Lagrangian cell-centered conservative scheme

This paper presents a Lagrangian cell-centered conservative gas dynamics scheme. The piecewise constant pressures of cells arising from the current time sub-cell densities and the current time isentropic speed of sound are introduced. Multipling the initial cell density by the initial sub-cell volumes obtains the sub-cell Lagrangian masses, and dividing the masses by the current time sub-cell volumes gets the current time sub-cell densities. By the current time piecewise constant pressures of cells, a scheme that conserves the momentum and total energy is constructed. The vertex velocities and the numerical fluxes through the cell interfaces are computed in a consistent manner due to an original solver located at the nodes. The numerical tests are presented, which are representative for compressible flows and demonstrate the robustness and accuracy of the Lagrangian cell-centered conservative scheme.     

拉格朗日高阶中心型守恒气体动力学格式

提出了拉格朗日高阶中心型守恒气体动力学格式。用产生于当前时刻子网格密度和当前时刻网格声速的子网格压力构造了子网格力,用加权本质无震荡方法构造的高阶子网格力构造了高阶空间通量,借助时间中点通量的泰勒展开完成了高阶时间通量离散,利用动量守恒条件使得格点速度以与网格面的数值通量相容的方式计算。编制了拉格朗日高阶中心型守恒气体动力学格式,对Saltzman活塞问题进行了数值模拟,数值结果表明,拉格朗日高阶中心型守恒气体动力学格式的有效性和精确性.     

A finite volume cell‐centered Lagrangian hydrodynamics approach for solids in general unstructured grids

A finite volume cell‐centered Lagrangian hydrodynamics approach, formulated in Cartesian frame, is presented for solving elasto‐plastic response of solids in general unstructured grids. Because solid materials can sustain significant shear deformation, evolution equations for stress and strain fields are solved in addition to mass, momentum, and energy conservation laws. The total stress is split into deviatoric shear stress and dilatational components. The dilatational response of the material is modeled using the Mie‐Grüneisen equation of state. A predicted trial elastic deviatoric stress state is evolved assuming a pure elastic deformation in accordance with the hypo‐elastic stress‐strain relation. The evolution equations are advanced in time by constructing vertex velocity and corner traction force vectors using multi‐dimensional Riemann solutions erected at mesh vertices. Conservation of momentum and total energy along with the increase in entropy principle are invoked for computing these quantities at the vertices. Final state of deviatoric stress is effected via radial return algorithm based on the J‐2 von Mises yield condition. The scheme presented in this work is second‐order accurate both in space and time. The suitability of the scheme is evinced by solving one‐ and two‐dimensional benchmark problems both in structured grids and in unstructured grids with polygonal cells. Copyright © 2013 John Wiley & Sons, Ltd.     

3D staggered Lagrangian hydrodynamics scheme with cell‐centered Riemann solver‐based artificial viscosity

The aim of the present work is the 3D extension of a general formalism to derive a staggered discretization for Lagrangian hydrodynamics on unstructured grids. The classical compatible discretization is used; namely, momentum equation is discretized using the fundamental concept of subcell forces. Specific internal energy equation is obtained using total energy conservation. The subcell force is derived by invoking the Galilean invariance and thermodynamic consistency. A general form of the subcell force is provided so that a cell entropy inequality is satisfied. The subcell force consists of a classical pressure term plus a tensorial viscous contribution proportional to the difference between the node velocity and the cell‐centered velocity. This cell‐centered velocity is an extra degree of freedom solved with a cell‐centered approximate Riemann solver. The second law of thermodynamics is satisfied by construction of the local positive definite subcell tensor involved in the viscous term. A particular expression of this tensor is proposed. A more accurate extension of this discretization both in time and space is also provided using a piecewise linear reconstruction of the velocity field and a predictor‐corrector time discretization. Numerical tests are presented in order to assess the efficiency of this approach in 3D. Sanity checks show that the 3D extension of the 2D approach reproduces 1D and 2D results. Finally, 3D problems such as Sedov, Noh, and Saltzman are simulated. Copyright © 2012 John Wiley & Sons, Ltd.     

A nodal Godunov method for Lagrangian shock hydrodynamics on unstructured tetrahedral grids

We present a nodal Godunov method for Lagrangian shock hydrodynamics. The method is designed to operate on three‐dimensional unstructured grids composed of tetrahedral cells. A node‐centered finite element formulation avoids mesh stiffness, and an approximate Riemann solver in the fluid reference frame ensures a stable, upwind formulation. This choice leads to a non‐zero mass flux between control volumes, even though the mesh moves at the fluid velocity, but eliminates volume errors that arise due to the difference between the fluid velocity and the contact wave speed. A monotone piecewise linear reconstruction of primitive variables is used to compute interface unknowns and recover second‐order accuracy. The scheme has been tested on a variety of standard test problems and exhibits first‐order accuracy on shock problems and second‐order accuracy on smooth flows using meshes of up to O(10⁶) tetrahedra. Copyright © 2014 John Wiley & Sons, Ltd.     

A high‐order fully explicit flux‐form semi‐Lagrangian shallow‐water model

A new approach is proposed for constructing a fully explicit third‐order mass‐conservative semi‐Lagrangian scheme for simulating the shallow‐water equations on an equiangular cubed‐sphere grid. State variables are staggered with velocity components stored pointwise at nodal points and mass variables stored as element averages. In order to advance the state variables in time, we first apply an explicit multi‐step time‐stepping scheme to update the velocity components and then use a semi‐Lagrangian advection scheme to update the height field and tracer variables. This procedure is chosen to ensure consistency between dry air mass and tracers, which is particularly important in many atmospheric chemistry applications. The resulting scheme is shown to be competitive with many existing numerical methods on a suite of standard test cases and demonstrates slightly improved performance over other high‐order finite‐volume models. Copyright © 2014 John Wiley & Sons, Ltd.     

An algorithm for isentropic flow

An efficient algorithm is presented for the solution of the equations of isentropic gas dynamics with a general convex gas law. The scheme is based on solving linearized Riemann problems approximately, and in more than one dimension incorporates operator splitting. In particular, only two function evaluations in each computational cell are required. The scheme is applied to a standard test problem in gas dynamics for a polytropic gas.     

A 3D finite volume scheme for solving the updated Lagrangian form of hyperelasticity

The finite volume discretization of nonlinear elasticity equations seems to be a promising alternative to the traditional finite element discretization as mentioned by Lee et al. [Computers and Structures (2013)]. In this work, we propose to solve the elastic response of a solid material by using a cell‐centered finite volume Lagrangian scheme in the current configuration. The hyperelastic approach is chosen for representing elastic isotropic materials. In this way, the constitutive law is based on the principle of frame indifference and thermodynamic consistency, which are imposed by mean of the Coleman–Noll procedure. It results in defining the Cauchy stress tensor as the derivative of the free energy with respect to the left Cauchy–Green tensor. Moreover, the materials being isotropic, the free‐energy is function of the left Cauchy–Green tensor invariants, which enable the use of the neo‐Hookean model. The hyperelasticity system is discretized using the cell‐centered Lagrangian scheme from the work of Maire et al. [J. Comput. Phys. (2009)]. The 3D scheme is first order in space and time and is assessed against three test cases with both infinitesimal displacements and large deformations to show the good accordance between the numerical solutions and the analytic ones. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of unsteady gas–liquid flows in a rectangular tank: Comparison of Euler–Eulerian and Euler–Lagrangian simulations

We study the dynamics of gas–liquid flows experimentally and computationally in a rectangular bubble column where the gas source is introduced at the corner. The flow in this reactor is complex and inherently unsteady in nature. The two-dimensional liquid phase velocity field is calculated by an Eulerian approach solving the unsteady Reynolds Averaged Navier Stokes equations. The conservation equations are closed using a two parameter turbulence model. The two-way coupling was accounted for by adding source terms in the conservation equations of the continuous phase to take into account the interaction with the dispersed phase. Bubble tracking is achieved through a Lagrangian approach. Here the equations of motion are solved taking into account the drag, pressure, buoyancy and gravity forces. The time-averaged flows along with the variables which characterize turbulence are analyzed for a wide range of gas flow-rates using Euler–Lagrangian simulations. These simulation predictions are validated with Euler–Eulerian simulations where the gas-phase distribution is captured as a void fraction and PIV experiments. The motion of bubbles induces turbulence in the flow. The applicability of two parameter models for turbulence like the standard k–ε model on time-averaged flow properties is addressed. From the results of the time averaged velocity field, turbulence intensity, turbulent viscosity and gas hold-up profiles, it is concluded that the Euler–Lagrangian model is applicable at lower gas flow-rates. The Euler–Eulerian approach was found to be valid at lower as well as higher gas flow-rates.     

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.     

Accurate and consistent particle tracking on unstructured grids

A new numerical method for particle tracking (Lagrangian particle advection) on 2‐D unstructured grids with triangular cells is presented and tested. This method combines key attributes of published methods, including streamline closure for steady flows and local mass conservation (uniformity preservation). The subgrid‐scale velocity reconstruction is linear, and this linear velocity field is integrated analytically to obtain particle trajectories. A complete analytic solution to the 2‐D system of ordinary differential equations (ODEs) governing particle trajectories within a grid cell is provided. The analytic solution to the linear system of locally mass‐conserving constraints that must be enforced to obtain the coefficients in the ODEs is also provided. Numerical experiments are performed to demonstrate that the new method has substantial advantages in accuracy over previously published methods and that it does not suffer from unphysical particle clustering. The method can be used not only in particle‐tracking applications but also as part of a semi‐Lagrangian advection scheme.Copyright © 2015 John Wiley & Sons, Ltd.     

Evaluation of an energy relaxation method for the simulation of unsteady,viscous, real gas flows

This study investigates a new energy relaxation method designed to capture the dynamics of unsteady, viscous, real gas flows governed by the compressible Navier–Stokes equations. We focus on real gas models accounting for inelastic molecular collisions and yielding temperature‐dependent heat capacities. The relaxed Navier–Stokes equations are discretized using a mixed finite volume/finite element method and a high‐order time integration scheme. The accuracy of the energy relaxation method is investigated on three test problems of increasing complexity: the advection of a periodic set of vortices, the interaction of a temperature spot with a weak shock, and finally, the interaction of a reflected shock with its trailing boundary layer in a shock tube. In all cases, the method is validated against benchmark solutions and the numerical errors resulting from both discretization and energy relaxation are assessed independently. Copyright © 2004 John Wiley & Sons, Ltd.     

Meshless Galerkin analysis of Stokes slip flow with boundary integral equations

This paper presents a novel meshless Galerkin scheme for modeling incompressible slip Stokes flows in 2D. The boundary value problem is reformulated as boundary integral equations of the first kind which is then converted into an equivalent variational problem with constraint. We introduce a Lagrangian multiplier to incorporate the constraint and apply the moving least‐squares approximations to generate trial and test functions. In this boundary‐type meshless method, boundary conditions can be implemented exactly and system matrices are symmetric. Unlike the domain‐type method, this Galerkin scheme requires only a nodal structure on the bounding surface of a body for approximation of boundary unknowns. The convergence and abstract error estimates of this new approach are given. Numerical examples are also presented to show the efficiency of the method. Copyright © 2009 John Wiley & Sons, Ltd.     

Neural networks for BEM analysis of steady viscous flows

This paper presents a new neural network‐boundary integral approach for analysis of steady viscous fluid flows. Indirect radial basis function networks (IRBFNs) which perform better than element‐based methods for function interpolation, are introduced into the BEM scheme to represent the variations of velocity and traction along the boundary from the nodal values. In order to assess the effect of IRBFNs, the other features used in the present work remain the same as those used in the standard BEM. For example, Picard‐type scheme is utilized in the iterative procedure to deal with the non‐linear convective terms while the calculation of volume integrals and velocity gradients are based on the linear finite element‐based method. The proposed IRBFN‐BEM is verified on the driven cavity viscous flow problem and can achieve a moderate Reynolds number of 1400 using a relatively coarse uniform mesh. The results obtained such as the velocity profiles along the horizontal and vertical centrelines as well as the properties of the primary vortex are in very good agreement with the benchmark solution. Furthermore, the secondary vortices are also captured by the present method. Thus, it appears that an ability to represent the boundary solution accurately can significantly improve the overall solution accuracy of the BEM. Copyright © 2003 John Wiley & Sons, Ltd.     

CHAPLYGIN EQUATION IN THREE-DIMENSIONAL NON-CONSTANT ISENTROPIC FLOW-THE THEORY OF FUNCTIONS OF A COMPLEX VARIABLE UNDER DIRAC-PAULI REPRESENTATION AND ITS APPLICATION IN FLUID DYNAMICS(Ⅲ)

This work is the continuation of the discussion of Ref. [1]. In this paper we resolve the equations of isentropic gas dynamics into two problems: the three-dimensional non-constant irrotational flow (thus the isentropic flow, too), and the three-dimensional non-constant indivergent flow (i. c. the in compressible isentropic flow). We apply the theory of functions of a complex variable under Dirac-Pauli representation and the Legendre transformation, transform these equations of two problems from physical space into velocity space, and obtain two general Chaplygin equations in this paper. The general Chaplygin equation is a linear difference equation, and its general solution can be expressed at most by the hypergeometric functions. Thus we can obtain the general solution of general problems for the three-dimensional non-constant isentropic flow of gas dynamics.     

An improved KGF‐SPH with a novel discrete scheme of Laplacian operator for viscous incompressible fluid flows

The kernel gradient free (KGF) smoothed particle hydrodynamics (SPH) method is a modified finite particle method (FPM) which has higher order accuracy than the conventional SPH method. In KGF‐SPH, no kernel gradient is required in the whole computation, and this leads to good flexibility in the selection of smoothing functions and it is also associated with a symmetric corrective matrix. When modeling viscous incompressible flows with SPH, FPM or KGF‐SPH, it is usual to approximate the Laplacian term with nested approximation on velocity, and this may introduce numerical errors from the nested approximation, and also cause difficulties in dealing with boundary conditions. In this paper, an improved KGF‐SPH method is presented for modeling viscous, incompressible fluid flows with a novel discrete scheme of Laplacian operator. The improved KGF‐SPH method avoids nested approximation of first order derivatives, and keeps the good feature of ‘kernel gradient free’. The two‐dimensional incompressible fluid flow of shear cavity, both in Euler frame and Lagrangian frame, are simulated by SPH, FPM, the original KGF‐SPH and improved KGF‐SPH. The numerical results show that the improved KGF‐SPH with the novel discrete scheme of Laplacian operator are more accurate than SPH, and more stable than FPM and the original KGF‐SPH. Copyright © 2015 John Wiley & Sons, Ltd.     

Acoustic upwinding for sub‐ and super‐sonic turbulent channel flow at low Reynolds number

A recently developed asymmetric implicit fifth‐order scheme with acoustic upwinding for the spatial discretization for the characteristic waves is applied to the fully compressible, viscous and non‐stationary Navier–Stokes equations for sub‐ and super‐sonic, mildly turbulent, channel flow (Re_τ=360). For a Mach number of 0.1, results are presented for uniform (32³, 64³ and 128³) and non‐uniform (expanding wall‐normal, 32³ and 64³) grids and compared to the (incompressible) reference solution found in (J. Fluid. Mech. 1987; 177 :133–166). The results for uniform grids on 128³ and 64³ nodes show high resemblance with the reference solution. Expanding grids are applied on 64³‐ and 32³‐node grids. The capability of the proposed technique to solve compressible flow is first demonstrated by increasing the Mach number to 0.3, 0.6 and 0.9 for isentropic flow on the uniform 64³‐grid. Next, the flow speed is increased to Ma=2. The results for the isothermal‐wall supersonic flows give very good agreement with known literature results. The velocity field, the temperature and their fluctuations are well resolved. This means that in all presented (sub‐ and super‐sonic) cases, the combination of acoustic upwinding and the asymmetric high‐order scheme provides sufficient high wave‐number damping and low wave‐number accuracy to give numerically stable and accurate results. Copyright © 2007 John Wiley & Sons, Ltd.     

A further work on multi‐phase two‐fluid approach for compressible multi‐phase flows

This paper is to continue our previous work Niu (Int. J. Numer. Meth. Fluids 2001; 36 :351–371) on solving a two‐fluid model for compressible liquid–gas flows using the AUSMDV scheme. We first propose a pressure–velocity‐based diffusion term originally derived from AUSMDV scheme Wada and Liou (SIAM J. Sci. Comput. 1997; 18 (3):633—657) to enhance its robustness. The scheme can be applied to gas and liquid fluids universally. We then employ the stratified flow model Chang and Liou (J. Comput. Physics 2007; 225 :240–873) for spatial discretization. By defining the fluids in different regions and introducing inter‐phasic force on cell boundary, the stratified flow model allows the conservation laws to be applied on each phase, and therefore, it is able to capture fluid discontinuities, such as the fluid interfaces and shock waves, accurately. Several benchmark tests are studied, including the Ransom's Faucet problem, 1D air–water shock tube problems, 2D shock‐water column and 2D shock‐bubble interaction problems. The results indicate that the incorporation of the new dissipation into AUSM⁺‐up scheme and the stratified flow model is simple, accurate and robust enough for the compressible multi‐phase flows. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Simple waves on a shear gas flow in a channel of constant cross section

The plane-parallel unsteady-state shear gas flow in a narrow channel of constant cross section is considered. The existence theorem of solutions in the form of simple waves of a set of equations of motion is proved for a class of isentropic flows with a monotone velocity profile over the channel depth. The exact solution described by incomplete beta-functions is found for a polytropic equation of state in a class of isentropic flows. Lavrent'ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 1, pp. 36–43, January–February, 1999.