Solid Wall Boundary Conditions Research Articles

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.

A weakly compressible SPH method for violent multi-phase flows with high density ratio

The weakly compressible SPH (WCSPH) method is known suffering from low computational efficiency, or unnatural voids and unrealistic phase separation when it is applied to simulate highly violent multi-phase flows with high density ratio, such as that between water and air. In this paper, to remedy these issues, we propose a multi-phase WCSPH method based on a low-dissipation Riemann solver and the transport-velocity formulation. The two-phase Riemann problem is first constructed to handle the pairwise interaction between fluid particles, then modified for the fluid-wall interaction to impose the solid wall boundary condition. Since the method uses the same artificial speed of sound for both heavy and light phases, the computational efficiency increases greatly. Furthermore, due to the transport-velocity formulation employed for the light phase and application of the two-phase Riemann problem, the unnatural voids and unrealistic phase separation are effectively eliminated. The method is validated with several 2- and 3D cases involving violent water-air flows, and demonstrates good robustness, improved or comparable accuracy, respectively, comparing to previous methods with the same choice of sound speed or those with much less computational efficiency.

Improved treatment of wall boundary conditions for a particle method with consistent spatial discretization

In recent years, consistent spatial discretization schemes for meshfree particle methods to numerically simulate incompressible flow have been studied by many researchers. This study is focused on the treatment of solid wall boundary conditions for one of those schemes, namely the least squares MPS (LSMPS) scheme, and proposes a new technique to deal with no-slip and free-slip wall boundary conditions. With the proposed treatment, wall geometries are expressed by surface meshes, i.e. polygons in 3D and line segments in 2D. Thus, complicated geometries can be handled easily. Based on a Taylor series expansion, wall boundary conditions are incorporated into the differential operators acting on fluid particles located in the vicinity of a wall, through a least squares approach. As a consequence, Neumann boundary conditions can be treated quite efficiently. To verify consistency of the proposed discretization scheme, a convergence study was carried out. As numerical examples, Couette flow, plane Poiseuille flow, gravity-driven flow in a 3D square duct, a rigid rotation problem, Taylor–Green vortices and lid-driven cavity flow have been calculated using the proposed boundary treatment, with both no-slip and free-slip conditions applied. As a result, the present method agreed well with the reference solutions, which verified its computational accuracy.

Conservative and entropy stable solid wall boundary conditions for the compressible Navier–Stokes equations: Adiabatic wall and heat entropy transfer

We present a novel technique for the imposition of non-linear entropy conservative and entropy stable solid wall boundary conditions for the compressible Navier–Stokes equations in the presence of an adiabatic wall, or a wall with a prescribed heat entropy flow. The procedure relies on the formalism and mimetic properties of diagonal-norm, summation-by-parts and simultaneous-approximation-term operators, and is a generalization of previous works on discontinuous interface coupling [1] and solid wall boundary conditions [2]. Using the method of lines, a semi-discrete entropy estimate for the entire domain is obtained when the proposed numerical imposition of boundary conditions are coupled with an entropy-conservative or entropy-stable discrete interior operator. The resulting estimate mimics the global entropy estimate obtained at the continuous level. The boundary data at the wall are weakly imposed using a penalty flux approach and a simultaneous-approximation-term technique for both the conservative variables and the gradient of the entropy variables. Discontinuous spectral collocation operators (mass lumped nodal discontinuous Galerkin operators), on high-order unstructured grids, are used for the purpose of demonstrating the robustness and efficacy of the new procedure for weakly enforcing boundary conditions. Numerical simulations confirm the non-linear stability of the proposed technique, with applications to three-dimensional subsonic and supersonic flows. The procedure described is compatible with any diagonal-norm summation-by-parts spatial operator, including finite element, finite difference, finite volume, discontinuous Galerkin, and flux reconstruction schemes.

A simple analytical model of complex wall in multibody dissipative particle dynamics

In the context of multibody dissipative particle dynamics (MDPD), a closed-form mathematical expression is developed to analytically model a complex wall. MDPD is a modified version of dissipative particle dynamics (DPD), a particle-based mesh free method. There have been several attempts to analytically model the influence of solid walls and non-periodic boundary conditions in the DPD approach. However, there is a limited number of studies for these boundary conditions associated with MDPD that capture static and dynamic fluid-structure interactions through direct modeling of fluid-solid particle interactions. This work, for the first time, employs an analytical model (integral approach) for the solid wall boundary condition in MDPD that brings substantial gain in computational efficiency and thus expands the scope of its applicability to curved or complex walls. Furthermore, a modified model of conservative force is used in the current investigation. The model is first normalized to address the discrepancies in wetting that exist in the present literature and is then validated through several benchmark studies and test cases, such as a Wenzel model. Moreover, comparisons between both the fully numerical and the semi-analytical (integral force model) approaches are drawn. Time efficiency, accuracy, density fluctuation in vicinity of solid wall, and limitations of the proposed model are thoroughly discussed.

High‐order ghost‐point immersed boundary method for viscous compressible flows based on summation‐by‐parts operators

SummaryA high‐order immersed boundary method is devised for the compressible Navier‐Stokes equations by employing high‐order summation‐by‐parts difference operators. The immersed boundaries are treated as sharp interfaces by enforcing the solid wall boundary conditions via flow variables at ghost points. Two different interpolation schemes are tested to compute values at the ghost points and a hybrid treatment is used. The first method provides the bilinearly interpolated flow variables at the image points of the corresponding ghost points and the second method applies the boundary condition at the immersed boundary by using the weighted least squares method with high‐order polynomials. The approach is verified and validated for compressible flow past a circular cylinder at moderate Reynolds numbers. The tonal sound generated by vortex shedding from a circular cylinder is also investigated. In order to demonstrate the capability of the solver to handle complex geometries in practical cases, flow in a cross‐section of a human upper airway is simulated.

On the use of a Prandtl-Glauert-Lorentz transformation for acoustic scattering by rigid bodies with a uniform flow

It is well-known that the convective wave equation with a uniform mean flow can be transformed into a standard wave equation without flow by a Prandtl-Glauert-Lorentz type transformation. This paper examines the boundary condition to be used in the transformed coordinates for acoustic scattering by rigid bodies. Recently, based on the energy conservation equation for acoustic waves propagating in a uniform flow, a new solid wall boundary condition, dubbed the Zero Energy Flux (ZEF) solid surface boundary condition, has been proposed. Instead of the commonly used solid surface boundary condition that the normal acoustic velocity be zero, the ZEF condition requires that the acoustic energy flux be zero. In this paper, we point out that when formulated in the acoustic velocity potential and under the ZEF boundary condition, the normal derivative of the velocity potential remains to be zero when computed in the transformed coordinates. As such, numerical methods developed for solving the standard wave equation can be directly applied to the convective wave equation through a use of the Prandtl-Glauert-Lorentz transformation. Utilization of the transformation is particularly beneficial for numerical methods based on the boundary integral formulation because of the simplified kernel function in the transformed coordinates, for both the time domain and the frequency domain approaches. Useful relations for applying the transformation are derived and numerical examples that demonstrate the effectiveness of the transformation and ZEF boundary condition are presented.

Global smooth axisymmetric subsonic flows with nonzero swirl in an infinitely long axisymmetric nozzle

In this paper, we consider steady full Euler system in three-dimensional axisymmetric nozzles. The existence and uniqueness of global smooth subsonic compressible flows with nonzero swirl component are established. With the upstream shear flow conditions and the solid wall boundary conditions, we obtain the asymptotic behavior and the critical mass flux. For general infinitely long axisymmetric nozzles, it is worth noting that this is a first result containing non-trivial swirl velocity.

Immersed boundary method for viscous compressible flows around moving bodies

A ghost-point immersed boundary method for the compressible Navier–Stokes equations with moving boundaries on fixed Cartesian grids is devised by employing high order summation-by-parts (SBP) difference operators. The immersed boundaries are treated as sharp interfaces by enforcing the solid wall boundary conditions via flow variables at ghost points using bilinearly interpolated flow variables at mirror points. Simulations for compressible viscous flows induced by a transversely oscillating cylinder in a free-stream and a harmonic in-line oscillating cylinder in an initially quiescent fluid are presented and compared with experiments and incompressible fluid flow simulations obtained with body-conforming grid methods.

Numerical investigation of the saturation process in an incompressible cavity flow

A numerical study of the saturation process inside a rectangular open cavity is presented. Previous experiments and linear stability analysis of the problem completely described the flow in its onset, as well as in a saturated regime, characterized by three-dimensional centrifugal modes. The morphology of the modes found in the experiments matched the ones predicted by linear analysis, but with a shift in frequencies for the oscillating modes. A three-dimensional incompressible direct numerical simulation (DNS) is employed for a detailed investigation of the saturation process inside a cavity with dimensions similar to the one used in the experiments, to further explain the behaviour of these modes. In this work, periodic boundary conditions are first imposed to better understand the effect of the saturation process far from the walls. Then, the effects of spanwise solid wall boundary conditions are investigated with a DNS reproducing the full dynamics of the experiments. The main flow structures are identified using the dynamic mode decomposition technique and compared with previous experimental and linear stability analysis results. The main reason for the aforementioned shift in frequency is explained in this paper, as it is a function of the velocity of the main recirculating vortex.

On a time domain boundary integral equation formulation for acoustic scattering by rigid bodies in uniform mean flow.

It has been well-known that under the assumption of a uniform mean flow, the acoustic wave propagation equation can be formulated as a boundary integral equation. However, the constant mean flow assumption, while convenient for formulating the integral equation, does not satisfy the solid wall boundary condition wherever the body surface is not aligned with the assumed uniform flow. A customary boundary condition for rigid surfaces is that the normal acoustic velocity be zero. In this paper, a careful study of the acoustic energy conservation equation is presented that shows such a boundary condition would in fact lead to source or sink points on solid surfaces. An alternative solid wall boundary condition, termed zero energy flux boundary condition, is proposed that conserves the acoustic energy and a time domain boundary integral equation is derived. Furthermore, stabilization of the integral equation by Burton-Miller type reformulation is presented. The stability is studied theoretically as well as numerically by an eigenvalue analysis. Numerical solutions are also presented that demonstrate the stability of the current formulation.

A fast linear complementarity problem solver for fluid animation using high level algebra interfaces for GPU libraries

We address the task of computing solutions for a separating solid wall boundary condition model. We present a parallel, easy to implement, fluid linear complementarity problem solver. All that is needed is the implementation of linear operators, using an existing high-level sparse algebra GPU library. No low-level GPU programming is necessary. This means we can rely on the efficiency of a tried-and-tested library, requiring minimal debugging compared to writing more low level GPU kernels. The solver exploits matrix-vector products as computational building blocks. We block the matrix-vector products in a way that allows us to evaluate the products, without having to assemble the full systems. Our work shows speedup factors ranging up to two orders of magnitudes for larger grid resolutions.

Nano-particle drag prediction at low Reynolds number using a direct Boltzmann–BGK solution approach

This paper outlines a novel approach for solution of the Boltzmann–BGK equation describing molecular gas dynamics applied to the challenging problem of drag prediction of a 2D circular nano-particle at transitional Knudsen number (0.0214) and low Reynolds number (0.25–2.0). The numerical scheme utilises a discontinuous-Galerkin finite element discretisation for the physical space representing the problem particle geometry and a high order discretisation for molecular velocity space describing the molecular distribution function. The paper shows that this method produces drag predictions that are aligned well with the range of drag predictions for this problem generated from the alternative numerical approaches of molecular dynamics codes and a modified continuum scheme. It also demonstrates the sensitivity of flow-field solutions and therefore drag predictions to the wall absorption parameter used to construct the solid wall boundary condition used in the solver algorithm. The results from this work has applications in fields ranging from diagnostics and therapeutics in medicine to the fields of semiconductors and xerographics.

CFD modeling of heat transfer and hydrodynamics in a draft tube conical spouted bed reactor under pyrolysis conditions: Impact of wall boundary condition

In the present study, the hydrodynamics and heat transfer parameters of a conical spouted bed with a draft tube were investigated under the pyrolysis conditions using computational fluid dynamics (CFD) technique. The Eulerian-Eulerian approach in conjunction with the kinetic theory of granular flow (KTGF) was applied, and some important parameters, such as pressure drop, solid volume fraction, particle velocity, and heat transfer coefficients between the wall and the bed were evaluated. The impact some solid-wall boundary conditions, such as specularity coefficient (φ), and particle-wall restitution coefficient (ew), have on the heat transfer and hydrodynamics of the bed were studied. The CFD predictions were in close quantitative agreement with measurements in terms of heat transfer coefficients and pressure drop. It was found that both specularity and particle-wall restitution coefficient have a significant influence on the pressure drop and heat transfer coefficients, with the effect on pressure drop being higher than on heat transfer coefficients.

Numerical studies on droplet impact to wettable solid boundary based on SPH method

Droplet impact on solid surface is a common natural phenomenon and industrial process which is also a complex polyphase conditions coupling process with gas, liquid and solid. And during the droplet’s deformation and movement, it is often affected by the wettability of solid wall. In this paper, a lagrange SPH numerical method, with the attraction of the van der Waals state equation added to simulate the surface tension of the droplet, is applicated to simulate the process of single droplet impact on wettable surface and investigate the influence mechanism of surface wettability to the droplet’s deformation and movement. Several improvements for traditional SPH methd are presented such as, a Lagrange wettable solid wall boundary condition of which solid wall pariticles’ hydrophilia and capillary action are unified supposed to be a adsorption force to the liquid particles of support domains. The force is deemed relevant to fluid pressure, saturation and solid wall’s hydrophilia. Then a stress correction method is proposed for the stress instability caused by the SPH kernel function. The method uses two different shapes of kernel functions to calculate the tensile and compressive stresses, respectively. After that, a SPH model compiled in Fortran language and based on the above two improved methods is established to simulate static and dynamic droplets’ deformation processes on different wettable wall. The following studies were carried out based on the simulation results. Firstly, the effect of stress correction method is investigated by the comparison between simulating results with the stress correction before and after used. Secondly, the validity of the wettable solid wall boundary condition is studied based on the simulation results of static droplet deformation. Finally, the influence of the wall wettability on the droplet movement process after the collision is analyzed. Researches show that, the stress correction method improves the stress instability problem in the traditional SPH method, even in the large deformation movement can still maintain a uniform particle distribution. In addition, the stress correction method can better simulate the surface tension of free surface flow, and can get a more smooth free surface. According to the change characteristics of droplet’s static contact angle, the mentioned wettable solid wall boundary condition can clearly reflect the wall’s wettability. Simulation of doplet impact on wettable wall shows good agreement with the experiment result. During rebound stage, a large enough wettability will induce droplet deforming into liquid column. The wettability have a little influence to droplet’s spreading stage, the wall friction force plays a major role. While during retraction and rebound stages, the influence of wettability is obvious. This study improves the theory of the interaction between free surface fluid and solid wall, and can provide a reference for further study of the effects between free surface fluids and wettable media or discrete particles.

Discussion of “On the use of Laplace's equation for pressure and a mesh-free method for 3D simulation of nonlinear sloshing in tanks.” By B. Boroomand, S. Bazazzadeh and S.M. Zandi

The authors used a meshless Lagrangian method to solve continuity and momentum equations of incompressible, inviscid and homogenous (ideal) fluid with the capability of considering container excitation and solving various wave related problems. Through algebraic manipulations, governing equations are written in terms of Laplace's equation for pressure deviation from hydrostatic as ∇2pH=0. The numerical solution procedure consists of building a series solution with each term being a general solution of Laplace's equation and calculating coefficients by imposing free surface and solid wall boundary conditions through collocation at several discrete points. This discussion questions the validity of using Laplace's equation for pressure for the case of ideal fluid; and by applying chain rule of differentiation we prove that positions of divergence operator and material derivative are not readily interchangeable, i.e. ∇·Du/Dt ≠ D(∇·u)/Dt.

Open boundary conditions for ISPH and their application to micro-flow

Open boundary conditions for incompressible Smoothed Particle Hydrodynamics (ISPH) are rare. For stable simulations with open boundary conditions, one needs to specify all boundary conditions correctly in the pressure force as well as in the linear equation system for pressure calculation. Especially for homogeneous or non-homogeneous Dirichlet boundary conditions for pressure there exist several possibilities but only a few lead to stable results. However, this isn't trivial for open boundary conditions. We introduce a new approach for open boundary conditions for ISPH to enable stable simulations. In contrast to existing models for weakly-compressible SPH, we can specify open pressure boundary conditions because in ISPH, pressure can be calculated independently of the density. The presented approach is based on the mirror particle approach already introduced for solid wall boundary conditions. Here we divide the mirror axis in several segments with time-dependent positions.We validate the presented approach for the example of Poiseuille flow and flow around a cylinder at different Reynolds numbers and show that we get good agreement with references. Then, we demonstrate that the approach can be applied to free surface flows. Finally, we apply the new approach to micro-flow through a random porous medium with a different number of in- and outlets and demonstrate its benefits.

Adaptive wavelet simulation of global ocean dynamics using a new Brinkman volume penalization

Abstract. In order to easily enforce solid-wall boundary conditions in the presence of complex coastlines, we propose a new mass and energy conserving Brinkman penalization for the rotating shallow water equations. This penalization does not lead to higher wave speeds in the solid region. The error estimates for the penalization are derived analytically and verified numerically for linearized one-dimensional equations. The penalization is implemented in a conservative dynamically adaptive wavelet method for the rotating shallow water equations on the sphere with bathymetry and coastline data from NOAA's ETOPO1 database. This code could form the dynamical core for a future global ocean model. The potential of the dynamically adaptive ocean model is illustrated by using it to simulate the 2004 Indonesian tsunami and wind-driven gyres.

A parallel VOF IB pressure-correction method for simulation of multiphase flows

This paper develops an improved pressure correction method, coupled with volume-of-fluid (VOF) and immersed boundary (IB) methods, to accelerate multiphase flow computations. A momentum interpolation method (MIM) is used to avoid the issue of a checkerboard pressure field on a non-staggered grid system. A transpose-free quasi-minimal residual (TFQMR) method is applied to reduce the CPU time when solving the Poisson equation for the pressure correction procedure. A suitable initial guess for the TFQMR method is investigated to enhance the stability of the matrix solver. The IB method is applied to treat the solid-wall boundary conditions while OpenMP is used to reduce CPU time. Two physical problems – dam breakage and a water wave impacting on a tall structure – are simulated and compared with known data. The improved numerical scheme has CPU speeds at least 8 times faster than the initial numerical scheme for simulating two-phase flows. With regard to the parallel computational efficiency, using six cores with the OpenMP allows the improved scheme to have 77% parallel efficiency. These modifications greatly enhance overall computational accuracy and efficiency.