Kenneth E. (1968). "Comparison of finite difference computations of natural convection". Journal of Research of the National Bureau of Standards: Mathematics and Mathematical Physics. 72B: 281–301.
- ISBN 9780131274983.
- ISBN 9780891165224.
- ISBN 9780891165224.
- ISSN 0045-7825.
- ISSN 0307-904X.
Further reading
- Patankar, Suhas V. (1980), Numerical Heat Transfer and Fluid Flow, Taylor & Francis Group, ISBN 9780891165224
- Wesseling, Pieter (2001), Principles of Computational Fluid Dynamics, Springer, ISBN 978-3-540-67853-3
- Date, Anil W. (2005), Introduction to Computational Fluid Dynamics, Cambridge University Press, ISBN 9780521853262