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The Ladyzhenskaya-Prodi-Serrin type L s , r criteria for the regularity of solutions to the incompressible Navier-Stokes equations are fundamental in the study of the millennium problem posted by the Clay Mathematical Institute about the incompressible N-S equations. This global L s , r norm is usually large and hence hard to control. Replacing the global L s , r norm with some kind of local norm is interesting. In this article, we introduce a local L s , r space and establish some localized criteria for the regularity of solutions to the equations. In fact, we obtain some a priori estimates of solutions to the equations depend only on some local L s , r type norms. These local norms, are small for reasonable initial value and shall remain to be small for global regular solutions. Thus, deriving the smallness or even the boundedness of the local L s , r type norms is necessary and sufficient to affirmatively answer the millennium problem. [ABSTRACT FROM AUTHOR]

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton's method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form.

In this paper, we present a methodology for the numerical solving of partial differential equations in 2D geometries with piecewise smooth boundaries via finite element method (FEM) using a Quantized Tensor Train (QTT) format. During the calculations, all the operators and data are assembled and represented in a compressed tensor format. We introduce an efficient assembly procedure of FEM matrices in the QTT format for curvilinear domains. The features of our approach include efficiency in terms of memory consumption and potential expansion to quantum computers. We demonstrate the correctness and advantages of the method by solving a number of problems, including nonlinear incompressible Navier–Stokes flow, in differently shaped domains. [ABSTRACT FROM AUTHOR]

We derived an explicit form of the Jacobian for discrete approximations of a nonlinear initial boundary value problems (IBVPs) in matrix-vector form. The Jacobian is used in Newton’s method to solve the corresponding nonlinear system of equations. The technique was exemplified on the incompressible Navier-Stokes equations discretized using summation-by-parts (SBP) difference operators and weakly imposed boundary conditions using the simultaneous approximation term (SAT) technique. The convergence rate of the iterations is verified by using the method of manufactured solutions. The methodology in this paper can be used on any numerical discretization of IBVPs in matrix-vector form, and it is particularly straightforward for approximations in SBP-SAT form. [ABSTRACT FROM AUTHOR]

We present a numerical method specifically designed for simulating three-dimensional fluid-structure interaction (FSI) problems based on the reference map technique (RMT). The RMT is a fully Eulerian FSI numerical method that allows fluids and large-deformation elastic solids to be represented on a single fixed computational grid. This eliminates the need for meshing complex geometries typical in other FSI approaches and greatly simplifies the coupling between fluid and solids. We develop a three-dimensional implementation of the RMT, parallelized using the distributed memory paradigm, to simulate incompressible FSI with neo-Hookean solids. As part of our method, we develop a field extrapolation scheme that works efficiently in parallel. Through representative examples, we demonstrate the method's suitability in investigating many-body and active systems, as well as its accuracy and convergence. The examples include settling of a mixture of heavy and buoyant soft ellipsoids, lid-driven cavity flow containing a soft sphere, and swimmers actuated via active stress.

In this paper, we first investigate the instantaneous radius of space analyticity for the solutions of 3D Navier–Stokes system with initial data in the Besov spaces B ˙ p , q s (R 3) for p ∈ ] 1 , ∞ [ , q ∈ [ 1 , ∞ ] and s ∈ [ - 1 + 3 p , 3 p [. Then for initial data u 0 ∈ L p (R 3) with p in ]3, 6[, we prove that 3D Navier–Stokes system has a unique solution u = u L + v with and v ∈ L ~ T ∞ ( B ˙ p , p 2 1 - 3 p ) ∩ L ~ T 1 ( B ˙ p , p 2 3 - 3 p ) for some positive time T. Furthermore, we derive an explicit lower bound for the radius of space analyticity of v, which in particular extends the corresponding results in Hu and Zhang (Chin Ann Math Ser B 43:749–772, 2022) with initial data in L p (R 3) for p ∈ [ 3 , 18 / 5 [. [ABSTRACT FROM AUTHOR]

The aim of this paper is to justify the rigorous derivation of the incompressible Navier–Stokes equations from the nonlinear Vlasov–Fokker–Planck (VFP) equation with a constant temperature. Under the incompressible Navier–Stokes scaling, we first establish the global existence of regular solutions to the rescaled nonlinear VFP equation near the Maxwellian, obtaining some uniform bound estimates. We then show the strong convergence of solution to the nonlinear VFP equation towards the incompressible Navier–Stokes system. [ABSTRACT FROM AUTHOR]

• The optimal convergence rate of L 2 -error for the velocity is completely proven. • The corresponding algorithm of our method is simple to be implemented by codes. • The remarkable advantage of our method is demonstrated via adequate numerical experiments. In this work, we develop a high-order pressure-robust method for the rotation form of the incompressible Navier–Stokes equations. The original idea is to change the velocity test functions in the discretization of trilinear and right hand side terms by using an H (div) -conforming velocity reconstruction operator. In order to match the rotation form and ease error analysis, a skew-symmetric discrete trilinear form containing the reconstruction operator is proposed, in which not only the velocity test function is changed. The corresponding well-posed discrete weak formulation stems straight from the classical inf-sup stable mixed conforming high-order finite elements, and it is proved to achieve the pressure-independent velocity errors. Optimal convergence rates of H 1 , L 2 -error for the velocity and L 2 -error for the Bernoulli pressure are completely established. Adequate numerical experiments are presented to demonstrate the theoretical results and remarkable performance of the proposed method. [ABSTRACT FROM AUTHOR]

We present a numerical method specifically designed for simulating three-dimensional fluid-structure interaction (FSI) problems based on the reference map technique (RMT). The RMT is a fully Eulerian FSI numerical method that allows fluids and large-deformation elastic solids to be represented on a single fixed computational grid. This eliminates the need for meshing complex geometries typical in other FSI approaches and greatly simplifies the coupling between fluid and solids. We develop a three-dimensional implementation of the RMT, parallelized using the distributed memory paradigm, to simulate incompressible FSI with neo-Hookean solids. As part of our method, we develop a field extrapolation scheme that works efficiently in parallel. Through representative examples, we demonstrate the method's suitability in investigating many-body and active systems, as well as its accuracy and convergence. The examples include settling of a mixture of heavy and buoyant soft ellipsoids, lid-driven cavity flow containing a soft sphere, and swimmers actuated via active stress. [ABSTRACT FROM AUTHOR]

An Eulerian method to numerically solve incompressible bifluid problems with high density ratio is presented. This method can be considered as an improvement of the Ghost Fluid method, with the specificity of a sharp second-order numerical scheme for the spatial resolution of the discontinuous elliptic problem for the pressure. The Navier-Stokes equations are integrated in time with a fractional step method based on the Chorin scheme and discretized in space on a Cartesian mesh. The bifluid interface is implicitly represented using a level-set function. The advantage of this method is its simplicity to implement in a standard monofluid Navier-Stokes solver while being more accurate and conservative than other simple classical bifluid methods. The numerical tests highlight the improvements obtained with this sharp method compared to the reference standard first-order methods. [ABSTRACT FROM AUTHOR]

In this paper, a new preconditioned iterative method is presented to solve a class of nonsymmetric nonsingular or singular saddle point problems. The implementation of the proposed preconditioned Krylov subspace method avoids solving inverse of Schur complement and only needs to solve one linear sub-system at each step, which implies that it may save considerable costs. Theoretical convergence analysis, including the bounds of eigenvalues and eigenvectors, the degree of the minimal polynomial of the preconditioned matrix, are discussed in details. Moreover, a novel algebraic estimation technique for finding a practical iteration parameter is presented, which is very effective and practical even for large scale problems. At last, some numerical examples are carried, showing that the theoretical results are valid and convincing. [ABSTRACT FROM AUTHOR]

The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the M a c h number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma , and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem of the 3D incompressible Navier–Stokes equations with nonlinear damping term $\alpha|\mathbf{u}|^{\beta-1}\mathbf{u}\ (\alpha>0\ \text{and}\ \beta\geq1)$. In [J. Math. Anal. Appl. 377(2011), 414–419], Zhang et al. obtained global strong solution for $\beta>3$ and the solution is unique provided that $33$.

In this work, physics-informed neural networks are applied to incompressible two-phase flow problems. We investigate the forward problem, where the governing equations are solved from initial and boundary conditions, as well as the inverse problem, where continuous velocity and pressure fields are inferred from scattered-time data on the interface position. We employ a volume of fluid approach, i.e. the auxiliary variable here is the volume fraction of the fluids within each phase. For the forward problem, we solve the two-phase Couette and Poiseuille flow. For the inverse problem, three classical test cases for two-phase modeling are investigated: (i) drop in a shear flow, (ii) oscillating drop and (iii) rising bubble. Data of the interface position over time is generated by numerical simulation. An effective way to distribute spatial training points to fit the interface, i.e. the volume fraction field, and the residual points is proposed. Furthermore, we show that appropriate weighting of losses associated with the residual of the partial differential equations is crucial for successful training. The benefit of using adaptive activation functions is evaluated for both the forward and inverse problem.

• We extend a third-order weighted vartiational rDG(P 1 P 2) to solve the incompressible flows on unstructured grids. • High-order DoFs are reconstructed by minimizing a weighted interfatial jump integration function using variational method. • This new rDG(P 1 P 2) method is able to achieve the designed optimal third order of accuracy. • The computational costs of this new rDG(P 1 P 2) method are significantly reduced compared to the standard DG(P 2) method. In this paper, a third-order reconstructed discontinuous Galerkin (DG) method based on a weighted variational minimization principle, which is denoted as P 1 P 2 (WVr) method, is presented for solving the incompressible flows on unstructured grids. In this method, the first-order degrees of freedom (DoFs) are obtained directly from the underlying second-order DG method, while the second-order DoFs are reconstructed through the weighted variational reconstruction. Specifically, we first introduce a weighted interfacial jump integration (WIJI) function which represents a measure of the jump between the reconstructed polynomial solutions from two neighboring cells. Then, we build the constitutive relations by minimizing this WIJI function using the variational method. A number of incompressible flow problems in both steady and unsteady forms are presented to assess the performance of the proposed P 1 P 2 (WVr) method. The numerical results demonstrate that the P 1 P 2 (WVr) method is able to achieve the designed optimal third-order accuracy at a significantly reduced computational costs. Moreover, when a suitable value of the weight parameter is chosen to be used, the P 1 P 2 (WVr) method outperforms the reconstructed DG methods based on either least-squares or Green-Gauss reconstruction for the simulations of incompressible flows. [ABSTRACT FROM AUTHOR]

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from [1] , we prove that if one component of the velocity remains small enough in a sub-space of H ˙ 1 2 "almost" scaling invariant, then the 3D Navier-Stokes equations are globally wellposed. In a second time, we investigate the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form L T p ( B ˙ 2 , ∞ α , 2 p − 1 2 − α) or L T p ( B ˙ q , ∞ 2 p + 3 q − 2). [ABSTRACT FROM AUTHOR]

We studied an initial boundary value problem for the full incompressible Navier-Stokes equations with viscosity μ and heat conductivity κ depending on temperature by the power law of Chapman-Enskog. With this focus, the existence of global-in-time strong solution, under some appropriate smallness assumptions on initial data, has been proved in this paper. Moreover, the large-time behavior and decay rate estimates of the strong solution are obtained. [ABSTRACT FROM AUTHOR]

A computational vascular fluid–structure interaction framework for the simulation of patient-specific cerebral aneurysm configurations is presented. A new approach for the computation of the blood vessel tissue prestress is also described. Simulations of four patient-specific models are carried out, and quantities of hemodynamic interest such as wall shear stress and wall tension are studied to examine the relevance of fluid–structure interaction modeling when compared to the rigid arterial wall assumption. We demonstrate that flexible wall modeling plays an important role in accurate prediction of patient-specific hemodynamics. Discussion of the clinical relevance of our methods and results is provided.

This paper presents a computational vascular fluid-structure interaction (FSI) methodology and its application to patient-specific aneurysm models of the middle cerebral artery bifurcation. A fully coupled fluid-structural simulation approach is reviewed, and main aspects of mesh generation in support of patient-specific vascular FSI analyses are presented. Quantities of hemodynamic interest such as wall shear stress and wall tension are studied to examine the relevance of FSI modeling as compared to the rigid arterial wall assumption. We demonstrate the importance of including the flexible wall modeling in vascular blood flow simulations by performing a comparison study that involves four patient-specific models of cerebral aneurysms varying in shape and size.

In this paper we present a seven-velocity three-dimensional (D3N7) vectorial lattice Boltzmann method (LBM) including various types of approximations to the pressure and propose a family of two-parameterized second-order boundary schemes with accuracy independent of the boundary location. In order to show the numerical stability of the D3N7 model, we construct a symmetrizer to handle the nonlinear approximations to the pressure. In the meantime, we relate the stability based on the vectorial model to that based on the conventional scalar model through an orthogonal similarity transformation. Finally, two 3-D examples with straight and curved boundaries numerically validate the D3N7 model, including linear and nonlinear approximations to the pressure, together with the proposed boundary schemes. [ABSTRACT FROM AUTHOR]

We introduce a residual-based stabilized formulation for incompressible Navier–Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf–sup stable spaces with H 1 -conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier–Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence. [ABSTRACT FROM AUTHOR]

We develop a novel Hybrid High-Order method for the incompressible Navier–Stokes problem robust for large irrotational body forces. The key ingredients of the method are discrete versions of the body force and convective contributions in the momentum equation formulated in terms of a globally divergence-free velocity reconstruction. Two key properties are mimicked at the discrete level, namely the invariance of the velocity with respect to irrotational body forces and the non-dissipativity of the convective term. A full convergence analysis is carried out, showing optimal orders of convergence under a smallness condition involving only the solenoidal part of the body force. The performance of the method is illustrated by a complete panel of numerical tests, including comparisons that highlight the benefits with respect to more standard formulations. [ABSTRACT FROM AUTHOR]

We consider the incompressible flow of the active hydrodynamics of liquid crystals with inhomogeneous density in the Beris-Edwards hydrodynamics framework. The Landau-de Gennes Q -tensor order parameter is used to describe the liquid crystalline ordering. Faedo-Galerkin's method is adopted to construct the solutions for the initial-boundary value problem. Two levels of approximations are used and the weak convergence is obtained through compactness estimates by new techniques due to the active terms. The existence of global weak solutions in dimension two and three is established in a bounded domain. [ABSTRACT FROM AUTHOR]

This study aims to link critical Reynolds numbers associated with either steady-state or temporal bifurcations to fluid flow regimes described by macroscopic laws (Darcy, Forchheimer) in a periodic 3D Kelvin foam, using direct numerical simulations. We first identify the permeability and inertial coefficient of the Darcy's law and the Forchheimer's one. We explicit the different flow regimes that are accounted for in the Forchheimer's framework (Darcian, weak inertia and strong inertia regimes, respectively). We present an original systematic way to determine the critical Reynolds number associated with both regime transitions. We calculate them over a wide range of porosities and present two power-law correlations that locate these regime transitions for engineering purposes. We have performed pore-scale resolved calculations for various porosities with the Asymptotic Numerical Method (ANM) to find steady-state bifurcations, if any, along with a Linear Stability Analysis (LSA) to find temporal (Hopf) bifurcations from steady-state base flows. All the computed bifurcations occur at Reynolds numbers in the vicinity of the transition from weak to strong inertia regimes, where a change of behavior takes place. On the other hand, no bifurcation has been found in the transition between Darcian and weak inertia regimes. [Display omitted] • Identification of macroscopic properties of a periodic Kelvin cell. • Flow regimes Transition identification. • 3D pore scale numerical simulations. • First time that the ANM and LSA method were used to automatically detect bifurcations in such system. • To correlate critical Reynolds of found bifurcations to the flow regimes. [ABSTRACT FROM AUTHOR]

An Eulerian method to numerically solve incompressible bifluid problems with high density ratio is presented. This method can be considered as an improvement of the Ghost Fluid method, with the specificity of a sharp second-order numerical scheme for the spatial resolution of the discontinuous elliptic problem for the pressure. The Navier–Stokes equations are integrated in time with a fractional step method based on the Chorin scheme and discretized in space on a Cartesian mesh. The bifluid interface is implicitly represented using a level-set function. The advantage of this method is its simplicity to implement in a standard monofluid Navier–Stokes solver while being more accurate and conservative than other simple classical bifluid methods. The numerical tests highlight the improvements obtained with this sharp method compared to the reference standard first-order methods.

Highlights • Gas-stirring of liquid steel can be modeled efficiently using single-phase flow models. • An improved model based on a comparison and redefinition of existing models is presented. • Numerical applications in 2d and 3d show a good agreement with experimental measurements. • The new single-phase model can be used within optimal flow control problems. Abstract Ladle stirring is an important step of the steelmaking process to homogenize the temperature and the chemical composition of the liquid steel and to remove inclusions before casting. Gas is injected from the bottom of the bath to induce a turbulent flow of the liquid steel. Multiphase modeling of ladle stirring can become computationally expensive, especially when used within optimal flow control problems. This note focuses therefore on single-phase flow models. It aims at improving the existing models from the literature. Simulations in a 2d axial-symmetrical configuration, as well as in a real 3d laboratory-scale ladle, are performed. The results obtained with the present model are in a relative good agreement with experimental data and suggest that it can be used as an efficient model in optimal flow control problems. [ABSTRACT FROM AUTHOR]

Abstract Two-dimensional Kelvin–Helmholtz instability problems are popular examples for assessing discretizations for incompressible flows at high Reynolds number. Unfortunately, the results in the literature differ considerably. This paper presents computational studies of a Kelvin–Helmholtz instability problem with high order divergence-free finite element methods. Reference results in several quantities of interest are obtained for three different Reynolds numbers up to the beginning of the final vortex pairing. A mesh-independent prediction of the final pairing is not achieved due to the sensitivity of the considered problem with respect to small perturbations. A theoretical explanation of this sensitivity to small perturbations is provided based on the theory of self-organization of 2D turbulence. Possible sources of perturbations that arise in almost any numerical simulation are discussed. [ABSTRACT FROM AUTHOR]

Many researchers have proposed special treatments for outlet boundary conditions owing to lack of information at the outlet. Among them, the simplest method requires a large enough computational domain to prevent or reduce numerical errors at the boundaries. However, an efficient method generally requires special treatment to overcome the problems raised by the outlet boundary condition used. For example, mass flux is not conserved and the fluid field is not divergence-free at the outlet boundary. Overcoming these problems requires additional computational cost. In this paper, we present a simple and efficient outflow boundary condition for the incompressible Navier–Stokes equations, aiming to reduce the computational domain for simulating flow inside a long channel in the streamwise direction. The proposed outflow boundary condition is based on the transparent equation, where a weak formulation is used. The pressure boundary condition is derived by using the Navier–Stokes equations and the outlet flow boundary condition. In the numerical algorithm, a staggered marker-and-cell grid is used and temporal discretization is based on a projection method. The intermediate velocity boundary condition is consistently adopted to handle the velocity–pressure coupling. Characteristic numerical experiments are presented to demonstrate the robustness and accuracy of the proposed numerical scheme. Furthermore, the agreement of computational results from small and large domains suggests that our proposed outflow boundary condition can significantly reduce computational domain sizes.

We describe our approach to treatment of surface tension in free surface flow simulations on adaptive octree-type grids. The approach is based on the semi-Lagrangian method for the transport and momentum equations and the pressure projection method to enforce the incompressibility constrain. The surface tension contributes to the Dirichlet boundary condition for the pressure equation at the projection step. The treatment of surface tension is based either on accurate finite difference calculation of the mean curvature or on a curvature estimation by the implicit solution of conservative mean curvature flow problem. The first method provides almost the second order accuracy in space for surface tension forces. The second method is characterized by greater stability and essentially larger time steps. Numerical experiments illustrate the main features of the methods. [ABSTRACT FROM AUTHOR]

Abstract Flow around a circular cylinder under the influence of a one dimensional gust is investigated in the present paper. In the present study the gusty flow which impinges on the cylinder is produced by superimposing a single harmonic sinusoidal transverse perturbation velocity component on uniform flow. A gust source region is defined for creating the perturbation velocity. Streamlines, vorticity and iso-velocity contours are used to physically interpret the flow field features. Temporal variations of force coefficients and stagnation-separation points on the cylinder surface are studied. Frequency spectrum of force coefficients reveal interesting facts about the variation of energy content in the gust frequency peak in comparison with vortex shedding frequency peak as a function of Reynolds number (Re) and the angular frequency of the gust (ω). Wherever possible, an attempt has been made to compare the flow features between uniform and gusty flow conditions. The present numerical work has been performed using 'CFRUNS' – a numerical scheme developed by Harichandan and Roy (2010) for solving incompressible two-dimensional Navier-Stokes equations using unstructured grid. [ABSTRACT FROM AUTHOR]

A unified fluid–structure interaction (FSI) formulation is presented for solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the generalized- α scheme are used for the spatial and temporal discretization. The membrane discretization is based on curvilinear surface elements that can describe large deformations and rotations, and also provide a straightforward description for contact. The fluid is described by the incompressible Navier–Stokes equations, and its discretization is based on stabilized Petrov–Galerkin FE. The coupling between fluid and structure uses a conforming sharp interface discretization, and the resulting non-linear FE equations are solved monolithically within the Newton–Raphson scheme. An arbitrary Lagrangian–Eulerian formulation is used for the fluid in order to account for the mesh motion around the structure. The formulation is very general and admits diverse applications that include contact at free surfaces. This is demonstrated by two analytical and three numerical examples exhibiting strong coupling between fluid and structure. The examples include balloon inflation, droplet rolling and flapping flags. They span a Reynolds-number range from 0.001 to 2000. One of the examples considers the extension to rotation-free shells using isogeometric FE. [ABSTRACT FROM AUTHOR]

grad–div stabilization is a classical remedy in conforming mixed finite element methods for incompressible flow problems, for mitigating velocity errors that are sometimes called poor mass conservation. Such errors arise due to the relaxation of the divergence constraint in classical mixed methods, and are excited whenever the spatial discretization has to deal with comparably large and complicated pressures. In this contribution, an analogue of grad–div stabilization for Discontinuous Galerkin methods is studied. Here, the key is the penalization of the jumps of the normal velocities over facets of the triangulation, which controls the measure-valued part of the distributional divergence of the discrete velocity solution. Our contribution is twofold: first, we characterize the limit for arbitrarily large penalization parameters, which shows that the stabilized nonconforming Discontinuous Galerkin methods remain robust and accurate in this limit; second, we extend these ideas to the case of non-simplicial meshes; here, broken grad–div stabilization must be used in addition to the normal velocity jump penalization, in order to get the desired pressure robustness effect. The analysis is performed for the Stokes equations, and more complex flows and Crouzeix–Raviart elements are considered in numerical examples that also show the relevance of the theory in practical settings. [ABSTRACT FROM AUTHOR]

We discretize the incompressible Navier–Stokes equations on a polytopal mesh by using mimetic reconstruction operators. The resulting method conserves discrete mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. To do this we introduce a dual mesh and show how the dual mesh can be completed to a cell-complex. We present existing mimetic reconstruction operators in a new symmetric way applicable to arbitrary dimension, use these to interpolate between primal and dual mesh and derive properties of these operators. Finally, we test both 2- and 3-dimensional versions of the method on a variety of complicated meshes to show its wide applicability. We numerically test the convergence of the method and verify the derived conservation statements. [ABSTRACT FROM AUTHOR]

We present the spatial discretization of the nonlinear convective flux in the incompressible Navier–Stokes equations using local boundary-value problems. The finite-volume discretization of the incompressible Navier–Stokes equations over staggered grids requires the approximation of the cell-face velocity components. In the proposed method, the cell-face velocity components are computed from local boundary-value problems, with the pressure gradient and the gradient of the transverse flux as source terms. The approximation is expressed as the sum of the homogeneous part, corresponding to the convection–diffusion operator, and the inhomogeneous part, corresponding to the source terms. The homogeneous part of the approximation is shown to be a weighted average of the central and the upwind approximation and thus, is first-order convergent over coarse grids and second-order over finer grids. We show that the inhomogeneous part of the approximation effectively removes the numerical diffusion introduced by the homogeneous part. The inclusion of the source terms in the local boundary-value problem results in a more accurate approximation that shows uniform second-order convergence and does not introduce any spurious oscillations. [ABSTRACT FROM AUTHOR]

A simple spatially discrete model problem consisting of mass points and dash-pots is presented which allows for the assessment of the properties of different projection schemes for the solution of the incompressible Navier–Stokes equations. In particular, the temporal accuracy, the stability and the numerical damping are investigated. The present study suggests that it is not possible to formulate a second order accurate projection/pressure-correction scheme which possesses any high-frequency damping. Motivated by this observation two new families of projection schemes are proposed which are developed from the generalised midpoint rule and from the generalised- α method, respectively, and offer control over high-frequency damping. Both schemes are investigated in detail on the basis of the model problem and subsequently implemented in the context of a finite element formulation for the incompressible Navier–Stokes equations. Comprehensive numerical studies of the flow in a lid-driven cavity and the flow around a cylinder are presented. The observations made are in agreement with the conclusions drawn from the model problem. [ABSTRACT FROM AUTHOR]

We present fully resolved computations of flow over periodic hills at the hill-Reynolds numbers ReH=5,600 and ReH=10,595 with the highest fidelity to date. The calculations are performed using spectral incompressible discontinuous Galerkin schemes of 8th and 7th order spatial accuracy, 3rd order temporal accuracy, as well as 34 and 180 million grid points, respectively. We show that the remaining discretization error is small by comparing the results to h- and p-coarsened simulations. We quantify the statistical averaging error of the reattachment length, as this quantity is widely used as an ‘error norm’ in comparing numerical schemes. The results exhibit good agreement with the experimental and numerical reference data, but the reattachment length at ReH=10,595 is predicted slightly shorter than in the most widely used LES references. In the second part of this paper, we show the broad range of capabilities of the numerical method by assessing the scheme for underresolved simulations (implicit large-eddy simulation) of the higher Reynolds number in a detailed h/p convergence study. [ABSTRACT FROM AUTHOR]