Numerical analysis of approximate de-convolution models of turbulence
In the approximation of turbulent flows it is usual and sensible to seek
to approximate flow averages rather than pointwise velocities and
pressures. However, after averaging the Navier-Stokes equations the
problem of closure arises and the central dilemma of turbulent flow
simulations occurs: exact equations for the pointwise flow values are
known but intractable to numerical solution while equations for flow
averages are not (yet) known and perhaps unknowable
When the averaging is performed by a local spacial filter,
Average(u) := (g*u)(x,t), g: chosen filter,
the computational strategy is known as large eddy simulation or LES. One
approach to solving the LES closure problem is through approximate
deconvolution :
given Average(u) solve the above equation approximately for u.
This problem is ill-posed and any strategy for solving ill-posed problems
gives a candidate LES-turbulence model for testing.
This talk will focus on one approximate de-convolution model developed
for image processing by van Cittert in 1934 and introduced into LES by
Stolz and Adams in the 1990s. The talk will :
give a careful presentation of the models,
explain their mathematical foundation, developed in joint work with
R. Lewandowski and work of A. Dunca and Y. Epshteyn,
show that the time-averaged consistency error of the model is small
uniformly in the Reynolds number (joint with R. Lewandowski) ,
delineate the energy cascade of the model that shows how the models
dynamics truncates solution scales (recent joint work with M. Neda) and
explain how the models energy balance is used to develop a rigorous
convergence theory for finite element approximations of the model (work of
C. Manica and S. Kaya).