Hyperviscosity for unstructured ALE meshes

This seminar explains an enhanced version of the well-known SPH (Smoothed Particle Hydrodynamics) open-source code DualSPHysics for the simulation of free-surface fluid flows, leading to the DualSPHysics+ code. The enhancements are made through incorporation of several schemes with respect to stability, accuracy and energy/volume conservation issues in simulating incompressible free-surface fluid flows within the weakly compressible SPH formalism. The Optimized Particle Shifting (OPS) scheme is implemented to improve the accuracy of particle shifting vectors in the free-surface region. To mitigate energy dissipation and maintain consistency, the artificial viscosity in δ-SPH is substituted with a Riemann stabilization term, leading to the δR-SPH. The Velocity divergence Error Mitigating (VEM) and Volume Conservation Shifting (VCS) schemes are adopted in DualSPHysics+ to mitigate the velocity divergence error and improve the volume conservation, and hence to enhance the resolution of the continuity equation. To further reduce both the instantaneous and accumulated errors in velocity divergence, a Hyperbolic/Parabolic Divergence Cleaning (HPDC) scheme is incorporated in addition to the VEM scheme. The implementations of the introduced schemes on both CPU and GPU-based versions of the DualSPHysics+ code along with details on the compilation, running and computational performance are presented. Validations in terms of accuracy, energy conservation and convergence of DualSPHysics+ are shown via several relevant benchmarks. It is demonstrated that a better velocity divergence error cleaning in both instantaneous and accumulated errors can be achieved by the combination of VEM and HPDC. Meanwhile, the excessive energy dissipation by the artificial viscosity is shown to be suppressed by adopting the Riemann stabilization term. Enhanced resolution of the continuity equation along with improved energy conservation of DualSPHysics+ advance the SPH-based simulation of incompressible free-surface fluid flows.

Variable property-based lattice Boltzmann flux solver for thermal flows in the low Mach number limit

A new code and methodology are introduced for solving the general relativistic magnetohydrodynamic (GRMHD) equations in fixed background spacetimes using time-explicit, finite-volume discretization. The code has options for solving the GRMHD equations using traditional artificial-viscosity (AV) or non-oscillatory central difference (NOCD) methods, or a new extended AV (eAV) scheme using artificial-viscosity together with a dual energy-flux-conserving formulation. The dual energy approach allows for accurate modeling of highly relativistic flows at boost factors well beyond what has been achieved to date by standard artificial viscosity methods. It provides the benefit of Godunov methods in capturing high Lorentz boosted flows but without complicated Riemann solvers, and the advantages of traditional artificial viscosity methods in their speed and flexibility. Additionally, the GRMHD equations are solved on an unstructured grid that supports local adaptive mesh refinement using a fully threaded oct-tree (in three dimensions) network to traverse the grid hierarchy across levels and immediate neighbors. A number of tests are presented to demonstrate robustness of the numerical algorithms and adaptive mesh framework over a wide spectrum of problems, boosts, and astrophysical applications, including relativistic shock tubes, shock collisions, magnetosonic shocks, Alfven wave propagation, blast waves, magnetized Bondi flow, and the magneto-rotational instability in Kerr black hole spacetimes.

In this paper, we justify the low Mach number limit for the three-dimensional full compressible Navier–Stokes–Korteweg equations rigorously within the framework of smooth solution. Under the assumptions of small density and temperature perturbation, we show that for sufficiently small Mach number, the initial-value problem of the three-dimensional full compressible Navier–Stokes–Korteweg equations admits a unique smooth solution on the time interval where the smooth solution of the corresponding incompressible Navier–Stokes equations exists. Moreover, we obtain the convergence of smooth solutions for the full compressible Navier–Stokes–Korteweg equations toward those for the incompressible Navier–Stokes equations with a convergence rate.

We design and analyse a first-order accurate implicit–explicit (IMEX) scheme for the two-dimensional wave equation system in the low Mach number limit. It has been shown by Dellacherie [S. Dellacherie, Analysis of Godunov-type schemes applied to the compressible Euler system at low Mach number. J. Comput. Phys., 229(4): 978–1016, 2010, [1]] that the standard Godunov-type numerical schemes suffer from a severe loss of accuracy at low Mach numbers. This inaccuracy arises due to the inability of schemes to preserve the incompressible space of constant densities and divergence-free velocities. Guided by this principle, we design an IMEX scheme which possess the invariance property, and analyse its stability. The proposed scheme has been shown to be stable under a usual CFL-like condition, independent of the Mach number. The results of numerical experiments confirm the accuracy and stability of the new scheme when applied to low Mach number problems.

An Arbitrary-Lagrangian-Eulerian hybrid finite volume/finite element method on moving unstructured meshes for the Navier-Stokes equations

Godunov-type schemes with an inertia term for unsteady full Mach number range flow calculations

This work focuses on the study of the asymptotic preserving property of a semi-implicit method. The semi-implicit method, which is a pressure-based method, has been successfully used to simulate two-phase flows in numerous industrial applications. This method is used in our studies due to the fact that pressure-based methods generally perform well at low Mach numbers. The semi-implicit method is applied to the homogeneous equilibrium model (HEM) in this work to simulate two-phase flows. We show that the semi-implicit method is asymptotic preserving, i.e. the discretization for a compressible model tends to a consistent discretization for the related incompressible model at the low Mach number limit. Finally, test cases are performed to show that the numerical method is able to deal with low Mach number flows, as well as flows with a wide range of Mach numbers.

Modeling flows with curved detonation waves

Instabilities in combustion systems have frequently been reported to occur when slight changes in operating conditions lead to significant and abrupt changes in flame shape or flame position, i.e., changes in the mode of flame stabilization. The present paper offers an explanation and mathematical model of this observation. The analysis rests on the assumption that changes in the mode of flame stabilization are accompanied by a significant variation of the pressure drop across burner and flame, such that the pressure drop-flow rate characteristic locally displays a negative slope. In the limit of low frequencies (Helmholtz mode), it is then straightforward to show that an oscillatory instability can result from such behavior. An analytical stability criterion is derived, relating the nondimensionalized gradient of the pressure drop characteristic to the Helmholtz number of the burner. The physical mechanism of the instability is explained, and it is observed that the Rayleigh criterion need not be satisfied for this kind of instability to occur. In order to extend the stability analysis to higher frequencies, the transfer matrix for a burner with nonmonotonic pressure drop is derived in the limit of low Mach number and negligible fluctuations of the rate of heat release. The transfer matrix is employed in stability analysis based on a linear acoustic model of a combustion system. Experimental results obtained with an externally premixed swirl burner are presented. The pressure drop characteristic, the observed onset of instability and the instability frequency match the model predictions very well.

We shortly review the incompressible limit of the barotropic Euler system of gas dynamics, also known as low Mach number limit, and the quasineutral limit of a simplified Euler–Poisson system. Then we develop a general pseudodifferential framework which is able to cover both examples, called generalised symmetrisable systems. This framework can also handle incompressible limits. As an application, we then discuss a barotropic Euler–Poisson system.

Numerical simulations of the flow and noise generated by two different automotive outside rear-view mirrors are investigated. The mirrors were mounted on the top of a specially designed table for simulations and wind tunnel tests. A decoupled two-stage methodology is applied that requires CFD simulations to collect pressure variations at solid surfaces followed by an acoustic post-processing of the data recorded. For the flow simulations, URANS (Unsteady RANS), DES (Detached Eddy Simulation) and LES (Large Eddy Simulation) turbulence modeling approaches are used. This study is part of an on-going project to develop a transient CFD procedure to accurately predict wind noise generated by the exterior surfaces of a moving ground vehicle. The Curle's formulation in low Mach number limit is used to calculate the acoustic pressure generated by the pressure fluctuations at solid walls. Reasonable agreement (within 5 dB) between measurements and predictions has been achieved both for the flow and sound characteristics (with denser mesh quality). It is found that the URANS and DES approaches underestimate surface pressure fluctuations and therefore, result in underestimated sound pressure levels. The LES approach on the other hand (particularly with the dynamic subgrid viscosity model) provides better agreement for the time-averaged surface pressure distributions and for the sound pressure levels.

We show how in the case of the whole domain, the use of dispersive estimates, allows in fluidinamic to overcome the problem connected with the formation of acoustic waves. It is well known in the theory of the incompressible limit or the low Mach number limit that the presence of acoustic waves, which propagates with high speed in the space domain may causes the lost of strong convergence. We show that similar problems arise also when we try to approximate the incompressible Navier Stokes equations by means of the artificial compressibility method, where we have to deal with acoustic waves for the pressure. In order to overcome such a diculty we use the dispersive properties of these waves. We exploit the wave equation structure of the pressure which allows us to use L p L q Strichartz type estimate, leading to the strong convergence of the approximating sequences, see [DM1], [D1]. Similar phenomena appear also in modeling the Debye screening eect for semiconductor devices. Those kind of systems are obtained by coupling the compressible Navier Stokes equation with an external force given by the electrostatic potential provided by the Poisson equation for the electric field. While studying the quasineutral limit for this system, the incompressible limit regime yields to introduce a suitable time scaling, which introduces a singularity by the coupling term with the electric field and it leads to formation of acoustic waves. In order to handle these diculties the previous model will be regarded as a semilinear wave equation and hence the dispersive analysis will lead to uniform estimates and then to the strong convergence, see [DM2]. Finally, we show same work in progress concerning the case of exterior domains (to a compact convex obstacle). In that case the analysis is harder due to the need of physical consistent boundary conditions.

This study explores the use of line-based finite difference techniques on unstructured grids employing high-order stencils for practical aerodynamic applications. The primary focus is on finite-difference explicit and compact spatial discretization up to sixth order. Problems studied include compressible flows containing discontinuities, which necessitated the use of a hybrid finite difference–finite volume approach, where the former is shown to be effective and accurate for smooth regions, while the latter is robust in capturing discontinuities and shocks. To differentiate between discontinuous and smooth zones, two distinct shock indicators are used, one based on pressure gradients and the other on the divergence of velocity. To achieve faster convergence, spatial discretization is applied with line-based alternating direction implicit (ADI) time integration. A key focus of this work is to demonstrate the proposed methodology in flow scenarios relevant to aircraft applications that also test various aspects of the methodology. Consequently, the cases studied are isentropic vortex convection, standing normal shock, transonic NACA0012 airfoil, supersonic cylinder, unsteady flow past tandem circular cylinders, and supersonic flow past a forward-facing step. Comparisons were also done against an existing line-based finite-volume technique for unstructured grids that employed Monotonic Upstream-Centered Scheme for Conservation Laws (MUSCL) and fifth-order Weighted Essentially NonOscillatory (WENO5) spatial reconstruction schemes. It is observed that the hybrid finite difference–finite volume approach performed well with excellent agreement against available results, sometimes with far fewer degrees of freedom. Furthermore, the implicit ADI scheme is effective over unstructured grids and provides a three-time speed-up over the traditional Runge–Kutta fourth-order scheme.

We have compared structured and unstructured grid-based 2D inversion algorithms for magnetotelluric (MT) and radiomagnetotelluric (RMT) data in terms of speed and accuracy. We have developed a new 2D inversion algorithm for MT and RMT data by using a finite-element (FE) method that uses unstructured triangle grids. We compare the inversion results of our unstructured grid-based algorithm with those of the conventional algorithm, which uses either a structured FE or structured finite-difference (FD) numerical solution technique. The imaging of the surface topography and the underground resistivity structures by the new algorithm requires fewer elements than those that use FE and FD structured grids. We also find that when unstructured grids are used, the quality of the mesh is increased and the numerical errors are significantly reduced. Thus, the program runs faster and can simulate the complex surface topography in a more stable setting than the classic inversion algorithms. Furthermore, we implement a new smoothing matrix format for the unstructured triangle grids for the inversion procedure. We use two samples of synthetic data for the MT and RMT frequencies as well as a sample of field RMT data collected across a fault zone for comparison. In our synthetic data experiment, we find that the resistivity values and the boundaries obtained from the inversion of the unstructured mesh are closer to those of the true a priori synthetic model. Results of the synthetic and field data verify the computational advantages (speed and accuracy) of our inversion algorithm with respect to the conventional structured grid-based inversion algorithms.