13 research outputs found
On the Prandtl-Kolmogorov 1-equation model of turbulence
We prove an estimate of total (viscous plus modelled turbulent) energy
dissipation in general eddy viscosity models for shear flows. For general eddy
viscosity models, we show that the ratio of the near wall average viscosity to
the effective global viscosity is the key parameter. This result is then
applied to the 1-equation, URANS model of turbulence for which this ratio
depends on the specification of the turbulence length scale. The model, which
was derived by Prandtl in 1945, is a component of a 2-equation model derived by
Kolmogorov in 1942 and is the core of many unsteady, Reynolds averaged models
for prediction of turbulent flows. Away from walls, interpreting an early
suggestion of Prandtl, we set \begin{equation*} l=\sqrt{2}k^{+1/2}\tau,
\hspace{50mm} \end{equation*} where selected time scale. In the near
wall region analysis suggests replacing the traditional ( wall
normal distance) with giving, e.g., \begin{equation*}
l=\min \left\{ \sqrt{2}k{}^{+1/2}\tau ,\text{ }0.41d\sqrt{\frac{d}{L}} \right\}
. \hspace{50mm} \end{equation*} This results in a simpler model
with correct near wall asymptotics. Its energy dissipation rate scales no
larger than the physically correct , balancing energy input with
energy dissipation
Local null controllability of a class of non-Newtonian incompressible viscous fluids
We investigate the null controllability property of systems that
mathematically describe the dynamics of some non-Newtonian incompressible
viscous flows. The principal model we study was proposed by O. A.
Ladyzhenskaya, although the techniques we develop here apply to other fluids
having a shear-dependent viscosity. Taking advantage of the Pontryagin Minimum
Principle, we utilize a bootstrapping argument to prove that sufficiently
smooth controls to the forced linearized Stokes problem exist, as long as the
initial data in turn has enough regularity. From there, we extend the result to
the nonlinear problem. As a byproduct, we devise a quasi-Newton algorithm to
compute the states and a control, which we prove to converge in an appropriate
sense. We finish the work with some numerical experiments