Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 127

Fig. 5.32 Flow variable solu-
tion using Central Differenc-
ing and Upper Differencing
schemes. Un-physical oscil-
lations are produced with the
Central Differencing scheme
whereas the Upper Differenc-
ing shows a more realistic
solution

discretisation schemes, improved results can be obtained if the grid spacing/mesh
resolution is refined (i.e. increase the number of control volumes).

False Diffusion Despite the Upper Differencing scheme being unconditionally
stable, it is essentially a backward differencing formula of first order accuracy. It
produces an artificial or false diffusion that is non-physical resulting in an incorrect
distribution of a flow variable ϕ through the flow domain for small Peclet numbers.
This occurs when the flow direction is not aligned with the grid lines and hence a
nonzero gradient exists in the direction perpendicular to the flow.
To demonstrate this, the transport equation for a pure 2D convection flow is
used,

du () φ dv () 0 φ

+ = (no diffusion term). When the flow moves horizontally

dx dy

through the domain in-line with the rectangular grid (Fig. 5.33 ), the Upper Differ-

ence solution provides a good representation and no false diffusion occurs. How-
ever, if the flow is now oriented at 45° to the grid lines, then false diffusion occurs.
In the absence of physical diffusion, the transported variable should exhibit a
uniform temperature of 0 below the dashed line, and 100 above the dashed line. If
we take the temperature values along the horizontal line half way up the domain
labelled as a-a' in Fig. 5.33b , the results show the Upper Difference scheme (using
a 5 × 5 control volume discretised domain), cannot capture the step profile of the
exact solution. Instead the variable, ϕ exhibits false diffusion behaviour spreading
across the domain rather than producing a step profile. Improvements to the result
can be obtained if the mesh is refined to a 10 × 10 control volume mesh (Fig. 5.34 ).
To illustrate why this occurs we consider the Upper Differenced discretised 1D
convection-diffusion equation where the diffusion term is set to zero:

φφ −

du () 0 φ

PW

(5.57)

=⇒ u = 0

dx ∆ x