Characteristic Boundary Conditions Research Articles

Numerical simulation of 3D seismic wave based on alternative flux finite-difference WENO scheme

Summary High-frequency non-physical oscillations may occur due to shock waves in seismic wavefield and dynamic rupture simulation. In this study, we introduced the alternative flux finite-difference weighted essentially non-oscillatory (AWENO) scheme to address potential shock wave issues in computational seismology effectively. The wavefield of the body-fitted curvilinear domain was accurately computed through conservative grid mapping, ensuring accurate implementation of free surface boundary conditions on irregular surfaces using characteristic boundary conditions and minimizing artificial boundary reflections with exponential decay absorbing layers. Finally, we compared our scheme with the GRTM for flat surfaces and the CGFDM3D-EQR for irregular surfaces to demonstrate its correctness and accuracy, and validated its non-oscillatory characteristics. The aforementioned scheme is anticipated to assume a significant function in simulating more intricate seismic wavefields or dynamic ruptures.

Theoretical analysis and numerical validation of the mechanisms controlling composition noise

A new definition of entropy fluctuation is provided in this paper, that allows properly separating entropy and mixture composition fluctuations. This decomposition into linearly independent variables prevents from overestimating compositional noise in indirect noise prediction. When considering quasi one-dimensional flow in nozzles, a new resulting system of linearized Euler equations is obtained. Two analytical solutions of this system of equations are investigated in this study, one dedicated to low-frequency perturbations and another one for all frequency perturbations. This new theory is then validated by comparing the model predictions with direct numerical simulation of nozzle flows in which composition fluctuations are pulsed. To do so, new Navier–Stokes characteristics boundary conditions to account for the new properly defined entropy wave are provided. Furthermore, non-reflecting inlet boundary conditions have been newly derived, relaxing on the axial mass-flow rate, static temperature and species mass fractions. Finally, a parametric study based on an air-kerosene (equivalent C10H20) mixture and an ideal framework shows that composition noise can reach a maximum of 10% of entropy noise for lean combustion and choked nozzle while for rich combustion, composition noise and entropy noise show comparable levels.

Navier-Stokes characteristic boundary conditions for high-enthalpy compressible flows in thermochemical non-equilibrium

This fundamental study presents Navier-Stokes characteristic boundary conditions (NSCBCs) for high-enthalpy hypersonic flows in thermochemical non-equilibrium. In particular, the relevant locally one-dimensional inviscid (LODI) relations are derived within a two-temperature framework for high-enthalpy hypersonic flows undergoing finite-rate thermochemical processes, including air dissociation and vibrational relaxation. Using these LODI relations, a set of NSCBCs are proposed and later demonstrated in canonical test cases, including the interaction of homogeneous isotropic turbulence with a shock wave subject to high-enthalpy thermochemical non-equilibrium effects.

Flame-acoustic interactions in high-pressure H2/air diffusion flames

Understanding flame-acoustics interaction (FAI) that can potentially trigger thermoacoustic instabilities is of critical importance to mitigate the serious side effects of pressure oscillations. While FAI has been studied extensively in premixed flames, comparatively less analysis has been conducted to explore FAI mechanisms in non-premixed flames, especially under high-pressure conditions. This study aims to numerically investigate FAI in high-pressure hydrogen/air counterflow diffusion flames, with special attention on the impact of pressure on FAI. To this end, a fully compressible unsteady counterflow solver combined with Navier-Stokes Characteristic Boundary Conditions (NSCBC) is employed to capture the reflection of the acoustic waves at the boundaries. The results show that for all ambient pressures ( 1 ≤ P a ≤ 60 bar) considered here, real-gas effects on flame structures are negligible and that increasing P a would lead to a power-law dependence of density gradient on P a with an exponent of 1.5. Furthermore, all tested cases show acoustic growth with time under fully reflecting boundary conditions, which is found to be pressure dependent. The flames become less resilient to FAI when the pressure (or density gradient) is increased. The results from the spectral analysis show that for all ambient pressures, there is only a single dominant frequency at the low-density fuel side and two dominant frequencies at the high-density oxidiser side. This indicates that the pressure oscillation is more sinusoidal at the H 2 side than that at the air side. Moreover, the phase space portraits of pressure signals indicate that the dynamics of the system are becoming periodic and approaching a limit cycle. It is found that the integrated heat release rate and the pressure fluctuations are partially in phase, which is responsible for the growth of high-frequency acoustic waves at both fuel and oxidiser sides. On the other hand, the significant density gradient contributes to the growth of low-frequency acoustic waves at the oxidiser side due to the high degree of acoustic reflectivity at the density boundary.

Numerical investigation of flow past a cylinder using cumulant lattice Boltzmann method

This paper presents simulations of flow past a circular cylinder within the subcritical Reynolds number (Re) range from 3900 to 2 × 105, utilizing the parameterized cumulant lattice Boltzmann model. In this study, a three-dimensional characteristic boundary condition for incompressible flow has been integrated into the lattice Boltzmann method at the outflow boundary to minimize spurious reflection. The flow field, wake statistics, hydrodynamic force, and power spectra results of Re = 3900 from the cumulant lattice Boltzmann model are exhaustively compared with the laboratory data and other numerical models. Relative to other numerical models employing turbulence closure, the cumulant lattice Boltzmann simulations demonstrate enhanced agreement with the experimental data even with relatively coarser grid resolution. The resolution-spanning feature for the cumulant lattice Boltzmann model in turbulent flows, without using explicit turbulence model, aligns with the previous benchmark case studies. The stability-preserving regularization process in the present model is analyzed. Results indicate that the influence of the regularization parameter is mitigated with improved grid resolution. A specific regularization parameter for flow around cylinder simulations is recommended. Variations in flow properties and hydrodynamic forces within the subcritical Reynolds number range of 3900 to 2 × 105 are analyzed. The results confirm that the parameterized cumulant lattice Boltzmann model can accurately simulate practical engineering flows, characterized by complex separation and recirculation, within the subcritical range. Moreover, the computational efficiency and parallel scalability are compared with other numerical methods.

Accelerate the numerical convergence of incompressible flows: Novel preconditioned characteristic boundary conditions

For the first time, the locally power-law preconditioning method (LPLPM) is used to formulate the preconditioned characteristic boundary conditions (CBCs). Then, it is implemented to solve the numerical modeling of unsteady and steady flows from viscous to turbulent regimes. The compatibility equations and Riemann invariants are mathematically derived and then utilized to the incompressible flow solvers as suitable boundary conditions. This method discretizes time derivative and governing equations' space terms by applying the four-stage, fourth-order Runge–Kutta method, and a finite volume, respectively. The preconditioning matrix in the LPLPM is automatically derived by local velocity sensors through a power-law formulation. The baseline k−ω is applied as an appropriate turbulence model. Several test cases are conducted around airfoils of Office National d'Etudes et de Recherches Aerospatiales, NACA0012 (National Advisory Committee for Aeronautics), and S809 at varied angles of attack of 0–20 and Reynolds numbers of 500 to 5.25 × 106 to examine the effectiveness and accuracy of the LPLPM employing preconditioned CBCs. A sensitivity analysis is also performed to examine how numerical parameters affect the simulation. The results show that using preconditioned CBCs in conjunction with LPLPM at the artificial boundary is precise, reliable, and computationally efficient in simulating viscous/turbulent flows. Furthermore, it is also concluded that the present approach considerably improves the convergence speed contrasted to the simplified boundary conditions.

Characteristic boundary layers in the vanishing viscosity limit for the Hunter-Saxton equation

The Hunter-Saxton equation models the propagation of weakly nonlinear orientation waves in a massive director field of the nematic liquid crystal. In this paper, we study the vanishing viscosity limit for an initial boundary value problem of the Hunter-Saxton equation with the characteristic boundary condition. By the formal multiscale analysis, we first derive the characteristic boundary layer profile, which satisfies a nonlinear parabolic equation. On the base of the Galerkin method along with a compactness argument, we then establish the global well-posedness of the boundary layer equation. Finally, we prove the global stability of the boundary layer profiles together with the optimal convergence rate of the vanishing viscosity limit by the energy method.

Topology optimization of isotropic elastic materials in the two-dimensional design domain with changes in characteristic boundary conditions

Topology optimization (TO) has become increasingly popular as a useful tool for designers and engineers during the initial stages of design. TO aims to optimize the geometry of a design to achieve a specific objective, which can range from discrete grid-like structures to continuum structures. In essence, the geometry is parameterized pixel-by-pixel, with the material density of each element or mesh point serving as a design variable. After that, the optimization problem is addressed using mathematical programming and analytic gradient calculation-based optimization approaches. In this paper, we investigate the material distribution when performing topology optimization for an isotropic material with boundary conditions including fixed structures, supports, or external forces changing. In addition, we investigate more cases where there are material holes in the design domain, meaning that the density of the material is zero. In this study, the modified SIMP method and filter sensitivity are used for topology optimization. The results of the study are the optimized structural domains and the change in compliance according to the number of iterations. The results indicate that the compliance value of most structures reaches convergence after optimization up to the 20th iteration. Moreover, if the force applied to the design domain is symmetrical, the optimal structure also exhibits symmetry. Thus, the distribution of material is concentrated at the positions of the supports. Topology optimization produces designs that both meet boundary conditions while saving material and reducing their mass. The results obtained are important data for structural optimization design for isotropic elastomeric materials. From there, it can be applied to real objects with different requirements and conditions

Hybrid Artificial Boundary Conditions for the Application of Blunt-Body Aerodynamic Noise Prediction

A hybrid artificial boundary condition (HABC) that combines the volume-based acoustic damping layer (ADL) and the local face-based characteristic boundary condition (CBC) is presented to enhance the absorption of acoustic waves near the computational boundaries. This method is applied to the prediction of aerodynamic noise from a circular cylinder immersed in uniform compressible viscous flow. Different ADLs are designed to assess their effectiveness whereby the effect of the mesh-stretch direction on wave absorption in the ADL is analysed. Large eddy simulation (LES) and FW-H acoustic analogy method are implemented to predict the far-field noise, and the sensitivities of each approach to the HABC are compared. In the LES computed propagation field of the fluctuation pressure and the frequency-domain results, the spurious reflections at edges are found to be significantly eliminated by the HABC through the effective dissipation of incident waves along the wave-front direction in the ADL. Thereby, the LES results are found to be in a good agreement with the acoustic pressure predicted using FW-H method, which is observed to be just affected slightly by reflected waves.

An implicit splitting scheme with characteristic boundary conditions for compressible reactive flows on unstructured grids

The computation of combustion instabilities requires efficient numerical solvers. For low Mach number compressible flows splitting schemes provide an adequate family of solution strategies. Here, the set of governing equations is separated into several subsystems, e.g. into an advective and acoustic subsystem. Advancements for such an algorithm are presented improving accuracy and stability. Moreover a new, gradient based approach is given for coupling the splitting scheme with characteristic boundary conditions on collocated unstructured grids. A large number of academic test cases is chosen to demonstrate the accuracy and performance of the proposed improved numerical scheme. In addition, the improved solver is tested in a complex, non-reactive LES of a swirl combustor on an unstructured grid.

Characteristic boundary condition for multispeed lattice Boltzmann model in acoustic problems

In computational aero-acoustics, the lattice Boltzmann method (LBM) has demonstrated a number of advantages such as low dissipation and high scalability. To recover the thermal equation and energy transport in a physically consistent way, the fourth order moments of LBM distribution function must be accurately captured, which in turn demands a multispeed velocity set. Accurate and economic realization of various boundary conditions with multispeed velocities has been a challenge in LBM. Here we present a boundary condition for open boundaries with arbitrarily specifiable sound reflection properties by embedding the characteristic non/partially reflecting boundary condition of finite difference (FD) method into multispeed LBM. The local one-dimensional inviscid analysis for Euler equations is used to build boundary treatment method for FD, and distribution functions are reconstructed from FD solution at open outlets as inner-domain LBM's boundary condition. The numeric scheme is validated in a number of test cases, including one-dimensional (1D) and two-dimensional (2D) propagation of acoustic, entropy and vorticity waves, and sound generation in flow past a 2D cylinder. Results show our method can achieve desired reflecting coefficient and reduce non-physical reflecting waves to a satisfactory level.

A Characteristic Boundary Condition for Multispeed Lattice Boltzmann Methods

A Characteristic Boundary Condition for Multispeed Lattice Boltzmann Methods

Lattice-Boltzmann modeling of the quiet and unstable PRECCINSTA burner modes

Recent studies have shown that Lattice-Boltzmann methods are indeed very promising in the field of reactive flows. More work is required, however, to demonstrate its ability to tackle complex reacting cases, as no study – to the authors’ knowledge – involves simultaneously high Reynolds flows (for which the collision kernel needs specific care), complex geometries (for which models are required at the wall boundaries), and non-uniform grids (where non-conform meshes need to be addressed).The present study intends to fill that gap, by investigating the well-known PRECCINSTA burner, including (i) characteristic boundary conditions, (ii) classical turbulent combustion modeling, (iii) multi-level grid refinements. Combining these elements, numerical simulations of the PRECCINSTA burner are carried out, both for the quiet (φ=0.83) and unstable regimes (φ=0.7). In both regimes, results are consistent with those obtained with classical (Navier–Stokes) solvers, but at a much lower cost. In particular, it is the first time that successful prediction of thermoacoustic instabilities in a complex burner is shown in the framework of Lattice-Boltzmann methods.

Time-dependent boundary conditions for data-driven coronal global and spherical wedge-shaped models

ABSTRACT The development of an efficient and accurate method for boundary condition treatments is of fundamental importance to data-driven magnetohydrodynamic (MHD) modelling of the global solar corona and solar active region. Particularly, in a 3D spherical wedge-shaped volume, suitable to the numerical study of solar active region, the transverse terms calls for a delicate treatment at the computational domain’s edges and corners, and properly prescribed conditions for boundaries joining regions of different flow properties, so as to take account of the joint effect of incoming and outgoing waves. To provide a solution to the determination of boundary conditions, in this paper a systematic tactics is formulated for handling edges and corners and the prescribed conditions for inner/outer/edge/corner boundaries are proposed through the combination (CBC-ILW) of the time-dependent characteristic boundary conditions (CBCs) and the inverse Lax-Wendroff (ILW) procedure. First, a data-driven 3D MHD simulation has been carried out to study the dynamic evolution of the solar corona from 1Rs to 6.7Rs during the period between 2018 May 16 and August 6. The simulated results of the global coronal evolution provide a good comparison with observed coronal images during the period investigated. Then, the validity of 3D MHD-CBC-ILW is verified for a 3D spherical wedge model, by producing almost the same results as those taken out of the global model on a 3D spherical wedge-shaped volume.

Lattice Boltzmann method with nonreflective boundary conditions for low Mach number combustion

The paper presents a lattice Boltzmann (LB) method for premixed and nonpremixed combustion simulations with nonreflective boundary conditions, in contrast to Navier–Stokes solvers or hybrid schemes. The current approach employs different sets of distribution functions for flow, temperature and species fields, which are fully coupled. The discrete equilibrium density distributions are obtained from the Hermite expansions thus thermal compressibility is included. The coupling among the momentum, energy and species transport enables the model to be applicable for reactive flows with chemical heat release. The characteristic boundary conditions are incorporated into the LB scheme to avoid numerical reflections. The multi-relaxation-time collision schemes are applied to all the LB solution procedures to improve numerical stability. With detailed thermodynamics and chemical mechanisms for hydrogen-air, the LB modelling framework is validated against both premixed flame propagation and nonpremixed counterflow diffusion flame benchmarks. Simulations of circular expanding premixed flames further demonstrate the capability of the new reactive LB method. The developed LB methodology retains the advantages of classic LB methods and extends the LB capability to low Mach number combustion with potential applications in mesoscale and microscale combustors, catalysis, fuel cells, batteries and so on.

Unified Integral Transforms and Non-Classical Eigenvalue Problems in Heat and Mass Transfer

Abstract The generalized integral transform technique (GITT) is reviewed as a computational–analytical methodology in linear and nonlinear convection–diffusion problems, based on eigenfunction expansions extracted from characteristic differential operators, coefficients, and boundary conditions present in the original partial differential problem formulation. Here, the emphasis is on the employment of nonclassical eigenvalue problems as the expansion basis, which do not fall into the more usual framework of Sturm–Liouville problems. The goal is to enable or improve the eigenfunction expansions convergence, by incorporating more information from the original operators into the chosen eigenvalue problem, while requiring the handling of such a more involved expansion base. In this concern, the proposed differential eigenvalue problem can itself be handled by the GITT, leading to an algebraic eigensystem analysis. Different classes of nonclassical eigenvalue problems are then reviewed and associated with typical applications in heat and mass transfer. Representative test cases are then chosen to illustrate the extended methodology and demonstrate the convergence rates attainable by this enhanced hybrid solution path.

Development and validation of Navier–Stokes characteristic boundary conditions applied to turbomachinery simulations using the lattice Boltzmann method

AbstractThis article reports a procedure to implement as well as to validate non‐reflecting boundary conditions applied for turbomachinery simulations, using Navier–Stokes characteristic boundary conditions in a compressible lattice Boltzmann solver. The implementation of both an inlet condition imposing total pressure, total temperature, and flow angles, as well as an outlet condition imposing a static pressure profile that allows the simulation to reach a simplified radial equilibrium, is described within the context of a lattice Boltzmann approach. The treatment at the boundaries relies on the characteristic methodology to derive conditions which are non‐reflecting in terms of acoustics and is also compatible with turbulence injection at the inlet. These properties are evaluated on test cases of increasing complexity, ranging from a simple 2D periodic domain to an S‐duct stage with turbulence injection.

A Non-Stiff Summation-By-Parts Finite Difference Method for the Scalar Wave Equation in Second Order Form: Characteristic Boundary Conditions and Nonlinear Interfaces

Curvilinear, multiblock summation-by-parts finite difference operators with the simultaneous approximation term method provide a stable and accurate framework for solving the wave equation in second order form. That said, the standard method can become arbitrarily stiff when characteristic boundary conditions and nonlinear interface conditions are used. Here we propose a new technique that avoids this stiffness by using characteristic variables to “upwind” the boundary and interface treatment. This is done through the introduction of an additional block boundary displacement variable. Using a unified energy, which expresses both the standard as well as characteristic boundary and interface treatment, we show that the resulting scheme has semidiscrete energy stability for the scalar anisotropic wave equation. The theoretical stability results are confirmed with numerical experiments that also demonstrate the accuracy and robustness of the proposed scheme. The numerical results also show that the characteristic scheme has a time step restriction based on standard wave propagation considerations and not the boundary closure.

Characteristic boundary conditions for magnetohydrodynamic equations

In the present study, a characteristic-based boundary condition scheme is developed for the compressible magnetohydrodynamic (MHD) equations in the general curvilinear coordinate system, which is an extension of the characteristic boundary scheme for the Navier–Stokes equations. The eigenstructure and the complete set of characteristic waves are derived for the ideal MHD equations in general curvilinear coordinates (ξ,η,ζ). The characteristic boundary conditions are derived and implemented in a high-order MHD solver where the sixth-order compact scheme is used for the spatial discretization. The fifth-order Weighted Essentially Non-Oscillatory (WENO) scheme is also employed for the spatial discretization of problems with discontinuities. In our MHD solver, the fourth-order Runge–Kutta scheme is utilized for time integration. The characteristic boundary scheme is first verified for the non-magnetic (i.e., B=0) Sod shock tube problem. Then, various in-house test cases are designed to examine the derived MHD characteristic boundary scheme for three different types of boundaries: non-reflecting inlet and outlet, solid wall, and single characteristic wave injection. The numerical examples demonstrate the accuracy and robustness of the MHD characteristic boundary scheme.

Approximation of Families of Characteristics of Energy Objects in the Class of Linear Differential Equations

A method is proposed for constructing a fast-computable mathematical model of a family of energy characteristics based on its comparison with a family of integral curves in the class of linear differential equations. The method uses an affine transformation and takes into account the two-level structure of the mathematical model of a family of characteristics (differential operator and boundary conditions). It allows reducing the number of approximating summands and increasing the adequacy by including additional information of a systemic nature about the object under study.