Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userNumerical Dissipation Based Error Estimators and Grid Adaptation for Large Eddy Simulation
120
0
0.0
(
0
)
Added by
Yao Jiang
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Grid adaptation for implicit Large Eddy Simulation (LES) is a non-trivial challenge due to the inherent coupling of the modeling and numerical errors. An attempt to address the challenge first requires a comprehensive assessment and then the development of error estimators to highlight regions that require refinement. Following the work of Schranner et al., a novel approach to estimate the numerical dissipation of the turbulent kinetic energy (TKE) equations is proposed. The presented approach allows the computation of the local numerical dissipation for arbitrary curvilinear grids through a post-processing procedure. This method, as well as empirical and kinetic-energy-based approaches, are employed to estimate the inherent numerical TKE. We incorporate the numerical TKE to evaluate an effective eddy viscosity, an effective Kolmogorov length scale, and an effective TKE to build a family of Index Quality (IQ) based error estimators. The proposed IQ based estimators are then assessed and utilized to show their effectiveness through an application of grid adaptation for the periodic hill test case and transitional flow over the SD 7003 airfoil. Numerical results are validated through a comparison against reference LES and experimental data. Flow over the adapted grids appear better abled to capture pertinent flow features and integrated functions, such as the lift and drag coefficients.
rate research
Read More
The well-known Prager-Synge identity is valid in $H^1(Omega)$ and serves as a foundation for developing equilibrated a posteriori error estimators for continuous elements. In this paper, we introduce a new inequality, that may be regarded as a generalization of the Prager-Synge identity, to be valid for piecewise $H^1(Omega)$ functions for diffusion problems. The inequality is proved to be identity in two dimensions. For nonconforming finite element approximation of arbitrary odd order, we propose a fully explicit approach that recovers an equilibrated flux in $H(div; Omega)$ through a local element-wise scheme and that recovers a gradient in $H(curl;Omega)$ through a simple averaging technique over edges. The resulting error estimator is then proved to be globally reliable and locally efficient. Moreover, the reliability and efficiency constants are independent of the jump of the diffusion coefficient regardless of its distribution.
We consider the numerical analysis of the inchworm Monte Carlo method, which is proposed recently to tackle the numerical sign problem for open quantum systems. We focus on the growth of the numerical error with respect to the simulation time, for which the inchworm Monte Carlo method shows a flatter curve than the direct application of Monte Carlo method to the classical Dyson series. To better understand the underlying mechanism of the inchworm Monte Carlo method, we distinguish two types of exponential error growth, which are known as the numerical sign problem and the error amplification. The former is due to the fast growth of variance in the stochastic method, which can be observed from the Dyson series, and the latter comes from the evolution of the numerical solution. Our analysis demonstrates that the technique of partial resummation can be considered as a tool to balance these two types of error, and the inchwormMonte Carlo method is a successful case where the numerical sign problem is effectively suppressed by such means. We first demonstrate our idea in the context of ordinary differential equations, and then provide complete analysis for the inchworm Monte Carlo method. Several numerical experiments are carried out to verify our theoretical results.
In the present work we show some results on the effect of the Smagorinsky model on the stability of the associated perturbation equation. We show that in the presence of a spectral gap, such that the flow can be decomposed in a large scale with moderate gradient and a small amplitude fine scale with arbitratry gradient, the Smagorinsky model admits stability estimates for perturbations, with exponential growth depending only on the large scale gradient. We then show in the context of stabilized finite element methods that the same result carries over to the approximation and that in this context, for suitably chosen finite element spaces the Smagorinsky model acts as a stabilizer yielding close to optimal error estimates in the $L^2$-norm for smooth flows in the pre-asymptotic high Reynolds number regime.
We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing $f(mathbf{A}) mathbf{b}$ when $mathbf{A}$ is a Hermitian matrix and $mathbf{b}$ is a given mathbftor. Assuming that $f : mathbb{C} rightarrow mathbb{C}$ is piecewise analytic, we give a framework, based on the Cauchy integral formula, which can be used to derive {em a priori} and emph{a posteriori} error bounds for Lanczos-FA in terms of the error of Lanczos used to solve linear systems. Unlike many error bounds for Lanczos-FA, these bounds account for fine-grained properties of the spectrum of $mathbf{A}$, such as clustered or isolated eigenvalues. Our results are derived assuming exact arithmetic, but we show that they are easily extended to finite precision computations using existing theory about the Lanczos algorithm in finite precision. We also provide generalized bounds for the Lanczos method used to approximate quadratic forms $mathbf{b}^textsf{H} f(mathbf{A}) mathbf{b}$, and demonstrate the effectiveness of our bounds with numerical experiments.
Global spectral analysis (GSA) is used as a tool to test the accuracy of numerical methods with the help of canonical problems of convection and convection-diffusion equation which admit exact solutions. Similarly, events in turbulent flows computed by direct numerical simulation (DNS) are often calibrated with theoretical results of homogeneous isotropic turbulence due to Kolmogorov, as given in Turbulence -U. Frisch, Cambridge Univ. Press, UK (1995). However, numerical methods for the simulation of this problem are not calibrated, as by using GSA of convection and/or convection-diffusion equation. This is with the exception in A critical assessment of simulations for transitional and turbulence flows-Sengupta, T.K., In Proc. of IUTAM Symp. on Advances in Computation, Modeling and Control of Transitional and Turbulent Flows, pp 491-532, World Sci. Publ. Co. Pte. Ltd., Singapore (2016), where such a calibration has been advocated with the help of convection equation. For turbulent flows, an extreme event is characterized by the presence of length scales smaller than the Kolmogorov length scale, a heuristic limit for the largest wavenumber present without being converted to heat. With growing computer power, recently many simulations have been reported using a pseudo-spectral method, with spatial discretization performed in Fourier spectral space and a two-stage, Runge-Kutta (RK2) method for time discretization. But no analyses are reported to ensure high accuracy of such simulations. Here, an analysis is reported for few multi-stage Runge-Kutta methods in the Fourier spectral framework for convection and convection-diffusion equations. We identify the major source of error for the RK2-Fourier spectral method using GSA and also show how to avoid this error and specify numerical parameters for achieving highest accuracy possible to capture extreme events in turbulent flows.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
