Stability limits of the single relaxation-time advection–diffusion lattice Boltzmann scheme

2017 ◽  
Vol 28 (12) ◽  
pp. 1750141 ◽  
Author(s):  
Seyed Ali Hosseini ◽  
Nasser Darabiha ◽  
Dominique Thévenin ◽  
Amir Eshghinejadfard
Keyword(s):  
Lattice Boltzmann ◽  
Scalar Fields ◽  
Passive Scalar ◽  
Wave Number ◽  
High Wave ◽  
Relaxation Coefficient ◽  
Numerical Noise ◽  
Wave Numbers ◽  
The Stability ◽  
Relaxation Coefficients

In many cases, multi-species and/or thermal flows involve large discrepancies between the different diffusion coefficients involved — momentum, heat and species diffusion. In the context of classical passive scalar lattice Boltzmann (LB) simulations, the scheme is quite sensitive to such discrepancies, as relaxation coefficients of the flow and passive scalar fields are tied together through their common lattice spacing and time-step size. This in turn leads to at least one relaxation coefficient, [Formula: see text] being either very close to 0.5 or much larger than unity which, in the case of the former (small relaxation coefficient), has been shown to cause instability. The present work first establishes the stability boundaries of the passive scalar LB method in the sense of von Neumann and as a result shows that the scheme is unconditionally stable, even for [Formula: see text], provided that the nondimensional velocity does not exceed a certain threshold. Effects of different parameters such as the distribution function and lattice speed of sound on the stability area are also investigated. It is found that the simulations diverge for small relaxation coefficients regardless of the nondimensional velocity. Numerical applications and a study of the dispersion–dissipation relations show that this behavior is due to numerical noise appearing at high wave numbers and caused by the inconsistent behavior of the dispersion relation along with low dissipation. This numerical noise, known as Gibbs oscillations, can be controlled using spatial filters. Considering that noise is limited to high wave numbers, local filters can be used to control it. In order to stabilize the scheme with minimal impact on the solution even for cases involving high wave number components, a local Total Variation Diminishing (TVD) filter is implemented as an additional step in the classical LB algorithm. Finally, numerical applications show that this filter eliminates the unwanted oscillations while closely reproducing the reference solution.

Asymptotic Stability of an Eikonal Transformation Based ADI Method for the Paraxial Helmholtz Equation at High Wave Numbers

2012 ◽  
Vol 12 (4) ◽  
pp. 1275-1292 ◽  
Author(s):  
Qin Sheng ◽  
Hai-Wei Sun
Keyword(s):  
Helmholtz Equation ◽  
Numerical Stability ◽  
Splitting Method ◽  
Computational Procedure ◽  
Matrix Analysis ◽  
Wave Number ◽  
Large Wave ◽  
High Wave ◽  
Wave Numbers ◽  
Large Wave Number

AbstractThis paper concerns the numerical stability of an eikonal transformation based splitting method which is highly effective and efficient for the numerical solution of paraxial Helmholtz equation with a large wave number. Rigorous matrix analysis is conducted in investigations and the oscillation-free computational procedure is proven to be stable in an asymptotic sense. Simulated examples are given to illustrate the conclusion.

The stability of pipe flow Part 1. Asymptotic analysis for small wave-numbers

1970 ◽  
Vol 43 (2) ◽  
pp. 279-290 ◽  
Author(s):  
W. P. Graebel
Keyword(s):  
Asymptotic Analysis ◽  
Poiseuille Flow ◽  
Reynolds Numbers ◽  
Transverse Component ◽  
Wave Number ◽  
Radial Velocity Component ◽  
Wave Numbers ◽  
Flow Part ◽  
The Stability ◽  
High Degree

The instability of Poiseuille flow in a pipe is considered for small disturbances. An asymptotic analysis is used which is similar to that found successful in plane Poiseuille flow. The disturbance is taken to travel in a spiral fashion, and comparison of the radial velocity component with the transverse component in the plane case shows a high degree of similarity, particularly near the critical point where the disturbance and primary flow travel with the same speed. Instability is found for azimuthal wave-numbers of 2 or greater, although the corresponding minimum Reynolds numbers are too small to compare favourably with either experiments or the initial restrictions on the magnitude of the wave-number.

Note on a heterogeneous shear flow

1964 ◽  
Vol 20 (2) ◽  
pp. 331-336 ◽  
Author(s):  
John W. Miles ◽  
Louis N. Howard
Keyword(s):  
Shear Flow ◽  
Unstable Mode ◽  
Eigenvalue Equation ◽  
Wave Number ◽  
Wave Numbers ◽  
Principal Branch ◽  
Small Density ◽  
Homogeneous Shear Flow ◽  
The Stability ◽  
Finite Band

Goldstein (1931) has considered the stability of a shear layer within which the velocity and the density vary linearly and outside which they are constant. Rayleigh (1880, 1887) had found that the corresponding, homogeneous shear flow is unstable in and only in a finite band of wave-numbers. Goldstein concluded that a small density gradient renders the flow unstable for all wave-numbers. This conclusion appears to depend on the acceptance of all possible branches of a multi-valued eigenvalue equation, and it is shown that the principal branch of this eigenvalue equation yields one and only one unstable mode if and only if the wave-number lies in a band that decreases from Rayleigh's band to zero as the Richardson number increases from 0 to ¼.

The stability of oscillatory internal waves

1967 ◽  
Vol 30 (4) ◽  
pp. 723-736 ◽  
Author(s):  
Russ E. Davis ◽  
Andreas Acrivos
Keyword(s):  
Internal Wave ◽  
Exponential Growth ◽  
Wave Number ◽  
Uniform Density ◽  
Primary Wave ◽  
Thin Region ◽  
Wave Numbers ◽  
The Stability ◽  
Minimum Amplitude ◽  
Interacting Disturbances

The stability of a periodic internal wave has been investigated experimentally and theoretically. From the analysis it is found that if a primary wave, with wave-number k0 and frequency ω0, is perturbed by two infinitesimal wave-like disturbances with wave-numbers k1 and k1 + k0 and frequencies ω1 and ω1 + ω0, exponential growth of these disturbances will take place under certain conditions. The analysis also indicates which resonantly interacting disturbances can induce an instability and, when viscous dissipation is accounted for, predicts the minimum amplitude for which a wave is unstable. Experimental results demonstrate that this type of instability can cause the breakdown of a first mode internal wave propagating in a fluid composed of two layers of uniform density separated by a thin region in which the density varies continuously.

Instabilities of convection rolls in a high Prandtl number fluid

1971 ◽  
Vol 47 (2) ◽  
pp. 305-320 ◽  
Author(s):  
F. H. Busse ◽  
J. A. Whitehead
Keyword(s):  
Prandtl Number ◽  
Three Dimensional ◽  
Wave Number ◽  
Stationary Convection ◽  
Wave Numbers ◽  
Theoretical Predictions ◽  
High Prandtl Number ◽  
The Stability ◽  
Convection Rolls ◽  
Rayleigh Numbers

An experiment on the stability of convection rolls with varying wave-number is described in extension of the earlier work by Chen & Whitehead (1968). The results agree with the theoretical predictions by Busse (1967a) and show two distinct types of instability in the form of non-oscillatory disturbances. The ‘zigzag instability’ corresponds to a bending of the original rolls; in the ‘cross-roll instability’ rolls emerge at right angles to the original rolls. At Rayleigh numbers above 23,000 rolls are unstable for all wave-numbers and are replaced by a three-dimensional form of stationary convection for which the name ‘bimodal convection’ is proposed.

scholarly journals Under-resolved and large eddy simulations of a decaying Taylor–Green vortex with the cumulant lattice Boltzmann method

2020 ◽  
Author(s):  
Martin Geier ◽  
Stephan Lenz ◽  
Martin Schönherr ◽  
Manfred Krafczyk
Keyword(s):  
Reynolds Number ◽  
Turbulence Model ◽  
Fourth Order ◽  
Lattice Boltzmann ◽  
Lattice Boltzmann Model ◽  
Wave Number ◽  
Scale Model ◽  
Good Correspondence ◽  
High Wave Number ◽  
High Wave

AbstractWe present a comprehensive analysis of the cumulant lattice Boltzmann model with the three-dimensional Taylor–Green vortex benchmark at Reynolds number 1600. The cumulant model is investigated in several different variants, using regularization, fourth-order convergent diffusion and fourth-order convergent advection with and without limiters. In addition, a cumulant model combined with a WALE sub-grid scale model is being evaluated. The turbulence model is found to filter out the high wave number contributions from the energy spectrum and the enstrophy, while the non-filtered cumulant methods show good correspondence to spectral simulations even for the high wave numbers. The application of the WALE turbulence model appears to be counter productive for the Taylor–Green vortex at a Reynolds number of 1600. At much higher Reynolds numbers ($${\hbox {Re}}=160{,}000$$ Re = 160 , 000 ) a deviation from the ideal Kolmogorov theory can be observed in the absence of an explicit turbulence model. Cumulant models with fourth-order convergent diffusion show much better results than single relaxation time methods.

A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number

2016 ◽  
Vol 16 (3) ◽  
pp. 429-445 ◽  
Author(s):  
Xiaobing Feng ◽  
Peipei Lu ◽  
Xuejun Xu
Keyword(s):  
Discontinuous Galerkin ◽  
Maxwell Equations ◽  
Impedance Boundary Condition ◽  
Wave Number ◽  
High Wave Number ◽  
Impedance Boundary ◽  
High Wave ◽  
Hybridizable Discontinuous Galerkin ◽  
Time Harmonic ◽  
The Stability

AbstractThis paper proposes and analyzes a hybridizable discontinuous Galerkin (HDG) method for the three-dimensional time-harmonic Maxwell equations coupled with the impedance boundary condition in the case of high wave number. It is proved that the HDG method is absolutely stable for all wave numbers ${\kappa>0}$ in the sense that no mesh constraint is required for the stability. A wave-number-explicit stability constant is also obtained. This is done by choosing a specific penalty parameter and using a PDE duality argument. Utilizing the stability estimate and a non-standard technique, the error estimates in both the energy-norm and the ${\mathbf{L}^{2}}$-norm are obtained for the HDG method. Numerical experiments are provided to validate the theoretical results and to gauge the performance of the proposed HDG method.

Measurements of intermittency of turbulent motion in a boundary layer

1959 ◽  
Vol 6 (2) ◽  
pp. 221-240 ◽  
Author(s):  
V. A. Sandborn
Keyword(s):  
Boundary Layer ◽  
Turbulent Energy ◽  
Turbulent Motion ◽  
Wave Number ◽  
Separation Region ◽  
High Wave Number ◽  
Large Wave ◽  
High Wave ◽  
Wave Numbers ◽  
Longitudinal Distance

Previous observations of turbulent motion at large wave-numbers have revealed the existence of an uneven distribution of turbulent energy. The spotty distribution of the turbulent motion at high wave-numbers is here studied experimentally for the turbulent boundary layer. The high wave-number intermittency is observed at all locations through and along the boundary layer from near transition to near separation.The flatness factors for the longitudinal turbulent component at different wave-numbers are measured to give a quantitative value for the intermittency at particular wave-numbers. Upstream of the separation region the flatness factors are found to depend on wave-number and longitudinal distance, but not on the distance from the wall. It appears that the intermittency develops in the transition region and does not diminish very rapidly with distance downstream. Near separation the flatness factors change radically in distribution near the wall, and are there no longer independent of distance from the wall.

Influence of viscosity temperature dependence on the spectral characteristics of the thermoviscous liquids flow stability equation

2019 ◽  
Vol 14 (1) ◽  
pp. 52-58 ◽  
Author(s):  
A.D. Nizamova ◽  
V.N. Kireev ◽  
S.F. Urmancheev
Keyword(s):  
Reynolds Number ◽  
Spectral Characteristics ◽  
Wave Number ◽  
Critical Parameters ◽  
Fluid Viscosity ◽  
Flat Channel ◽  
Stability Equation ◽  
The Stability ◽  
Thermoviscous Fluid ◽  
Sommerfeld Equation

The flow of a viscous model fluid in a flat channel with a non-uniform temperature field is considered. The problem of the stability of a thermoviscous fluid is solved on the basis of the derived generalized Orr-Sommerfeld equation by the spectral decomposition method in Chebyshev polynomials. The effect of taking into account the linear and exponential dependences of the fluid viscosity on temperature on the spectral characteristics of the hydrodynamic stability equation for an incompressible fluid in a flat channel with given different wall temperatures is investigated. Analytically obtained profiles of the flow rate of a thermovisible fluid. The spectral pictures of the eigenvalues of the generalized Orr-Sommerfeld equation are constructed. It is shown that the structure of the spectra largely depends on the properties of the liquid, which are determined by the viscosity functional dependence index. It has been established that for small values of the thermoviscosity parameter the spectrum compares the spectrum for isothermal fluid flow, however, as it increases, the number of eigenvalues and their density increase, that is, there are more points at which the problem has a nontrivial solution. The stability of the flow of a thermoviscous fluid depends on the presence of an eigenvalue with a positive imaginary part among the entire set of eigenvalues found with fixed Reynolds number and wavenumber parameters. It is shown that with a fixed Reynolds number and a wave number with an increase in the thermoviscosity parameter, the flow becomes unstable. The spectral characteristics determine the structure of the eigenfunctions and the critical parameters of the flow of a thermally viscous fluid. The eigenfunctions constructed in the subsequent works show the behavior of transverse-velocity perturbations, their possible growth or decay over time.

Infrared spectra and substituent effects in 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones

1983 ◽  
Vol 48 (2) ◽  
pp. 586-595 ◽  
Author(s):  
Alexander Perjéssy ◽  
Pavol Hrnčiar ◽  
Ján Šraga
Keyword(s):  
Substituent Effects ◽  
Phenyl Group ◽  
Wave Number ◽  
Vibrational Coupling ◽  
Wave Numbers ◽  
Stretching Vibration ◽  
Stretching Vibrations ◽  
The Relationship ◽  
Derivatives Of ◽  
Effect Of Structure

The wave numbers of the fundamental C=O and C=C stretching vibrations, as well as that of the first overtone of C=O stretching vibration of 2-(3-, and 4-substituted phenylmethylene)-1,3-cycloheptanediones and 1,3-cycloheptanedione were measured in tetrachloromethane and chloroform. The spectral data were correlated with σ+ constants of substituents attached to phenyl group and with wave number shifts of the C=O stretching vibration of substituted acetophenones. The slope of the linear dependence ν vs ν+ of the C=C stretching vibration of the ethylenic group was found to be more than two times higher than that of the analogous correlation of the C=O stretching vibration. Positive values of anharmonicity for asymmetric C=O stretching vibration can be considered as an evidence of the vibrational coupling in a cyclic 1,3-dicarbonyl system similarly, as with derivatives of 1,3-indanedione. The relationship between the wave numbers of the symmetric and asymmetric C=O stretching vibrations indicates that the effect of structure upon both vibrations is symmetric. The vibrational coupling in 1,3-cycloheptanediones and the application of Seth-Paul-Van-Duyse equation is discussed in relation to analogous results obtained for other cyclic 1,3-dicarbonyl compounds.

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