Form Of Conservation Equations Research Articles

A fully-coupled algorithm with implicit surface tension treatment for interfacial flows with large density ratios

The stability of most surface-tension-driven interfacial flow simulations is governed by the capillary time-step constraint. This concerns particularly small-scale flows and, more generally, highly-resolved liquid-gas simulations with moderate inertia. To date, the majority of interfacial-flow simulations are performed using an explicit surface-tension treatment, which restrains the performance of such simulations. Recently, an implicit treatment of surface tension able to breach the capillary time-step constraint using the volume-of-fluid (VOF) method was proposed, based on a fully-coupled pressure-based finite-volume algorithm. To this end, the interface-advection equation is incorporated implicitly into the linear flow solver, resulting in a tight coupling between all implicit solution variables (colour function, pressure, velocity). However, this algorithm is limited to uniform density and viscosity fields. Here, we present a fully-coupled algorithm for interfacial flows with implicit surface tension applicable to interfacial flows with large density and viscosity ratios. This is achieved by solving the continuity and momentum equations in conservative form, whereby the density is treated implicitly with respect to the colour function, and the advection term of the interface-advection equation is discretised using the THINC/QQ algebraic VOF scheme, yielding a consistent discretisation of the advective terms. This new algorithm is tested by considering representative surface-tension-dominated interfacial flows, including the Laplace equilibrium of a stationary droplet and the three-dimensional Rayleigh-Plateau instability of a liquid filament. The presented results demonstrate that interfacial flows with large density and viscosity ratios can be simulated and energy conservation is ensured, even with a time step larger than the capillary time-step constraint, provided that other time-step restrictions are satisfied.

On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms

This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in J. Differ. Equ.319 (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in Arch. Ration. Mech. Anal.217 (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.

Momentum boundary layers in transcritical channel flows

We present well-resolved large-eddy simulations (LES) of a channel flow solving the fully compressible Navier–Stokes equations in conservative form. An adaptive look-up table method is used for thermodynamic and transport properties. We apply a physically consistent subgrid-scale turbulence model, that is based on the Adaptive Local Deconvolution Method (ALDM) for implicit LES. The wall temperatures are set to enclose the pseudo-boiling temperature at a supercritical pressure, leading to strong property variations within the channel geometry. In total seven cases are computed covering different Reynolds numbers, pseudo-boiling positions and pressure values. The hot wall at the top and the cold wall at the bottom produce asymmetric mean velocity and temperature profiles which result in different momentum layer thicknesses. All cases feature a turbulent mass flux which is essential in turbulence modelling. Furthermore, we analyse the turbulent kinetic energy budgets, perform a quadrant and octant analysis of the Reynolds shear stress and employ an invariant map to study the anisotropy in transcritical turbulent channel flows.

A fast ADI based matrix splitting preconditioning method for the high dimensional space fractional diffusion equations in conservative form

In this paper, we consider the high-dimensional two-sided space fractional diffusion equations with monotonic variable diffusion coefficients, which are derived from the fractional Fick's law. We apply the implicit Euler scheme to discretize the temporal derivative and certain difference operator to discretize the space fractional derivatives. For the discretized linear systems, we apply the alternating direction implicit (ADI) scheme to split the linear systems into two sub-systems of linear equations. For both coefficient matrices of the sub-systems of linear equations, we propose the lopsided scaled diagonal and Toeplitz splitting preconditioner. The generalized minimal residual (GMRES) method combined with the proposed preconditioner is applied to solve both sub-systems of linear equations. The spectral distributions of the preconditioned matrices are analyzed, and the theoretical results are given. Numerical results demonstrate that the proposed preconditioner is efficient in accelerating the convergence rate of the GMRES method for solving the discretized system of linear equations.

On the theoretical foundation of overset grid methods for hyperbolic problems II: Entropy bounded formulations for nonlinear conservation laws

We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes a two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. In addition to the interface coupling, which is required, entropy dissipation and coupling can optionally be added through the use of linear penalties within the overlap region.

On the Theoretical Foundation of Overset Grid Methods for Hyperbolic Problems Ii: Entropy Bounded Formulations for Nonlinear Conservation Laws

We derive entropy conserving and entropy dissipative overlapping domain formulations for systems of nonlinear hyperbolic equations in conservation form, such as would be approximated by overset mesh methods. The entropy conserving formulation imposes two-way coupling at the artificial interface boundaries through nonlinear penalty functions that vanish when the solutions coincide. The penalty functions are expressed in terms of entropy conserving fluxes originally introduced for finite volume schemes. Entropy dissipation and additional coupling in the overlap region are added through the use of linear penalties.

Machine Learning for Conservative-to-Primitive in Relativistic Hydrodynamics

The numerical solution of relativistic hydrodynamics equations in conservative form requires root-finding algorithms that invert the conservative-to-primitive variables map. These algorithms employ the equation of state of the fluid and can be computationally demanding for applications involving sophisticated microphysics models, such as those required to calculate accurate gravitational wave signals in numerical relativity simulations of binary neutron stars. This work explores the use of machine learning methods to speed up the recovery of primitives in relativistic hydrodynamics. Artificial neural networks are trained to replace either the interpolations of a tabulated equation of state or directly the conservative-to-primitive map. The application of these neural networks to simple benchmark problems shows that both approaches improve over traditional root finders with tabular equation-of-state and multi-dimensional interpolations. In particular, the neural networks for the conservative-to-primitive map accelerate the variable recovery by more than an order of magnitude over standard methods while maintaining accuracy. Neural networks are thus an interesting option to improve the speed and robustness of relativistic hydrodynamics algorithms.

Numerical Study on Subsonic-Supersonic Laval Nozzle Using MacCormack Scheme

Due to the wide application of the Delaware nozzle in many fields, the flow characteristics of the subsonic-supersonic isentropic flow in the De-Laval nozzle are analyzed numerically. The flow in the nozzle can be simplified to a quasi-one-dimensional flow problem. First, the MacCormack format is employed to discretize the control equations in conservative form. Then, the results with and without artificial viscosity are compared. Grid independence is also discussed. The results show that the numerical solution and the theoretical solution agree very well, indicating that the numerical simulation results are very reliable. In addition, a higher pressure will reduce the peak and valley values of Mach number, pressure, density, temperature and velocity in the nozzle, and these extreme values of subsonic supersonic isentropic nozzle appear earlier. Additionally, the shock wave is accurately captured, and the shock wave is behind the throat. This research is of great significance to understand the flow characteristics in the nozzle.

1D Finite Volume Scheme for Simulating Gas–Solid Reactions in Porous Spherical Particles with Application to Biomass Pyrolysis

Models for gas–solid reactions in porous particles typically consist of a set of mass and energy balances in the form of conservation equations. For spherical or close to spherical particles, these equations are formulated in 1D spherical coordinates. In the case where accumulation of gas inside the particle is significant, the balance equations contain convective terms. The present work presents a simple numerical scheme based on flux limited finite volume methods for discretizing conservation equations for convective and diffusive transport of mass and energy in radial direction in a porous sphere. The velocity is governed by Darcy’s law coupled to an equation of state. The proposed scheme is applied to a series of test problems that admit full or partial analytical solutions. For the cases where only partial analytical solutions are available, a Comsol model is adopted for comparison. It is found that the scheme is able to resolve step gradients without generating oscillations and that it properly handles changes in the sign of the convective velocity. Applying the scheme for solving a common model for biomass pyrolysis reveals the importance of convective gas transport in the pyrolysis of thermally thick biomass particles.

A High-Order Maximum-Principle-Satisfying Discontinuous Galerkin Method for the Level Set Problem

Level set (LS) method is a widely used interface capturing method. In the simulations of incompressible two-phase flows, in order to avoid discontinuities at interfaces, the LS function is usually taken as a smeared-out Heaviside function bounded on [0, 1] and advected by a given velocity field $$\mathbf {u}$$ obtained from the solution of the incompressible Navier-Stokes equations. In the incompressible limit $$\nabla \cdot \mathbf {u}=0$$ , the advection equation for the LS function can be written and discretized in conservative form. However, due to numerical errors, the resulting velocity field is in general not divergence free which leads to the solution of the advection equation in conservative form does not satisfy the maximum principle. To overcome this issue, in this work, we develop a high-order discontinuous Galerkin (DG) method to directly solve the advection equation for the LS function in non-conservative form. Moreover, we prove that by applying a linear scaling limiter, the proposed method together with a strong stability preserving (SSP) time discretization scheme can satisfy the strict maximum principle under a suitable CFL condition. Numerical simulations of several well-known benchmark problems, including the application to incompressible two-phase flows, are presented to demonstrate the high-order accuracy and maximum-principle-satisfying property of the proposed method.

Large-eddy simulation of turbulent channel flow at transcritical states

We present well-resolved large-eddy simulations (LES) of a channel flow solving the fully compressible Navier–Stokes equations in conservative form. An adaptive look-up table method is used for thermodynamic and transport properties. A physically consistent subgrid-scale turbulence model is incorporated, that is based on the Adaptive Local Deconvolution Method (ALDM) for implicit LES. The wall temperatures are set to enclose the pseudo-boiling temperature at a supercritical pressure, leading to strong property variations within the channel geometry. The hot wall at the top and the cold wall at the bottom produce asymmetric mean velocity and temperature profiles which result in different momentum and thermal boundary layer thicknesses. Different turbulent Prandtl number formulations and their components are discussed in context of strong property variations.

The turbulent bubble break-up cascade. Part 1. Theoretical developments

Abstract

Numerical simulations of transonic flows with non-unique solutions

Inviscid, compressible flows are described by a first-order system of hyperbolic partial differential equations in conservation form, and are solved numerically with simple schemes. These schemes show the ability to reproduce well-known solutions and benchmarks for the two-dimensional Euler equations. Finally, the schemes are used to explore the anomaly of non-unique solutions in the transonic regime. Furthermore, numerical solutions for the isentropic Euler equations with γ = 2, which are analogous to the shallow water equations, also exhibit non-uniqueness. In the appendix, results computed with OVERFLOW also exhibit non-unique solutions for the same geometries considered in the text.

A statically condensed discontinuous Galerkin spectral element method on Gauss-Lobatto nodes for the compressible Navier-Stokes equations

We present a static-condensation method for time-implicit discretizations of the Discontinuous Galerkin Spectral Element Method on Gauss-Lobatto points (GL-DGSEM). We show that, when solving the compressible Navier-Stokes equations, it is possible to reorganize the linear system that results from the implicit time-integration of the GL-DGSEM as a Schur complement problem, which can be efficiently solved using static condensation. The use of static condensation reduces the linear system size and improves the condition number of the system matrix, which translates into shorter computational times when using direct and iterative solvers.The statically condensed GL-DGSEM presented here can be applied to linear and nonlinear advection-diffusion partial differential equations in conservation form with the standard and the split-form variants of the DGSEM. To test it we solve the compressible Navier-Stokes equations with direct and Krylov subspace solvers, and we show for a selected problem that using the statically condensed GL-DGSEM leads to speed-ups of up to 200 when compared to the time-explicit GL-DGSEM, and speed-ups of up to three when compared with the time-implicit GL-DGSEM that solves the global system.The GL-DGSEM has gained increasing popularity in recent years because it satisfies the summation-by-parts property, which enables the construction of provably entropy stable schemes, and because it is computationally very efficient. In this paper, we show that the GL-DGSEM has an additional advantage: It can be statically condensed.

Conservative discontinuous Galerkin methods for the nonlinear Serre equations

In this paper, we develop three conservative discontinuous Galerkin (DG) schemes for the one-dimensional nonlinear dispersive Serre equations, including two conserved schemes for the equations in conservative form and a Hamiltonian conserved scheme for the equations in non-conservative form. One of the schemes owns the well-balanced property via constructing a high order approximation to the source term for the Serre equations with a non-flat bottom topography. By virtue of the Hamiltonian structure of the Serre equations, we introduce an Hamiltonian invariant and then develop a DG scheme which can preserve the discrete version of such an invariant. Numerical experiments in different cases are performed to verify the accuracy and capability of these DG schemes for solving the Serre equations.

Structural Safety Analysis for an Oscillating Water Column Wave Power Conversion System Installed in Caisson Structure

In this study, an alternative way, a so called caisson based type of oscillating water column (OWC) wave energy converting system was proposed to capture and convert wave energy. Since the caisson structure is constructed to protect the coastal line or ports, it is important to know if a built-in associated OWC system will be a burden to affect the safety of the structure or it is safe enough to work appropriately. In this study, three steps of structural analysis were performed: firstly, the analysis for the structural safety of the whole caisson structure; secondly, performing the mechanic analysis for the chamber of the associated OWC system; and finally, performing the analysis for the wave induced air-pressure in the chamber under the design conditions of a local location during the wave-converting operation. For the structural safety analysis, a typical structural model associated with caisson breakwater was built and analyzed while the shape of the structure, material applied to the construction, and associated boundary conditions were all set-up according to the wave and structures. The motion and the strain distribution of the caisson structure subjected to designated waves of 50-year return period were evaluated and compared to the safety requirement by the code. For the analysis of the energy converting performance, a numerical method by using a theorem of unsteady Navier–Stokes equations in conservation form was used to analyze the proposed OWC model when the structure subjected to an incident wave of a 10-year return period.

A state-space formulation of a discontinuous Galerkin method for thermoacoustic stability analysis

A hybrid approach for thermoacoustic stability analysis is formulated in a state-space framework. The approach distinguishes between regions of the computational domain with and without important interactions between acoustics and mean flow or unsteady heat release, respectively. The former regions are modeled by a discontinuous Galerkin finite element method (DG-FEM) for the linearized Navier-Stokes equations in conservative form. The latter are represented by reduced-order models of acoustic wave propagation or dissipation, and provide complex-valued, frequency dependent impedance boundary conditions for the DG-FEM domain. The flow-flame coupling is modeled by a flame transfer function that governs a volumetric source term for the fluctuating heat release rate.The respective (sub-)models are formulated and interconnected in a state-space framework, which facilitates the monolithic formulation of hybrid thermoacoustic models. Moreover, the state-space interconnect framework makes it possible to formulate thermoacoustic stability analysis as a linear eigenvalue problem – even if flame transfer function or acoustic boundary conditions depend in a non-trivial manner on frequency.The approach is first verified against analytical solutions for a duct with mean flow across a thin heat source, similar to a Rijke tube. Then the thermoacoustic eigenmodes of a premixed, swirl-stabilized combustor are computed in order to validate the method against experimental data for a configuration of applied interest. For this second validation case, a detailed comparison against predictions of a low-order network-model is also presented.

Magnetohydrodynamic simulations of a Uranus-at-equinox type rotating magnetosphere

Context. As proven by measurements at Uranus and Neptune, the magnetic dipole axis and planetary spin axis can be off by a large angle exceeding 45°. The magnetosphere of such an (exo-)planet is highly variable over a one-day period and it does potentially exhibit a complex magnetic tail structure. The dynamics and shape of rotating magnetospheres do obviously depend on the planet’s characteristics but also, and very substantially, on the orientation of the planetary spin axis with respect to the impinging, generally highly supersonic, stellar wind. Aims. On its orbit around the Sun, the orientation of Uranus’ spin axis with respect to the solar wind changes from quasi-perpendicular (solstice) to quasi-parallel (equinox). In this paper, we simulate the magnetosphere of a fictitious Uranus-like planet plunged in a supersonic plasma (the stellar wind) at equinox. A simulation with zero wind velocity is also presented in order to help disentangle the effects of the rotation from the effects of the supersonic wind in the structuring of the planetary magnetic tail. Methods. The ideal magnetohydrodynamic (MHD) equations in conservative form are integrated on a structured spherical grid using the Message-Passing Interface-Adaptive Mesh Refinement Versatile Advection Code (MPI-AMRVAC). In order to limit diffusivity at grid level, we used background and residual decomposition of the magnetic field. The magnetic field is thus made of the sum of a prescribed time-dependent background field B0(t) and a residual field B1(t) computed by the code. In our simulations, B0(t) is essentially made of a rigidly rotating potential dipole field. Results. The first simulation shows that, while plunged in a non-magnetised plasma, a magnetic dipole rotating about an axis oriented at 90° with respect to itself does naturally accelerate the plasma away from the dipole around the rotation axis. The acceleration occurs over a spatial scale of the order of the Alfvénic co-rotation scale r*. During the acceleration, the dipole lines become stretched and twisted. The observed asymptotic fluid velocities are of the order of the phase speed of the fast MHD mode. In two simulations where the surrounding non-magnetised plasma was chosen to move at supersonic speed perpendicularly to the rotation axis (a situation that is reminiscent of Uranus in the solar wind at equinox), the lines of each hemisphere are symmetrically twisted and stretched as before. However, they are also bent by the supersonic flow, thus forming a magnetic tail of interlaced field lines of opposite polarity. Similarly to the case with no wind, the interlaced field lines and the attached plasma are accelerated by the rotation and also by the transfer of kinetic energy flux from the surrounding supersonic flow. The tailwards fluid velocity increases asymptotically towards the externally imposed flow velocity, or wind. In one more simulation, a transverse magnetic field, to both the spin axis and flow direction, was added to the impinging flow so that magnetic reconnection could occur between the dipole anchored field lines and the impinging field lines. No major difference with respect to the no-magnetised flow case is observed, except that the tailwards acceleration occurs in two steps and is slightly more efficient. In order to emphasise the effect of rotation, we only address the case of a fast-rotating planet where the co-rotation scale r* is of the order of the planetary counter-flow magnetopause stand-off distance rm. For Uranus, r*≫ rm and the effects of rotation are only visible at large tailwards distances r ≫ rm.

Physics-informed neural networks for high-speed flows

In this work we investigate the possibility of using physics-informed neural networks (PINNs) to approximate the Euler equations that model high-speed aerodynamic flows. In particular, we solve both the forward and inverse problems in one-dimensional and two-dimensional domains. For the forward problem, we utilize the Euler equations and the initial/boundary conditions to formulate the loss function, and solve the one-dimensional Euler equations with smooth solutions and with solutions that have a contact discontinuity as well as a two-dimensional oblique shock wave problem. We demonstrate that we can capture the solutions with only a few scattered points clustered randomly around the discontinuities. For the inverse problem, motivated by mimicking the Schlieren photography experimental technique used traditionally in high-speed aerodynamics, we use the data on density gradient ∇ρ(x,t), the pressure p(x∗,t) at a specified point x=x∗ as well as the conservation laws to infer all states of interest (density, velocity and pressure fields). We present illustrative benchmark examples for both the problem with smooth solutions and Riemann problems (Sod and Lax problems) with PINNs, demonstrating that all inferred states are in good agreement with the reference solutions. Moreover, we show that the choice of the position of the point x∗ plays an important role in the learning process. In particular, for the problem with smooth solutions we can randomly choose the position of the point x∗ from the computational domain, while for the Sod or Lax problem, we have to choose the position of the point x∗ from the domain between the initial discontinuous point and the shock position of the final time. We also solve the inverse problem by combining the aforementioned data and the Euler equations in characteristic form, showing that the results obtained by using the Euler equations in characteristic form are better than that obtained by using the Euler equations in conservative form. Furthermore, we consider another type of inverse problem, specifically, we employ PINNs to learn the value of the parameter γ in the equation of state for the parameterized two-dimensional oblique wave problem by using the given data of the density, velocity and the pressure, and we identify the parameter γ accurately. Taken together, our results demonstrate that in the current form, where the conservation laws are imposed at random points, PINNs are not as accurate as traditional numerical methods for forward problems but they are superior for inverse problems that cannot even be solved with standard techniques.

Closure to “Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme” by Aboudou Seck, Musandji Fuamba, and René Kahawita

Closure to “Finite-Volume Solutions to the Water-Hammer Equations in Conservation Form Incorporating Dynamic Friction Using the Godunov Scheme” by Aboudou Seck, Musandji Fuamba, and René Kahawita