Application of a thermodynamically compatible two‐phase flow model to the high‐resolution simulations of compressible gas–magma flow

This paper reports on the application and development of a fully hyperbolic and fully conservative two‐phase flow model for the simulation of gas and magma flow within volcanic processes. The model solves a set of mixture conservation equations for the gas and magma two‐phase flow with velocity non‐equilibrium. In this model, the effect of the relative velocity is introduced by a kinetic constitutive equation with other equations for volume and mass fractions of the gas phase. The model is examined numerically by the widely used finite volume Godunov methods of centered‐type. Using the Riemann problem, we numerically simulate wave propagation and the development of shocks and rarefactions in volcanic eruptions. These simulations are of magma fragmentation type where the relative velocity continues to dominate. A series of test cases whose solution contains features relevant to gas–magma mixtures are conducted. In particular, numerical results indicate that the model implementation predicts key features of the relative velocity within volcanic processes without any mathematical or physical simplifications. Simulation results are sharply and accurately provided without any spurious oscillations in all of the flow variables. The numerical methods and results are also compared with other numerical methods available in the literature. It is found that the provided resolutions are more accurate for the considered test cases. Copyright © 2014 John Wiley & Sons, Ltd.     

On Fokker–Planck Equations for Turbulent Reacting Flows. Part 2. Filter Density Function for Large Eddy Simulation

The application of large eddy simulation (LES) to turbulent reacting flow calculations is faced with several closure problems. Suitable parametrizations for filtered reaction rates for instance are hardly available in general. A way to overcome these problems is investigated here. This is done by extending LES equations for filtered velocities and scalars (mass fractions of species and temperature) to equations that involve subgrid scale (SGS) fluctuations. Such equations are called filter density function (FDF) methods because they determine the FDF, which is essentially the probability density function of SGS variables. The FDF model considered involves only three parameters: C ₀ that controls the generation of velocity fluctuations and two parameters which determine the relaxation of velocity and scalar fluctuations. The consideration of this model may be seen as the analysis of a limiting case: the implications of the most simple equations for the dynamics of SGS fluctuations are investigated in this way. These equations were proved recently by various simulations. Here, the FDF model is used analytically to improve simpler methods. Existing models for the SGS stress tensor in velocity LES equations and the diffusion coefficient in scalar FDF equations are generalized in this way. The advantages of these models compared to existing ones are pointed out. These investigations provide further evidence for the suitability of the FDF model considered and they provide its parameters. A theoretical value C ₀ = 19/12 is derived, which agrees very well with the results of direct numerical simulation. This estimate implies the same value for the universal Kolmogorov constant of the energy spectrum, which is consistent with the results of many measurements. The other two model parameters can be obtained then by dynamic procedures. Therefore, the closure problems of LES equations are overcome in this way such that adjustable parameters are not involved.     

尖端具有吸附性接触的界面裂纹对瞬态弹性波作用下的动态响应

建立并研究一类接触型界面裂纹模型对瞬态弹性波作用下的动态响应问题。文中利用积分变换和积分方程法推导了确定这类问题的奇异积分方程组。采用围道积分技术和切比雪夫多项式展开技术，得到了待定系数的非线性代数方程组。最后给出了裂纹尖端接触区大小和接触应力随时间变化的数值结果，揭示了这种接触裂纹的动力学特性及物理上的合理性。     

多体动力学超定运动方程广义-α求解新算法

完整约束多体系统第一类Lagrange方程建模得到的运动方程是指标-3形式的微分-代数方程(differental-algebraic equations,DAEs).如果同时考虑速度约束,将得到超定运动方程,该方程是指标-2的超定微分-代数方程(over-determined differential-algebraic equations,ODAEs).基于结构动力学中常用的广义-α方法,将其拓展,求解包含速度约束的超定运动方程,相对于其他求解指标-2 ODAEs的算法,新的算法没有增加离散得到的非线性方程组方程的数目.通过数值实验验证算法,并说明其求解ODAEs不存在精度降阶的现象,仍然具有二阶精度,同时算法的数值耗散也是可以控制的.最后新方法与其他求解多体系统ODAEs形式运动方程算法的CPU时间进行了比较分析.     

Uniformly convergent numerical method for singularly perturbed convection-diffusion type problems with nonlocal boundary condition

In this article, we consider a class of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary conditions. A uniformly convergent numerical method is constructed via nonstandard finite difference and numerical integration methods to solve the problem. The nonlocal boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ϵ -uniformly convergent.     

Investigation of the stability of a corona discharge continuum model in the shortwave approximation

The stability of a negative corona discharge between two spherical electrodes the inner of which is a hydrodynamic source is investigated. A continuum discharge model consisting of equations for the electrons and positive and negative ions written with allowance for electrokinetic reactions and electrodynamic equations is used. The steady-state (undisturbed) solution of this electrohydrodynamic system of equations is found using numerical methods and its stability is analyzed in the shortwave approximation for various structural zones of the corona discharge, namely, the ionization zone, the zone of attachment of electrons to neutral molecules, and the unipolar charge zone (negative ion zone). The perturbation growth rates in these zones are determined. It is shown that in the ionization zone the corona discharge considered is unstable.     

Self-similar degeneration of the turbulent wake behind a body towed in a passively stratified medium

A mathematical model of the far turbulent wake behind a towed body in a passively stratified medium, based on the known semi-empirical e-? model of turbulence, is considered. A grouptheoretical analysis of the model is performed. With the help of the method of B-determining equations, the model is reduced to a system of ordinary differential equations, which is solved numerically. The resultant solution is compared with a self-similar solution obtained by direct numerical integration of the differential equations at large distances from the body.     

Modeling of simultaneous heat and mass transfer processes in ammonia–water absorption systems from general correlations

This paper presents a general differential mathematical model to analyze the simultaneous heat and mass transfer processes that occur in different components of an ammonia–water absorption system: absorber, desorber, rectifier, distillation column, condenser and evaporator. Heat and mass transfer equations are considered, taking into account the heat and mass transfer resistances in the liquid and vapour phases. The model considers the different regions: vapour phase, liquid phase and an external heating or cooling medium. A finite difference numerical method has been considered to solve the resulting set of nonlinear differential equations and an iterative algorithm is proposed for its solution. A map of possible solutions of the mass transferred composition z is presented when varying the interface temperature, which enables to establish a robust implementation code. The analysis is focused on the processes presented in ammonia–water absorption systems. The model is applied to analyze the ammonia purification process in an adiabatic packed rectification column and the numerical results show good agreement with experimental data.     

Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

The finite element absolute nodal coordinate formulation (ANCF) is often used in modeling very flexible bodies in multibody system (MBS) applications. This formulation leads to a constant mass matrix, allowing for an efficient sparse matrix implementation. Nonetheless, the use of the ANCF finite elements to model stiff structures can lead to high frequencies associated with ANCF coupled deformation modes, as discussed in the literature. Implicit numerical integration methods can be effectively used to develop efficient procedures for the solution of MBS differential/algebraic equations. Most existing implicit integration algorithms, however, require numerical differentiation of the equations of motion, and some of these integration methods do not ensure that the kinematic algebraic constraint equations are satisfied at all levels (position, velocity, and acceleration). Because of these limitations, existing implicit integration methods can be less accurate and less efficient when used to solve large scale MBS applications. In order to circumvent this problem, the two-loop implicit sparse matrix numerical integration (TLISMNI) method was proposed for the solution of MBS differential/algebraic equations. The TLISMNI method does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation. This paper discusses TLISMNI implementation issues including the step size selection, the error control, and the effect of the numerical damping. The relation between the step size selection and the structure stiffness is also discussed. The use of the computer implementation described in this paper is demonstrated by solving very stiff structure problems using the Hilber?CHughes?CTaylor (HHT) method, which includes numerical damping. An eigenvalue analysis and Fast Fourier Transform (FFT) are performed in order to identify the fundamental modes of deformation and demonstrate that the contributions of these fundamental modes can be erroneously damped out when some other implicit integration methods are used. The TLISMNI method, on the other hand, captures the contributions of these fundamental modes. The results, obtained using the TLISMNI method, are compared with the results obtained using other methods including the implicit HHT-I3 and the explicit Adams integration methods. The results obtained show that the TLISMNI method can be five times faster than the other two methods when no numerical damping is considered.     

Analysis of the possibilities of the methods for calculating turbulent jet noise

Two methods for calculating the noise of turbulent exhaust jets of civil aircraft nozzles are considered. The first method is chiefly intended for engineering mass-volume calculations and is based on the solution of the averaged Navier-Stokes equations closed by a two-equation turbulence model. The second method uses direct numerical simulation of large eddies in a turbulent jet and the Kirchhoff surface for calculating noise spectra and radiation patterns in the far field. The possibilities and certain important restrictions of these methods are analyzed. The results obtained using these methods are compared with experimental data.     

Prandtl-Meyer solutions of viscoplasticity equations with negative strain rate sensitivity

We present a further study of a viscoplasticity model with nonmonotonic strain rate dependence ensuring the complete integrability of the two-dimensional equilibrium and consistency equations. The considered nonlinear equations change their type from hyperbolic to elliptic at a certain critical value of the strain rate intensity; the type change is accompanied by the formation of an interphase in the solid. This model is of interest for describing spatial autowave processes in active continua, and the integrability of equations allows one to construct efficient methods for the numerical solution of boundary value problems and ensures the existence of closed-form solutions. The present paper shows that the considered material function satisfies a criterion for the separation of the system of these equations into two noninteracting subsystems. We derive kinematic equations on the characteristics. We obtain and analyze centered self-similar solutions (Prandtl-Meyer solutions) in the domain of hyperbolicity of the equations, which describe flows in convergent and divergent channels.     

Approximate Analytical Solutions for Flow of a Third-Grade Fluid Through a Parallel-Plate Channel Filled with a Porous Medium

The flow of a non-Newtonian fluid through a porous media in between two parallel plates at different temperatures is considered. The governing momentum equation of third-grade fluid with modified Darcy’s law and energy equation have been derived. Approximate analytical solutions of momentum and energy equations are obtained by using perturbation techniques. Constant viscosity, Reynold’s model viscosity, and Vogel’s model viscosity cases are treated separately. The criteria for validity of approximate solutions are derived. A numerical residual error analysis is performed for the solutions. Within the validity range, analytical and numerical solutions are in good agreement.     

Integral boundary layer relations in the theory of wave flows for capillary liquid films

A generalized method of deriving the model equations is considered for wave flow regimes in falling liquid films. The viscous liquid equations are used on the basis of integral boundary layer relations with weight functions. A family of systems of evolution differential equations is proposed. The integer parameter n of these systems specifies the number of a weight function. The case n = 0 corresponds to the classical IBL (Integral Boundary Layer) model. The case n ≥ 1 corresponds to its modifications called the WIBL (Weighted Integral Boundary Layer) models. The numerical results obtained in the linear and nonlinear approximations for n = 0, 1, 2 are discussed. The numerical solutions to the original hydrodynamic differential equations are compared with experimental data. This comparison leads us to the following conclusions: as a rule, the most accurate solutions are obtained for n = 0 in the case of film flows on vertical and inclined solid surfaces and the accuracy of solutions decreases with increasing n. Hence, the classical IBL model has an advantage over the WIBL models.     

Application of the DuFort-Frankel and Saul'ev methods with time splitting to the formulation of a three dimensional hydrodynamic sea model

A three dimensional hydrodynamic sea model of an arbitrary sea area is formulated using sigma co-ordinates in the vertical. The solution of the equations using finite difference grids in the horizontal and the vertical is described. Discretization of the vertical viscosity term in the hydrodynamic equations using the DuFort-Frankel and Saul'ev methods is developed. Some numerical instabilities occur with the DuFort-Frankel method which can be overcome by splitting the hydrodynamic equations into equations describing the mean flow and equations describing the deviations from it. The computational advantages of solving these equations with different time steps are discussed. The accuracy and stability of the various methods is demonstrated for wind induced flow in a simple rectangular basin having dimensions representing the North Sea.     

Nonlinear dispersion of long internal waves

The propagation of long weakly nonlinear waves in an atmospheric waveguide is considered. A model system of Kadomtsev-Petviashvili equations [1], which describes the propagation of such waves, is derived. In the case of one excited wave mode the system of model equations goes over into the Kadomtsev-Petviashvili equation, in which, however, the variables x and t are interchanged. The reasons for this are clarified. In the two-dimensional case an approximate solution of the model equations is constructed, and steady nonlinear waves and their interaction in a collision are considered. The results of a numerical verification of the stability of the approximate steady solutions and of the solution to the problem of decay of the wave into quasisolitons are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 151–157, May–June, 1988.     

Simulation of planar flexible multibody systems with clearance and lubricated revolute joints

Modeling of clearance joints plays an important role in the analysis and design of multibody mechanical systems. Based on the absolute nodal coordinate formulation (ANCF), a new computational methodology for modeling and analysis of planar flexible multibody systems with clearance and lubricated revolute joints is presented. A planar absolute nodal coordinate formulation based on the locking-free shear deformable beam element is implemented to discretize the flexible bodies. A continuous contact-impact model is used to evaluate the contact force, in which energy dissipation in the form of hysteresis damping is considered. A force transition model from hydrodynamic lubrication forces to dry contact forces is introduced to ensure continuity in the joint reaction force. A comprehensive study with different lubrication force models has also been carried out. The generalized-α method is used to solve the equations of motion and several efficient methods are incorporated in the proposed model. Finally, the methodology is validated by two numerical examples.     

Finite element modelling of local scour below a pipeline under steady currents

Local scour has been identified as the main factor that causes failures of structures in offshore engineering. Numerous research efforts have been devoted to local scour around offshore pipelines in the past. In this paper, a finite element numerical model is established for simulating local scour below offshore pipelines in steady currents. The flow is simulated by solving the unsteady Reynolds-averaged Navier–Stokes equations with a standard k ? ? turbulent model closure. A sand slide scheme is proposed for the scour calculation, and bed load is considered in the proposed scour model. To account for changes in bed level, the moving mesh method is adopted to capture the water–sediment interface (bed), and the change of bed level is calculated by solving Exner–Polya equation. All the equations are discretised within the two-step Taylor–Galerkin algorithm in this paper. It is found that the sand slide model works well for the simulation of the scour, and the numerical results are shown to be in good agreement with the available experimental data.     

MULTIGRID CONVERGENCE ACCELERATION FOR TURBULENT SUPERSONIC FLOWS

A multigrid convergence acceleration technique has been developed for solving both the Navier–Stokes and turbulence transport equations. For turbulence closure a low-Reynolds-number q–ω turbulence model is employed. To enable convergence, the stiff non-linear turbulent source terms have to be treated in a special way. Further modifications to standard multigrid methods are necessary for the resolution of shock waves in supersonic flows. An implicit LU algorithm is used for numerical time integration. Several ramped duct test cases are presented to demonstrate the improvements in performance of the numerical scheme. Cases with strong shock waves and separation are included. It is shown to be very effective to treat fluid and turbulence equations with the multigrid method. A comparison with experimental data demonstrates the accuracy of the q–ω turbulence closure for the simulation of supersonic flows. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 1019–1035, 1997.     

Numerical simulation of the flow field in a thermal plasma reactor

A numerical simulation is presented for a thermal plasma reactor with particle-trajectory model in this paper.Turbulance is considered by using simple SGS model.Thegoverning equations are solved by means of the algorithm of SIMPLER.The calculatedresults give the velocity and the temperature fields within plasma reactor,and thetrajectories of the injected particles.     

Numerical simulation of the unsteady behaviour of cavitating flows

A 2D numerical model is proposed to simulate unsteady cavitating flows. The Reynolds‐averaged Navier–Stokes equations are solved for the mixture of liquid and vapour, which is considered as a single fluid with variable density. The vapourization and condensation processes are controlled by a barotropic state law that relates the fluid density to the pressure variations. The numerical resolution is a pressure‐correction method derived from the SIMPLE algorithm, with a finite volume discretization. The standard scheme is slightly modified to take into account the cavitation phenomenon. That numerical model is used to calculate unsteady cavitating flows in two Venturi type sections. The choice of the turbulence model is discussed, and the standard RNG k–εmodel is found to lead to non‐physical stable cavities. A modified k–εmodel is proposed to improve the simulation. The influence of numerical and physical parameters is presented, and the numerical results are compared to previous experimental observations and measurements. The proposed model seems to describe the unsteady cavitation behaviour in 2D geometries well. Copyright © 2003 John Wiley & Sons, Ltd.