Numerical simulation of nonequilibrium flows by using the state-to-state approach in commercial software

Nonequilibrium flows of a two-component oxygen mixture O₂/O behind a shock wave are studied with due allowance for the state-to-state vibrational and chemical kinetics. The system of gas-dynamic equations is supplemented with kinetic equations including contributions of VT (TV)-exchange and dissociation processes. A method of the numerical solution of this system with the use of the ANSYS Fluent commercial software package is proposed, which is used in a combination with the authors’ code that takes into account nonequilibrium kinetics. The computed results are compared with parameters obtained by solving the problem in the shock-fitting formulation. The vibrational temperature is compared with experimental data. The numerical tool proposed in the present paper is applied to study the flow around a cylinder.     

Numerical simulation of gas flows on the basis of quasi-hydrodynamic equations

Based on seven well-known test problems, reflecting the outstanding characteristics of nonstationary flows of an inviscid gas, the possibilities of using a quasi-hydrodynamic algorithm for the numerical solution of Euler equations are investigated.     

Radiative transfer in finite cylindrical media using transformed integral equations

A modified direct integration method is presented to solve three-dimensional radiative transfer in emitting, absorbing and linear-anisotropic scattering finite cylindrical media. This scheme effectively avoids an integral singularity in the coupled Fredholm type integral equations of radiative transfer. The scheme leads to faster and more accurate results, which are needed in combined mode and non-gray problems. The calculated incident radiation and heat fluxes agree well with published results by discrete ordinates method. Using the transformed integral equations, the effects of boundary emission and reflection can also be easily handled.     

激光激发黏弹表面波有限元数值模拟

研究黏弹性材料中激光激发的Rayleigh波的传播特征. 考虑到黏弹性材料的黏性特征,在频域内建立黏弹性材料中激光激发Rayleigh波的有限元数值模型. 在验证有限元频域数值模型正确性的基础上,模拟脉冲激光作用在黏弹性材料上激发出Rayleigh波,进而讨论激光激发的黏弹Rayleigh波的传播特征,并比较黏弹性材料与弹性材料中激光激发的Rayleigh波差异,同时分析了材料的黏性劲度参量变化对Rayleigh波特征的影响. 关键词： 表面波 激光超声 有限元方法 黏弹性     

Numerical solution of fractional differential equations via a Volterra integral equation approach

The main focus of this paper is to present a numerical method for the solution of fractional differential equations. In this method, the properties of the Caputo derivative are used to reduce the given fractional differential equation into a Volterra integral equation. The entire domain is divided into several small domains, and by collocating the integral equation at two adjacent points a system of two algebraic equations in two unknowns is obtained. The method is applied to solve linear and nonlinear fractional differential equations. Also the error analysis is presented. Some examples are given and the numerical simulations are also provided to illustrate the effectiveness of the new method.     

A compact finite difference scheme for the fractional sub-diffusion equations

In this paper, a compact finite difference scheme for the fractional sub-diffusion equations is derived. After a transformation of the original problem, the L1 discretization is applied for the time-fractional part and fourth-order accuracy compact approximation for the second-order space derivative. The unique solvability of the difference solution is discussed. The stability and convergence of the finite difference scheme in maximum norm are proved using the energy method, where a new inner product is introduced for the theoretical analysis. The technique is quite novel and different from previous analytical methods. Finally, a numerical example is provided to show the effectiveness and accuracy of the method.     

A family of dynamic finite difference schemes for large-eddy simulation

A family of dynamic low-dispersive finite difference schemes for large-eddy simulation is developed. The dynamic schemes are constructed by combining Taylor series expansions on two different grid resolutions. The schemes are optimized dynamically during the simulation according to the flow physics and dispersion errors are minimized through the real-time adaption of the dynamic coefficient. In case of DNS-resolution, the dynamic schemes reduce to the standard Taylor-based finite difference schemes with formal asymptotic order of accuracy. When going to LES-resolution, the schemes seamlessly adapt to dispersion-relation preserving schemes. The schemes are tested for large-eddy simulation of Burgers’ equation and numerical errors are investigated as well as their interaction with the subgrid model. Very good results are obtained.     

A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions

One-dimensional fractional anomalous sub-diffusion equations on an unbounded domain are considered in our work. Beginning with the derivation of the exact artificial boundary conditions, the original problem on an unbounded domain is converted into mainly solving an initial-boundary value problem on a finite computational domain. The main contribution of our work, as compared with the previous work, lies in the reduction of fractional differential equations on an unbounded domain by using artificial boundary conditions and construction of the corresponding finite difference scheme with the help of method of order reduction. The difficulty is the treatment of Neumann condition on the artificial boundary, which involves the time-fractional derivative operator. The stability and convergence of the scheme are proven using the discrete energy method. Two numerical examples clarify the effectiveness and accuracy of the proposed method.     

反射声波成像测井的有限元模拟

应用有限元法对新兴的反射声波成像测井进行了数值模拟研究，计算了井旁附近存在不同倾角声阻抗不连续界面对声波的反射，使用的井内激励源为幅度加权相控线阵声波辐射器。从计算出的声全波波形上可以清楚地看到沿井壁传播的折射波和来自井旁界面的反射纵波。应用偏移叠加等数据处理技术对数值模拟波形进行了处理，从偏移剖面上反演出的反射界面的尺寸、位置和倾角等几何特性与数值模拟输入参数基本一致。     

Numerical simulation of premixed combustion using an enriched finite element method

In this paper we present a novel discretization technique for the simulation of premixed combustion based on a locally enriched finite element method (FEM). Use is made of the G-function approach to premixed combustion in which the domain is divided into two parts, one part containing the burned and another containing the unburned gases. A level-set or G-function is used to define the flame interface separating burned from unburned gases. The eXtended finite element method (X-FEM) is employed, which allows for velocity and pressure fields that are discontinuous across the flame interface. Lagrange multipliers are used to enforce the correct essential interface conditions in the form of jump conditions across the embedded flame interface. A persisting problem with the use of Lagrange multipliers in X-FEM has been the discretization of the Lagrange multipliers. In this paper the distributed Lagrange multiplier technique is adopted. We will provide results from a spatial convergence analysis showing good convergence. However, a small modification of the interface is required to ensure a unique solution. Finally, results are presented from the application of the method to the problems of moving flame fronts, the Darrieus–Landau instability and a piloted Bunsen burner flame.     

A finite difference beam propagation algorithm based on generalized transmission line equations

A wide angle beam propagation algorithm is presented, which is based on generalized transmission line (GTL) equations. Besides the discretization, no further approximation is introduced. In principle, a full vectorial analysis is possible, and anisotropic material can be taken into account. The algorithm has been verified for planar 2D-waveguide devices. The wide angle characteristic has been examined by determining the propagation of tilted Gaussian beams in homogeneous media. Further studies were performed for tilted waveguides and for taper structures. Results for the latter devices were compared with other methods showing a very good agreement.     

Numerical simulation of non-viscous liquid pinch-off using a coupled level set-boundary integral method

Simulations of the pinch-off of an inviscid fluid column are carried out based upon a potential flow model with capillary forces. The interface location and the time evolution of the free surface boundary condition are both approximated by means of level set techniques on a fixed domain. The interface velocity is obtained via a Galerkin boundary integral solution of the 3D axisymmetric Laplace equation. A short-time analytical solution of the Raleigh–Taylor instability in a liquid column is available, and this result is compared with our numerical experiments to validate the algorithm. The method is capable of handling pinch-off and after pinch-off events, and simulations showing the time evolution of the fluid tube are presented.     

A Bayesian approach to characterising multi-phase flows using magnetic resonance: application to bubble flows

Magnetic Resonance (MR) imaging is difficult to apply to multi-phase flows due to both the inherently short T?* characterising such systems and the relatively long time taken to acquire the data. We develop a Bayesian MR approach for analysing data in k-space that eliminates the need for image acquisition, thereby significantly extending the range of systems that can be studied. We demonstrate the technique by measuring bubble size distributions in gas-liquid flows. The MR approach is compared with an optical technique at a low gas fraction (～2%), before being applied to a system where the gas fraction is too high for optical measurements (～15%).     

A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations

An implicit finite difference method with non-uniform timesteps for solving fractional diffusion and diffusion-wave equations in the Caputo form is presented. The non-uniformity of the timesteps allows one to adapt their size to the behaviour of the solution, which leads to large reductions in the computational time required to obtain the numerical solution without loss of accuracy. The stability of the method has been proved recently for the case of diffusion equations; for diffusion-wave equations its stability, although not proven, has been checked through extensive numerical calculations.     

A high-order WENO-Z finite difference based particle-source-in-cell method for computation of particle-laden flows with shocks

A high-order particle-source-in-cell (PSIC) algorithm is presented for the computation of the interaction between shocks, small scale structures, and liquid and/or solid particles in high-speed engineering applications. The improved high-order finite difference weighted essentially non-oscillatory (WENO-Z) method for solution of the hyperbolic conservation laws that govern the shocked carrier gas flow, lies at the heart of the algorithm. Finite sized particles are modeled as points and are traced in the Lagrangian frame. The physical coupling of particles in the Lagrangian frame and the gas in the Eulerian frame through momentum and energy exchange, is numerically treated through high-order interpolation and weighing. The centered high-order interpolation of the fluid properties to the particle location is shown to lead to numerical instability in shocked flow. An essentially non-oscillatory interpolation (ENO) scheme is devised for the coupling that improves stability. The ENO based algorithm is shown to be numerically stable and to accurately capture shocks, small flow features and particle dispersion. Both the carrier gas and the particles are updated in time without splitting with a third-order Runge–Kutta TVD method. One and two-dimensional computations of a shock moving into a particle cloud demonstrates the characteristics of the WENO-Z based PSIC method (PSIC/WENO-Z). The PSIC/WENO-Z computations are not only in excellent agreement with the numerical simulations with a third-order Rusanov based PSIC and physical experiments in [V. Boiko, V.P. Kiselev, S.P. Kiselev, A. Papyrin, S. Poplavsky, V. Fomin, Shock wave interaction with a cloud of particles, Shock Waves, 7 (1997) 275–285], but also show a significant improvement in the resolution of small scale structures. In two-dimensional simulations of the Mach 3 shock moving into forty thousand bronze particles arranged in the shape of a rectangle, the long time accuracy of the high-order method is demonstrated. The fifth-order PSIC/WENO-Z method with the fifth-order ENO interpolation scheme improves the small scale structure resolution over the third-order PSIC/WENO-Z method with a second-order central interpolation scheme. Preliminary analysis of the particle interaction with the flow structures shows that sharp particle material arms form on the side of the rectangular shape. The arms initially shield the particles from the accelerated flow behind the shock. A reflected compression wave, however, reshocks the particle arm from the shielded area and mixes the particles.     

The HyperCASL algorithm: A new approach to the numerical simulation of geophysical flows

We describe a major extension to the Contour-Advective Semi-Lagrangian (CASL) algorithm [D.G. Dritschel, M.H.P. Ambaum, A contour-advective semi-Lagrangian numerical algorithm for simulating fine-scale conservative dynamical fields, Quart. J. Roy. Meteorol. Soc. 123 (1997) 1097–1130; D.G. Dritschel, M.H.P. Ambaum, The diabatic contour advective semi-Lagrangian algorithm, Mon. Weather Rev. 134 (9) (2006) 2503–2514]. The extension, called ‘HyperCASL’ (HCASL), uses Lagrangian advection of material potential vorticity contours like CASL, but a Vortex-In-Cell (VIC) method for the treatment of diabatic forcing or damping. In this way, HyperCASL is fully Lagrangian regarding advection. A grid is used as in CASL to deal with ‘inversion’ (computing the velocity field from the potential vorticity field).     

A new mixed subgrid-scale model for large eddy simulation of turbulent drag-reducing flows of viscoelastic fluids

A mixed subgrid-scale(SGS) model based on coherent structures and temporal approximate deconvolution(MCT) is proposed for turbulent drag-reducing flows of viscoelastic fluids. The main idea of the MCT SGS model is to perform spatial filtering for the momentum equation and temporal filtering for the conformation tensor transport equation of turbulent flow of viscoelastic fluid, respectively. The MCT model is suitable for large eddy simulation(LES) of turbulent dragreducing flows of viscoelastic fluids in engineering applications since the model parameters can be easily obtained. The LES of forced homogeneous isotropic turbulence(FHIT) with polymer additives and turbulent channel flow with surfactant additives based on MCT SGS model shows excellent agreements with direct numerical simulation(DNS) results. Compared with the LES results using the temporal approximate deconvolution model(TADM) for FHIT with polymer additives, this mixed SGS model MCT behaves better, regarding the enhancement of calculating parameters such as the Reynolds number.For scientific and engineering research, turbulent flows at high Reynolds numbers are expected, so the MCT model can be a more suitable model for the LES of turbulent drag-reducing flows of viscoelastic fluid with polymer or surfactant additives.