these simulation numerique et modelisation de l'ecoulement autour ...

A.4 The 4 models implemented in AVBP
For example, on a 1D uniform mesh, of mesh size ∆x, this yields :
R k∈Ωleft = smu4 [(
∆x 1
2 ∆t 2 (w k − w k−2
2∆x
R k∈Ωright = smu4 [(
∆x 1
2 ∆t 2 (w k+1 − w k−1
2∆x
Adding these 2 contributions gives :
+ w )
k+1 − w k−1
) ·
2∆x
+ w )
k+2 − w k
) ·
2∆x
( −∆x
2
) ( )]
wk−1 + w k
−
− w k
2
( ) (
(A.19)
)]
∆x wk + w k+1
−
− w k
2
2
(A.20)
dw k = smu4 ∆x
16∆t (w k−2 − 4w k−1 + 6w k − 4w k+1 + w k+2 )
(A.21)
which can be interpreted :
dw k = κ AV ∫
(∆∆ F D
k,∆x w) dx
(A.22)
with :
κ AV = smu4.∆x4
16∆t
= smu4.∆x3 |u + c|
16 CFL
and
∆∆ F k,∆x D w = w k−2 − 4w k−1 + 6w k − 4w k+1 + w k+2
∆x 4
(A.23)
where ∆∆ F k,∆x D is exactly the classical FD bi-Laplacian operator evaluated at k and of size ∆x.
This shows that κ AV can be seen as an “artificial” 4 th order hyper-viscosity, which is controlled by the
user-defined parameter smu4. Just like smu2, the smu4 parameter is dimensionless.
A.4 The 4 models implemented in AVBP
The four AV models available in AVBP are :
– “Honey” model (iavisc=-1),
– “Jameson” model (iavisc=1),
– “Colin” model (iavisc=2),
– and “SLK” (Schönfeld–Lartigue–Kaufmann) model (iavisc=3).
Where by “model” we denote a combination of several parameters :
– the choice of the sensor and the variable which is used for the sensor,
– the way the 2 nd order and the 4 th order operators are combined,
– and finally on which variables the operators are applied.
A.4.1
“Honey” model
The concept of sensor is not relevant in this case : ζ HON = 1 everywhere. Consequently, 2 nd order
AV is applied everywhere in the field. Of course, in this case, the use of 4 th order AV is not necessary and
smu4 is set to zero. It is yet highly recommended to put smu4 = 0 in your run.dat file, to remember
that no 4 th order AV is used in this case, although AVBP does it automatically. . .
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