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

Annexe A
Artificial Viscosity Models
A.1 Introduction
The numerical discretization methods in AVBP are spatially centered. These types of schemes are
known to be naturally subject to small-scale oscillations in the vicinity of steep solution variations. This
is why it is common practice to add a so-called artificial viscosity (AV) term to the discrete equations,
to avoid these spurious modes (also known as “wiggles”) and in order to smooth very strong gradients.
We describe here the different AV methods used in AVBP. These AV models are characterized by the
“linear preserving” property which leaves unmodified a linear solution on any type of element. The
models are based on a combination of a “shock capturing” term (called 2 nd order AV) and a “background
dissipation” term (called 4 th order AV). In AVBP, adding AV is done in two steps :
– first a sensor detects if AV is necessary, as a function of the flow characteristics,
– then a certain amount of 2 nd and 4 th AV is applied, depending on the sensor value and on userdefined
parameters.
A.2 The sensors
A sensor ζ Ωj is a scaled parameter which is defined for every cell Ω j of the domain that takes values
from zero to one. ζ Ωj = 0 means that the solution is well resolved and that no AV should be applied while
ζ Ωj = 1 signifies that the solution has strong local variations and that AV must be applied. This sensor
is obtained by comparing different evaluations (on different stencils) of the gradient of a given scalar
(pressure, total energy, mass fractions, . . .). If these gradients are identical, then the solution is locally
linear and the sensor is zero. On the contrary, if these two estimations are different, local non-linearities
are present, and the sensor is activated. The key point is to find a suitable sensor-function that is non-zero
only at places where stability problems occur.
Two sensors are available in AVBP : the so-called ‘Jameson-sensor’ (ζΩ J j
) Jameson et al. (1981) and the
‘Colin-sensor’ (ζ C Ω j
) Colin (2000) which is an upgrade of the previous one.