Velocity-pressure Formulation Research Articles

Lattice Boltzmann simulation of breaking waves

Water wave breaking at the ocean’s surface is a turbulent phenomenon and encompasses a wide range of physical processes such as generation of bubbles/drops with different spatio-temporal scales. In the present study, breaking of periodic waves in deep water regime has been simulated using phase-field lattice Boltzmann method (LBM). Compared to direct Navier–Stokes equation based numerical models, the present LB model is Poisson-free and explicit in time integration. Both the flow field and interface are solved using LBM with two sets of distribution functions. The normalized pressure-velocity formulation is used for the flow field, whereas the conservative Allen–Cahn equation is used to capture the interface. An impulsively started wave approach with Stokes third order wave profile has been used to initiate the breaking process. Through systematic variation of Weber number and initial steepness, the effect of surface tension on the kinematic, geometric, and energy dissipation characteristics has been studied for three breaker types, viz., spilling, weak plunging, and plunging breakers. The results are in good agreement with the past experimental observations and direct numerical simulation. In particular, the obtained breaking strength for different Weber numbers agrees well with the semi-empirical relation used in a wind-wave forecasting models.

An H-div finite element method for the Stokes equations on polytopal meshes

In this paper, we introduce an H(div)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H({\ ext {div}})$$\\end{document} finite element method on polygonal and polyhedral meshes for solving the Stokes equations in the primary velocity–pressure formulation. An H(div)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H({\ ext {div}})$$\\end{document} finite element on polygons or polyhedra is introduced to approximate the velocity so that the method is pressure robust and produces exact divergence free solutions, when the discontinuous Pk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_k$$\\end{document} finite element is adopted for the pressure approximation. In addition, this method is stabilizer-free, in handling the H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document} non-conformity. Optimal order error estimates are established for the method. Numerical tests are conducted to confirm the theory.

A finite element model for concentration polarization and osmotic effects in a membrane channel

AbstractIn this article, we study a mathematical model that represents the concentration polarization and osmosis effects in a reverse osmosis cross‐flow channel with dense membranes at some of its boundaries. The fluid is modeled using the Navier–Stokes equations and the solution‐diffusion is used to impose the momentum balance on the membrane. The scheme consist of a conforming finite element method with the velocity–pressure formulation for the Navier–Stokes equations, together with a primal scheme for the convection–diffusion equations. The Nitsche's method is used to impose the permeability condition across the membrane. Several numerical experiments are performed to show the robustness of the method. The resulting model accurately replicates the analytical models and predicts similar results to previous works. It is found that the submerged configuration has the highest permeate production, but also has the greatest pressure loss of all three configurations studied.

Accounting for mesoscale geometry and intra-yarn fiber volume fraction distribution on 3D angle-interlock fabric permeability

Understanding resin flows within 3D interlock fabrics involves addressing a dual-scale flow problem. Indeed, the fluid can flow through inter-yarn channels, characterized by a unit cell mesoscale morphology, but also within yarns considered as homogeneous equivalent porous media. The former is assumed to follow the Stokes’ law while the latter be related to the Darcy’s law, directly linked to the microscopic intra-yarn fiber volume fraction (FVF) field. In this work, a coupled Stokes–Darcy steady-state flow within a 3D woven textile, modeled at mesoscale, is solved through a finite element monolithic approach with a mixed velocity-pressure formulation, stabilized by the Variational MultiScale Method (VMS) and implemented in the Z-set software. The fabric permeability tensor is then computed without any assumptions on its nature and analyzed both through its diagonal components, its eigenvectors and eigenvalues. The main contribution of this study lies in examining the influence of variations of the inter-yarn porous medium morphology, through textile compaction and geometrical reduction to prevent edge flows, and the intra-yarn permeability (from complete field to unique value) on the overall fabric permeability.

Optimization of system parameters during conjugate mixed convective flow in a square domain with an oscillating spinning cylinder

A computational analysis has been executed to analyze the combined conduction-mixed convection heat transfer of a rotationally oscillating solid cylinder in a differentially heated square box filled with air. The conjugate mixed convective flow initiates the heat transfer process, where the left-side boundary is isothermally kept to a higher temperature, and the right-side boundary is maintained at a lower temperature. Conduction heat transfer takes place inside the solid cylinder. Navier-Stokes and heat energy conservation equations model the system in the dimensionless pressure-velocity formulation. All these equations are solved via the Galerkin finite element approach. Three different combinations of Grashof (103-105), Reynolds (32–316), and Richardson (0.1–10) numbers are examined to systematically investigate the variations of governing parameters on instantaneous Nusselt numbers and the respective time-averaged values along the hot wall. In each combination, the impacts of the oscillating amplitude and frequency and the variation of cylinder diameter are examined to perform the optimization study. Power spectrum analysis is also done using the Fast Fourier Transform in the frequency domain to visualize the principal frequency of the system. The instantaneous values of the Nusselt number exhibit a wavering pattern over time owing to the recurrent waning and waxing of the thermal boundary layer. For all the cases, the maximum diameter and oscillating amplitude of the cylinder are found to maximize the heat transfer. However, the optimized frequency of the oscillation strongly depends on the selection of the governing parameters. In addition, the principal thermal frequency of the system is determined to be independent of the oscillation frequency.

A mixed pressure-velocity formulation to model flow in heterogeneous porous media with physics-informed neural networks

Current implementations of Physics Informed Neural Networks (PINNs) can experience convergence problems in simulating fluid flow in porous media with highly heterogeneous domains. The objective of this work is to develop an accurate implementation of PINNs to model fluid flow in heterogeneous porous media that avoids alternative approaches in the literature that tend to impose unphysical continuity of the hydraulic conductivity field. The proposed implementation is inspired by the mixed formulation of the governing equations, which separates the mass continuity equation and Darcy's law. This methodology has been used with the mixed finite element method and is known to provide accurate pressure and velocity fields in highly heterogeneous domains. In this work, a similar methodology is applied to PINNs, where the training loss function is based on the decoupled continuity equation and Darcy's law. The separation of the continuity equation and Darcy's law allows for calculating the residual term in the learning loss function without evaluating the spatial derivatives of the discontinuous hydraulic conductivity and associated non-differentiable functions. This approach provides more accurate automatic differentiation of the neural networks for pressure and velocity fields. The new implementation of PINNs can be applied to simulate flow in porous media, regardless of the type and level of heterogeneity with a discontinuous hydraulic conductivity distribution. A variety of structures of the new implementation are investigated. Numerical experiments show that the structure based on different neural networks for the pressure and each component of the velocity field provides the highest accuracy with equivalent training time as other structures. The proposed methodology presents a novel approach to broaden the scope of PINNs in modeling fluid flow in porous media, while preserving the precise representation of the domain's discontinuous features.

Stokes problem with the Coulomb stick–slip boundary conditions in 3D: Formulations, approximation, algorithms, and experiments

The paper deals with the approximation and numerical realization of the Stokes system in 3D with Coulomb’s slip boundary conditions. The weak velocity–pressure formulation leads to an implicit inequality type problem which is discretized by the P1+bubble/P1 elements. To regularize the discrete non-smooth slip term and to release the discrete impermeability condition the duality approach is used. For numerical realization of the resulting saddle-point problem two strategies are proposed, namely (i) its fixed-point formulation solved by the method of successive approximations (ii) the direct numerical solution of the saddle-point problem. The semi-smooth Newton method is used to solve non-smooth equations appearing in both these approaches.

Influence of time-dependent wall temperature fluctuation on conjugate mixed convection inside a chamber with an oscillating cylinder

The current work is an attempt to examine the mixed convective heat flow in a square chamber subjected to time-dependent sinusoidal wall temperature containing a heat-conducting rotationally oscillating circular cylinder at the center. The right wall is kept at a cold temperature and the left wall is subjected to time-varying wall temperature, whereas the remaining walls are thermally insulated. A sinusoidal oscillating velocity is applied on the surface of the oscillating cylinder. This problem has practical relevance for several applications, notably solar collectors, microelectronic cooling systems, chemical reactors, radiators, and heaters. The main goal of this study is to promote heat transfer by investigating optimum system characteristics. The pressure-velocity formulation of the Navier-Stokes and heat energy equations is used to model the fluid domain inside the chamber, and heat transfer inside the cylinder is modeled by the two-dimensional heat conduction equation. Those equations are discretized using the Galerkin finite element method after converting them into non-dimensional forms. The instantaneous and time-averaged Nusselt numbers of the heated wall are examined for three different frequencies of the cylinder oscillation and sinusoidal wall temperature variation. Parametric simulation is carried out for three different cases within a mixed convection regime through a combination of Reynolds (31.62–316.23), Grashof (103–105), and Richardson (0.1–10) numbers. For each different combination, the optimum values of cylinder frequency are determined. Due to the thermal boundary layer's repeated contraction and expansion, the instantaneous Nusselt number demonstrates an oscillating pattern over time. Moreover, the maximum wall temperature frequency is found to maximize the heat transfer process.

No pressure? Energy-consistent ROMs for the incompressible Navier-Stokes equations with time-dependent boundary conditions

This work presents a novel reduced-order model (ROM) for the incompressible Navier-Stokes equations with time-dependent boundary conditions. This ROM is velocity-only, i.e. the simulation of the velocity does not require the computation of the pressure, and preserves the structure of the kinetic energy evolution.The key ingredient of the novel ROM is a decomposition of the velocity into a field with homogeneous boundary conditions and a lifting function that satisfies the mass equation with the prescribed inhomogeneous boundary conditions. This decomposition is inspired by the Helmholtz-Hodge decomposition and exhibits orthogonality of the two components. This orthogonality is crucial to preserve the structure of the kinetic energy evolution. To make the evaluation of the lifting function efficient, we propose a novel method that involves an explicit approximation of the boundary conditions with POD modes, while preserving the orthogonality of the velocity decomposition and thus the structure of the kinetic energy evolution.We show that the proposed velocity-only ROM is equivalent to a velocity-pressure ROM, i.e., a ROM that simulates both velocity and pressure. This equivalence can be generalized to other existing velocity-pressure ROMs and reveals valuable insights in their behaviour.Numerical experiments on test cases with inflow-outflow boundary conditions confirm the correctness and efficiency of the new ROM, and the equivalence with the velocity-pressure formulation.

Interior penalty discontinuous Galerkin methods for the velocity-pressure formulation of the Stokes spectral problem

Interior penalty discontinuous Galerkin methods for the velocity-pressure formulation of the Stokes spectral problem

Divergence-conforming isogeometric collocation methods for the incompressible Navier–Stokes equations

We develop two isogeometric divergence-conforming collocation schemes for incompressible flow. The first is based on the standard, velocity–pressure formulation of the Navier–Stokes equations, while the second is based on the rotational form and includes the vorticity as an unknown in addition to the velocity and pressure. We describe the process of discretizing each unknown using B-splines that conform to a discrete de Rham complex and collocating each governing equation at the Greville abcissae corresponding to each discrete space. Results on mapped domains are obtained by mapping the equations back to a parametric domain using structure-preserving transformations. Numerical results show the promise of the method, including accelerated convergence rates of the three field, vorticity–velocity–pressure scheme when compared to the two field, velocity–pressure scheme.

New numerical method for solving optimal control problem for the stationary Navier-Stokes equations

<abstract><p>The purpose of this article is to propose a new numerical method to solve the problem of optimal control of the steady-state Navier-Stokes equations via the velocity-pressure formulation. We apply a new technique to derive a system of equations from which the solution can be calculated. The spectral method is used to discretize the problem. An extended relaxation method is proposed in the numerical part to ensure the correct convergence of the system. Finally, numerical results are provided to confirm the effectiveness of this approach.</p></abstract>

Acoustic Analysis of Partially Flexible Cavity with Opening

In this study, the SPL at the boundary and within the domain of a rectangular-shaped container with a flexible laminated composite panel with opening have been investigated. Finite Element Analysis (FEM) for the flexible panel has been done and coupled with the acoustic domain using the Boundary Element Method (BEM) through the mobility relation. A MATLAB program has been developed to find flexible panel behavior by FEM and BEM to calculate the sound pressure level for a cavity. Eight-noded isoparametric serendipity elements have been used to model the boundary. A pressure-velocity formulation has been adopted to model the acoustic domain with radiation impedance for window boundary. It has been shown that the presence of a thin flexible plate and opening, drastically changes the SPL pattern inside and at the boundary compared to a rigid cavity due to the relative movement of flexible panel and energy dissipation through the opening by radiation.

A multipoint vorticity mixed finite element method for incompressible Stokes flow

We propose a mixed finite element method for Stokes flow with one degree of freedom per element and facet of simplicial grids. The method is derived by considering the vorticity–velocity–pressure formulation and eliminating the vorticity locally through the use of a quadrature rule. The discrete solution is pointwise divergence-free and the method is pressure robust. The theoretically derived convergence rates are confirmed by numerical experiments.

Fast Multipole Boundary Element Method (FMM/BEM) for the solution of the Navier-Stokes in primitive variables based on the Burton and Miller formulation in two-dimensions

In the present work the fast multipole boundary element method (FMM/BEM) is presented for the solution of incompressible Navier-Stokes equations in 2 dimensions. This is a velocity-pressure primitive variable formulation abbreviated as uv-p. The Stokes-let fundamental solution is used as a weight function in the momentum equation eliminating the pressure from free variable. This way, the LBB conditions are bypassed removing any restriction in the type of grid, placement of nodes, or interpolation. Most of the numerical methods cannot use constant elements for all fields to solve the Navier-Stokes. BEM is the only known so far semi-analytical method capable of such an achievement. The continuity is enforced with the correction of velocity, using a scalar potential, the gradient of which is computed directly with hypersingular equations. The stress hypersingular equation is added to the boundary nodes rendering the well known Burton and Miller formulation and recommendation about the coupling constant is given. Accurate results for cavity Re = 1000 with constant elements and Re=10000 with linear quadrilaterals are obtained and presented. The reached degrees of freedom (dof) are beyond 1 million. Two more problems with mixed BCs are solved. The flow past cylinder is a kind of problem in which both singular and hypersingular equations must be used to ensure convergence. The main advantage of the method is the direct computation of the wall normal stresses or tractions, which is of cornerstone importance in many realistic applications. The tractions are obtained directly from the solution of the system of equations and can be integrated on the surface of a bluff body, to obtain forces and drag and lift coefficients. The pressure is computed in a post-processing phase with hypersingular equation derived in the framework of the present method. Smooth pressure field is obtained in all cases, even when the problem is solved with unstructured grids and constant elements. The method permits the use of non-conforming grids, and it can easily be combined with other numerical methods to obtain wall tractions and stresses via hypersingular equations. Moments for pressure and convective terms are derived.

Stokes-Cahn-Hilliard formulation in sliding bi-periodic frames for the simulation of two-phase flows

A direct numerical simulation technique is proposed in this study for droplet emulsions in simple shear flow in Newtonian-Newtonian liquid systems. We combine the sliding bi-periodic frame with the diffuse interface method in the finite element framework to treat the multi-drop problem without wall effects. We employ the standard velocity-pressure formulation in a creeping regime and the diffuse interface method with the Galerkin weak formulation. Sliding bi-periodic frame constraints are implemented by Lagrangian multiplier. The results are presented for the morphological development of a single drop, two drops, and multiple drops in a sliding bi-periodic frame. To preserve the identical solution for a given relative configuration of droplets (with different droplet locations) within the context of the bi-periodic frame, a phase-translation method is proposed. The method corrects the inconsistency in the time integration, which yields discrepancy in flow solution particularly for problems with thin interfaces. By applying this procedure, accurate solutions for a thin interface are obtained regardless of location of the identical relative configuration of droplets. The effect of phase-field translation is also analyzed for the coalescence of two drops. For the first time, the sliding bi-periodic frame has been implemented with the combination of the standard finite element method and the diffuse interface method, which can be easily extended to more complex rheological liquids to solve industrially important problems.

Weak Versus Strong Wall Boundary Conditions for the Incompressible Navier-Stokes Equations

The pressure-velocity formulation of the incompressible Navier-Stokes equations is solved using high-order finite difference operators satisfying a summation-by-parts property. Two methods for imposing Dirichlet boundary conditions (one strong and one weak) are presented and proven stable using the energy method. Additionally, novel diagonal-norm second-derivative finite difference operators are derived with highly improved boundary accuracy. Accuracy and convergence measurements are presented and verified against theoretical expectations. Numerical experiments also show that subtle effects close to solid walls are more efficiently captured with strong boundary condition imposition methods rather than weak (less degrees of freedom required).

A hybrid Lagrangian–Eulerian flow solver applied to elastically mounted cylinders in tandem arrangement

The fluid–structure interaction of cylinders in tandem arrangement is used as validation basis of a multi-domain Lagrangian–Eulerian hybrid flow solver. The solver is built on a Lagrangian approximation of the entire flow-field using particles, in which grid based Eulerian flow solutions are overlaid, one for every solid body. The Eulerian grids are body fitted but of limited width and may overlap in cases of close proximity of the bodies. The Eulerian and Lagrangian solutions are strongly (implicitly) interconnected in two ways: The Lagrangian solution provides the conditions over the outer boundaries of the Eulerian grids, while the Eulerian solutions update the flow properties of the particles that are within their own domains. Also, implicit is the coupling of the solver with the structural dynamics in case the cylinders are elastically supported. The Lagrangian solver is based on the density–dilatation–vorticity–pressure formulation and makes use of the Particle Mesh method to obtain the flow velocity field while the Eulerian one on the density–velocity–pressure formulation. The hybrid solver is first validated in the case of an isolated rigid cylinder at Re=100. Then the case of a single elastically mounted cylinder at Re=200 is considered, followed by the case of two cylinders in tandem arrangement that are either rigid or elastically mounted. Good agreement with results produced with spectral element and immersed boundary methods is found indicating the capabilities of the hybrid predictions. Also the flexibility of the method in handling complex multi-body fluid–structure interaction problems is demonstrated by allowing grid-overlapping.

Finite Element Discretization Methods for Velocity-Pressure and Stream Function Formulations of Surface Stokes Equations

In this paper we study parametric trace finite element (TraceFEM) and parametric surface finite element (SFEM) discretizations of a surface Stokes problem. These methods are applied both to the Stokes problem in velocity-pressure formulation and in stream function formulation. A class of higher order methods is presented in a unified framework. Numerical efficiency aspects of the two formulations are discussed and a systematic comparison of TraceFEM and SFEM is given. A benchmark problem is introduced in which a scalar reference quantity is defined and numerically determined.

Analysis of high Reynolds free surface flows

ABSTRACT In this paper, we will combine an upwind radial basis function-finite element with direct velocity–pressure formulation to study the two-dimensional Navier-Stokes equations with free surface flows. We will examine this formulation in an improved mixed-order finite element and localized radial basis function method. A particle tracking method and the arbitrary Lagrangian-Eulerian scheme will then be applied to simulate the two-dimensional high Reynolds free surface flows. An upwind improved finite element formulation based on a localized radial basis function differential quadrature (LRBFDQ) method is used to deal with high Reynolds number convection dominated flows. This study successfully obtained very high Reynolds number free surface flows, up to Re = 500 000. Finally, we will demonstrate and discuss the capability and feasibility of the proposed model by simulating two complex free surface flow problems: (1) a highly nonlinear free oscillation flow and (2) a large amplitude sloshing problem. Using even very coarse grids in all computing scenarios, we have achieved good results in accuracy and efficiency.