Non-physical Oscillations Research Articles

Analysis of unsteady reacting flows and impact of chemistry description in Large Eddy Simulations of side-dump ramjet combustors

Among all the undesired phenomena observed in ramjet combustors, combustion instabilities are of foremost importance and predicting them using Large Eddy Simulation (LES) is an active research field. While acoustics are naturally captured by compressible LES provided that the proper boundary conditions are applied, combustion/chemistry modelling remains a critical issue and its impact on numerical predictions must still be assessed for complex applications. To do so, two different ramjet LES’s are compared here. The first simulation is based on a standard one-step chemistry known to over-estimate the laminar flame speed in fuel rich conditions. The second simulation uses the same scheme but introduces a correction of reaction rates for rich flames to match a detailed mechanism provided by Peters (1993) [1]. Even though the two chemical schemes are very similar and very few points burn in rich regimes, distinct limit-cycles are obtained with LES depending on which scheme is used. Results obtained with the standard one-step chemistry exhibit high frequency self-sustained oscillations. Multiple flame fronts are stabilized in the vicinity of the shear layer developing at the exit of the air inlets. When compared to the experiment, the fitted one-step scheme yields better predictions than the standard scheme. With the fitted scheme, the flame is detached from the air inlets and stabilizes in the regions identified in the experiment (Ristori et al. (2005) [2], Heid and Ristori (2003) [3], Heid and Ristori (2005) [4], Ristori et al. (1999) [5]). LES and experiments exhibit all main low-frequency modes including the first longitudinal acoustic mode. The high frequencies excited with the standard scheme are damped with the fitted scheme. The chemical scheme is found, for this ramjet burner, to have a strong impact on the predicted stability: approximate chemical schemes even in a limited range of equivalence ratio can lead to the occurence of non-physical combustion oscillations.

ENTROPY AND WEAK SOLUTIONS IN THE THERMAL MODEL FOR THE COMPRESSIBLE EULER EQUATIONS

Among the existing lattice Boltzmann models (LBMs) for compressible flow, the one by Kataoka and Tsutahara [KT model, Phys. Rev. E69, 056702 (2004)] has a simple and rigorous theoretical background. The drawback of this KT model is that it can cause numerical instability if the local Mach number exceeds 1. The precise mechanism of this instability has not yet been clarified. In this paper, we derive entropy functions whose local equilibria are suitable to recover the Euler-like equations in the framework of the lattice Boltzmann method for the KT model. Numerical examples are also given, which are consistent with the above theoretical arguments, and show that the entropy condition is not fully guaranteed in KT model. The negative entropy may be the inherent cause for the nonphysical oscillations in the vicinity of the shock. In contrast to these Karlin's microscopic entropy approach, the corresponding subsidiary entropy condition in the LBM calculation could also be deduced explicitly from the macroscopic version, which provides some insights on the numerical instability of the LBM for shock calculation.

A technique for improving the accuracy of quadrangular mixed finite elements for Darcy’s flow on heterogeneous domains

The mixed hybrid finite element (MHFE) method is well suited for the resolution of the Darcy’s flow on anisotropic and heterogeneous domains. However, the accuracy of the method can be altered for highly distorted meshes with non-convex quadrangles. Moreover, the standard quadrangular MHFE method can lead to non-physical oscillations for transient simulations since it does not respect the maximum principle. In this work, we derive a new realization of the MHFE method on general convex or non-convex quadrangular elements. Each element is fictitiously divided into triangles. The mass balance and flux law are then discretized over each triangle and aggregated to eliminate interior degrees of freedom at the quadrangular element level. The method is combined with the mass lumping procedure for triangles to improve the monotonicity of the discretization. The material properties as well as the pressure and the divergence of the flux are allowed to vary inside the quadrangular element to better describe heterogeneous domains. The obtained matrix is symmetric and positive definite and has the same stencil than the standard approach. The numerical experiments show the performances of this formulation compared to the standard one for heterogeneous domains and non-convex elements. An example is also provided for transient flow simulations where the unphysical oscillations are avoided with the new approach.

A smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations

The effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods. The problem with this representation is that the volume force exhibits non-physical oscillations in moving boundary simulations. A smoothing technique for discrete delta functions has been developed in this paper to suppress the non-physical oscillations in the volume forces. We have found that the non-physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions. It has been shown that the smoothed discrete delta functions constructed in this paper have one-order higher derivative than the regular ones. Moreover, not only the smoothed discrete delta functions satisfy the first two discrete moment conditions, but also their derivatives satisfy one-order higher moment condition than the regular ones. The smoothed discrete delta functions are tested by three test cases: a one-dimensional heat equation with a moving singular force, a two-dimensional flow past an oscillating cylinder, and the vortex-induced vibration of a cylinder. The numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non-physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations.

Adaptive anisotropic meshing for steady convection-dominated problems

Obtaining accurate solutions for convection–diffusion equations is challenging due to the presence of layers when convection dominates the diffusion. To solve this problem, we design an adaptive meshing algorithm which optimizes the alignment of anisotropic meshes with the numerical solution. Three main ingredients are used. First, the streamline upwind Petrov–Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor by an anisotropic centroidal Voronoi tessellation algorithm. Our algorithm is tested on a variety of two-dimensional examples and the results shows that the algorithm is robust in detecting layers and efficient in avoiding non-physical oscillations in the numerical approximation.

Stability analysis of implicit time discretizations for the Compton-scattering Fokker–Planck equation

The Fokker–Planck equation is a widely used approximation for modeling the Compton scattering of photons in high energy density applications. In this paper, we perform a stability analysis of three implicit time discretizations for the Compton-Scattering Fokker–Planck equation. Specifically, we examine (i) a Semi-Implicit (SI) scheme that employs backward-Euler differencing but evaluates temperature-dependent coefficients at their beginning-of-time-step values, (ii) a Fully Implicit (FI) discretization that instead evaluates temperature-dependent coefficients at their end-of-time-step values, and (iii) a Linearized Implicit (LI) scheme, which is developed by linearizing the temperature dependence of the FI discretization within each time step. Our stability analysis shows that the FI and LI schemes are unconditionally stable and cannot generate oscillatory solutions regardless of time-step size, whereas the SI discretization can suffer from instabilities and nonphysical oscillations for sufficiently large time steps. With the results of this analysis, we present time-step limits for the SI scheme that prevent undesirable behavior. We test the validity of our stability analysis and time-step limits with a set of numerical examples.

Finite element method for modeling radiative transfer in semitransparent graded index cylindrical medium

Both Galerkin finite element method (GFEM) and least squares finite element method (LSFEM) are developed and their performances are compared for solving the radiative transfer equation of graded index medium in cylindrical coordinate system (RTEGC). The angular redistribution term of the RTEGC is discretized by finite difference approach and after angular discretization the RTEGC is formulated into a discrete-ordinates form, which is then discretized based on Galerkin or least squares finite element approach. To overcome the RTEGC-led numerical singularity at the origin of cylindrical coordinate system, a pole condition is proposed as a special mathematical boundary condition. Compared with the GFEM, the LSFEM has very good numerical properties and can effectively mitigate the nonphysical oscillation appeared in the GFEM solutions. Various problems of both axisymmetry and nonaxisymmetry, and with medium of uniform refractive index distribution or graded refractive index distribution are tested. The results show that both the finite element approaches have good accuracy to predict the radiative heat transfer in semitransparent graded index cylindrical medium, while the LSFEM has better numerical stability.

Ultrafast Laser-Induced Elastodynamics in Single Crystalline Silicon Part I: Model Formulation

The various responses of a silicon wafer excited by a femtosecond pulsed laser are investigated. A multi-time scale axisymmetric model that governs the transport dynamics in silicon is presented based on the relaxation-time approximation of the Boltzmann equation. Temperature-dependent multi-phonons, free-carrier absorptions, and the recombination and impact ionization processes are considered using a set of balance equations. The mechanical response of the lattice is described by momentum equations. To solve the model of 17 coupled time-dependent partial differential equations without having to be concerned with non-physical oscillations in the solution, an implicit finite difference scheme on a staggered grid is developed. The staggered finite difference scheme allows velocities and first-order spatial derivative terms to be calculated at locations midway between two consecutive grid points, and shear stresses to be evaluated at the center of each element. A multi-time-scale approach involving the use of varying time steps ranging from 5 fs to 5 ps is implemented to successfully obtain time integration results up to 10 ns.

Direct Modeling of Motional Eddy Currents in Highly Saturated Solid Conductors by the Magnetic Equivalent Circuit Method

Motional eddy currents are induced in conductors due to a localized vrarrtimesBrarr electric field. Direct modeling of this term by finite-element-based analysis may lead to nonphysical oscillations. In this paper, two modified magnetic equivalent circuit (MEC) methods to calculate motional eddy currents in highly saturated moving solid conductors are presented. The comparison between the results of the proposed methods with the finite-element method (FEM) proves high accuracy of the proposed methods in a wide velocity range for highly saturated solid conductors. The mesh sensitivity analysis shows the higher stability of the calculations of the proposed methods, compared with FEM, as no numerical oscillations occur.

Collocation Methods for Hyperbolic Partial Differential Equations with Singular Sources

A numerical study is given on the spectral methods and the high order WENO finite difference scheme for the solution of linear and nonlinear hyperbolic partial differential equations with stationary and non-stationary singular sources. The singular source term is represented by the d-function. For the approximation of the d-function, the direct projection method is used that was proposed in (6). The d-function is constructed in a consistent way to the derivative operator. Non- linear sine-Gordon equation with a stationary singular source was solved with the Chebyshev collocation method. The d-function with the spectral method is highly oscillatory but yields good results with small number of collocation points. The results are compared with those computed by the second order finite difference method. In modeling general hyperbolic equations with a non-stationary singu- lar source, however, the solution of the linear scalar wave equation with the non- stationary singular source using the direct projection method yields non-physical oscillations for both the spectral method and the WENO scheme. The numerical artifacts arising when the non-stationary singular source term is considered on the discrete grids are explained.

An optimal‐order estimate for MMOC‐MFEM approximations to porous medium flow

AbstractMathematical models used to describe porous medium flow lead to coupled systems of time‐dependent partial differential equations. Standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid‐orientation effect. The MMOC‐MFEM time‐stepping procedure, in which the modified method of characteristics (MMOC) is used to solve the transport equation and a mixed finite element method (MFEM) is used for the pressure equation, simulates porous medium flow accurately even if large spatial grids and time steps are used. In this article we prove an optimal‐order error estimate for a family of MMOC‐MFEM approximations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

An upwind finite volume element method based on quadrilateral meshes for nonlinear convection‐diffusion problems

AbstractConsidering an upwind finite volume element method based on convex quadrilateral meshes for computing nonlinear convection‐diffusion problems, some techniques, such as calculus of variations, commutating operator, and the theory of prior error estimates and techniques, are adopted. Discrete maximum principle and optimal‐order error estimates in H1 norm for fully discrete method are derived to determine the errors in the approximate solution. Thus, the well‐known problem [(Li et al., Generalized difference methods for differential equations: numerical analysis of finite volume methods, Marcel Dekker, New York, 2000), p 365.] has been solved. Some numerical experiments show that the method is a very effective engineering computing method for avoiding numerical dispersion and nonphysical oscillations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009

On 1D diffusion problems with a gradient-dependent diffusion coefficient

0021-9991/$ see front matter 2008 Elsevier Inc doi:10.1016/j.jcp.2008.06.032 * Corresponding author. Tel.: +1 609 243 2635; fa E-mail address: jardin@pppl.gov (S.C. Jardin). In solving the 1D (flux surface averaged) transport equations for the temperatures, magnetic fields, and densities in the ‘‘evolving equilibrium” description of a tokamak [1], one increasingly encounters highly nonlinear thermal conductivity and diffusivity functions, such as GLF23 [2], that have a strong and non-analytic dependence on the temperature gradients. These arise from a subsidiary microstability based calculation in which the growth rates and hence transport coefficients are sensitive functions of these gradients [3]. When these nonlinear functions are interfaced with an existing transport framework that uses a standard implicit time advancement algorithm such as Crank-Nicolson or backward Euler [4], large non-physical oscillations can develop and, as a result, non-convergent solutions can occur. Here we describe a relatively simple modification to these implicit algorithms that cures this difficulty. To illustrate the method, we start with a simple diffusion equation in cylindrical polar coordinates:

Gas-liquid interface treatment in underwater explosion problem using moving least squares-smoothed particle hydrodynamics

In this study, we investigate the discontinuous-derivative treatment at the gas-liquid interface in underwater explosion (UNDEX) problems by using the Moving Least Squares-Smoothed Particle Hydrodynamics (MLS-SPH) method, which is known as one of the particle methods suitable for problems where large deformation and inhomogeneity occur in the whole domain. Because the numerical oscillation of pressure arises from derivative discontinuity in the UNDEX analysis using the standard SPH method, the MLS shape function with Discontinuous-derivative Basis Function (DBF) that is able to represent the derivative discontinuity of field function is utilized in the MLS-SPH formulation in order to suppress the nonphysical pressure oscillation. The effectiveness of the MLS-SPH with DBF is demonstrated in comparison with the standard SPH and conventional MLS-SPH though a shock tube problem and benchmark standard problems of UNDEX of a trinitrotoluene (TNT) charge.

Locally conservative, stabilized finite element methods for variably saturated flow

Standard Galerkin finite element methods for variably saturated groundwater flow have several deficiencies. For instance, local oscillations can appear around sharp infiltration fronts without the use of mass-lumping, and velocity fields obtained from differentiation of pressure fields are discontinuous at element boundaries. Here, we consider conforming finite element discretizations based on a multiscale formulation along with recently developed, local postprocessing schemes. The resulting approach maintains the basic flexibility and appeal of traditional finite element methods, while controlling nonphysical oscillations and producing element-wise mass-conservative velocity fields. Accuracy and efficiency of the proposed schemes are evaluated through a series of steady-state and transient variably saturated groundwater flow problems in homogeneous as well as heterogeneous domains. The schemes are formulated for a generic nonlinear advection–diffusion equation and are thus applicable to many other flow models.

A Pseudospectral Method with Edge Detection-Free Postprocessing for Two-Dimensional Hyperbolic Heat Transfer

Under the governing equations of hyperbolic heat transfer, thermal disturbances travel with a finite speed of propagation and are visible as sharp discontinuities in the solution profiles. As a result of the well-known Gibbs phenomenon, the numerical solution of hyperbolic heat transfer problems by high-order numerical methods such as pseudospectral methods feature nonphysical numerical oscillations. For pseudospectral methods, postprocessing methods have been developed to lessen or even eliminate the effects of the Gibbs phenomenon. The most powerful postprocessing methods require that the exact location of the discontinuities be known. Because of the reflection and interaction of thermal waves, the solutions of multidimensional hyperbolic heat transfer problems often have very complex features that make accurately locating discontinuities difficult or even impossible. The application of an edge detection-free postprocessing method that is effective for this type of problem is discussed.

Noncoincident validation of Aura MLS observations using the Langley Research Center Lagrangian chemistry and transport model

We evaluate Aura Microwave Limb Sounder version 2.2 measurements of O3, HCl, and H2O with version 19 HALOE observations using a model‐assisted, noncoincident intercomparison technique. Air parcels in the Langley Research Center Lagrangian chemistry and transport model (LCTM) are initialized from HALOE observations made during three different 3‐week periods in 2004 and early 2005. The LCTM tracks the dynamical and photochemical evolution of the ensemble of air parcels following initialization. We show that the LCTM predictions agree closely with subsequent HALOE measurements not used to initialize the model, demonstrating the model's capability to accurately propagate information from the HALOE observations. HALOE‐initialized LCTM O3 predictions agree very well in the stratosphere with MLS measurements, with coincident zonal mean differences generally less than 10%. We find low biases in HCl of 10–30% relative to the MLS observations. We also find low biases with respect to MLS H2O of up to 15% in the upper and middle stratosphere and ∼30% low biases in the lower stratosphere/upper troposphere midlatitudes. We note some nonphysical vertical oscillations of MLS H2O in the tropical and midlatitude lower stratosphere that are not present in the HALOE‐initialized LCTM output. The oscillations are smaller in version 2.2 than in previous versions. These results are generally consistent with published HALOE/MLS intercomparisons of these species using a traditional coincident validation technique. This consistency suggests that the traditional coincident validation results are applicable on daily temporal and zonal spatial scales.

A coupling of mixed and discontinuous Galerkin finite-element methods for poroelasticity

In this paper, we formulate a finite-element procedure for approximating the coupled fluid and mechanics in Biot’s consolidation model of poroelasticity. We approximate the flow variables by a mixed finite-element space and the displacement by a family of discontinuous Galerkin methods. Theoretical convergence error estimates are derived and, in particular, are shown to be independent of the constrained specific storage coefficient, c o . This suggests that our proposed algorithm is a potentially effective way to combat locking, or the nonphysical pressure oscillations, which sometimes arise in numerical algorithms for poroelasticity.

A Hyperbolic Two-Step Model-Based Finite-Difference Method for Studying Thermal Deformation in a 3-D Thin Film Exposed to Ultrashort Pulsed Lasers

In this article we extend our previous research and study thermal deformation in a three-dimensional thin metal film exposed to ultrashort pulsed lasers based on a hyperbolic two-step model. A finite-difference scheme on a staggered grid is employed for solving the energy equations and the dynamic equations of motion. In particular, a fourth-order compact scheme is employed for evaluating the stress derivatives in the dynamic equations of motion. This method allows us to avoid nonphysical oscillations in the solution. Its performance is demonstrated by investigating thermal deformation in a gold thin film.

Lie Symmetry Preservation and Shock-Capturing Methods

It is known that some traditional shock-capturing schemes, including Godunov- or Roe-type schemes, generate downstream oscillatory wave patterns that cannot be effectively dampened by the dissipation methods they provide. The theme of this paper is to understand the formation and behavior at these upstream/downstream patterns from the point view of the symmetry inherent in the PDE. This paper investigates symmetry breaking and spatial oscillations in finite difference solutions for the inviscid Burgers and one-dimensional Navier-Stokes shock equations using the Lax-Wendroff, TVD, and nonoscillatory, nonfree parameter dissipative schemes. Symmetry breaking is considered as a candidate for the inherent cause of the nonphysical oscillations in the vicinity of the shock.