## Implicit Large Eddy Simulations
## IntroductionImplicit Large Eddy Simulation (ILES) methods employ second-order or higher finite-volume schemes to capture the inviscid cascade of kinetic energy through the inertial range, while the inherent numerical dissipation acts as an implicit subgrid model, forming a natural form of LES. However, the absence of a physical viscosity prohibits conventional characterization of these methods, specifically how kinetic energy is dissipated at the grid scale and how to define a relevant Reynolds number.
## AnalysisKolmogorov's 1941 papers achieve this characterization for real-world viscous fluids in terms of a universal equilibrium range determined uniquely by the rate of energy dissipation and the physical viscosity. An analogous approach can be taken for ILES. Assuming the resolution is sufficiently high (analogous to the Reynolds number), the characteristic parameters are the energy dissipation rate, the integral length scale, and the computational cell width (which replaces viscosity). Therefore, dimensional analysis can be used to write an expression for the kinetic energy spectrum in an ILES fluid The first similarity hypothesis can be stated such that for scales much smaller than the integral length scale, the turbulent statistics are of universal form determined uniquely by the energy dissipation rate and the computational cell width. Similarly, the second similarity hypothesis can be stated such that for scales within the universal equilibrium range that are larger than the computational cell width, the turbulent statistics are independent of the cell width. Therefore, an inertial range exists for an ILES fluid. The important point is independence from the small scales means that the inertial range in an ILES fluid should be consistent with that of a real-world viscous fluid. An effective Kolmogorov length scale can be derived, which dimensional analysis suggests should follow the relation
for some dimensionless function Π is a measure of the velocity gradients in the flow.
Complete similarity suggests that this function Π
## Results
The above theory was tested in the context of maintained homogeneous isotropic turbulence using
IAMR.
Simulations were run from 32
In each case, it was found that the value of Π Please refer to the paper for more details. Any questions should be directed to Andy Aspden. ## ReferencesA. J. Aspden, N. Nikiforakis, S. B. Dalziel and J. B. Bell. "Analysis of Implicit LES Methods" Communications in Applied Mathematics and Computational Science, Volume 3, pp.103-126, 2008. [CAMCoS] |