High-resolution Central Scheme Research Articles

Numerical solution of extended black-oil model incorporating capillary effects based on a high-resolution central scheme

The purpose of this paper is to present an efficient numerical method to solve the extended black-oil model’s equations with considering the capillary pressure effects. The governing equations are formulated based on a global pressure concept to account for the capillary effects. The developed equations are conservative and suitable for the control volume approach. To solve the equations, an implicit pressure-explicit saturation scheme is used. The component transport equations are decomposed to a pure convection term driven by the global pressure and a pure diffusion part affected by the capillary pressures. A high-resolution central scheme is employed to discretize the convection term of the component transport equations. The results of the proposed model are compared with several semi-analytical and numerical solutions that were previously published. It is shown that the suggested model is computationally efficient and easy to be implemented for solving convection-diffusion problems. The results illustrate reasonable accuracy in comparison with the reference solutions. As a result of simplicity, low computational cost, and decent accuracy, this method is appropriate for solving complicated problems of reservoir engineering.

Nonlinear coupled constitutive relation model for monatomic gas in the OpenFOAM framework

In high-speed rarefied gas flow simulations, the linear constitutive relations of the classical Navier-Stokes-Fourier (NSF) equations are no longer successfully applied. In the present work, the two-dimensional nonlinear coupled constitutive relations (NCCR) model of the monatomic gas in an algebraic-type form is successfully implemented in the rhoCentralFoam in OpenFOAM to simulate the high-speed rarefied gas flows. Methods: This solver initially resolves the NSF equations using high-resolution central schemes in the finite volume method. The implicit NCCR model is solved by the iterative process with the initial values from the linear constitutive relations of the NSF model. After the convergence, these nonlinear constitutive relations are embedded in the NSF equations to continue the solution. Results: This new modified solver would be validated for the argon gas flows past a circular cylinder in cross-flow at Mach of 5 and Kn = 0.05 and the sharp leading edge flat plate at Mach of 4 and Kn = 0.0042. Conclusion: The simulation results show that the NCCR model agrees well with the DSMC data for the temperature contours for both cases and the temperature and velocity along the shock stand-off distance and is better than the NSF model in the high-speed rarefied gas simulations for the circular cylinder case.

High resolution central scheme using a new upwind slope limiter for hyperbolic conservation laws

In this paper, a new high resolution central-upwind scheme for hyperbolic conservation laws is proposed, based on a new upwind biased slope limiter. The new scheme is defined as a correction of KT central scheme by applying the new slope limiter. The new limiter uses more upwind information for the piecewise linear interpolations, so that it has an upwind nature. The main features of the new scheme are high resolution and nonoscillatory. At the same time, it retains the efficiency and simplicity of central scheme. The total-variation diminishing (TVD) property and maximum principle of the new scheme are proved. The numerical experiments, including one-dimensional, two-dimensional Riemann problems and double Mach reflection problem for Euler equations, show the desired resolution and robustness of the new scheme.

Well-balanced finite volume multi-resolution schemes for solving the Ripa models

In this article, fifth order well-balanced finite volume multi-resolution weighted essentially non-oscillatory (FV MR-WENO) schemes are constructed for solving one-dimensional and two-dimensional Ripa models. The Ripa system generalizes the shallow water model by incorporating horizontal temperature gradients. The presence of temperature gradients and source terms in the Ripa models introduce difficulties in developing high order accurate numerical schemes which can preserve exactly the steady-state conditions. The proposed numerical methods are capable to exactly preserve the steady-state solutions and maintain non-oscillatory property near the shock transitions. Moreover, in the procedure of derivation of the FV MR-WENO schemes unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical schemes. Further, the results obtained from considered numerical schemes are compared with those of a high resolution central upwind scheme and available exact solutions of the Ripa model.

Central upwind scheme based immersed boundary method for compressible flows around complex geometries

The present work combines easy to implement high resolution central upwind scheme (CUS) with a ghost cell based discrete forcing immersed boundary method (IBM) and proposes a novel CUS-IBM finite volume solver to simulate high-speed compressible inviscid/viscous flows around complex geometries on Cartesian grids. Proposed CUS-IBM solver has been tested for a variety of inviscid and viscous compressible flow problems, ranging from transonic to supersonic flow speeds (0.625 < M < 3), involving flow over different geometries, i.e., bump, airfoil and cylinder. Results from our computations clearly show that solutions computed with the proposed method are in good agreement with the numerical data available in literature for the various inviscid/viscous and external/internal compressible flow problems. The present study demonstrates that the proposed CUS-IBM exhibits nominal second-order accuracy in space and is an easy to implement, robust and promising alternative to the methods involving Riemann solver and characteristics decomposition.

Coupling of CFD and population balance modelling for a continuously seeded helical tubular crystallizer

A comprehensive model coupling the computational fluid dynamics with the population balance equation (CFD−PBE) for a continuous seeded mixed suspension mixed product removal (MSMPR)-helical tubular (HT) crystallizer assembly has been developed. The model accurately depicts the fluid dynamics, mass transfer, heat transfer and crystal size distribution (CSD) in the MSMPR-HT crystallizer. The crystallization system considered was to produce metastable α-form of L-glutamic acid. The PBE was discretized using a high-resolution central scheme and solved simultaneously with CFD equations in COMSOL Multiphysics® utilizing user-defined scalars. A good agreement was observed between the model and experimental data for the metastable α-form crystals of L-glutamic acid with uniform particles and a mean crystal size of 160 µm. The continuous crystallization under different stirrer speeds, saturated solution flow rates, inlet saturated solution concentrations and temperature was investigated. It was found the higher stirrer speeds, saturated solution flow rate and temperature lead to uniform distribution with smaller crystal size. Moreover, the mean crystal size becomes larger and the CSD wider when the saturated solution concentration increase.

Effect of Mixing on the Particle Size Distribution of Paracetamol Continuous Cooling Crystallization Products Using a Computational Fluid Dynamics–Population Balance Equation Simulation

Continuous crystallization is a widely viewed topic as it has many advantages in industrial production. However, it is very resource intensive and time-consuming to determine operation conditions through trial and error. Computer simulation appears to an effective method in continuous crystallization design. The continuous cooling crystallization of paracetamol in a 5 L crystallizer was modeled to investigate the effect of mixing on the particle size distribution (PSD). A novel coupled computational fluid dynamics–population balance equation (CFD–PBE) simulation was developed in which the PBE was discretized using a high-resolution central scheme and solved simultaneously with CFD equations in ANSYS FLUENT 15.0 utilizing user-defined scalars. The influences of both internal grid and external (spatial) grid on the final PSD were studied, and an appropriate grid was chosen. The spatial distribution of velocity, temperature, slurry density, and nucleation rate were simulated. The continuous cooling crystallization under different stirrer speeds, bath temperatures, and residence times was investigated. It was found that (a) higher stirrer speeds lead to uniform distribution but smaller PSD, and (b) as the average temperature gets higher, or the residence time gets longer, the influence of mixing on final PSD gets smaller.

Kurganov-Tadmor高解像度中心差分スキームを用いたチャネル乱流のDNS

Kurganov-Tadmor高解像度中心差分スキームを用いたチャネル乱流のDNS

Central upwind scheme for a compressible two-phase flow model.

In this article, a compressible two-phase reduced five-equation flow model is numerically investigated. The model is non-conservative and the governing equations consist of two equations describing the conservation of mass, one for overall momentum and one for total energy. The fifth equation is the energy equation for one of the two phases and it includes source term on the right-hand side which represents the energy exchange between two fluids in the form of mechanical and thermodynamical work. For the numerical approximation of the model a high resolution central upwind scheme is implemented. This is a non-oscillatory upwind biased finite volume scheme which does not require a Riemann solver at each time step. Few numerical case studies of two-phase flows are presented. For validation and comparison, the same model is also solved by using kinetic flux-vector splitting (KFVS) and staggered central schemes. It was found that central upwind scheme produces comparable results to the KFVS scheme.

A Numerical Scheme for Three-Dimensional Front Propagation and Control of Jordan Mode

As an example of a front propagation, we study the propagation of a three-dimensional nonlinear wavefront into a polytropic gas in a uniform state and at rest. The successive positions and geometry of the wavefront are obtained by solving the conservation form of equations of a weakly nonlinear ray theory. The proposed set of equations forms a weakly hyperbolic system of seven conservation laws with an additional vector constraint, each of whose components is a divergence-free condition. This constraint is an involution for the system of conservation laws, and it is termed a geometric solenoidal constraint. The analysis of a Cauchy problem for the linearized system shows that when this constraint is satisfied initially, the solution does not exhibit any Jordan mode. For the numerical simulation of the conservation laws we employ a high resolution central scheme. The second order accuracy of the scheme is achieved by using MUSCL-type reconstructions and Runge–Kutta time discretizations. A constrained transport–type technique is used to enforce the geometric solenoidal constraint. The results of several numerical experiments are presented, which confirm the efficiency and robustness of the proposed numerical method and the control of the Jordan mode.

High resolution central schemes for multi-dimensional non-linear acoustic simulation of silencers in internal combustion engines

Because of their small numerical viscosity even when very small time steps are enforced, central schemes look very suitable for acoustic simulations of silencers in internal combustion engines. In this work, a high resolution central scheme has been used with ad-hoc developed boundary conditions for the generation of different acoustic perturbations (white noise, sweep, impulse) in the OpenFOAM ® technology. The temporal solution, carried out by a first-order integration of the conservation laws by the explicit Euler’s method, has been first transferred into the frequency domain using FFT and then it has been processed to evaluate the transfer function of different geometries of silencers for internal combustion engines. The results obtained from the simulations have been compared with experimental data.

Central Schemes and Second Order Boundary Conditions for 1D Interface and Piston Problems in Lagrangian Coordinates

We study high-resolution central schemes in Lagrangian coordinates for the one-dimensional system of conservation laws describing the evolution of two gases in slab geometry separated by an interface. By using Lagrangian coordinates, the interface is transformed to a fixed coordinate in the computational domain and, as a consequence, the movement of the interface is obtained as a byproduct of the numerical solution. The main contribution is the derivation of a special equation of state to be imposed at the interface in order to avoid non-physical oscillations. Suitable boundary conditions at the piston that guarantee second order convergence are described. We compare the solution of the piston problem to other results available in the literature and to a reference solution obtained within the adiabatic approximation. A shock-interface interaction problem is also treated. The results on these tests are in good agreement with those obtained by other methods. AMS subject classifications: 65M06, 65M99, 76T05

A high order kinetic flux-vector splitting method for the reduced five-equation model of compressible two-fluid flows

We present a high order kinetic flux-vector splitting (KFVS) scheme for the numerical solution of a conservative interface-capturing five-equation model of compressible two-fluid flows. This model was initially introduced by Wackers and Koren (2004) [21]. The flow equations are the bulk equations, combined with mass and energy equations for one of the two fluids. The latter equation contains a source term in order to account for the energy exchange. We numerically investigate both one- and two-dimensional flow models. The proposed numerical scheme is based on the direct splitting of macroscopic flux functions of the system of equations. In two space dimensions the scheme is derived in a usual dimensionally split manner. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge–Kutta time stepping method. For validation, the results of our scheme are compared with those from the high resolution central scheme of Nessyahu and Tadmor [14]. The accuracy, efficiency and simplicity of the KFVS scheme demonstrate its potential for modeling two-phase flows.

High-resolution numerical simulation of 2D nonlinear wave structures in electromagnetic fluids with absorbing boundary conditions

Here we show how the full set of governing equations for the dynamics of charged-particle fluids in an electromagnetic field may be solved numerically in order to model nonlinear wave structures propagating in two dimensions. We employ a source-term adaptation and two-fluid extension of the second-order high-resolution central scheme of Balbas et al. (2004) [1]. The model employed is a 2D extension of that used by Baboolal and Bharuthram (2007) [5] in studies of 1D shocks and solitons in a two-fluid plasma under 3D electromagnetic fields. Further, we outline the use of free-flow boundary conditions to obtain stable wave structures over sufficiently long modelling times. As illustrative results, we examine the formation and evolution of shock-like and soliton structures of the magnetosonic mode.

Simulating coalescing compact binaries by a new code (SACRA)

We report our new code, named SACRA (SimulAtor for Compact objects in Relativistic Astrophysics) for numerical relativity simulations in which an adaptive mesh refinement algorithm is implemented. In this code, the Einstein equations are solved in the Baumgarte-Shapiro-Shibata-Nakamura formalism with a fourth-order finite differencing, and the hydrodynamic equations are solved by a third-order high-resolution central scheme. The fourth-order Runge-Kutta scheme is adopted for integration in time. To test the code, simulations for coalescence of black hole-black hole, neutron star-neutron star (NS-NS), and black hole-neutron star (BH-NS) binaries are performed, and also, properties of BHs formed after the merger and gravitational waveforms are compared among those three cases. For the simulations of black hole-black hole binaries, we adopt the same initial conditions as those by Buonanno et al. [1] and compare numerical results. We find reasonable agreement except for a slight disagreement possibly associated with the difference in choice of gauge conditions and numerical schemes. For an NS-NS binary, we performed simulations employing both SACRA and Shibata's previous code, and find reasonable agreement between two numerical results for the final outcome and qualitative property of gravitational waveforms. We also find that the convergence is relatively slow for numerical results of NS-NS binaries, and again realize that long-term numerical simulations with several resolutions and grid settings are required for validating the results. For a BH-NS binary, we compare numerical results with our previous ones, and find that gravitational waveforms and properties of the BH formed after the merger agree well with those of our previous ones, although the disk mass formed after the merger is less than 0.1% of the total rest mass, which disagrees with the previous result. We also report numerical results of a long-term simulation (with $\ensuremath{\sim}4$ orbits) for a BH-NS binary for the first time. All these numerical results show behavior of convergence, and extrapolated numerical results for time spent in the inspiral phase agree with post-Newtonian predictions in a reasonable accuracy. These facts validate the results by SACRA.

A time-accurate explicit multi-scale technique for gas dynamics

We present a new time-accurate algorithm for the explicit numerical integration of the compressible Euler equations of gas dynamics. This technique is based on the discrete-event simulation (DES) methodology for nonlinear flux-conservative PDEs [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194]. DES enables adaptive distribution of CPU resources in accordance with local time scales of the underlying numerical solution. It distinctly stands apart from multiple (local) time-stepping algorithms in that it requires neither selecting a global synchronization time step nor pre-determining a sequence of time-integration operations for individual parts of a heterogeneous numerical system. In this paper we extend the DES methodology in three important directions: (i) we apply DES to a system of coupled gas dynamics equations discretized via a central-upwind scheme [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282; A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740]; (ii) we introduce a new Preemptive Event Processing (PEP) technique, which automatically enforces synchronous execution of events with sufficiently close update times; (iii) we significantly improve the accuracy of the previous algorithm [Y.A. Omelchenko, H. Karimabadi, Self-adaptive time integration of flux-conservative equations with sources, J. Comput. Phys. 216 (1) (2006) 179–194] by applying locally second-order-in-time flux-conserving corrections to the solution obtained with the forward Euler scheme. The performance of the new technique is demonstrated in a series of one-dimensional gas dynamics test problems by comparing numerical solutions obtained in event-driven and equivalent time-stepping simulations.

A low dissipation essentially non-oscillatory central scheme

Here we present a new, semidiscrete, central scheme for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The method presented in this paper is an extension of the centrally weighted non-oscillatory schemes ( Cweno) presented in [A. Kurganov, E. Tadmor, New high resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comp. Phys. 160 (2000) 241–282; A. Kurganov, D. Levy, A third-order semidiscrete central scheme for conservation laws and convection–diffusion equations, SIAM J. Sci. Comput. 22 (2000) 1461–1488; A. Kurganov, S. Noelle, G. Petrova, Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations, SIAM J. Sci. Comput. 23 (3) (2001) 707–740]. The method suggested in this manuscript is derived independently of the order of the scheme. The gain in this new method is a decreased dissipation especially for high Mach-number flows, which are frequently encountered, e.g., in astrophysical contexts or turbulent systems.

An improved algorithm for simulating three-dimensional, viscoelastic turbulence

We present a new finite-difference formulation to update the conformation tensor in dumbbell models (e.g., Oldroyd-B, FENE-P, Giesekus) that guarantees positive eigenvalues of the tensor (i.e., the tensor remains positive definite) and prevents over-extension for finite-extensible models. The formulation is a generalization of the second-order, central difference scheme developed by Kurganov and Tadmor [A. Kurganov, E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations, J. Comput. Phys. 160 (2000) 241–282] that guarantees a scalar field remains everywhere positive. We have extended the algorithm to guarantee a tensor field remains everywhere positive definite following an update. Extensive testing of the algorithm shows that the volume average of the conformation tensor is conserved. Furthermore, volume averages of the conformation tensor in homogeneous turbulent shear flow made over the Eulerian grid are in quantitative agreement with Lagrangian averages made over fluid particles moving throughout the domain, highlighting the accuracy of the treatment of the convective terms.

Simulation of Mixing Effects in Antisolvent Crystallization Using a Coupled CFD-PDF-PBE Approach

Antisolvent crystallization is widely used in the production of pharmaceuticals. Although it has been observed experimentally that the crystal size distribution is strongly influenced by the imperfect mixing of the antisolvent with the solution, these effects have not been adequately quantified. In this work, a turbulent computational fluid dynamics (CFD) code was coupled with a multienvironment probability density function (PDF) model, which captures the micromixing in the subgrid scale, and the population balance equation, which models the evolution of the crystal size distribution. The population balance equation (PBE) was discretized along the internal coordinate using a high-resolution central scheme. The presence of solids was addressed by treating the suspension as a pseudo-homogeneous phase with a spatial variation in the effective viscosity. This coupled CFD-PDF-PBE algorithm was applied to an antisolvent crystallization process in an agitated semibatch vessel, where the rising liquid level was modeled by a dynamic mesh. The effects of agitation speed, addition mode, and scale-up on the local primary nucleation and size-dependent growth and dissolution rates, as well as the crystal size distribution, were numerically investigated.

An investigation of the internal structure of shock profiles for shock capturing schemes

The theoretical understanding of discrete shock transitions obtained by shock capturing schemes is very incomplete. Previous experimental studies indicate that discrete shock transitions obtained by shock capturing schemes can be modeled by continuous functions, so called continuum shock profiles. However, the previous papers have focused on linear methods. We have experimentally studied the trajectories of discrete shock profiles in phase space for a range of different high resolution shock capturing schemes, including Riemann solver based flux limiter methods, high resolution central schemes and ENO type methods. In some cases, no continuum profiles exists. However, in these cases the point values in the shock transitions remain bounded and appear to converge toward a stable limit cycle. The possibility of such behavior was anticipated in Bultelle, Grassin and Serre, 1998, but no specific examples, or other evidence, of this behavior have previously been given. In other cases, our results indicate that continuum shock profiles exist, but are very complicated. We also study phase space orbits with regard to post shock oscillations.