A high‐order PIC method for advection‐dominated flow with application to shallow water waves

A hybrid Eulerian‐Lagrangian particle‐in‐cell–type numerical method is developed for the solution of advection‐dominated flow problems. Particular attention is given over to the high‐order transfer of flow properties from the particles to the grid. For smooth flows, the method presented is of formal high‐order accuracy in space. The method is applied to solve the nonlinear shallow water equations resulting in a new, and novel, shock capturing shallow water solver. The approach is able to simulate complex shallow water flows, which can contain an arbitrary number of discontinuities. Both trivial and nontrivial bottom topography is considered, and it is shown that the new scheme is inherently well balanced, exactly satisfying the ‐property. The scheme is verified against several one‐dimensional benchmark shallow water problems. These include cases that involve transcritical flow regimes, shock waves, and nontrivial bathymetry. In all the test cases presented, very good results are obtained.     

Handling contacts in an Eulerian frame: a finite element approach for fluid structures with contacts

ABSTRACT

Eulerian variational formulations for deformable solids, with or without fluids around them, end up, after implicit time discretisation, as large non-linear systems for the velocities in the moving domains. Handling moving domains and moving boundaries requires careful meshing procedures; on the other hand, the detection of contact is particularly simple with a distance function. Then at every time step, a variational inequality can be used to update the velocities. This article gives new implementation details and two new complex simulations: a very soft bouncing ball in an axisymmetric flow and a disk hit by a club.     

Numerical simulation of flow over an airfoil in heavy rain via a two-way coupled Eulerian–Lagrangian approach

Airfoil performance degradation in heavy rain has attracted many aeronautical researchers’ eyes. In this work, a two-way momentum coupled Eulerian–Lagrangian approach is developed to study the aerodynamic performance of a NACA 0012 airfoil in heavy rain environment. Scaling laws are implemented for raindrop particles. A random walk dispersion approach is adopted to simulate raindrop dispersion due to turbulence in the airflow. Raindrop impacts, splashback and formed water film are modeled with the use of a thin liquid film model. The steady-state incompressible air flow field and the raindrop trajectory are calculated alternately through a curvilinear body-fitted grid surrounding the airfoil by incorporating an interphase momentum coupling term. Our simulation results of aerodynamic force coefficients agree well with the experimental results and show significant aerodynamic penalties at low angles of attack for the airfoil in heavy rain. An about 3° rain-induced increase in stall angle of attack is predicted. The loss of boundary momentum by raindrop splashback and the effective roughening of the airfoil surface due to an uneven water film are testified to account for the degradation of airfoil aerodynamic efficiency in heavy rain environment.     

An approach formulated in terms of conserved variables for the characterisation of propellant combustion in internal ballistics

A model formulated in terms of conserved variables is proposed for its use in the study of internal ballistic problems of pyrotechnical mixtures and propellants. It is a transient two‐phase flow model adapted from the non‐conservative Gough model. This conversion is mathematically attractive because of the wide range of numerical methods for this kind of systems that may be applied. We propose the use of the AUSM+, AUSM + up and Rusanov schemes as an efficient alternative for this type of two‐phase problem. A splitting technique is applied, which solves the system of equations in several steps. A second‐order approach based on Monotonic Upstream‐Centred Scheme for Conservation Laws (MUSCL) is also used. Some tests are used to validate the code, namely a shock wave test, a contact discontinuity problem and an internal ballistics problem. In this last case, one‐dimensional numerical results are compared with experimental data of 155‐mm gunshots. Copyright © 2015 John Wiley & Sons, Ltd.     

Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time

We consider the approximation of the unsteady Stokes equations in a time dependent domain when the motion of the domain is given. More precisely, we apply the finite element method to an Arbitrary Lagrangian Eulerian (ALE) formulation of the system. Our main results state the convergence of the solutions of the semi-discretized (with respect to the space variable) and of the fully-discrete problems towards the solutions of the Stokes system.     

High-order ALE method for the Navier–Stokes equations on a moving hybrid unstructured mesh using flux reconstruction method

The flux reconstruction (FR) formulation can unify several popular discontinuous basis high-order methods for fluid dynamics, including the discontinuous Galerkin method, in a simple, efficient form. An arbitrary Lagrangian–Eulerian (ALE) extension to the high-order FR scheme is developed here for moving mesh fluid flow problems. The ALE Navier–Stokes equations are derived by introducing a grid velocity. The conservation law are spatially discretised on hybrid unstructured meshes using Huynh’s scheme (Huynh 2007) on anisotropic elements (quadrilaterals) and using Correction Procedure via Reconstruction scheme on isotropic elements (triangles). The temporal discretisation uses both explicit and implicit treatments. The mesh movement is described by node positions given as a time series, instead of an analytical formula. The geometric conservation law is tested using free stream preservation problem. An isentropic vortex propagation test case is performed to show the high-order accuracy of the developed method on both moving and fixed hybrid meshes. Flow around an oscillating cylinder shows the capability of the method to solve moving boundary viscous flow problems, with the numeric method further verified by comparison of the result on a smoothly deforming mesh and a rigid moving mesh.     

Two-phase micro- and macro-time scales in particle-laden turbulent channel flows

The micro-and macro-time scales in two-phaseturbulent channel flows are investigated using the direct numerical simulation and the Lagrangian particle trajectorymethods for the fluid-and the particle-phases,respectively.Lagrangian and Eulerian time scales of both phases are calculated using velocity correlation functions.Due to flowanisotropy,micro-time scales are not the same with the theoretical estimations in large Reynolds number(isotropic) turbulence.Lagrangian macro-time scales of particle-phaseand of fluid-phase seen by particles are both dependent onparticle Stokes number.The fluid-phase Lagrangian integral time scales increase with distance from the wall,longerthan those time scales seen by particles.The Eulerian integral macro-time scales increase in near-wall regions but decrease in out-layer regions.The moving Eulerian time scalesare also investigated and compared with Lagrangian integraltime scales,and in good agreement with previous measurements and numerical predictions.For the fluid particles themicro Eulerian time scales are longer than the Lagrangianones in the near wall regions,while away from the walls themicro Lagrangian time scales are longer.The Lagrangianintegral time scales are longer than the Eulerian ones.Theresults are useful for further understanding two-phase flowphysics and especially for constructing accurate predictionmodels of inertial particle dispersion.     

Comparisons of some difference forms for compressible flow in cylindrical geometry on arbitrary Lagrangian and Eulerian framework

The study of cylindrically symmetric compressible fluid is interesting from both theoretical and numerical points of view. In this paper, the typical spherical symmetry properties of the numerical schemes are discussed, and an area weighted scheme is extended from a Lagrangian method to an arbitrary Lagrangian and Eulerian (ALE) method. Numerical results are presented to compare three discrete configurations, i.e., the control volume scheme, the area weighted scheme, and the plane scheme with the addition of a geometrical source. The fact that the singularity arises from the geometrical source term in the plane scheme is illustrated. A suggestion for choosing the discrete formulation is given when the strong shock wave problems are simulated.     

Space-time correlations in turbulent flow: A review

This paper reviews some of the principal uses, over almost seven decades, of correlations, in both Eulerian and Lagrangian frames of reference, of properties of turbulent flows at variable spatial locations and variable time instants. Commonly called space—time correlations, they have been fundamental to theories and models of turbulence as well as for the analyses of experimental and direct numerical simulation turbulence data.