Abstracts - KTH Mechanics

ALer ay T ype Regular ization for Low-dimensional
P OD-Galer kin Systems
F. S a b e t g h a d a m ∗ , S . S h a r a fa t m a n d jo o r †
In recent years, Leray regularization of the Navier-Stokes equations has been used
fairly extensive to manipulating the energy spectrum and improving the dissipation
characteristics in the turbulent flow simulations. The main advantage of the method
is that the resulting equations, usually called the Navier-Stokes-α equations, preserve
the structures and statistics of the large scales without need to fully resolve the fine
scales 1 .
On the other hand, one of the most popular low-dimensional modelling of turbulent
flows is the POD-Galerkin method. However, in many cases in the turbulent flow
simulations, the resulting POD-Galerkin ODE systems have been observed to be
unstable or , in the stable cases, can converge to wrong attractors. To improve the
behavior of these low-dimensional models, a variety of methods have been suggested
and tested. Methods like definition of the POD modes in H 1 Sobolev space instead of
L 2 space, optimization of the POD eigenmodes 2 , non-linear Galerkin projection and
introducing the eddy viscosity or spectral vanishing viscosity 3 .
In this article, inspired by the Leray regularization, a new method of improving
behavior of the POD-Galerkin systems is suggested and assessed. The method consists
of replacement of the non-linear terms, in the Galerkin projection of the Navier-Stokes
equations, by their filtered ones. Because of using a filter with an α width, the method
can be called the POD-α modelling. The method is assessed for a two-dimensional
turbulent flow and the primary results show steeper decreasing in the energy spectrum
for high wave numbers in comparison to the classical POD-Galerkin model and more
stability of the model for a variety of chosen snapshots.
∗ A ssistan t P r of., S cien ce & Resear ch b r an ch ,IA U ,T eh r an ,Ir an .
† P h Dstu d en t, S cien ce & Resear ch b r an ch ,IA U ,T eh r an ,Ir an .
1 C h en etal, P h y sica D 133 (1999) 66-83.
2 C ou p let etal, J ou r n al of C om p u tation al P h y sics 207 (2005) 192-220.
3 S ir isu p an d Kar n iad ak is, J ou r n al of C om p u tation al P h y sics 194 (2004) 92-116.
^
E (k,t)
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classical POD-Galerkin
^
k
POD-α modelling
Figure 1: comparison of the energy spectrums for the POD and POD-α models
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