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

A.3 The operators
– ζ C Ω j
is very small when both ∆ k 1 and ∆k 2 are small compared to S Ω j
. This corresponds to low
amplitude numerical errors (when ∆ k 1 and ∆k 2 have opposite signs) or smooth gradients that are
well resolved by the scheme (when ∆ k 1 and ∆k 2 have the same sign).
– ζΩ C j
is small when ∆ k 1 and ∆k 2 have the same sign and the same order of magnitude, even if they
are quite large. This corresponds to stiff gradients well resolved by the scheme.
– ζΩ C j
is big when ∆ k 1 and ∆k 2 have opposite signs and one of the two term is large compared to the
other. This corresponds to a high-amplitude numerical oscillation.
– ζΩ C j
is big when either ∆ k 1 or ∆k 2 is of the same order of magnitude as S Ω j
. This corresponds to a
non-physical situation that originates from a numerical problem.
The exact definition of the Colin-sensor is :
ζΩ C j
= 1 ( ( )) Ψ − Ψ0
1 + tanh
− 1 ( ( )) −Ψ0
1 + tanh
2
δ 2
δ
with :
(
∆ k )
Ψ = max 0,
k∈Ω j |∆ k ζk
J | + ɛ 1 S k
( )
∆ k = |∆ k 1 − ∆ k 2| − ɛ k max |∆ k 1|, |∆ k 2|
⎛
)
max
(|∆ k
ɛ k 1
= ɛ 2
⎝1 − ɛ |, |∆k 2 | ⎞
3
|∆ k 1 | + |∆k 2 | + S ⎠
k
(A.6)
(A.7)
(A.8)
(A.9)
The numerical values used in AVBP are :
Ψ 0 = 2.10 −2 δ = 1.10 −2 ɛ 1 = 1.10 −2 ɛ 2 = 0.95 ɛ 3 = 0.5 (A.10)
WARNING :
Note, that these definitions of Ψ and ɛ k apply only for the Navier-Stokes variables. For species, the
reference value is not S k but 1, which is the maximum value of a species mass fraction :
(
∆ k )
Ψ = max 0,
k∈Ω j |∆ k ζk
J | + ɛ 1
and
⎛
)
max
(|∆ k
ɛ k 1
= ɛ 2
⎝1 − ɛ |, |∆k 2 | ⎞
3
|∆ k 1 | + |∆k 2 | + 1 ⎠
(A.11)
A.3 The operators
There are two AV operators in AVBP : a 2 nd order operator and a 4 th order operator. All AV models
in AVBP are a blend of these two operators. These operators have the following properties :
– 2 nd order operator : it acts just like a “classical” viscosity. It smoothes gradients, and introduces
artificial dissipation. It is thus associated to a sensor which determines where it must be applied.
Doing this, the numerical scheme keeps its order of convergence in the zones where the sensor is
inactive, while ensuring stability and robustness in the critical regions. Historically, it was used to
control shocks, but it can actually smooth any physical gradient.
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