Mesh Velocity Research Articles

A high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for one-dimensional compressible multi-material flows

In this paper, a high-order physical-constraints-preserving arbitrary Lagrangian–Eulerian (ALE) discontinuous Galerkin scheme is proposed for one-dimensional compressible multi-material flows. Our scheme couples a conservative equation related to the volume-fraction model with the Euler equations for describing the dynamics of fluid mixture. The mesh velocity in the ALE framework is obtained by using an adaptive mesh method that can automatically concentrate the mesh nodes near the regions with large gradient values and greatly reduce the numerical dissipation near material interfaces. Using this adaptive mesh, the resolution of solution near some special regions such as material interfaces can be improved effectively by our scheme. With the appropriate time step condition and using a bound-preserving and positivity-preserving limiter, our scheme can ensure the positivity of density and pressure and the boundness of volume-fraction, which further ensures the computational robustness and degree of confidence of simulations under large density or pressure ratios and so on. In general, our scheme can be applied to the simulations of compressible multi-material flows efficiently with the essentially non-oscillatory property and physical-constraints-preserving (bound-preserving and positivity-preserving) property, and its steps are more concise compared to some other methods such as the indirect ALE methods. Some examples are tested to demonstrate the accuracy, essentially non-oscillatory property and physical-constraints-preserving property of our scheme.

An Effective Arbitrary Lagrangian-Eulerian-Lattice Boltzmann Flux Solver Integrated with the Mode Superposition Method for Flutter Prediction

An Effective Arbitrary Lagrangian-Eulerian-Lattice Boltzmann Flux Solver Integrated with the Mode Superposition Method for Flutter Prediction

A high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin scheme for compressible multi-material flows

In this paper, a high-order limiter-free arbitrary Lagrangian–Eulerian discontinuous Galerkin (ALE-DG) scheme is proposed for simulating one-dimensional compressible multi-material flows. The system is discretized by the DG method, and a kind of Taylor’s expansion basis functions in the general cell is used to construct the interpolation polynomials of the variables. The mesh velocity at the node recognized as the material interface is set to approximate fluid velocity, which makes our scheme can capture the material interface accurately and clearly. For controlling the oscillations and reducing the numerical dissipation, a reconstruction based on the boundary variation diminishing (BVD) algorithm and Tangent of Hyperbola for INterface Capturing (THINC) function is employed to minimize the jump between the values at the left and right sides of cell boundaries. Due to the essentially monotone and bounded properties of THINC function, the difficulties caused by solving discontinuous solutions and complexities of designing limiters can be avoided. Finally, several examples are presented to demonstrate the accuracy and good performance of our scheme such as the essentially non-oscillatory property and so on for simulating the compressible multi-material flows.

A velocity-based moving mesh virtual element method

We present a velocity-based moving mesh virtual element method for the numerical solution of PDEs involving boundaries which are free to move. The virtual element method is used for computing both the mesh velocity and a conservative Arbitrary Lagrangian-Eulerian solution transfer on general polygonal meshes. The approach extends the linear finite element method to polygonal mesh structures, achieving the same degree of accuracy. In the context of moving meshes, a major advantage of the virtual element approach is the ease with which nodes can be inserted on mesh edges. Demonstrations of node insertion techniques are presented to show that moving polygonal meshes can be simply adapted for situations where a boundary encounters a solid object or another moving boundary, without reduction in degree of accuracy.

Shock capturing for a high-order ALE discontinuous Galerkin method with applications to fluid flows in time-dependent domains

In recent decades, the arbitrary Lagrangian–Eulerian (ALE) approach has been one of the most popular choices to deal with the fluid flows having moving boundaries. The ALE finite volume (FV) method is well-known for its capability of shock capturing. Several ALE discontinuous Galerkin (DG) methods are developed to fully explore the advantages of high-order methods, especially in complex flows. The DG scheme is highly efficient and has high resolution in smooth parts of the flow while the FV scheme has advantage on the robustness against shocks. In this paper, we present a novel algorithm to combine both the ALE FV and ALE DG methods into one stable and efficient hybrid approach. The main challenge for a mixed ALE FV method and ALE DG method is to reduce the inconsistency between both discretizations. Of particular concern are the same mesh velocity distribution between DG and FV elements and a continuous mesh velocity at the coupling interfaces. Since the computational efficiency of the proposed algorithm is of major concern, performance aspects will be considered during the construction of the scheme. To investigate the accuracy of the new scheme, several benchmark test cases with prescribed movements are included. Afterwards, the algorithm is applied to more general and complex scenarios to show its ability to cope with complex setups. In our paper, the scheme is implemented into a loosely-coupled fluid–structure interaction (FSI) framework. With a shock-driven cylinder movement in a channel flow and a transonic flow-induced airfoil vibration, we demonstrate our method for FSI applications.

An arbitrary Lagrangian–Eulerian discontinuous Galerkin method for two‐dimensional compressible flows on adaptive quadrilateral meshes

AbstractIn this article, a discontinuous Galerkin (DG) scheme on the adaptive quadrilateral meshes is proposed to simulate two‐dimensional compressible flows in the direct arbitrary Lagrangian–Eulerian (ALE) framework. In our scheme, the Euler equations are discretized in the reference element with the help of a bilinear map. A kind of Taylor expansion basis functions in the reference element is used to construct the interpolation polynomials of variables. We describe the property that the material derivatives of the basis functions used in the DG discretization are equal to zero, with which the scheme is simplified. Furthermore, the mesh velocity in our ALE framework is obtained by implementing the approach of mesh movement based on the variational principle from [Adaptive mesh methods for one‐ and two‐dimensional hyperbolic conservation laws. SIAM J Numer Anal. 2003;41:487–515]. This approach of mesh movement automatically concentrates the mesh nodes near the regions with large gradient values of the variables and can greatly improve the resolution of the solution near these regions. In addition, a WENO (weighted essentially non‐oscillatory) reconstruction helps our scheme remove the numerical oscillations. Some numerical examples are presented to demonstrate the accuracy, high resolution, and robustness of our scheme.

Effect of pitched short blades on the flow characteristics in a stirred tank with long-short blades impeller

This work focuses on the design improvement of the long-short blades (LSB) impeller by using pitched short blades (SBs) to regulate the flow field in the stirred vessel. After mesh size evaluation and velocity field validation by the particle image velocimetry, large eddy simulation method coupled with sliding mesh approach was used to study the effect of the pitched SBs on the flow characteristics. We changed the inclined angles of the SBs from 30° to 60° and compared the flow characteristics when the impeller was operated in the down-pumping and up-pumping modes. In the case of down-pumping mode, the power number is relatively smaller and vortexes below the SBs are suppressed, leading to turbulence intensification in the bottom of the vessel. Whereas in the case of up-pumping mode, the axial flow rate in the center increased significantly with bigger power number, resulting in more efficient mass exchange between the axial and radial flows in the whole vessel. The LSB with 45° inclined angle of the SBs in the up-pumping mode has the most uniform distributions of flow field and turbulent kinetic energy compared with other impeller configurations.

One‐ and two‐step semi‐Lagrangian integrators for arbitrary Lagrangian–Eulerian‐finite element two‐phase flow simulations

AbstractSemi‐Lagrangian (SL) schemes are of utmost relevance to simulate two‐phase flows where advection dominates. The combination of SL schemes with the finite element (FE) method and arbitrary Lagrangian–Eulerian (ALE) dynamic meshes yields a strong ingredient to deal with applications in Earth sciences, environmental engineering, and oil and gas industry whose numerical background is incipient. This article is intended to propose a second‐order time multistep SL/ALE/FE scheme to track particle trajectories traveling amidst single‐ or two‐phase incompressible flows both in fixed and moving mesh situations. The novel scheme is a BDF2‐like implementation that considers the mesh velocity as part of its backward‐in‐time integration. Trajectory departure points are searched through extrapolation and followed by a multistep interpolation corrector that works both for constant and varying time steps. A series of two‐phase benchmark flow simulations are carried out by using the cubic element to compare the performance of the new integrator against single‐step approximations as well as to analyze enhancements in representing the two‐phase flow dynamics. We use the 2D axisymmetric Navier–Stokes equations as underlying model. Error analyses, convergence tests, quantitative, and qualitative comparisons are presented and discussed to highlight the superior conservative feature of the novel scheme.

Optimal convergence of arbitrary Lagrangian–Eulerian iso-parametric finite element methods for parabolic equations in an evolving domain

Abstract An optimal-order error estimate is presented for the arbitrary Lagrangian–Eulerian (ALE) finite element method for a parabolic equation in an evolving domain, using high-order iso-parametric finite elements with flat simplices in the interior of the domain. The mesh velocity can be a linear approximation of a given bulk velocity field or a numerical solution of the Laplace equation with specified boundary value matching the velocity of the boundary. The optimal order of convergence is obtained by comparing the numerical solution with the ALE-Ritz projection of the exact solution, and by establishing an optimal-order estimate for the material derivative of the ALE-Ritz projection error.

Left atrial appendage shape impacts on the left atrial flow hemodynamics: A numerical hypothesis generating study on two cases

Background and objectivesThe left atrial appendage (LAA) is the most common region for thrombus formation in atrial fibrillation (AF). Morphological parameters such as shape, size, and LAA volume can cause insufficient effectiveness of available therapeutic options. This study aimed to examine blood flow inside LAA and its removal effects. Computational fluid dynamic (CFD) simulations were carried out on two patients with different morphologies. MethodsTwo patients' CT was used to reconstruct the 3D geometries of the left atrium (LA) and left atrial appendage (LAA). Then, the geometries were refined in the mentioned software, and the LAA in some models was removed. Next, in generated 3D volume mesh, sinus rhythm (SR) and atrial fibrillation (AF) outflow velocity were imposed at the mitral valve as boundary conditions. Finally, CFD simulation was conducted to analyzing blood flow within LA with/without LA. ResultsThe results confirmed that velocity and vorticity decreased under AF conditions inside the LA domain for both patients. However, removing LAA may cause unpredictable consequences, due to different shape and volume of LAA. LAA removal had insignificant effects on velocity and vorticity within LA in SR-mitral outflow. However, removing LAA increased the blood flow rate by 9.15% and vorticity by 7.27% for patient one under AF rhythm (SR)-outflow. In contrast, for patient two, LAA removal in both AF and SR decreased velocity and vorticity within the LA domain. In SR-mitral outflow, velocity dropped by 18.8 %, and vorticity by 13.2%. Also, under AF velocity and vorticity decreased by 23.33% and 18.6% respectively. Meanwhile, the results indicated that the vorticity magnitude increased inside the LAA under AF associated with the risk of thrombus formation, particularly for patient one under AF. The distal part of LAA in both patients was the most common region for blood stasis because of the lowest velocity magnitude. ConclusionOverall, the morphology of LAA could be the critical parameter to determine the possibility of thrombosis formation, particularly under AF conditions. High volume, low blood flow velocity and two-lobe-appendage are more likely to have blood stasis. Furthermore, the morphology difference can affect the LAA removal result and make it more complicated. So, it could be challenging to generalize LAA removal as a therapeutic option for different patients. The implication of this CFD observation needs more investigation.

Dynamics of a buoyant pulsating bubble near two crossed walls

The dynamics of a buoyant pulsating bubble near two crossed perpendicular rigid boundaries (a horizontal and a vertical wall) are studied using the boundary element method combined with the method of mirror images. The Kelvin impulse and the elastic mesh velocity method are used to calculate the direction and volume of the liquid jet generated during bubble collapse. The numerical results show good agreement with experiments. An increase in buoyancy causes a local high-pressure zone at the root of the jet to move toward the bottom of the bubble, causing the jet to rotate upward toward the vertical wall. At a certain position, with the change in buoyancy, the dimensionless bubble volume at the instant of jet impact reaches a minimum when the jet direction is horizontal, with a peak in the dimensionless jet velocity occurring. A comprehensive parametric study of jet characteristics, including jet direction, velocity, and relative volume (the volume ratio of the jet to the bubble at the instant of jet impact), is carried out in terms of buoyancy and the standoff distances to the two walls. The Blake criterion can be used to judge whether a bubble jet is pointing obliquely upward or downward, provided that it deviates significantly from the horizontal direction. Depending on the buoyancy, the jet characteristics at different standoff distances are found to exhibit three distinct patterns of behavior. Finally, we discuss the changes in the jet velocity and relative volume as the buoyancy is varied.

On delayed transition to turbulence in an eccentric stenosis model for clean vs. noisy high-fidelity CFD

Recent comparisons between experiments and computational fluid dynamics (CFD) simulations of flow in the Food and Drug Administration (FDA) standardized nozzle geometry have highlighted the potential sensitivity of axisymmetric CFD models to small perturbations induced by mesh and inlet velocity, particularly for Reynolds numbers (Re) in the transitional regime. This evokes the classic experiment of Reynolds on transition to turbulence in a straight pipe, which can be delayed, apparently indefinitely, if special care is taken to control for external influences. Such idealized experiments are, however, extremely difficult to perform and, in the context of cardiovascular modeling, belie the “noise” inherent in typical experimental and physiological systems. Previous high-fidelity CFD of a canonical eccentric (i.e., non-axisymmetric) stenosis model showed transition occurring for steady flow at Re ~ 700–800, with modest delay caused by the introduction of shear-thinning rheology. On the other hand, recent experimental measurements of steady flowing blood and blood-mimicking fluids in this same stenosis model report transition for Re ~ 400–500. Taking a cue from the FDA nozzle controversy, the present study demonstrates that the addition of small-magnitude random noise at the inlet brings the eccentric-stenosis CFD results more in-line with experiments, and reveals a more gradual transition towards turbulence. This highlights that, even in non-axisymmetric idealized geometries, unnaturally “clean” high-fidelity CFD may impede not only good agreement with experiments, but also understanding of the onset and character of blood flow instabilities as they may exist, naturally, in the vasculature.

Model of interaction of a rigid mesh with a deformable target

We present an analytical model of the high-velocity impact interaction of a rigid mesh with a semi-infinite deformable target, which is modeled as a rigid-plastic body. We consider the so-called “normal” impact of the mesh on the target: we assume that at the initial moment and subsequent moments of time the mesh is parallel to the target surface while the mesh velocity vector is perpendicular to this surface. The model reproduces the most interesting case in which the mesh aperture is comparable to or less than the diameter of the strings from which the mesh is woven. The aim of the study was to investigate the dependence of the mesh penetration depth on the impact velocity $$V_{0}$$ and on the geometric parameter of the mesh γ that is equal to the ratio of the string diameter to the mesh period. Two versions of the model are considered: with and without taking into accounts the fragmentation of the ejected material of the target. The results obtained on the basis of the proposed model are compared with the numerical solutions which were obtained using the LS-DYNA package. The example of the penetration of a steel mesh into an aluminum-alloy target with impact velocities of 1–3 km/s is analyzed. It is shown that the model that takes into account fragmentation agrees well with the numerical simulations for the mesh parameter interval $$\gamma_{p} < \gamma < 1$$ , in which the lower boundary decreases with increasing impact velocity: $$\gamma_{p}$$ = 0.49, 0.29, 0.2 for $$V_{0}$$ = 1, 2, 3 km/s, respectively.

An a priori error analysis for a projection based variational multiscale finite element method for Oseen problems in a time-dependent domain

Stability and error estimates for a projection based variational multiscale finite element scheme for Oseen problem in a time-dependent domain are derived in this paper. The use of Geometric Conservation Law (GCL) provides an unconditional stable scheme, whereas a restriction on the time-step needs to be imposed to obtain stability estimates independent of the mesh velocity when GCL is violated. Further, a priori error estimate is derived for the semi-discrete problem obtained with the backward Euler time discretization.

On the Thrust Generation from an Elliptic Airfoil in Plunging and Translating Motion at Low Reynolds Numbers

In this paper, numerical simulation elliptic airfoil model, which mimics the biological locomotion, is studied. Elliptic airfoil undergoes a combined plunging and translating at low Reynolds number is simulated by using body fitted coordinate system. The moving mesh in the physical domain is mapped to a regular fixed mesh in the computational domain through a time dependent transformation between the physical and computational co-ordinates. The governing equations of laminar incompressible flow are transformed in the computational plane by incorporating the time dependent transformation, which naturally accounts for the mesh velocities. The transformed equations are discretized on the structured, collocated, o-type elliptic grid using the finite difference methodology. The unsteady equations are marched in time by using a semi-implicit pressure correction (projection) scheme. Along with the time marching of the governing equations, utilizing the mesh velocities and the forward Eulertime integration also moves the mesh points. The effect of Reynolds number (Re) is investigated on the flapping flight propulsion is investigated. It is found that there exists a critical Reynolds number (Rec) for every frequency after which there exists a thrust force. The effect of Rec is related to transformation of neutral wake to thrust generating wake. It is also found that the optimal frequency corresponds to a reduced frequency parameter of 0.7 where a lock in exists. It is also found that this Stc is independent of Re and the mode of vortex shedding is same at Re = 100 and 200 for Stc = 0.7. Further, it is shown that the mode of vortex shedding present is always helpful in thrust generation.

An energy-stable scheme for a 2D simple fluid-particle interaction problem

We develop an energy-stable scheme for simulating fluid-particle interaction problems governed by a coupled system consisting of the incompressible Navier-Stokes (NS) equations defined in a time-dependent fluid domain and Newton's second law for particle motion. A modified temporary arbitrary Lagrangian-Eulerian (tALE) method is designed based on a bijective mapping between the fluid regions at different time steps. In the proposed numerical scheme, the tALE mesh velocity, the incompressible NS equations, and Newton's second law are solved simultaneously. We prove that under certain conditions, the new time discretization scheme satisfies an energy law. For the space discretization, the extended finite element method (XFEM) is used to solve the problem on a fixed Cartesian mesh. The developed method is first-order accurate in time and space without being momentum conservative. To verify the accuracy and stability of our numerical scheme, we present numerical experiments including the fitting of the Jeffery orbit by rotating of an ellipse and the free-falling of an elliptic particle in water.

High-order ALE gas-kinetic scheme with WENO reconstruction

In this paper, a high-order multi-dimensional gas-kinetic scheme is presented for both inviscid and viscous flows in arbitrary Lagrangian-Eulerian (ALE) formulation. Compared with the traditional ALE method, the flow variables are updated in the finite volume framework, and the rezoning and remapping steps are not required. The two-stage fourth-order method is used for the temporal discretization, and the second-order gas-kinetic solver is applied for the flux evaluation. In the two-stage method, the spatial reconstruction is performed at the initial and intermediate stages, and the computational meshes are determined by the mesh velocity. In the moving mesh procedure, the mesh may distort severely and the mesh quality is reduced. To achieve the accuracy and improve the robustness, the newly developed WENO method [40] on quadrilateral meshes is adopted at each stage. The Gaussian quadrature is used for flux calculation. For each Gaussian point, the WENO reconstruction is performed in local moving coordinate, where the variation of mesh velocity along cell interface is taken into account. Numerical examples are presented to validate the performance of current scheme, where the mesh adaptation method and the cell centered Lagrangian method are used to provide mesh velocities.

A surface moving mesh method based on equidistribution and alignment

A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the Jacobian matrix of the affine mapping between the reference element and any simplicial surface element is not square. The development starts with revealing the relation between the area of a surface element in the Euclidean or Riemannian metric and the Jacobian matrix of the corresponding affine mapping, formulating the equidistribution and alignment conditions for surface meshes, and establishing a meshing energy function based on the conditions. The moving mesh equation is then defined as the gradient system of the energy function, with the nodal mesh velocities being projected onto the underlying surface. The analytical expression for the mesh velocities is obtained in a compact, matrix form, which makes the implementation of the new method on a computer relatively easy and robust. Moreover, it is analytically shown that any mesh trajectory generated by the method remains nonsingular if it is so initially. It is emphasized that the method is developed directly on surface meshes, making no use of any information on surface parameterization. It utilizes surface normal vectors to ensure that the mesh vertices remain on the surface while moving, and also assumes that the initial surface mesh is given. The new method can apply to general surfaces with or without explicit parameterization since the surface normal vectors can be computed based on the current mesh. A selection of two- and three-dimensional examples are presented.

Comparative Analysis of Building Representations in TELEMAC-2D for Flood Inundation in Idealized Urban Districts

This study aims to better understand the impact of different building representations and mesh resolutions on urban flood simulations using the TELEMAC-2D model in idealized urban districts. A series of numerical models based on previous laboratory experiments was established to simulate urban flooding around buildings, wherein different building layouts (aligned and staggered) were modeled for different building representations: building–hole (BH), building–block (BB), and building–resistance (BR) methods. A sensitivity analysis of the Manning coefficient for building grids indicated that the unit-width discharge and water depth in building grids reduce as the Manning coefficient is less than 104 m − 1 / 3 · s . The simulated depths via the BH, BB, and BR methods were compared with the measured data in terms of three accuracy indicators: root mean square error, Pearson product–moment correlation coefficient, and Nash–Sutcliffe efficiency. Observing apparent discrepancies based on the hydrographs was difficult; however, some slight distinctions were observed based on the aforementioned three indicators. The sensitivity of 1, 2, and 5 cm mesh resolutions was also analyzed: results obtained using 1 cm resolution were better than those obtained using other resolutions. The complex flow regime around buildings was also investigated based on mesh resolution, velocity, and Froude number according to our results. This study provides key data regarding urban flood model benchmarks, focusing on the effect of different building representations and mesh resolutions.

On the Geometric Conservation Law for the Non Linear Frequency Domain and Time-Spectral methods

The aim of this paper is to present and validate two new procedures to enforce the Geometric Conservation Law (GCL) on a moving grid for an Arbitrary Lagrangian Eulerian (ALE) formulation for the Euler equations discretized in time for either the Non Linear Frequency Domain (NLFD) or Time-Spectral (TS) methods. The equations are spatially discretized by a structured finite-volume scheme on a hexahedral mesh. The derived methodologies follow a general approach where the positions and the velocities of the grid points are known at each time step. The integrated face mesh velocities are derived either from the Approximation of the Exact Volumetric Increments (AEVI) relative to the undeformed mesh or exactly computed based on a Trilinear Mapping (TRI-MAP) between the physical space and the computational domain. The accuracy of the AEVI method highly depends on the computation of the volumetric increments and limits the temporal-order of accuracy of the deduced integrated face mesh velocities to between one and two. Thus defeating the purpose of the NLFD method which possesses spectral rate of convergence. However, the TRI-MAP method has proven to be more computationally efficient, ensuring the satisfaction of the GCL once the convergence of the temporal derivative of the cell volume is reached in Fourier space. The methods are validated numerically by verifying the conservation of uniform flow and by comparing the integrated face mesh velocities to the exact values derived from the mapping. Their performances for aerodynamic simulation are evaluated for a pitching NACA0012 airfoil.