Boundary Treatment Method Research Articles

A coupled smoothed particle hydrodynamics–peridynamics model for two-dimensional hydroelastic water entry

In this paper, a numerical model coupling the smoothed particle hydrodynamics (SPH) with peridynamics (PD) is developed to solve the problem of the hydroelastic water entry. By combining the geometric nonlinear peridynamics with the weakly compressible SPH, the proposed model can efficiently simulate the nonlinear deformation of the structure and the deformation of a fluid free surface. The transmission of information between the fluid and solid phases is implemented by a dummy particle boundary treatment method. By simulating benchmark tests, the ability of the proposed model to solve nonlinear fluid–structure coupling problems is verified. In order to improve the computational efficiency, the graphics processing unit parallel acceleration scheme is applied to the SPH-PD model. Compared with the serial scheme, the speedup effect of the parallel scheme in large-scale computation is verified. As an application of the numerical model, the water entry of the flexible panel at a constant entry velocity is studied. The mechanism of hydroelastic effect is explained by analyzing the variations in the fluid pressure field, slamming force, and panel deformation during water entry. In addition, the influences of plate rigidity, impact velocity, and deadrise angle on the hydroelastic effect are investigated.

A compact open boundary treatment for a pure thermal plume by direct numerical simulation

This study introduces a novel open boundary treatment method for simulating thermal plumes in natural convection, utilizing a combination of the Perfectly Matched Layer (PML) and local one-dimensional inviscid (LODI) methods within a compressible flow solver for Direct Numerical Simulation (DNS). This innovative approach significantly reduces the size of the computational domain yet effectively captures the complex transition from laminar to turbulent flow. Validation of the proposed methodology includes comparisons with existing empirical formulations, experimental data, and well-resolved DNS results, incorporating both time-averaged and spectral analyses. Additionally, the buoyancy-induced laminar-turbulent transition is investigated to enhance understanding of the dynamics and behaviors of thermal plumes. The findings offer insights into optimizing computational efficiency without compromising the accuracy needed to analyze intricate fluid dynamics in thermal engineering applications.

Segment-based wall treatment model for heat transfer rate in smoothed particle hydrodynamics

We propose a new smoothed particle hydrodynamics (SPH) model that applies a segment-based wall boundary treatment method (SBWM) for heat transfer applications. We begin by focusing on natural convection simulations, where accurately modeling heat-transferring wall boundaries is crucial as they are the energy source driving the flow. A conventional SPH approach that handles such tasks is the boundary particle (BP) method, which constructs wall boundaries by placing multiple layers of particles on and behind the walls. Despite its capability of imposing accurate boundary conditions, the BP approach becomes a non-trivial task when the fluid domain involves complex boundaries. Moreover, computational costs may significantly increase because of the increased number of SPH particles necessary for modeling walls. Therefore, we utilize the recent development of SBWM to efficiently model energy-transferring wall boundaries. Specifically, SBWM is applied to the energy conservation equation for the wall heat transfer model, using the boundary truncation terms derived in this work. The SBWM-SPH method is verified in various numerical examples, comparing the results with BP-SPH and finite volume method as well as experimental data in the literature. Our study finally extends to investigating a heat exchanger with an optimized shape, demonstrating how SBWM-SPH effectively handles practical issues associated with the BP method while providing accurate heat transfer calculations for the wall.

A Parametric Kinetic Solver for Simulating Boundary-Dominated Turbulent Flow Phenomena

Boundary layer flow plays a very important role in shaping the entire flow feature near and behind obstacles inside fluids. Thus, boundary treatment methods are crucial for a physically consistent fluid simulation, especially when turbulence occurs at a high Reynolds number, in which accurately handling thin boundary layer becomes quite challenging. Traditional Navier-Stokes solvers usually construct multi-resolution body-fitted meshes to achieve high accuracy, often together with near-wall and sub-grid turbulence modeling. However, this could be time-consuming and computationally intensive even with GPU accelerations. An alternative and much faster approach is to switch to a kinetic solver, such as the lattice Boltzmann model, but boundary treatment has to be done in a cut-cell manner, sacrificing accuracy unless grid resolution is much increased. In this paper, we focus on simulating the boundary-dominated turbulent flow phenomena with an efficient kinetic solver. In order to significantly improve the cut-cell-based boundary treatment for higher accuracy without excessively increasing the simulation resolution, we propose a novel parametric boundary treatment model, including a semi-Lagrangian scheme at the wall for non-equilibrium distribution functions, together with a purely link-based near-wall analytical mesoscopic model by analogy with the macroscopic wall modeling approach, which is yet simple to compute. Such a new method is further extended to handle moving boundaries, showing increased accuracy. Comprehensive analyses are conducted, with a variety of simulation results that are both qualitatively and quantitatively validated with experiments and real life scenarios, and compared to existing methods, to indicate superiority of our method. We highlight that our method not only provides a more accurate way for boundary treatment, but also a valuable tool to control boundary layer behaviors. This has not been achieved and demonstrated before in computer graphics, which we believe will be very useful in practical engineering.

Characteristic boundary condition for multispeed lattice Boltzmann model in acoustic problems

In computational aero-acoustics, the lattice Boltzmann method (LBM) has demonstrated a number of advantages such as low dissipation and high scalability. To recover the thermal equation and energy transport in a physically consistent way, the fourth order moments of LBM distribution function must be accurately captured, which in turn demands a multispeed velocity set. Accurate and economic realization of various boundary conditions with multispeed velocities has been a challenge in LBM. Here we present a boundary condition for open boundaries with arbitrarily specifiable sound reflection properties by embedding the characteristic non/partially reflecting boundary condition of finite difference (FD) method into multispeed LBM. The local one-dimensional inviscid analysis for Euler equations is used to build boundary treatment method for FD, and distribution functions are reconstructed from FD solution at open outlets as inner-domain LBM's boundary condition. The numeric scheme is validated in a number of test cases, including one-dimensional (1D) and two-dimensional (2D) propagation of acoustic, entropy and vorticity waves, and sound generation in flow past a 2D cylinder. Results show our method can achieve desired reflecting coefficient and reduce non-physical reflecting waves to a satisfactory level.

Dry and wet boundary treatment and improvement of a TVD‐MacCormack scheme in shallow water flow

AbstractTo solve shallow water equation, this paper proposes a simple and easy‐to‐operate dry‐wet boundary treatment method based on the total variation diminishing (TVD)‐ MacCormack scheme. The method requires to judge dry and wet nodes before the calculation of prediction step and correction step, respectively. Then, the dry nodes are fictitiously celled and the topographic variables are reset. Moreover, previous researches show that when the same differential scheme was used, the left and right traveling waves showed over predicted computational fluxes during the downstream dry bed flow evolution, which led to distorted values and non‐real physical phenomena. To solve the problem, the difference scheme for prediction step and correction step is modified, and a new finite difference scheme improvement method is proposed. Finally, the numerical solutions are compared with the analytical solution results by five classical cases to verify the rationality of the proposed method in this paper.

Passive fluidic control on aero-optics of transonic flow over turrets with rough walls

In the transonic flow over a hemisphere-on-cylinder turret, strong aero-optical effects can be caused by local shock/boundary-layer interactions and separation shear layers at the turret's zenith. The effects of an annular rough wall on the passive control of fluid and aero-optics are investigated by experimental measurements and numerical simulations. The local shock/boundary-layer interaction and separated shear layer at the zenith of the turret are recorded using shadowing and Mach–Zehnder interferometer measurements. The aero-optics are measured using a Shack–Hartmann wavefront sensor. The experimental results show that the annular rough wall on the turret weakens the local shock wave, moves the flow separation point forward, and reduces the wavefront distortion at the zenith. The rough wall functions for the shear stress transport (SST) k-ω turbulence model proposed by B. Aupoix [“Roughness corrections for the k–ω shear stress transport model: Status and proposals,” J. Fluids Eng. 137, 021202 (2014)] and C.-H. Lee [“Rough boundary treatment method for the shear-stress transport k–ω model,” Eng. Appl. Comput. Fluid 12, 261–269 (2018)] are used to further study the control effect of different roughnesses. Numerical simulations based on both rough wall functions show good agreement with the experimental measurements. For various transonic flows, the steady wavefront distortions at the zenith with the rough wall at roughness ks=1 mm are 21%–50% smaller than those with smooth walls. The smaller the supersonic region, the more effective the rough wall is in reducing wavefront distortion.

Numerical analysis of 3-D inner flow field for ladder-shaped multiple propellant rocket motor

In order to study the characteristics of ladder-shaped multiple propellant rocket motor, it was conducted that the numerical simulation of the full 3-D inner flow field for ladder-shaped multiple propellant rocket motor working process using FLUENT calculation software. The secondary development was carried out through UDF interface programming, and the changing rules of the added mass in propellant burning surface and dynamic boundary with the burning rate were set, so that the dynamic process of propellant combustion is accurately simulated. The numerical simulation results demonstrated the spatial distribution information of parameters, such as pressure and flow rate at each moment in the inner flow field, and the pressure-time curves of three monitoring points in front, center and rear of the combustion chamber. Then, several key characteristic moments were selected for specific analysis, and the pressure-time curve of the monitoring point was compared with the experimental data to show that the calculated results were basically accurate. The pressure, flow rate and the pressure difference between the internal and external surfaces of the propellant in the rocket motor inner flow field were analyzed. At present, the dynamic boundary treatment method that the burning rate of the propellant varies regularly with the surface pressure has not been applied in the 3-D calculation model. The research results of this paper can provide reference for the numerical simulation of similar structure of solid rocket motors.

Research on smoke simulation with vortex shedding.

The Lagrangian vortex method has the advantage of producing highly detailed simulations of fluids such as turbulent smoke. However, this method has two problems: the construction of the velocity field from the vorticity field is inefficient, and handling the boundary condition is difficult. We present a pure Lagrangian vortex method, including a nested grid to accelerate the construction of the velocity field, and a novel boundary treatment method for the vorticity field. Based on a tree structure, the nested grid algorithm considerably improves the efficiency of the velocity computation while producing visual results that are comparable with the original flow. Based on the vortex-generating method, the least square method is used to compute the vorticity strength of the new vortex elements. Further, we consider the mutual influence between the generated vortex particles. We demonstrate our method’s benefits by using a vortex ring and various examples of interaction between the smoke and obstacles.

An improved smoothed particle hydrodynamics method for simulating crack propagation and coalescence in brittle fracture of rock materials

A new meshless numerical method, namely the improved smoothed particle hydrodynamics (ISPH) method, is proposed to simulate the fracture process of rock samples with pre-existing flaws under uniaxial compression. By improving the kernel function of SPH method, the brittle fracture characteristics of materials are presented. In ISPH, a prefabricated node segment (PNS) method is used to generate arbitrarily complex flaws, which avoids the division and distortion of traditional particles. In order to further improve the numerical accuracy, a new solid boundary treatment method called coupled dynamic buffer SBT algorithm is applied to ISPH, which is verified to be effective and robust via two benchmark samples. The effect of the length of a pre-existing flaw on the fracture mode and compressive strength of the rock sample is studied by using the elastic-brittle constitutive model. The numerical prediction results are in good agreement with the experimental observations. In addition, the effect of the unconnected rate on the crack propagation and coalescence mode of a rock sample with two pre-existing flaws under uniaxial compression is also studied. The numerical simulation results are in good agreement with the experimental results, which further verify the accuracy and superiority of the ISPH method.

An improved impermeable solid boundary scheme for Meshless Local Petrov–Galerkin method

Meshless methods have become an essential numerical tool for simulating a wide range of flow–structure interaction problems. However, the way by which the impermeable solid boundary condition is implemented can significantly affect the accuracy of the results and computational cost. This paper develops an improved boundary scheme through a weak formulation for the boundary particles based on Pressure Poisson’s Equation (PPE). In this scheme, the wall boundary particles simultaneously satisfy the PPE in the local integration domain by adopting the Meshless Local Petrov–Galerkin method with the Rankine source solution (MLPG_R) integration scheme (Ma, 2005b) and the Neumann boundary condition, i.e., normal pressure gradient condition, on the wall boundary which truncates the local integration domain. The new weak formulation vanishes the derivatives of the unknown pressure at wall particles and is discretized in the truncated support domain without extra artificial treatment. This improved boundary scheme is validated by analytical solutions, numerical benchmarks, and experimental data in the cases of patch tests, lid-driven cavity, flow over a cylinder and monochromic wave generation. Second-order convergent rate is achieved even for disordered particle distributions. The results show higher accuracy in pressure and velocity, especially near the boundary, compared to the existing boundary treatment methods that directly discretize the pressure Neumann boundary condition.

Boundary treatment of linear multistep methods for hyperbolic conservation laws

When using high-order schemes to solve hyperbolic conservation laws in bounded domains, it is necessary to properly treat boundary conditions so that the overall accuracy and stability are maintained. In [1, 2] a finite difference boundary treatment method is proposed for Runge-Kutta methods of hyperbolic conservation laws. The method combines an inverse Lax-Wendroff procedure and a WENO type extrapolation to achieve desired accuracy and stability. In this paper, we further develop the boundary treatment method for high-order linear multistep methods (LMMs) of hyperbolic conservation laws. We test the method through both 1D and 2D benchmark numerical examples for two third-order LMMs, one with a constant time step and the other with a variable time step. Numerical examples show expected high order accuracy and excellent stability. In addition, the approach in [3] may be adopted to deal with an exceptional case where eigenvalues of the flux Jacobian matrix change signs at the boundary. These results demonstrate that the combined boundary treatment method works very well for LMMs of hyperbolic conservation laws.

Fast and versatile fluid-solid coupling for turbulent flow simulation

The intricate motions and complex vortical structures generated by the interaction between fluids and solids are visually fascinating. However, reproducing such a two-way coupling between thin objects and turbulent fluids numerically is notoriously challenging and computationally costly: existing approaches such as cut-cell or immersed-boundary methods have difficulty achieving physical accuracy, or even visual plausibility, of simulations involving fast-evolving flows with immersed objects of arbitrary shapes. In this paper, we propose an efficient and versatile approach for simulating two-way fluid-solid coupling within the kinetic (lattice-Boltzmann) fluid simulation framework, valid for both laminar and highly turbulent flows, and for both thick and thin objects. We introduce a novel hybrid approach to fluid-solid coupling which systematically involves a mesoscopic double-sided bounce-back scheme followed by a cut-cell velocity correction for a more robust and plausible treatment of turbulent flows near moving (thin) solids, preventing flow penetration and reducing boundary artifacts significantly. Coupled with an efficient approximation to simplify geometric computations, the whole boundary treatment method preserves the inherent massively parallel computational nature of the kinetic method. Moreover, we propose simple GPU optimizations of the core LBM algorithm which achieve an even higher computational efficiency than the state-of-the-art kinetic fluid solvers in graphics. We demonstrate the accuracy and efficacy of our two-way coupling through various challenging simulations involving a variety of rigid body solids and fluids at both high and low Reynolds numbers. Finally, comparisons to existing methods on benchmark data and real experiments further highlight the superiority of our method.

SPH simulations of water entry problems using an improved boundary treatment

Water entry problem is common in ocean engineering, including complex fluid–structure interaction (FSI) phenomena at the interfaces. As a Lagrangian and mesh-free particle method, the smoothed particle hydrodynamics (SPH) method owns great advantages in dealing with FSI problems. However, solid boundary treatment has always been one of the main issues affecting the accuracy of the SPH method. Through combining the dummy particle boundary and the normal flux method, an improved solid boundary treatment technology is proposed in this paper. After fully validated by a series of 2D and 3D water entry cases, the boundary treatment method is applied to simulate the water entry of a lifeboat. The results indicate that the SPH model in this paper can provide a reliable numerical prediction for water entry problems with 3D complex geometries.

Smooth particle hydrodynamics simulation of dam-break impacting different obstacles

Aiming at the problems of smoothed particle hydrodynamics (SPH) method boundary imposing difficulties and particles easily penetrating solid walls, an improved complex boundary processing method is proposed. First, the improved SPH method is applied to numerically simulate the dam-break flow problem, and the numerical results are compared with the experimental results to verify the effectiveness of the method; then, the flux is added to the continuity equation to improve the pressure; finally, Numerical simulation is carried out to study the flow state of the dam-break water under different conditions of impacting on different obstacles, and the obstacles are considered as columnar, columnar with holes, wedge-shaped, semi-circular, 1/4 lower right circle obstacles and 1/4 upper right circle obstacles. The results show that: the stable pressure field can be obtained by adding flux; the improved boundary treatment method can effectively prevent particles from penetrating the solid wall, which is suitable for any complex boundary; SPH method has significant advantages in dealing with strongly nonlinear extreme deformation flow problems such as dam-break.

Boundary treatment of implicit-explicit Runge-Kutta method for hyperbolic systems with source terms

In this paper, we develop a high order finite difference boundary treatment method for the implicit-explicit (IMEX) Runge-Kutta (RK) schemes solving hyperbolic systems with possibly stiff source terms on a Cartesian mesh. The main challenge is how to obtain the solutions at ghost points resulting from the wide stencil of the interior high order scheme. We address this problem by combining the idea of using the RK schemes at the boundary and an inverse Lax-Wendroff (ILW) procedure in [29,30]. Specifically, we only apply the ILW procedure in the starting stage of the RK method. In the intermediate stages, we solve out the intermediate solutions as well as their first-order spatial derivatives at the boundary by using the RK schemes, which are then used to compute solutions at ghost points by Taylor expansions. Our method is different from the widely used approach for the explicit RK schemes by imposing boundary conditions at intermediate stages [14,29,30] which does not apply to IMEX schemes. In addition, the intermediate boundary conditions are available for explicit RK schemes only up to third order while our method applies to IMEX and explicit RK schemes of arbitrary order. The stability and the desired accuracy of our boundary treatment with IMEX RK method up to fourth order are demonstrated numerically through 1D examples and 2D reactive Euler equations.

Boundary treatment of high order Runge-Kutta methods for hyperbolic conservation laws

In [4], we developed a boundary treatment method for implicit-explicit (IMEX) Runge-Kutta (RK) methods for solving hyperbolic systems with source terms. Since IMEX RK methods include explicit ones as special cases, this boundary treatment method naturally applies to explicit methods as well. In this paper, we examine this boundary treatment method for the case of explicit RK schemes of arbitrary order applied to hyperbolic conservation laws. We show that the method not only preserves the accuracy of explicit RK schemes but also possesses good stability. This compares favorably to the inverse Lax-Wendroff method in [5,6] where analysis and numerical experiments have previously verified the presence of order reduction [5,6]. In addition, we demonstrate that our method performs well for strong-stability-preserving (SSP) RK schemes involving negative coefficients and downwind spatial discretizations. It is numerically shown that when boundary conditions are present and the proposed boundary treatment is used, that SSP RK schemes with negative coefficients still allow for larger time steps than schemes with all nonnegative coefficients. In this regard, our boundary treatment method is an effective supplement to SSP RK schemes with/without negative coefficients for initial-boundary value problems for hyperbolic conservation laws.

Estimating pressure fields from planar velocity data around immersed bodies; a finite element approach

A finite element approach for calculating pressure fields from planar velocity data is examined in this paper and compared with widely cited procedures. In the current approach, the order of velocity derivatives in the source function is reduced via a finite element procedure where the pressure Poisson equation is solved. This method benefits from the homogeneous spread of information in the Poisson solvers while preventing the magnification of velocity uncertainties. These combined features enable the method to limit the propagation of uncertainties into the computed pressure field. A cut-cell approach along fluid–solid boundaries is proposed, which enables information on moving, immersed boundaries to be passed into the pressure estimation procedure, thereby significantly reducing the pressure errors along the boundary. The finite element method is validated using three canonical laminar flows where the “ground-truth” pressure and velocity fields are obtained from analytical solutions and numerical simulations. The accuracy of the finite element approach is first investigated as a function of imposed noise in the velocity field data without any boundary treatments. The resulting pressure field is compared with existing pressure estimation procedures. The finite element method performs agreeably for all flow fields of interest, with approximately a 7% decrease in the pressure computation error for one of the flows, when compared to existing approaches. Next, the inclusion of boundary treatments is considered, where the implementation of the cut-cell approach is shown to reduce pressure estimation errors at solid–fluid boundaries when compared with existing boundary treatment methods. The findings are most pronounced for moving boundaries, where an order of magnitude decrease in the pressure estimation error along the moving surface is observed. This is because, when paired with the finite element approach, the cut-cell method incorporates the surface dynamics in the pressure solution, whereas existing solution approaches do not have this capability.

Numerical simulation of column charge explosive in rock masses with particle flow code

The detonation of a column charge and the damage process in a rock mass are simulated in this study by the particle flow code method. The expansion loading method and the dynamic boundary treatment method are established according to the discrete element mechanism. Mechanical parameters are calibrated on the basis of the Starfield superposition principle and the dynamic compensation method. Dynamic stiffness and microscopic parameters are used for numerical simulations. Numerical test results are consistent with lab experimental test results. The velocity attenuation law is compared with that of theoretical results, and the blasting crater action index is compared with that of field test results, to verify the rationality of the numerical explosion model of the column charge. The mechanism of the rock breaking process under column charge explosion is analyzed. Stress wave propagation, detonation wave propagation, and crack extension are investigated. The stress wave gradient law is obtained, and the effects of blasting rock breaking, crack extension, and explosion effect outside the blasting crater area are determined with different initiation modes. The research method can explain the blasting stress wave propagation law and describe the dynamical failure process in rock masses. This study can provide a reliable numerical analysis method for follow-up blasting research and offer practical guidance on engineering blasting.

Establishment of Numerical Wave Flume Based on the Second-Order Wave-Maker Theory

With growing computational power, the first-order wave-maker theory has become well established and is widely used for numerical wave flumes. However, existing numerical models based on the first-order wave-maker theory lose accuracy as nonlinear effects become prominent. Because spurious harmonic waves and primary waves have different propagation velocities, waves simulated by using the first-order wave-maker theory have an unstable wave profile. In this paper, a numerical wave flume with a piston-type wave-maker based on the second-order wave-maker theory has been established. Dynamic mesh technique was developed. The boundary treatment for irregular wave simulation was specially dealt with. Comparisons of the free-surface elevations using the first-order and second-order wave-maker theory prove that second-order wave-maker theory can generate stable wave profiles in both the spatial and time domains. Harmonic analysis and spectral analysis were used to prove the superiority of the second-order wave-maker theory from other two aspects. To simulate irregular waves, the numerical flume was improved to solve the problem of the water depth variation due to low-frequency motion of the wave board. In summary, the new numerical flume using the second-order wave-maker theory can guarantee the accuracy of waves by adding an extra motion of the wave board. The boundary treatment method can provide a reference for the improvement of nonlinear numerical flume.