5.4 CFD-DEM Coupled Simulation and Development
327
Fig. 5.70 Comparison of
different spatial distribution
functions
erf(x) =
2
√ π
x
0
exp(−y
2
)dy
(5.181)
Even though the error function is a non-elementary function, it is available in
conventional programming languages, such as C/C++, Fortran, and Java. There is a
removable discontinuity at the origin of the coordinate for the diffusion function and
re-defined as D(0) = lim
|r|→0 +
D(r).
From the diffusion mechanism, it is proven that 0 < D(r) < 1, lim
|r|→∞
D(r) = 0,
and
R 3
D(r)dV =
R 3
H(r)dV.
(5.182)
The degree of smoothing relative to the step function H(r) can be defined as
η =
2τ
2τ + 1
(5.183)
When η goes to 0, the sequence of diffusion function converges to H(r). It is proven
mathematically that
lim
η→0 +
D η (r) − H(r)
2 = lim
τ →0 +
R 3
( D(r) − H(r))
2 dV = 0.
(5.184)
Figure 5.70 shows the diffusion functions at different smoothing degrees and the
Gaussian function. The diffusion function at η = 0.5 is applied in the CFD-DEM
framework, which is in good agreement with the Gaussian function. The function
decreases significantly from 0.72 at the origin to almost 0 at r = 2.5.