In this article, a suite of physically inconsistent properties of the Navier-Stokes equations, associated with the lack of mass diffusion and the definition of velocity, are presented. We show that these inconsistencies are consequences of the Lagrangian derivation that models viscous stresses rather than diffusion. A new model for compressible and diffusive (viscous and heat conducting) flows of an ideal gas, is derived in a purely Eulerian framework. We propose that these equations supersede the Navier-Stokes equations. A few numerical experiments demonstrate some differences and similarities between the new system and the Navier-Stokes equations.
We report a detailed study of Eulerian and Lagrangian statistics from high resolution Direct Numerical Simulations of isotropic weakly compressible turbulence. Reynolds number at the Taylor microscale is estimated to be around 600. Eulerian and Lagrangian statistics is evaluated over a huge data set, made by $1856^3$ spatial collocation points and by 16 million particles, followed for about one large-scale eddy turn over time. We present data for Eulerian and Lagrangian Structure functions up to ten order. We analyse the local scaling properties in the inertial range and in the viscous range. …
A positivity-preserving fractional algorithm is presented for solving the four-equation homogeneous relaxation model (HRM) with an arbitrary number of ideal gases and a liquid governed by the stiffened gas equation of state. The fractional algorithm consists of a time step of the hyperbolic five-equation model by Allaire et al. and an algebraic numerical thermal relaxation step at an infinite relaxation rate. Interpolation and flux limiters are proposed for the use of high-order Cartesian finite difference or finite volume schemes in a general form such that the positivity of the partial densi…
Recently the generation of eddy currents by interacting surface waves was observed experimentally. The phenomenon provides the possibility for manipulation of particles which are immersed in the fluid. The analysis shows that the amplitude of the established eddy currents produced by stationary surface waves does not depend on the fluid viscosity in the free surface case. The currents become parametrically larger being inversely proportional to the square root of the fluid viscosity in the case when the fluid surface is covered by an almost incompressible thin liquid (i.e. shear elasticity is …
We present a two-way coupled fluid-structure interaction scheme for rigid bodies using a two-population lattice Boltzmann formulation for compressible flows. Arbitrary Lagrangian-Eulerian formulation of the discrete Boltzmann equation on body-fitted meshes is used in a combination with polynomial blending functions. The blending function approach localizes mesh deformation and allows treating multiple moving bodies with a minimal computational overhead. We validate the model with several test cases of vortex induced vibrations of single and tandem cylinders and show that it can accurately desc…
DNS of compressible multiphase flows through the Eulerian approach
Scuola Normale Superiore di Pisa（比萨高等师范学院）；Istituto Nazionale di Vulcanologia e Geofisica（意大利火山学与地球物理研究所）；Istituto Nazionale di Geofisica e Vulcanologia（意大利地球物理与火山学研究所）
A mechanical interaction of compressible viscoelastic fluids with viscoelastic solids in Kelvin-Voigt rheology using the concept of higher-order (so-called 2nd-grade multipolar) viscosity is investigated in a quasistatic variant. The no-slip contact between fluid and solid is considered and the Eulerian-frame return-mapping technique is used for both the fluid and the solid models, which allows for a "monolithic" formulation of this fluid-structure interaction problem. Existence and a certain regularity of weak solutions is proved by a Schauder fixed-point argument combined with a suitable reg…
Direct numerical simulation of a compressible multiphase flow through the fast Eulerian approach
Scuola Normale Superiore di Pisa（比萨高等师范学院）；Istituto Nazionale di Vulcanologia e Geofisica（火山学与地球物理学国家研究所）；Istituto Nazionale di Geofisica e Vulcanologia（地球物理学与火山学国家研究所）
A new smoothed particle hydrodynamics method based on high-order moving-least-square targeted essentially non-oscillatory scheme for compressible flows
ASHEE: a compressible, equilibrium-Eulerian model for volcanic ash plumes
Scuola Normale Superiore, Pisa, Italy（比萨高等师范学院）；Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Pisa, Italy（意大利国家地球物理与火山学研究所）；Dipartimento di Matematica, Università degli Studi di Pisa, Italy（比萨大学数学系）
The fluid structure interaction of cylinders in tandem arrangement is used as validation basis of a multi-domain Lagrangian-Eulerian hybrid flow solver. In this hybrid combination, separate grids of limited width are defined around every solid body, on which the Eulerian flow equations are solved using finite volume approximations. In order to interconnect the domains defined by the grids, the entire flow is described in Lagrangian coordinates and the corresponding equations are solved via particle approximations in fully coupled mode with the solutions within the Eulerian grids. The flow solv…
We investigate the two dimensional incompressible Navier-Stokes(NS) and the continuity equations in Cartesian coordinates and Eulerian description for non-Newtonian fluids. As a non-Newtonian viscosity we consider the Ladyzenskaya model with a non-linear velocity dependent stress tensor. The key idea is the multi-dimensional generalization of the well-known self-similar Ansatz, which was already used for non-compressible and compressible viscous flow studies. The geometrical interpretations of the trial function is also discussed. We compare our recent results to the former Newtonian ones.
We present a coupled Eulerian-Lagrangian method to simulate cloud cavitation in a compressible liquid. The method is designed to capture the strong, volumetric oscillations of each bubble and the bubble-scattered acoustics. The dynamics of the bubbly mixture is formulated using volume-averaged equations of motion. The continuous phase is discretized on an Eulerian grid and integrated using a high-order, finite-volume weighted essentially non-oscillatory (WENO) scheme, while the gas phase is modeled as spherical, Lagrangian point-bubbles at the sub-grid scale, each of whose radial evolution is …
A hybrid Eulerian-Lagrangian solver RYrhoCentralFoam is developed based on OpenFOAM to simulate detonative combustion in two-phase gas-liquid mixtures. For Eulerian gas phase, RYrhoCentralFoam enjoys second order of accuracy in time and space discretizations and is based on finite volume method on polyhedral cells. The following developments are made based on the standard compressible flow solver rhoCentralFoam in OpenFOAM: (1) multi-component species transport, (2) detailed fuel chemistry for gas phase combustion, and (3) Lagrangian solver for gas-droplet two-phase flows and sub-models for li…
We investigate the statistical properties of Lagrangian tracers transported by a time-correlated compressible renewing flow. We show that the preferential sampling of the phase space performed by tracers yields significant differences between the Lagrangian statistics and its Eulerian counterpart. In particular, the effective compressibility experienced by tracers has a non-trivial dependence on the time correlation of the flow. We examine the consequence of this phenomenon on the clustering of tracers, focusing on the transition from the weak- to the strong-clustering regime. We find that the…
The conservation laws of continuum mechanic written in an Eulerian frame make no difference between fluids and solids except in the expression of the stress tensors, usually with Newton's hypothesis for the fluids and Helmholtz potentials of energy for hyperelastic solids. By taking the velocities as unknown , monolithic methods for fluid structure interactions (FSI) are built. In this article such a formulation is analyzed when the fluid is compressible and the fluid is incompressible. The idea is not new but the progress of mesh generators and numerical schemes like the Characteristics-Galer…
We investigate the incompressible and compressible heat conducting boundary layer with applying the two-dimensional self-similar Ansatz. Analytic solutions can be found for the incompressible case which can be expressed with special functions. The parameter dependencies are studied and discussed in details. In the last part of our study we present the ordinary differential equation (ODE) system which is obtained for compressible boundary layers.
Nonlinear WKB is a multiscale technique for studying locally-plane-wave solutions of nonlinear partial differential equations (PDE). Its application comprises two steps: (1) replacement of the original PDE with an extended system separating the large scales from the small, and (2) reduction of the extended system to its slow manifold. In the context of variational fluid theories with particle relabeling symmetry, nonlinear WKB in the mean Eulerian frame is known to possess a variational structure. This much has been demonstrated using, for instance, the theoretical apparatus known as the gener…
We establish the global-in-time existence of solutions of the Cauchy problem for the full Navier-Stokes equations for compressible heat-conducting flow in multidimensions with initial data that are large, discontinuous, spherically symmetric, and away from the vacuum. The solutions obtained here are of global finite total relative-energy including the origin, while cavitation may occur as balls centred at the origin of symmetry for which the interfaces between the fluid and the vacuum must be upper semi-continuous in space-time in the Eulerian coordinates. On any region strictly away from the …
Multiphase, compressible and viscous flows are of crucial importance in a wide range of scientific and engineering problems. Despite the large effort paid in the last decades to develop accurate and efficient numerical techniques to address this kind of problems, current models need to be further improved to address realistic applications. In this context, we propose a numerical approach to the simulation of multiphase, viscous flows where a compressible and an incompressible phase interact in the low-Mach number regime. In this frame, acoustics is neglected but large density variations of the…
A non-oscillatory energy-splitting method for the computation of compressible multi-fluid flows
School of Mathematical Sciences, Beijing Normal University（北京师范大学数学科学学院）；Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics（应用物理与计算数学研究所计算物理实验室）
This paper investigates the application of mesh adaptation techniques in the Non-Ideal Compressible Fluid Dynamic (NICFD) regime, a region near the vapor-liquid saturation curve where the flow behavior significantly departs from the ideal gas model, as indicated by a value of the fundamental derivative of gasdynamics less than one. A recent interpolation-free finite-volume adaptive scheme is exploited to modify the grid connectivity in a conservative way, and the governing equations for compressible inviscid flows are solved within the Arbitrary Lagrangian Eulerian framework by including speci…
Compressible Navier-Stokes Flow in Schrödinger-Type Variables
Canadian Institute for Theoretical Astrophysics（加拿大理论天体物理研究所）；Department of Astrophysical Sciences, Princeton University（普林斯顿大学天体物理学系）；Department of Physics, University of Toronto（多伦多大学物理系）；D. A. Dunlap Department of Astronomy, University of Toronto（多伦多大学D.A.邓普天文学系）；Perimeter Institute for Theoretical Physics（基础物理珀尔迈特研究所）
We study the impact of compressibility on two-dimensional turbulent flows, such as those modeling astrophysical disks. We demonstrate that the direction of cascade undergoes continuous transition as the Mach number Ma increases, from inverse at zero Ma, to direct at infinite Ma. Thus, at Ma of order one comparable amounts of energy flow from the scale of the pumping to large and small scales, in accord with previous data. For supersonic turbulence with large Ma, the cascade is direct, as in three dimensions, which results in multifractal density field. For compressible flows of conducting flui…
It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous), nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. For proving such results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a priz…
