Solid Wall Boundary Conditions Research Articles

Numerical simulation of melt flow and heat transfer in casting filling process based on SPH

There is a great influence of the melt filling process on the quality of castings. Smoothed particle hydrodynamics (SPH) that materials are approximated by free particles rather than by fixed grids is applied to accurately predict fluid flows involving complex free surfaces. In this paper we present a mathematical model of melt flow and heat transfer by using SPH method. A novel approach is used in the governing equations to ensure stable numerical schemes and homogeneous particle distributions. The SPH heat equation takes into account the thermal release during phase transition and is more suitable for alloy solidification. The solid wall boundary conditions are slightly modified to satisfy the filling simulations. Several case studies are carried out to predict significant details about the filling order and flow structures in the mold cavity. The velocity and temperature distributions during different stages of melt filling are also given. The results show that this proposed model allows us to understand the predisposition of defects such as gas porosity and weld lines in the castings. These predictions can be used as inputs for improving process parameters, venting and cooling systems design.

Grad's 13 moments-based gas kinetic flux solver with triangle unstructured meshes for simulating continuum and rarefied fluid flows

A numerical framework based on the gas kinetic flux solver (GKFS) with unstructured meshes of triangle cells has been proposed and assessed in this work. Like conventional GKFS, the present scheme only updates the cell-average macroscopic conservative variables in time, thereby saving computational costs. At each cell interface, the numerical normal flux is computed with the GKFS scheme proposed by Liu et al. [J. Comput. Phys. 415, 109548 (2020)], in which the distribution functions at the surrounding points are reconstructed with the Grad's 13 moments (G13). Additionally, a solid wall boundary condition treatment has been proposed in the present G13-GKFS framework, which allows one to model from purely specular to perfectly thermalized solid walls. The present framework has been validated with the steady cylindrical Couette flow, the lid-driven cavity flow, the unsteady Rayleigh flow, and the rarefied flow around the NACA0012 airfoil test-cases, where good agreements are found with references for a quite wide range of flow regimes, from continuum to transitional flow regimes.

Stall Inception Prediction of Transonic Compressor with Wire Mesh Casing Treatment

A wire mesh casing treatment is proposed and numerically tested on the transonic axial compressor (Rotor 37). A theoretical stall inception prediction model was developed to account for the wire mesh casing treatment, which was modeled as an impedance boundary condition. This model was validated on a low-speed compressor, replicating the experimental trend with reliable precision. The impedance boundary condition effectively attenuates disturbances and confines the distribution range of perturbations compared to the solid wall boundary condition. Furthermore, the installation of the wire mesh casing treatment induced a downstream shift in the passage shock wave and enhanced the flow capacity in the blade tip region. The effects of the casing treatment on different installation locations were investigated, and the results indicated that the maximum stall margin improvement occurred when the casing treatment was positioned at the midchord location. The integrated design of the casing treatment and blade profile, along with the established prediction model, is also discussed. This model demonstrates reliable precision when compared to the experimental results and addresses the need for rapid design iteration during the initial design phase.

Aerodynamics simulations of three-dimensional inviscid flow using curvilinear discontinuous Galerkin method on unstructured meshes

Over the last decades, the discontinuous Galerkin (DG) method has demonstrated its excellence in accurate, higher-order numerical simulations for a wide range of applications in aerodynamics simulations. However, the development of practical, computationally accurate flow solvers for industrial applications is still in the focus of active research, and applicable boundary conditions and fluxes are also very important parts. Based on curvilinear DG method, we have developed a flow solver that can be used for solving the three-dimensional subsonic, transonic and hypersonic inviscid flows on unstructured meshes. The development covers the geometrical transformation from the real curved element to the rectilinear reference element with the hierarchical basis functions and their gradient operation in reference coordinates up to full third order. The implementation of solid wall boundary conditions is derived by the contravariant velocities, and an enhanced algorithms of Harten-Lax-van Leer with contact (HLLC) flux based on curved element is suggested. These new techniques do not require a complex geometric boundary information and are easy to implement. The simulation of subsonic, transonic and hypersonic flows shows that the linear treatment can limit the accuracy at high order and demonstrates how the boundary treatment involving curved element overcomes this restriction. In addition, such a flow solver is stable on a reasonably coarse meshes and finer ones, and has good robustness for three-dimensional flows with various geometries and velocities. For engineering practice, a reasonable accuracy can be obtained at reasonably coarse unstructured meshes.

Application of Gas-Kinetic Scheme for Continuum and Near-Continuum Flow on Unstructured Mesh

A gas-kinetic scheme (GKS) with kinetic boundary condition based on unstructured mesh is present here. In the GKS method, the solid wall boundary conditions can be constructed by virtue of the gas distribution function, which is similar to the diffuse-scattering rule used in the other kinetic schemes. The kinetic boundary condition has a concise form and easy to implement. The use of unstructured mesh expands the adaptability of GKS to simulate the flows with complex geometry. The kinetic boundary condition can recover to the non-slip boundary condition in the continuum regime. In the slip regime, the slip velocity can be accurately predicted by kinetic boundary condition, which turns into the slip boundary condition. The use of kinetic boundary condition improves the calculation results of GKS in near-continuum flow. A series of test cases, from incompressible to compressible flow with a wide range of Knudsen number, are investigated to demonstrate the performance of kinetic boundary condition in near-continuum flow, which can provide a reference for the construction and optimisation for GKS-based multi-scale hybrid algorithms.

Impacts of solid wall boundary conditions in the lattice Boltzmann method on turbulent outdoor flow: A case study of a single 1:1:2 building model

Solid wall boundary condition is an important topic in outdoor wind environment simulation represented by single building based on lattice Boltzmann method (LBM), but there is a lack of quantitative comparative analysis of simulation results under these boundary conditions. This paper conducted LBM-based large-eddy simulation (LBM-LES) of turbulent flow around a 1:1:2 single building, applying both half-way bounce-back and interpolation-based bounce-back (the Bouzidi–Firdaouss–Lallemand scheme, BFL) to solid wall boundaries. The simulation accuracy of different solid wall boundary conditions was examed with consideration of grid resolutions and relative positions between grid points and solid boundaries. Data obtained via experiments and the finite volume method-based large-eddy simulation (FVM-LES) technique were used to examine the flow structure and distribution of the time-averaged velocity and turbulent kinetic energy (TKE). The computational costs were also discussed. Results demonstrated that when the all the boundaries fell on the grid and orthogonal to the grid, the half-way bounce-back scheme attained a high accuracy. On the contrary, the BFL scheme achieved a higher accuracy if the solid wall boundaries did not fall on the grid. Under both boundary schemes, the grid resolutions of b/32 and b/16 (b: building width) could satisfy the acceptable range, but the b/32 grid resolution facilitated a better performance near the wall. The BFL scheme was more sensitive to the grid resolution and the accuracy improved faster with increasing grid resolution than half-way. As for the calculation costs, the calculation time of BFL is more than twice that of half-way bounce-back. However, BFL is still recommended if the wall boundaries are curved and complex in the actual built environment.

Comparative study of the single and double specularity coefficients on two-fluid modeling of a pseudo-2D gas–solid fluidized bed

The specularity coefficient is an unmeasurable parameter in the most popular wall boundary model during the two-fluid modeling of dense gas–solid flows. Using multiphaseEulerFoam solver, the influence of different specularity coefficient setting strategies on the gas–solid flow inside a pseudo-2D fluidized bed has been explored. It is found that the single specularity coefficient plays a regulatory role in the quantitative prediction. Increasing the specularity coefficient would cause a fluidization transition from freely bubbling to slugging, and the bed characteristics such as pressure drop and bed expansion present monotonic nonlinear changes. The double specularity coefficients approach is shown to significantly improve the predictive accuracy through verifying with the measured particle velocities, bubble diameter and rise velocity. In addition, the lognormal bubble size distribution and Gaussian bubble rise velocity distribution are observed. The specularity coefficient for walls in thickness direction is crucial and its different effects are unignorable. Overall, the present study provides a practical strategy of double specularity coefficients for the solid wall boundary conditions during two-fluid modeling.

Wall-layer boundary condition method for laminar and turbulent flows in weakly-compressible SPH

A new method of applying the solid wall boundary condition in a weakly compressible Smooth Particle Hydrodynamics (SPH) method is proposed and tested in a few basic benchmark flows. It makes use of approximate equations of motion and the wall similarity for the pressure and the velocity of the flow near solid walls avoiding the use of the particle interpolation formula which otherwise require special treatment for the truncated smoothing region near the boundary. It is intended to be used in the calculation and the simulation of turbulent flows and test calculations have been done with the Reynolds Averaged Navier Stokes (RANS) equations and spatially-filtered equations of motion and continuity equation with appropriate treatment of turbulence effects. The test calculations of fully developed open channel flow without resolving the viscous sublayer and the transient dam break flows without fully resolving the turbulent fluctuations show that the behaviour seen in the corresponding experimental results is reproduced. The present formulation appears to be an alternative boundary condition formulation to be used in the SPH calculation of practical engineering and environmental flows.

On a New Spatial Discretization for a Regularized 3D Compressible Isothermal Navier–Stokes–Cahn–Hilliard System of Equations with Boundary Conditions

We construct a new spatial finite-difference discretization for a regularized 3D Navier–Stokes–Cahn–Hilliard system of equations. The system can be attributed to phase field type models and describes flows of a viscous compressible isothermal two-component two-phase fluid with surface effects; the potential body force is also taken into account. In the discretization, the main sought functions are defined on one and the same mesh, and an original approximation of the solid wall boundary conditions (homogeneous with the discretization of equations) is suggested. The discretization has an important property of the total energy dissipativity allowing one to eliminate completely the so-called spurious currents. The discrete total mass and component mass conservation laws hold as well, and the discretization is also well-balanced for the equilibrium solutions. To ensure that the concentration C remains within the physically meaningful interval (0, 1), the non-convex part of the Helmholtz free energy is taken in a special logarithmic form (the Flory–Huggins potential). The speed of sound can depend on C that leads to different equilibrium mass densities of the “pure” phases. The results of numerical 3D simulations are also presented including those with a gravitational-type force. The positive role of the relaxation parameter is discussed too.

Nonlocal operator method for the Cahn-Hilliard phase field model

In this paper we propose a Nonlocal Operator Method (NOM) for the solution of the Cahn-Hilliard (CH) equation exploiting the higher order continuity of the NOM. The method is derived based on the method of weighted residuals and implemented in 2D and 3D. Periodic boundary conditions and solid-wall boundary conditions are considered. For these boundary conditions, the highest order in the NOM scheme is 2 and 3, respectively. The proposed NOM makes use of variable support domains allowing for adaptive refinement. The generalized α-method is employed for time integration and the Newton-Raphson method to iterate nonlinearity. The performance of the proposed method is demonstrated for several two and three dimensional benchmark problems. We also implemented a CH equation with 6th order partial differential derivative and studied the influence of higher order coefficients on the pattern evolution of the phase field.

Improvements in MLPG formulation for 3D wave interaction with fixed structures

This paper presents new developments in meshless local Petrov–Galerkin with Rankine source (MLPG_R) particle based method for studying interaction of waves with fixed structures in a numerical wave-tank. A new 3D formulation of the Lagrangian flow problem for incompressible fluid with optimised solution strategy is presented. The pressure Poisson equation is solved in local weak-form with integration done semi-analytically using a new symmetric expression. The wave-generation is done using one-way coupling with a 2D fully-nonlinear potential theory based finite-element model. Further a simple identification method for free-surface particles is proposed, which is shown to work well in vicinity of the structure. The solid-wall boundary condition is treated using ghost and mirror particles for accurate calculation of gradients. The waterline on domain boundary faces is treated using a tangentially moving side-wall approach which makes this particle based scheme capable of capturing small amplitude waves and focusing waves. The paper briefly presents experimental setup used for studying the interaction of a fixed emergent cylinder with uni-directional regular and focusing waves in 3D. The numerical model is validated against results from this experiment. An analysis is conducted on parameters related to local integration domain, wave-making coupling algorithm, particle distribution and time-step. This work highlights the use of hybrid approach for efficient and accurate simulation of waves-structure interaction.

Numerical modeling for compressible two-phase flows and application to near-field underwater explosions

In this study, an accurate shock- and interface-capturing method using curvilinear body-fitted structured grids is introduced to simulate compressible multiphase flows with shockwaves. A five-equation model—proficient in capturing unsteady shocks in compressible multiphase flows without nonphysical spurious oscillations—was enhanced by extending it to a multidimensional general curvilinear coordinate system. A new eigensystem was constructed for a monotonic upstream-centered scheme for conservation laws/Godunov-type finite volume scheme that can capture strong shocks in compressible multiphase flows. The developed method was validated using a typical test involving a free-field underwater explosion that produces a strong shockwave in addition to bubble interfaces. Numerical results were compared and found to agree well with the empirical equation and previously published results. Further, the interaction of a shockwave with a wedge was computed to evaluate the shockwave propagation characteristics and appropriate treatment of solid wall boundary conditions. The numerical results show good agreement with the experimental data. Finally, the impact and propagation characteristics of shockwaves in two more complex cases, involving one and two explosions near a rigid cylinder, were numerically analyzed, and the results indicate high impacts of primary blast waves on the structure.

An inverse Lax-Wendroff procedure for hyperbolic conservation laws with changing wind direction on the boundary

In this paper, we reconsider the inverse Lax-Wendroff (ILW) procedure, which is a numerical boundary treatment for solving hyperbolic conservation laws, and propose a new approach to evaluate the values on the ghost points. The ILW procedure was firstly proposed to deal with the “cut cell” problems, when the physical boundary intersects with the Cartesian mesh in an arbitrary fashion. The key idea of the ILW procedure is repeatedly utilizing the partial differential equations (PDEs) and inflow boundary conditions to obtain the normal derivatives of each order on the boundary. A simplified ILW procedure was proposed in [28] and used the ILW procedure for the evaluation of the first order normal derivatives only. The main difference between the simplified ILW procedure and the proposed ILW procedure here is that we define the unknown u and the flux f(u) on the ghost points separately. One advantage of this treatment is that it allows the eigenvalues of the Jacobian f′(u) to be close to zero on the boundary, which may appear in many physical problems. We also propose a new weighted essentially non-oscillatory (WENO) type extrapolation at the outflow boundaries, whose idea comes from the multi-resolution WENO schemes in [32]. The WENO type extrapolation maintains high order accuracy if the solution is smooth near the boundary and it becomes a low order extrapolation automatically if a shock is close to the boundary. This WENO type extrapolation preserves the property of self-similarity, thus it is more preferable in computing the hyperbolic conservation laws. We provide extensive numerical examples to demonstrate that our method is stable, high order accurate and has good performance for various problems with different kinds of boundary conditions including the solid wall boundary condition, when the physical boundary is not aligned with the grids.

CFD simulation and experimental validation of nanoparticles fluidization in a conical spouted bed

In the present study, hydrodynamic parameters of micro/nanoparticles in a conical spouted bed were investigated using experimental and computational fluid dynamics (CFD) techniques. The Eulerian–Eulerian approach, in conjunction with the kinetic theory of granular flow (KTGF) has been used, and the effects of solid wall boundary condition parameters such specularity coefficient (φ), and particle-wall restitution coefficient (ew), on the hydrodynamics of the bed were studied. The effect of drag force on the fluidization behavior of nanoparticles was investigated using Gidaspow and RUC drag models, and the results showed the superiority of the RUC model. Experiments were carried out for measuring the pressure drop in a lab-scale conical spouted bed. The CFD predictions were in good agreement with the experimental data in terms of pressure drop and particle distribution.

An accurate triangular spectral element method-based numerical simulation for acoustic problems in complex geometries

An accurate triangular spectral element method (TSEM) is developed to simulate acoustic problems in complex computational domains. With Fekete points and Koornwinder-Dubiner polynomials introduced, triangular elements are used in the present method to substitute quadrilateral elements in traditional spectral element method (SEM). The efficiency of discretizing complex geometry is enhanced while high accuracy of SEM is remained. The weak form of the second-order governing equations derived from the linearized Euler equations (LEEs) are solved, and perfectly matched layer (PML) boundary condition is implemented. Three benchmark problems with analytical solutions are employed to testify the exponential convergence rate, convenient implementation of solid wall boundary condition and capable discretization in complex geometries of the present method respectively. An application on Helmholtz resonator (HR) is presented as well to demonstrate the possibility of using the present method in practical engineering. The numerical resonance frequency of HR reaches an excellent agreement with the theoretical result.

Segregation behavior of a ternary mixture of titanium dioxide particles in a pseudo two-dimensional fluidized bed

The segregation behavior of a mixture of silica-coated titanium dioxide (TiO2) particles of three different sizes in a pseudo two-dimensional fluidized bed was studied experimentally by the freeze–sieving method and numerically by the multi-fluid model (MFM). Three-dimensional computational fluid dynamics (CFD) simulations were carried out to evaluate the effects of the solid wall boundary conditions on particle segregation in terms of specularity and particle–wall restitution coefficients. The quantitative indexes of segregation tendency and segregation degree were used to determine the axial segregation of the mixture in triangular coordinates. The simulation results revealed that the axial segregation increased with the specularity coefficient, whereas the particle–wall restitution coefficient had a minor effect on axial segregation. Comparison of the simulation results with experimental data showed that the appropriate value of the specularity coefficient used in the CFD model depended on superficial gas velocity. The study of the effects of superficial gas velocity on segregation behavior demonstrated that the greatest segregation was obtained at minimum fluidization velocity and the segregation decreased as the gas velocity gradually increased.

An efficient, open source, iterative ISPH scheme

In this paper a simple, robust, and general purpose approach to implement the Incompressible Smoothed Particle Hydrodynamics (ISPH) method is proposed. This approach is well suited for implementation on CPUs and GPUs. The method is matrix-free and uses an iterative formulation to setup and solve the pressure-Poisson equation. A novel approach is used to ensure homogeneous particle distributions and improved boundary conditions. This formulation enables the use of solid wall boundary conditions from the weakly-compressible SPH schemes. The method is fast and runs on GPUs without the need for complex integration with sparse linear solvers. We show that this approach is sufficiently accurate and yet efficient compared to other approaches. Several benchmark problems that illustrate the robustness, performance, and wide range of applicability of the new scheme are demonstrated. An open source implementation is provided and the manuscript is fully reproducible. Program summaryProgram Title: SISPHProgram Files doi:http://dx.doi.org/10.17632/kw5f75wcw7.1Licensing provisions: BSD 3-ClauseProgramming language: PythonExternal routines/libraries: PySPH (https://github.com/pypr/pysph), scipy (https://pypi.org/project/scipy/), matplotlib (https://pypi.org/project/matplotlib/), compyle (https://pypi.org/project/compyle/), automan (https://pypi.org/project/automan/).Nature of problem: Incompressible SPH solvers solve a sparse system of equations to obtain a solution to the pressure-Poisson equation. This is often implemented by solving a sparse matrix, and can be complex to implement on GPUs. SPH accuracy deteriorates when particles become disordered. Ensuring homogeneous particle distributions requires particle shifting algorithms which requires tuning of parameters. We propose the use of matrix-free algorithm and an accurate and general purpose way to ensure particle homogeneity.Solution method: We use an explicit iterative approach to solve the pressure-Poisson equations which makes it easy to implement in parallel on GPUs. This also makes it easy to implement the boundary conditions. We propose an accurate way to ensure homogeneous particle distribution by using a background pressure.Additional comments: The source code for this repository can be found at https://gitlab.com/pypr/sisph.

Kinetic relaxation to entropy based coupling conditions for isentropic flow on networks

We consider networks for isentropic gas and prove existence of weak solutions for a large class of coupling conditions. First, we construct approximate solutions by a vector-valued BGK model with a kinetic coupling function. Introducing so-called kinetic invariant domains and using the method of compensated compactness justifies the relaxation towards the isentropic gas equations. We will prove that certain entropy flux inequalities for the kinetic coupling function remain true for the traces of the macroscopic solution. These inequalities define the macroscopic coupling condition. Our techniques are also applicable to networks with arbitrary many junctions which may possibly contain circles. We give several examples for coupling functions and prove corresponding entropy flux inequalities. We prove also new existence results for solid wall boundary conditions and pipelines with discontinuous cross-sectional area.

Solid wall and open boundary conditions in hybrid recursive regularized lattice Boltzmann method for compressible flows

Complex geometries and open boundaries have been intensively studied in the nearly incompressible lattice Boltzmann method (LBM) framework. Therefore, only few boundary conditions for the high speed fully compressible LBM have been proposed. This paper deals with the definition of efficient boundary conditions for the compressible LBM methods, with the emphasis put on the newly proposed hybrid recursive regularized D3Q19 LBM (HRR-LBM) with applications to compressible aerodynamics. The straightforward simple extrapolation-based far-field boundary conditions, the characteristic boundary conditions, and the absorbing sponge layer approach are extended and estimated in the HRR-LBM for the choice of open boundaries. Moreover, a cut-cell type approach to handle the immersed solid is proposed to model both slip and no-slip wall boundary conditions with either isothermal or adiabatic behavior. The proposed implementations are assessed considering the simulation of (i) isentropic vortex convection with subsonic to supersonic inflow and outflow conditions, (ii) two-dimensional (2D) compressible mixing layer, (iii) steady inviscid transonic flow over a National Advisory Committee for Aeronautics (NACA) 0012 airfoil, (iv) unsteady viscous transonic flow over a NACA 0012 airfoil, and (v) three-dimensional (3D) transonic flows over a German Aerospace Center (DLR) F6 full aircraft configuration.

Modeling of droplet impact on a heated solid surface with a diffuse interface model

A Diffuse Interface Model (DIM) is employed to model droplet impact on a heated solid surface. The DIM uses an especially constructed solid wall boundary condition which enables simulations with different wetting conditions of the solid surface. The model is also extended to include the effects of surface roughness on the behavior of the contact line dynamics. Multiple simulations are carried out to demonstrate the capabilities of the presented model. The simulation results demonstrate the influence of the wetting properties of the solid, with a higher cooling rate for hydrophilic than for hydrophobic wetting conditions. Surface roughness of the solid surface increases the cooling rate of the solid by enhancing the heat transfer between solid and fluid.