No-penetration Boundary Condition Research Articles

Passage from the Boltzmann equation with diffuse boundary to the incompressible Euler equation with heat convection

We derive the incompressible Euler equations with heat convection with the no-penetration boundary condition from the Boltzmann equation with the diffuse boundary in the hydrodynamic limit for the scale of large Reynold number. Inspired by the recent framework in [30], we consider the Navier-Stokes-Fourier system with no-slip boundary conditions as an intermediary approximation and develop a Hilbert-type expansion of the Boltzmann equation around the global Maxwellian that allows the nontrivial heat transfer by convection in the limit. To justify our expansion and the limit, a new direct estimate of the heat flux and its derivatives in the Navier-Stokes-Fourier system is established adopting a recent Green's function approach in the study of the inviscid limit.

SPH-DEM modeling of the seismic response of shallow foundations resting on liquefiable sand

In this study, the seismic response of shallow foundations resting on a liquefiable soil layer is modeled using a coupled smoothed particle hydrodynamics (SPH)-discrete element method (DEM) scheme. In this framework, the soil deposit is represented by an assembly of DEM particles and the fluid domain is lumped into a set of SPH particles carrying local fluid properties. The averaged forms of Navier-Stokes equations dictate the motion of the fluid-particle mixture and the interphase forces are estimated using well-known semi-empirical equations. A saturated soil-foundation system with an average contact pressure of 50 kPa was created using the coupled scheme. The foundation block was composed of a collection of DEM particles glued together by high-stiffness bonds. No-penetration boundary condition was applied to all sides of the foundation block to allow for fluid-foundation interaction. The model was subjected to a strong base acceleration and the response was analyzed and compared to the free-field. The ground settlement in the soil-foundation system mostly originated from co-seismic deviatoric deformations while volumetric strains were the main contributing factor at the free-field. In addition, the impact of soil permeability on the seismic response of the soil-foundation system was examined by changing the pore fluid viscosity. According to the results, as the soil permeability decreased, smaller excess pore pressures developed beneath the footing thanks to the slower migration of pore fluid from the sides and bottom, and a larger magnitude of soil strength and stiffness was maintained in the expansive zone. As a result, the system with the lowest permeability experienced the smallest foundation settlement while the foundation acceleration amplitude was the highest in this case. The results also showed that the percentage of post-shaking settlement appreciably increased in the lower-permeability deposits. The results of this study were compared with the published centrifuge studies that showed good qualitative consistency. • A SPH-DEM scheme is utilized to model the seismic response of shallow foundations on liquefiable soils. • Averaged form of Navier-Stokes equations are used to describe the pore fluid motion in the mixture. • The foundation block was composed of a collection of DEM particles glued together by high-stiffness bonds. • No-penetration boundary condition was applied to all foundation sides to allow for fluid-foundation interaction. • The results of this study qualitatively agree with published centrifuge studies.

Incompressible Euler Limit from Boltzmann Equation with Diffuse Boundary Condition for Analytic Data

A rigorous derivation of the incompressible Euler equations with the no-penetration boundary condition from the Boltzmann equation with the diffuse reflection boundary condition has been a challenging open problem. We settle this open question in the affirmative when the initial data of fluid are well-prepared in a real analytic space, in 3D half space. As a key of this advance, we capture the Navier-Stokes equations of $$\begin{aligned} \textit{viscosity} \sim \frac{\textit{Knudsen number}}{\textit{Mach number}} \end{aligned}$$satisfying the no-slip boundary condition, as an intermediary approximation of the Euler equations through a new Hilbert-type expansion of the Boltzmann equation with the diffuse reflection boundary condition. Aiming to justify the approximation we establish a novel quantitative \(L^p\)-\(L^\infty \) estimate of the Boltzmann perturbation around a local Maxwellian of such viscous approximation, along with the commutator estimates and the integrability gain of the hydrodynamic part in various spaces; we also establish direct estimates of the Navier-Stokes equations in higher regularity with the aid of the initial-boundary and boundary layer weights using a recent Green’s function approach. The incompressible Euler limit follows as a byproduct of our framework.

Direct effects of boundary permeability on turbulent flows: observations from an experimental study using zero-mean-shear turbulence

Abstract

An analysis of the spatio-temporal resolution of the immersed boundary method with direct forcing

The immersed boundary method (IBM) with direct forcing is very popular in the simulation of rigid particulate flows. In the IBM, an interaction force is introduced at the interface between fluid and particle in order to approximate the no-slip boundary condition. The interaction force is calculated through dividing the velocity difference (or error) between fluid and particle at the interface by the time step. Here, a dynamic equation for the velocity difference is derived. Analyses on the dynamic equation provide a few new findings: (i) The interaction force is the solution of a least-squares error problem, with the direct implication that the Lagrangian marker distribution has no effect on the large scale flow structure once the distribution of Lagrangian markers become saturated along the interface (i.e., each marker remains properly correlated with all its neighbors); (ii) The Lagrangian volume-weight is a relaxation factor to control how fast the velocity error decays to the ideal value of zero; (iii) The optimal choice of the Lagrangian volume-weight is the largest value permissible by a stability condition. A comprehensive convergence analysis with regard to the spatial and temporal resolution is presented for the velocity error and also for the shear-stress and surface pressure. In three simple canonical problems, it is analytically and numerically shown that the IBM results converge to the theoretical solutions obtained with precise imposition of no-slip and no-penetration boundary conditions. It is observed that it is not necessary to match the Lagrangian marker volume-weight to that of the local Eulerian cell volume and in fact this matching leads to lower than optimal computational efficiency. However, it is found that extremely high Eulerian grid resolution and small time step have to be used to obtain high precision simulation results. Especially, the time step should be inversely proportional to the particle Reynolds number for low Reynolds number flows. For high frequency oscillation problems, the grid size needs to be reduced by a factor of the square root of the frequency, and the time step to be reduced by a factor of the frequency. The theoretical findings here can be used to alleviate the technical difficulties in simulating non-spherical particles by not requiring the Lagrangian marker distribution to match the Eulerian grids and also in the implementation of IBM on non-uniform Eulerian grids. The present work also provides simple practical guidance on the choice of temporal and spatial resolution so as to control the simulation error a priori.

A Novel Approach for Direct Numerical Simulation of Hydraulic Fracture Problems

Hydraulic fracturing is a non-linear and multi-physics problem involving the break up of a solid medium due to the action of hydrodynamic forces. Fluid and solid mechanics are involved at the same time together with fracture mechanics. Despite its relevance in many scientific and engineering fields, the theoretical and numerical description of hydraulic fracturing remains a challenging matter and the capabilities of existing models for applications are still limited. In this context, we propose a novel numerical approach to the Direct Numerical Simulation of hydraulic fracturing based on the Navier–Stokes equations coupled with peridynamic theory of solid mechanics through a multi-direct Immersed Boundary Method. The main advantage of this approach consists in the reliable crack-detection and tracking capabilities of peridynamics together with the capability of the Immersed Boundary Method of managing no-slip and no-penetration boundary conditions on complex and time-evolving interfaces. A massive-parallel solver based on this model has been implemented. We present a detailed theoretical description of the proposed methodology as well as the results of an extensive validation campaign for the new solver. Different benchmarking tests are provided together with the qualitative results of a simulation reproducing the fracture of a solid structure in a laminar, unsteady flow.

Analysis of Cross-Diffusion Systems for Fluid Mixtures Driven by a Pressure Gradient

The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy's law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.

Entropy Stability and the No-Slip Wall Boundary Condition

We present an entropy stable numerical scheme subject to no-slip wall boundary conditions. To enforce entropy stability only the no-penetration boundary condition and a temperature condition are needed at a wall, and this leads to an $L^2$ bound on the conservative variables. In this article, we take the next step and design a finite difference scheme that also bounds the velocity gradients. This necessitates the use of the full no-slip conditions.

Meshless analysis of Darcy flow with a variational multiscale interpolating element-free Galerkin method

In this study, a variational multiscale interpolating element-free Galerkin (VMIEFG) method is developed for numerical analysis of Darcy flow. In the VMIEFG method, the improved interpolating moving least squares method is used to construct shape functions with interpolating properties. Then, the no-penetration boundary condition of Darcy flow can be satisfied exactly. Meanwhile, the utilization of Hughes’ variational multiscale formulation eliminates the numerical oscillation in the standard element-free Galerkin method. Thus, the VMIEFG method permits equal-order basis for both the velocity and pressure, and the restraint of the Babuška-Brezzi condition is violated completely. Numerical results demonstrate that the present VMIEFG method has high accuracy and stability for solving Darcy flow.

Boundary-conforming free-surface flow computations: Interface tracking for linear, higher-order and isogeometric finite elements

The simulation of certain flow problems requires a means for modeling a free fluid surface; examples being viscoelastic die swell or fluid sloshing in tanks. In a finite-element context, this type of problem can, among many other options, be dealt with using an interface-tracking approach with the Deforming-Spatial-Domain/Stabilized-Space-Time (DSD/SST) formulation. A difficult issue that is connected with this type of approach is the determination of a suitable coupling mechanism between the fluid velocity at the boundary and the displacement of the boundary mesh nodes. In order to avoid large mesh distortions, one goal is to keep the nodal movements as small as possible; but of course still compliant with the no-penetration boundary condition. Standard displacement techniques are full velocity, velocity in a specific coordinate direction, and velocity in normal direction. In this work, we investigate how the interface-tracking approach can be combined with isogeometric analysis for the spatial discretization. If NURBS basis functions of sufficient order are used for both the geometry and the solution, both a continuous normal vector as well as the velocity are available on the entire boundary. This circumstance allows the weak imposition of the no-penetration boundary condition. We compare this option with an alternative that relies on strong imposition at discrete points. Furthermore, we examine several coupling methods between the fluid equations, boundary conditions, and equations for the adjustment of interior control point positions.

Planar Limits of Three-Dimensional Incompressible Flows with Helical Symmetry

Helical symmetry is invariance under a one-dimensional group of rigid motions generated by a simultaneous rotation around a fixed axis and translation along the same axis. The key parameter in helical symmetry is the step or pitch, the magnitude of the translation after rotating one full turn around the symmetry axis. In this article we study the limits of three-dimensional helical viscous and inviscid incompressible flows in an infinite circular pipe, with respectively no-slip and no-penetration boundary conditions, as the step approaches infinity. We show that, as the step becomes large, the three-dimensional helical flow approaches a planar flow, which is governed by the so-called two-and-half Navier–Stokes and Euler equations, respectively.

VORTEX METHOD SIMULATION OF BLASIUS' FLAT-PLATE BOUNDARY LAYER

The vortex method has been extensively used to simulate external flows around bluff and streamlined bodies. It relies on the discretization of the vorticity field into a cloud of vortex blobs to simulate the convective−diffusive transport of vorticity. Vortex blobs are generated in the neighborhood of the solid wall in order to satisfy the no-slip and the no-penetration boundary conditions, and they move in a Lagrangian manner to solve the vorticity transport equation. Despite the tremendous development that this powerful mesh-free technique has recently achieved, the numerical implementation of the wall boundary conditions is currently under intense investigation, since it is intimately connected to the vorticity generation process in the vicinity of the body surface. In this paper we describe an efficient two-dimensional vortex method algorithm, with emphasis on a new model for the vortex creation near the surface that increases the accuracy of the simultaneous implementation of the wall boundary conditions. We employ the adaptive fast multipole method to calculate the induced velocities and the corrected core-spreading method to simulate the vorticity diffusion in the boundary layer and wake. The method is second-order accurate in space when Gaussian vortex blobs are used and second-order accurate in time when the Adams-Bashforth scheme is used to march the integration process in time. The algorithm is tested against the well-known two-dimensional, incompressible boundary-layer flow over a flat plate. The agreement between the numerical results and the exact Blasius solution indicates that the algorithm provides an excellent representation of the vorticity field.

Real-time ink simulation using a grid-particle method

This paper presents an effective method to simulate the ink diffusion process in real time that yields realistic visual effects. Our algorithm updates the dynamic ink volume using a hybrid grid-particle method: the fluid velocity field is calculated with a low-resolution grid structure, whereas the highly detailed ink effects are controlled and visualized with the particles. To facilitate user interaction and extend this method, we propose a particle-guided method that allows artists to design the overall states using the coarse-resolution particles and to preview the motion quickly. To treat coupling with solids and other fluids, we update the grid-particle representation with no-penetration boundary conditions and implicit interaction conditions. To treat moving “ink-emitting” objects, we introduce an extra drag-force model to enhance the particle motion effects; this force might not be physically accurate, but it proves effective for producing animations. We also propose an improved ink rendering method that uses particle sprites and motion blurring techniques. The simulation and the rendering processes are efficiently implemented on graphics hardware at interactive frame rates. Compared to traditional fluid simulation methods that treat water and ink as two mixable fluids, our method is simple but effective: it captures various ink effects, such as pinned boundaries (Nelson, 2005 [1]) and filament patterns (Shiny et al., 2010 [2]), while still running in real time, it allows easy control of the animation, it includes basic solid–fluid interactions, and it can address multiple ink sources without complex interface tracking. Our method is attractive for animation production and art design.

Stability of detonation in a circular pipe with porous walls

A stability analysis is carried out taking into account slightly porous walls in an idealized detonation confined to a circular pipe. The analysis is carried out using the normal-mode approach and corrections are obtained to the underlying impenetrable wall case results to account for the effect of the slight porosity. The porous walls are modelled by an acoustic boundary condition for the perturbations linking the normal velocity and the pressure components and thus replacing the conventional no-penetration boundary condition at the wall. This new boundary condition necessarily complicates the derivation of the stability problem with respect to the impenetrable wall case. However, exploiting the expressly slight porosity, the modified temporal stability can be determined as a two-point boundary value problem similar to the case of a non-porous wall.

Efficient implementation of high order inverse Lax–Wendroff boundary treatment for conservation laws

In [20], two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax–Wendroff (ILW) procedure for inflow boundary conditions and high order extrapolation for outflow boundary conditions. However, the algebra of the ILW procedure is quite heavy for two dimensional (2D) hyperbolic systems, which makes it difficult to implement the procedure for order of accuracy higher than three. In this paper, we first discuss a simplified and improved implementation for this procedure, which uses the relatively complicated ILW procedure only for the evaluation of the first order normal derivatives. Fifth order WENO type extrapolation is used for all other derivatives, regardless of the direction of the local characteristics and the smoothness of the solution. This makes the implementation of a fifth order boundary treatment practical for 2D systems with source terms. For no-penetration boundary condition of compressible inviscid flows, a further simplification is discussed, in which the evaluation of the tangential derivatives involved in the ILW procedure is avoided. We test our simplified and improved boundary treatment for Euler equations with or without source terms representing chemical reactions in detonations. The results demonstrate the designed fifth order accuracy, stability, and good performance for problems involving complicated interactions between detonation/shock waves and solid boundaries.

Subgrid-Scale Modeling of Turbulent Flow Around Circular Cylinder by Mesh-Free Vortex Method

:This paper proposes a numerical model for the two-dimensional, incompressible, unsteady, turbulent flow applied to an impulsively started flow around a circular cylinder in the Reynolds number range from 1×104 to 6×105. A Lagrangian mesh-free vortex method blended with the Large Eddy Simulation theory is employed to simulate the large-scale motion, whereas the turbulent subgrid-scale motion is modeled with an eddy viscosity coefficient, expressed in terms of the Second-Order Velocity Structure Function. The filtered vorticity field is calculated by a superposition of Lamb vortices that are generated near the body surface such that circulation is conserved and the no-slip boundary condition is explicitly imposed at a finite number of points on the cylinder. The no-penetration boundary condition is satisfied exactly on the entire cylinder surface through the application of the circle theorem. The vorticity transport equation is solved using the convective-diffusive operator-splitting algorithm, where vorticity diffusion is simulated with the random walk method and the convective motion of the vortices is integrated in time using the second-order Adams-Bashforth scheme. Numerous simulations for high Reynolds numbers are carried out to determine the numerical parameters of the model. Results for the drag coefficient and the Strouhal number as a function of the Reynolds number present satisfactory agreement with other results from the literature.

Effect of Particle Clusters on Carrier Flow Turbulence: A Direct Numerical Simulation Study

Experiments indicate that particle clusters that form in fluidized–bed risers can enhance gas-phase velocity fluctuations. Direct numerical simulations (DNS) of turbulent flow past uniform and clustered configurations of fixed particle assemblies at the same solid volume fraction are performed to gain insight into particle clustering effects on gas-phase turbulence, and to guide model development. The DNS approach is based on a discrete-time, direct-forcing immersed boundary method (IBM) that imposes no-slip and no-penetration boundary conditions on each particle’s surface. Results are reported for mean flow Reynolds number Rep = 50 and the ratio of the particle diameter dp to Kolmogorov scale is 5.5. The DNS confirm experimental observations that the clustered configurations enhance the level of fluid-phase turbulent kinetic energy (TKE) more than the uniform configurations, and this increase is found to arise from a lower dissipation rate in the clustered particle configuration. The simulations also reveal that the particle-fluid interaction results in significantly anisotropic fluid-phase turbulence, the source of which is traced to the anisotropic nature of the interphase TKE transfer and dissipation tensors. This study indicates that when particles are larger than the Kolmogorov scale (dp > η), modeling the fluid-phase TKE alone may not be adequate to capture the underlying physics in multiphase turbulence because the Reynolds stress is anisotropic. It also shows that multiphase turbulence models should consider the effect of particle clustering in the dissipation model.

Investigation of Unsteady Aerodynamic Characteristics of a Seagull Wing in Level Flight

Unsteady aerodynamic characteristics of a seagull wing in level flight are investigated using a boundary element method. A new no-penetration boundary condition is imposed on the surface of the wing by considering its deformation. The geometry and kinematics of the seagull wing are reproduced using the functions and data in the previously published literature. The proposed method is validated by comparing the computed results with the published data in the literature. The unsteady aerodynamics characteristics of the seagull wing are investigated by changing flapping frequency and advance ratio. It is found that the peak values of aerodynamic coefficients increase with the flapping frequency. The thrust and drag generations are complicated functions of frequency and wing stroke motions. The lift is inversely proportional to the advance ratio. The effects of several flapping modes on the lift and induced drag (or thrust) generation are also investigated. Among three single modes (flapping, folding and lead & lag), flapping generates the largest lift and can produce thrust alone. For three combined modes, both flapping/folding and flapping/lead & lag can produce lift and thrust larger than the flapping-alone mode can. Folding is shown to increase thrust when combined with flapping, whereas lead & lag has an effect of increasing the lift when also combined with flapping. When three modes are combined together, the bird can obtain the largest lift among the investigated modes. Even though the proposed method is limited to the inviscid flow assumption, it is believed that this method can be used to the design of flapping micro aerial vehicle.

Three-component micro-PIV using the continuity equation and a comparison of the performance with that of stereoscopic measurements

Recently, a number of techniques have been presented for the determination of the third “out-of-plane” velocity component in micro particle image velocimetry (micro-PIV) data. In particular, the conventional macroscopic stereo-PIV technique has been converted to the micro scale by the use of stereo-microscopy. In this work a different technique is investigated, which uses conventional, two-component micro-PIV to generate velocity data on a number of planes. The in-plane velocity gradients are then calculated, which can be used in the continuity equation to produce the out-of-plane velocity gradients. These, together with the no-penetration boundary condition, can then be used to calculate the out-of-plane velocities. An algorithm is presented that is capable of handling up to one invalid vector per column of data by using a combination of first order and second order projections of the velocity. The advantage of the continuity based technique is that it uses the existing two-component micro-PIV technology, which at present is in a more advanced stage of development then stereo-microscopy based micro-PIV. The technique is investigated using a flow similar to one used previously to assess stereoscopic micro-PIV (Meas Sci Technol 17:2175–2185, 2006). This allows a comparison of the performance of the two techniques. The results show that the continuity based data agrees well with an independent computational fluid dynamics solution and has a smaller experimental uncertainty than the stereoscopic technique at a better spatial resolution. There are, however, potential limitations to the continuity based technique. These include the two-dimensionality of the data, which is limited to the planes on which the original images were taken, and the dependence of the technique on the data close to surfaces, where the experimental errors are often greatest. Stereoscopic micro-PIV does not have these limitations so, whilst at present it appears that continuity based techniques may be more accurate, there is sufficient potential for stereoscopic techniques to justify their further development.

Discrete vortex model of flow over a square cylinder

Abstract The discrete vortex method (DVM) is used to numerically study the velocity field in the near wake of a 2-D cylinder of square cross-section. Vortex panels are used to model the cylinder surface, at which a no-penetration boundary condition was imposed. The DVM uses a series of individual vortex elements to model the separated shear layers of the cylinder. The discrete-vortex elements interact to form clusters in the wake region, which represent the larger vortical structures present in such flows. By careful choice of numerical parameters, good agreement is obtained with experimental measurements for the mean and fluctuating velocity fields in the near-wake region. The method is especially effective in modelling the unsteady, quasi-periodic motions associated with the large-scale vortex structures in the wake.