Small Time Step Size Research Articles

A new 2D ESPH bedload sediment transport model for rapidly varied flows over mobile beds

A 2D Eulerian meshless bedload sediment transport model is developed using smoothed particle hydrodynamics (SPH) to simulate rapidly varied flows over mobile beds. In the developed model, we adopt a weakly coupled numerical approach to explicitly solve the governing equations, including 2D shallow water equations for fluid flow motion and the Exner equation for bed sediment movement at the same time step. A defined virtual bedload velocity in a weakly coupled approach is involved in the calculation of time step sizes. However, nonphysical virtual bedload velocities, which often occur in cases with nearly flat beds, require extremely small time step sizes. A formulation of the lower and upper bounds of information propagation speeds for an HLLC approximate Riemann solver is thus proposed to remedy the problem. Four case studies involving dam break flows in prismatic and erodible channels, knickpoint migration, dam erosion due to overtopping flow and dam-break flow in an erodible channel with an abrupt expansion are adopted to validate the developed model. In addition, to compare the performances of different particle interaction configurations under Cartesian uniform particle arrangements, four and eight interacting particles are obtained by changing the smoothing length. Against the measured results, the case of eight interacting particles shows more accurate predictions of erosion peaks along the measured cross-sections because the lateral flows are significant. The good agreement between the simulated and measured results shows that the developed 2D Eulerian SPH bedload sediment transport model with eight interacting particles is well suited for simulating complicated flow-induced bed erosion.

Partially explicit splitting method for a multi-physics problem

Multi-physics are widely studied and used in numerical analysis and simulation. Physics-based splitting methods are introduced, in which different methods are applied to solve each individual aspect of a physical system in multiple steps. The implicit methods are typically used for temporal discretization and are unconditionally stable such that the time step size can be large during the computation. However, the implicit methods are more complicated to solve, especially for nonlinear problems. In contrast, the explicit methods are relatively easier to compute, while they may require a much smaller time step size. In this paper, we propose a partially explicit scheme with physics-based splitting. We take the convection diffusion equation as an example. In this scheme, the convection equation is solved using the exact solution, and then we solve the diffusion equation via finite element methods. Some degrees of freedom in the diffusion equation are treated implicitly while others are treated explicitly. We can change the large-scale implicit system into one smaller-scale implicit system and one smaller-scale explicit system, which provides saving in computation. We analyze the stability and the order of accuracy. We also present some numerical results to show the performance of our proposed scheme.

A time-consistent stabilized finite element method for fluids with applications to hemodynamics

Several finite element methods for simulating incompressible flows rely on the streamline upwind Petrov–Galerkin stabilization (SUPG) term, which is weighted by tau _{text{SUPG}}. The conventional formulation of tau _{text{SUPG}} includes a constant that depends on the time step size, producing an overall method that becomes exceedingly less accurate as the time step size approaches zero. In practice, such method inconsistency introduces significant error in the solution, especially in cardiovascular simulations, where small time step sizes may be required to resolve multiple scales of the blood flow. To overcome this issue, we propose a consistent method that is based on a new definition of tau _{text{SUPG}}. This method, which can be easily implemented on top of an existing streamline upwind Petrov–Galerkin and pressure stabilizing Petrov–Galerkin method, involves the replacement of the time step size in tau _{text{SUPG}} with a physical time scale. This time scale is calculated in a simple operation once every time step for the entire computational domain from the ratio of the L2-norm of the acceleration and the velocity. The proposed method is compared against the conventional method using four cases: a steady pipe flow, a blood flow through vascular anatomy, an external flow over a square obstacle, and a fluid–structure interaction case involving an oscillatory flexible beam. These numerical experiments, which are performed using linear interpolation functions, show that the proposed formulation eliminates the inconsistency issue associated with the conventional formulation in all cases. While the proposed method is slightly more costly than the conventional method, it significantly reduces the error, particularly at small time step sizes. For the pipe flow where an exact solution is available, we show the conventional method can over-predict the pressure drop by a factor of three. This large error is almost completely eliminated by the proposed formulation, dropping to approximately 1% for all time step sizes and Reynolds numbers considered.

Implicit coupling methods for nonlinear interactions between a large‐deformable hyperelastic solid and a viscous acoustic fluid of infinite extent

AbstractThis paper addresses the challenges in studying the interaction between high‐intensity sound waves and large‐deformable hyperelastic solids, which are characterized by nonlinearities of the hyperelastic material, the finite‐amplitude acoustic wave, and the large‐deformable fluid–solid interface. An implicit coupling method is proposed for predicting nonlinear structural‐acoustic responses of the large‐deformable hyperelastic solid submerged in a compressible viscous fluid of infinite extent. An arbitrary Lagrangian–Eulerian (ALE) formulation based on an unsplit complex‐frequency‐shifted perfectly matched layer method is developed for long‐time simulation of the nonlinear acoustic wave propagation without exhibiting long‐time instabilities. The solid and acoustic fluid domains are discretized using the finite element method, and two different options of staggered implicit coupling procedures for nonlinear structural‐acoustic interactions are developed. Theoretical formulations for stability analysis of the implicit methods are provided. The accuracy, robustness, and convergence properties of the proposed methods are evaluated by a benchmark problem, that is, a hyperelastic rod interacting with finite‐amplitude acoustic waves. The numerical results substantiate that the present methods are able to provide long‐time steady‐state solutions for a nonlinear coupled hyperelastic solid and viscous acoustic fluid system without numerical constraints of small time step sizes and long‐time instabilities. The methods are applied to investigate nonlinear dynamic behaviors of coupled hyperelastic elliptical ring and acoustic fluid systems. Physical insights into 2:1 and 4:2:1 internal resonances of the hyperelastic elliptical ring and period‐doubling bifurcations of the structural and acoustic responses of the system are provided.

Development of a Composite Implicit Time Integration Scheme for Three-Dimensional Discontinuous Deformation Analysis

Discontinuous deformation analysis (DDA) is a discontinuum-based and implicit method for investigating the deformational behavior of block systems. The constant acceleration integration (CAI) scheme characterized by unconditional stability is employed in the traditional DDA. In this study, the problems of the CAI scheme regarding the time step in DDA are pointed out. A too large or too small time step size adopted in the CAI scheme will have adverse effects on the DDA computation. To overcome the weaknesses, an alternative composite implicit time integration (CITI) scheme, which is a combination of the trapezoidal rule and the three-point backward Euler method, is implemented in the three-dimensional (3D) DDA method. Verification examples and slope numerical simulations are presented to illustrate the accuracy and effectiveness of the proposed methodology. The results showed that the CITI scheme can overcome the numerical error caused by the large time step size, and the algorithm damping is closely related to the time step size and the selected splitting parameter.

A new graph-theoretic approach for the study of the surface of a thin sheet of a viscous liquid model

In this study, we considered the model of the surface of a thin sheet of viscous liquid which is known as the Buckmaster equation (BME), and presented a new graph-theoretic polynomial collocation method named the Hosoya polynomial collocation method (HPCM) for the solution of nonlinear Buckmaster equation. In the literature, the majority of the developed numerical methods considered small time step sizes like 0.01s and 0.05s to obtain relatively accurate approximations for the nonlinear BME. This study focused on optimizing the time step sizes by adopting bigger time steps sizes like 1.0s,3.0s, and 5.0s, etc without adversely affecting accuracy. First, using the Gram- Schmidt process, we generated the orthonormal functions from the Hosoya polynomial of the path graph. Then developed the functional integration matrix using orthonormal Hosoya polynomials of path graphs. With this active matrix-involved method, the nonlinear BMEs are transformed into a system of nonlinear equations and solved the equations by Newton’s method through the Mathematica software for unknown coefficients. The exactness of the proposed strategy is tested with two numerical examples. The acquired results contrasted with the current analytical solutions to these problems. Also provided the convergence analysis, comparison of error norms, graphical plots of correlation of HPCM results, and the results of other numerical methods in the literature to validate the productivity and accuracy of the newly developed HPCM.

Convergent sequential fully implicit method for reservoir simulation

We propose a stable time stepping algorithm for sequential fully implicit method (SFIM) in solving the coupled flow and transport equations for multi-phase flow in reservoir simulation. In developing a stable sequential fully implicit method, we are concerned with the flow and transport systems that yield a stable, unique solution for a sufficiently small time step size. The sequential fully implicit method comprises three major steps: (1) pressure solution, (2) conservative total velocity calculation, and (3) saturation solution from the nonlinear transport equation. These three major steps are coupled by outer-loop iterations.In solving the transport equation, the fractional flow formulation with fixed total velocity is employed. We assume that the transport equation can be always solved with an under-relaxation method for a given conservative velocity field. For example, by analyzing the S-shaped fractional flow curve, Jenny et al. (2009) [17] developed an unconditionally stable under-relaxation method for the iterative solution of two-phase flow with weak gravity effect. In multi-phase (two or three phases) flow with gravity effect, the fluxes between two cells are split into viscous and gravitational fluxes. The fractional flow for viscous flux is a function of the upwind cell saturation, whereas that for gravitational flux is a function of two saturations of the neighboring cells. For a simple representation of multi-variable functions in Newton iterations, the fractional flow curves are parameterized along the Newton iterative direction. Since the fractional flow curves for viscous and gravitational flows are distinctly different, we devise two under-relaxation methods for multi-phase flow with gravity effect.In SFIM, the algorithm can be optimized based on the total velocity changes during a time step. In general, one iteration of saturation is computationally less expensive than that for one pressure calculation. As a result, if the total velocity is not a strong function of saturations (e.g., viscous flow with small gravity number), for one outer-loop calculation, the transport equation is iterated to convergence. When the total velocity is a strong function of saturations (e.g., lock-exchange problem), saturations are computed only for one iterative update for each outer-loop iteration. It ensures that the solution path will closely follow the trust region of pressure and saturation in Newton's method. We demonstrate convergence of SFIM with numerical examples: gravity segregation, a linear displacement model, reservoir models with wells, a lock-exchange problem, etc.

High fidelity transient solver in STREAM based on multigroup coarse-mesh finite difference method

This study incorporates a high-fidelity transient analysis solver based on multigroup CMFD in the MOC code STREAM. Transport modeling with heterogeneous geometries of the reactor core increases computational cost in terms of memory and time, whereas the multigroup CMFD reduces the computational cost. The reactor condition does not change at every time step, which is a vital point for the utilization of CMFD. CMFD correction factors are updated from the transport solution whenever the reactor core condition changes, and the simulation continues until the end. The transport solution is adjusted once CMFD achieves the solution. The flux-weighted method is used for rod decusping to update the partially inserted control rod cell material, which maintains the solution's stability. A smaller time-step size is needed to obtain an accurate solution, which increases the computational cost. The adaptive step-size control algorithm is robust for controlling the time step size. This algorithm is based on local errors and has the potential capability to accept or reject the solution. Several numerical problems are selected to analyze the performance and numerical accuracy of parallel computing, rod decusping, and adaptive time step control. Lastly, a typical pressurized LWR was chosen to study the rod-ejection accident.

An Adaptive Time-Stepping Control Algorithm for Molten Salt Reactor Transient Analyses

This paper presents an adaptive time-stepping (ATS) control algorithm developed for transient analyses of Molten Salt Reactors (MSRs) with flowing fuel. To enhance the computational efficiency while maintaining a targeted accuracy, an ATS algorithm with the proportional-integral-derivative (PID) controller and error monitoring was developed and implemented into the PROTEUS-NODAL code. The performance of the ATS algorithm was tested for a wide range of the Molten Salt Fast Reactor (MSFR) numerical benchmark and the Molten Salt Reactor Experiments (MSRE). The test results show over 20 times speedups for very rapid transients and very slow, long-time transients relative to the reference solution obtained with a small time-step size. In addition, extensive parametric studies were performed on the local error estimates and controller parameters and derived a set of recommended values of initial time step, tolerance and gain for four different transient categories.

CDFIB: A convolutional direct forcing immersed boundary method for solving incompressible flows with time-varying geometries

When incorporating the immersed boundary method to model solid obstacles in incompressible flows, the conventional direct forcing immersed boundary (DFIB) method may face the issue of spurious force oscillations (SFO) due to the mass imbalance when integrating drag/lift forces around the solid obstacle. Although the SFO can be controlled with large time step size when employing conventional DFIB method, the method remains prone to large SFO when the time step size is small. When we deal with flow-structure interaction (FSI) scenarios, the solid bodies are driven by hydrodynamic forces that are produced by flow fluids. Unphysical acceleration/deceleration may happen during the simulations due to SFO, thus making the simulations unstable or even diverge. To alleviate the SFO, we thus propose a novel method for modeling the solid bodies in incompressible fluid flows by performing the convolution kernel, dubbed as cDFIB method. cDFIB method inherits the merits of DFIB method, while effectively suppressing SFO for both small and large time step size. From the numerical results of the SFO oscillating cylinder problem, the SFO is proportional to O(Δx) and O(Δt) for present cDFIB method. This property allows cDFIB method to solve challenging problems where employing large time step size becomes infeasible.

Bending fluctuations in semiflexible, inextensible, slender filaments in Stokes flow: Toward a spectral discretization.

Semiflexible slender filaments are ubiquitous in nature and cell biology, including in the cytoskeleton, where reorganization of actin filaments allows the cell to move and divide. Most methods for simulating semiflexible inextensible fibers/polymers are based on discrete (bead-link or blob-link) models, which become prohibitively expensive in the slender limit when hydrodynamics is accounted for. In this paper, we develop a novel coarse-grained approach for simulating fluctuating slender filaments with hydrodynamic interactions. Our approach is tailored to relatively stiff fibers whose persistence length is comparable to or larger than their length and is based on three major contributions. First, we discretize the filament centerline using a coarse non-uniform Chebyshev grid, on which we formulate a discrete constrained Gibbs-Boltzmann (GB) equilibrium distribution and overdamped Langevin equation for the evolution of unit-length tangent vectors. Second, we define the hydrodynamic mobility at each point on the filament as an integral of the Rotne-Prager-Yamakawa kernel along the centerline and apply a spectrally accurate "slender-body" quadrature to accurately resolve the hydrodynamics. Third, we propose a novel midpoint temporal integrator, which can correctly capture the Ito drift terms that arise in the overdamped Langevin equation. For two separate examples, we verify that the equilibrium distribution for the Chebyshev grid is a good approximation of the blob-link one and that our temporal integrator for overdamped Langevin dynamics samples the equilibrium GB distribution for sufficiently small time step sizes. We also study the dynamics of relaxation of an initially straight filament and find that as few as 12 Chebyshev nodes provide a good approximation to the dynamics while allowing a time step size two orders of magnitude larger than a resolved blob-link simulation. We conclude by applying our approach to a suspension of cross-linked semiflexible fibers (neglecting hydrodynamic interactions between fibers), where we study how semiflexible fluctuations affect bundling dynamics. We find that semiflexible filaments bundle faster than rigid filaments even when the persistence length is large, but show that semiflexible bending fluctuations only further accelerate agglomeration when the persistence length and fiber length are of the same order.

Dimensional analysis of vapor bubble growth considering bubble–bubble interactions in flash boiling microdroplets of highly volatile liquid electrofuels

Electrofuels (e-fuels) produced from renewable electricity and carbon sources have gained significant attention in recent years as promising alternatives to fossil fuels for the transportation sector. However, the highly volatile e-fuels, such as short-chain oxymethylene ethers (OMEx) are prone to flash vaporization phenomena, which is associated with the formation and growth of vapor bubbles, followed by explosive bursting of the liquid jet. This phenomenon is important in many practical applications, for example, superheated liquid sprays in gasoline direct injection engines as well as cryogenic engines. The simulation of a flash boiling spray of such highly volatile liquid fuels is numerically challenging due to several reasons, including (1) the complexity of the bubble growth process in the presence of multiple vapor bubbles and (2) the need to use an extremely small time step size to accurately capture the underlying physics associated with the flash boiling process. In this paper, we first present a bubble growth model in flash boiling microdroplets considering bubble–bubble interactions along with the finite droplet size effects. A dimensional analysis of the newly derived Rayleigh–Plesset equation (RPE) with bubble–bubble interactions is then performed for Reynolds numbers of different orders of magnitude to estimate the relative importance of different forces acting on the bubble surface. Based on this analysis, a simplified nondimensional semi-analytical solution for bubble growth, which also includes the bubble–bubble interactions, is derived to estimate the bubble growth behavior with reasonable accuracy using the larger time step sizes for a wide range of operating conditions. The derived semi-analytical solution is shown to be a good approximation for describing the bubble growth rate over the whole lifetime of the bubble, thus making it useful for simulations of superheated sprays with large numbers of droplets and even more bubbles. The bubble–bubble interactions are found to have a significant impact on the bubble growth dynamics and result in delaying the onset of droplet bursting due to the slower growth of the vapor bubble compared to the bubble growth without bubble–bubble interactions. Furthermore, in a comparison with DNS results, the proposed bubble growth model is shown to reasonably capture the impact of bubble interactions leading to smaller volumetric droplet expansion.

Nonlinear static analysis of extreme structural behavior: Overcoming convergence issues via an unconditionally stable explicit dynamic approach

Simulating extreme structural behavior is of vital importance as it is a major means to study the damage and failure mechanism of engineering structures. However, convergence issues frequently occur when simulating structural response under extreme loading using traditional nonlinear solution strategies, e.g., the Newton–Raphson family of methods. Though commercial finite element software provides explicit dynamic methods to overcome this issue, they are only conditionally stable and require a small time step size. In this paper, a new explicit dynamic approach to simulate static problems involving extreme structural behavior is presented. The multi-support excitation pattern is employed to apply a nodal displacement history of the problem in a dynamic way, then the unconditionally stable explicit KR-α method is used to eliminate the convergence issues. Three examples are presented to illustrate the effectiveness of the proposed dynamic approach, and a parametric study is conducted to investigate the influence of the algorithmic parameters. The results indicate that the proposed method provides the same accuracy as the traditional static method without any convergence issues.

Partially explicit time discretization for time fractional diffusion equation

Time fractional PDEs have been used in many applications for modeling and simulations. Many of these applications are multiscale and contain high contrast variations in the media properties. It requires very small time step size to perform detailed computations. On the other hand, in the presence of small spatial grids, very small time step size is required for explicit methods. Explicit methods have many advantages as we discuss in the paper. In this paper, we propose a partial explicit method for time fractional PDEs. The approach solves the forward problem on a coarse computational grid, which is much larger than spatial heterogeneities, and requires only a few degrees of freedom to be treated implicitly. Via the construction of appropriate spaces and careful stability analysis, we can show that the time step can be chosen not to depend on the contrast or scale as the coarse mesh size. Thus, one can use larger time step size in an explicit approach. We present stability theory for our proposed method and our numerical results confirm the stability findings and demonstrate the performance of the approach.

Variational Quantum Simulation of Chemical Dynamics with Quantum Computers.

Classical simulations of real-space quantum dynamics are challenging due to the exponential scaling of computational cost with system dimensions. Quantum computers offer the potential to simulate quantum dynamics with polynomial complexity; however, existing quantum algorithms based on the split-operator techniques require large-scale fault-tolerant quantum computers that remain elusive in the near future. Here, we present variational simulations of real-space quantum dynamics suitable for implementation in noisy intermediate-scale quantum (NISQ) devices. The Hamiltonian is first encoded onto qubits using a discrete variable representation and binary encoding scheme. We show that direct application of a real-time variational quantum algorithm based on the McLachlan's principle is inefficient as the measurement cost grows exponentially with the qubit number for general potential energy, and an extremely small time-step size is required to achieve accurate results. Motivated by the insights that many chemical dynamics occur in the low-energy subspace, we propose a subspace expansion method by projecting the total Hamiltonian, including the time-dependent driving field, onto the system low-energy eigenstate subspace using quantum computers, and the exact quantum dynamics within the subspace can then be solved classically. We show that the measurement cost of the subspace approach grows polynomially with dimensionality for general potential energy. Our numerical examples demonstrate the capability of our approach, even under intense laser fields. Our work opens the possibility of simulating chemical dynamics with NISQ hardware.

A Stable and High Accurate Numerical Simulator for Convection-Diffusion Equation

This study presents a simulator to obtain numerical solution of convection-diffusion equation. It includes explicit, fully implicit and semi-implicit time discretization techniques. Although the explicit method is efficient for some simple conduction problems, it needs stability criteria. Otherwise, it is unstable especially for high Courant number and large time step size. When using the explicit method, the time step size is chosen with care to get well-posed numerical solution. This requirement sets a serious constraint for the explicit method because choosing small time step size causes the simulation time to become quite long. Even though the implicit method is stable for large time step size and high Courant number, it leads to numerical dispersion like the explicit method. On the other hand, second-order accurate semi-implicit method significantly reduces numerical dispersion. In addition to time discretization techniques, this simulator contains several space discretization methods such as first-order upstream and UMIST (University of Manchester Institute of Science and Technology) techniques. The proposed numerical simulator is suitable for easily using the different combinations of time and space discretization methods. Secondly, this study is to propose the use of semi-implicit time discretization technique with UMIST space discretization method to minimize numerical dispersion and suppress unphysical oscillation. In spite of the fact that the UMIST method suppresses unphysical oscillation, it causes a small and undesired oscillation at flood front for very large Courant number. Third objective of this study is to propose minor modification on the UMIST method to eliminate this unphysical oscillation.

Modeling neighborhood-scale shallow geothermal energy utilization: a case study in Berlin

Nowadays, utilizing shallow geothermal energy for heating and cooling buildings has received increased interest in the building sector. Among different technologies, large borehole heat exchanger arrays are widely employed to supply heat to various types of buildings. Recently, a 16-borehole array was constructed to extract shallow geothermal energy to provide heat to a newly-developed public building in Berlin. To guarantee the quality of the numerical model and reveal its sensitivity to different subsurface conditions, model simulations were conducted for 25 years with two finite element simulators, namely the open-source code OpenGeoSys and the widely applied commercial software FEFLOW. Given proper numerical settings, the simulation results from OpenGeoSys and FEFLOW are in good agreement. However, further analysis reveals differences with respect to borehole inflow temperature calculation implemented in the two software. It is found that FEFLOW intrinsically uses the outflow temperature from the previous time step to determine the current inflow temperature, which makes it capable of much faster simulation by avoiding iterations within a single time step. In comparison, OpenGeoSys always updates the inflow and outflow temperature based on their current time step values. Because the updates are performed after each iteration, it delivers more accurate results with the expense of longer simulation time. Based on this case study, OpenGeoSys is a valid alternative to FEFLOW for modeling ground source heat pump systems. For modellers working in this field, it is thus recommended to adopt small enough time step size, so that potential numerical error can be avoided.

On the entropy conserving/stable implicit DG discretization of the Euler equations in entropy variables

The aim of this paper is to investigate the behavior of a high-order accurate Discontinuous Galerkin entropy conserving/stable scheme in space for unsteady compressible inviscid flows. In order to ensure an entropy conserving/stable scheme in space, several entropy conserving/stable numerical fluxes are considered. For the time discretization, high-order accurate linearly implicit Rosenbrock-type Runge–Kutta schemes are used. These schemes cannot be provably entropy conserving/stable, but, for an enough small time step size, the error due to the time integration has negligible contribution, and entropy conserving/stable properties can be fulfilled. The properties of the fully discrete system of equations are assessed in a series of numerical experiments of growing complexity, i.e.: (i) the isentropic vortex convection problem, (ii) the double shear layer, (iii) the Sod shock tube and (iv) the Taylor–Green vortex. For each one of these test-case the accuracy and the related order of convergence of the time integration schemes and of the numerical fluxes employed will be investigated, as well as the comparison between the accuracy provided by the entropy and the primitive set of variables and their stability properties. Furthermore, same relevant issues related to the use of entropy conserving fluxes are addressed, like the necessity to use higher accurate quadrature rules, the spatial sub-optimal order of convergence, and the behavior of the fully discrete system of equations when long-time simulations are performed.

A stable discontinuous Galerkin method based on high‐order dual splitting scheme without additional stabilization term for incompressible flows

AbstractIn this work, we consider equal‐order discontinuous Galerkin (DG) solver for incompressible Navier–Stokes equations based on high‐order dual splitting scheme. In order to stay stable, the time step size of this method has been reported that is strictly limited. The upper bound of time step size is restricted by Courant–Friedrichs–Lewy (CFL) condition (Hesthaven and Warburton, 2007) and lower bound is required to be larger than the critical value which depends on Reynolds number and spatial resolution (Ferrer et al., 2014). For high‐Reynolds‐number flow problems, if the spatial resolution is low, the critical value may be larger than CFL condition, then instability will occur for any time step size. Therefore, sufficiently high spatial resolution is indispensable in order to maintain stability, which increases the computational cost. To overcome these difficulties and develop a robust solver for high‐Reynolds‐number flow problem, it is necessary to further study the instability problem at small time steps. We numerically investigate the effect of the pressure gradient term in projection step and the velocity divergence term in pressure Poisson equation on the stability for small time step size, respectively, and conclude that the DG formulation of the pressure gradient term has a more significant effect on the stability of the scheme than that of the velocity divergence term. Integration by parts of these terms is essential in order to improve the stability of the scheme. Based on this discretization format, an appropriate penalty parameter for pressure Poisson equation is utilized so as to provide the scheme with an inf‐sup stabilization. Moreover, the lid‐driven cavity flow is considered to verify that this numerical algorithm enhances the stability without additional stabilization term at small time step size and high‐Reynolds number for equal‐order polynomial approximations.

CFD modeling of liquid entrainment through vertical T-junction of fourth stage automatic depressurization system (ADS-4)

In this study, a three-dimensional (3D) transient computational fluid dynamics (CFD) model is presented to investigated liquid entrainment in gas–liquid flow through vertical T-junction of fourth stage automatic depressurization system (ADS-4) in AP1000. A specialized CFD model for the liquid entrainment phenomenon study, which is based on coupled Eulerian-Eulerian VOF (volume of fluid) formulation with suitable interfacial drag and standard κ-ε turbulence model for each phase was proposed. The entrainment rate was calculated and results were validated with ADETEL experimental data, which was built at XJTU-NuTheL. The good agreement between CFD calculations and experimental data demonstrated that the proposed CFD model could reasonably predict entrainment within the studied range. The effect of vertical to horizontal branch diameter ratio and gas mass flow rates on liquid entrainment were also studied. In addition, the liquid volume fraction distributions and velocity field were investigated to develop an understanding of the entrainment process. The relatively high demand for computational resources due to very small timestep size and small grid size to accommodate high flow velocities was found to be a challenge.