Oseen Equations Research Articles

Augmenting the grad-div stabilization for Taylor–Hood finite elements with a vorticity stabilization

Abstract The least squares vorticity stabilization (LSVS), proposed in N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzmán, A. Linke, and C. Merdon (“A pressure-robust discretization of Oseen’s equation using stabilization in the vorticity equation,” SIAM J. Numer. Anal., vol. 59, no. 5, pp. 2746–2774, 2021) for the Scott–Vogelius finite element discretization of the Oseen equations, is studied as an augmentation of the popular grad-div stabilized Taylor–Hood pair of spaces. An error analysis is presented which exploits the situation that the velocity spaces of Scott–Vogelius and Taylor–Hood are identical. Convection-robust error bounds are derived under the assumption that the Scott–Vogelius discretization is well posed on the considered grid. Numerical studies support the analytic results and they show that the LSVS-grad-div method might lead to notable error reductions compared with the standard grad-div method.

An Upwind Weak Galerkin Scheme for Convection-Dominated Oseen Equations

An Upwind Weak Galerkin Scheme for Convection-Dominated Oseen Equations

A multi-region analysis of unsteady Oseen's equation for accelerating flow past a circular cylinder

It is difficult to find a valid high-order (of Reynolds number) analytical approximation to the uniform viscous flow past an infinite circular cylinder by applying perturbation techniques. In this study, we show that by accelerating flow past a circular cylinder with taking unsteady Stokes' solution as an initial approximation, higher-order approximation solutions to unsteady Oseen's equation can be obtained by an iteration scheme of perturbation, which exactly satisfy the boundary conditions. As a matter of fact, the nonlinear (convective) term linearized by Oseen's approximation is overweighted especially near the body surface. To eliminate the overweight, a multi-region analysis is proposed in this study to improve analytical unsteady Oseen's solution so as to extend the valid range of Reynolds numbers. In other words, the flow region is hypothetically divided into several annular regions, and the linearized convective term in each region is modified by multiplying with a Carrier's coefficient c (0 < c ≤ 1). The results show that, when the accelerating parameter in the range of 0.5 ≤ a ≤ 4, the maximum effective Reynolds number Reeff of the fourth-order five-region solution is both much larger than that of Stokes' and Oseen's ones. For example, when a = 0.5, Reeff = 22.26 is roughly eight times that of Stokes' solution (Reeff = 2.67), and nine times that of Oseen's solution (Reeff = 2.41). Moreover, when the instantaneous Reynolds number Re(t) is smaller than the Reeff, the flow separation angle and the wake length are both consistent with the numerical results obtained by accurately solving the full Navier-Stokes equations. In addition, the flow properties, including the drag coefficients, streamline patterns, and the pressure coefficients, as well as the vorticity distributions also agree well with the numerical results. This study represents a significant extension of our previous study for unsteady Stokes' equation [Xu et al., Phys. Fluids 35, 033608 (2023)] to unsteady Oseen's equation and its generalization.

Augmented Lagrangian Acceleration of Global-in-Time Pressure Schur Complement Solvers for Incompressible Oseen Equations

This work is focused on an accelerated global-in-time solution strategy for the Oseen equations, which highly exploits the augmented Lagrangian methodology to improve the convergence behavior of the Schur complement iteration. The main idea of the solution strategy is to block the individual linear systems of equations at each time step into a single all-at-once saddle point problem. By elimination of all velocity unknowns, the resulting implicitly defined equation can then be solved using a global-in-time pressure Schur complement (PSC) iteration. To accelerate the convergence behavior of this iterative scheme, the augmented Lagrangian approach is exploited by modifying the momentum equation for all time steps in a strongly consistent manner. While the introduced discrete grad-div stabilization does not modify the solution of the discretized Oseen equations, the quality of customized PSC preconditioners drastically improves and, hence, guarantees a rapid convergence. This strategy comes at the cost that the involved auxiliary problem for the velocity field becomes ill conditioned so that standard iterative solution strategies are no longer efficient. Therefore, a highly specialized multigrid solver based on modified intergrid transfer operators and an additive block preconditioner is extended to solution of the all-at-once problem. The potential of the proposed overall solution strategy is discussed in several numerical studies as they occur in commonly used linearization techniques for the incompressible Navier–Stokes equations.

The scalar Oseen problem in two-dimensional exterior domains

In the present paper, we investigate the scalar Oseen equation in a two-dimensional exterior domain Ω, with Dirichlet boundary condition. This problem represents a linearized form of the steady Navier-Stokes equations. In order to better control the behavior at infinity of functions, the use of weighted Sobolev spaces as a functional framework turns out to be adequate. The established results are related to the existence and the uniqueness of weak solutions on Lp-theory for any real 1<p<∞. Our analysis is mainly based on the principle that linear exterior problems can be solved by combining their properties in bounded domains which are also established in this paper and their properties well known in the whole space R2.

Hypocoercivity in Algebraically Constrained Partial Differential Equations with Application to Oseen Equations

AbstractThe long-time behavior of solutions to different versions of Oseen equations of fluid flow on the 2D torus is analyzed using the concept of hypocoercivity. The considered models are isotropic Oseen equations where the viscosity acts uniformly in all directions and anisotropic Oseen-type equations with different viscosity directions. The hypocoercivity index is determined (if it exists) and it is shown that similar to the finite dimensional case of ordinary differential equations and differential-algebraic equations it characterizes its decay behavior.

The nonconforming virtual element method for Oseen’s equation using a stream-function formulation

We approximate the solution of the stream function formulation of the Oseen equations on general domains by designing a nonconforming Morley-type virtual element method. Under a suitable assumption on the continuous problem’s coefficients, the discrete scheme is well-posed. By introducing an enriching operator, we derive an a priori estimate of the error in a discrete H2 norm. By post-processing the discrete stream function, we compute the discrete velocity and vorticity fields. Furthermore, we recover an approximate pressure field by solving a Stokes-like problem in a nonconforming Crouzeix–Raviart-type virtual element space that is in a Stokes-complex relation with the Morley-type space of the virtual element approximation. Finally, we confirm our theoretical estimates by solving benchmark problems that include a convex and a nonconvex domain.

Feasibility Study of Drill Cuttings Slip Velocity Estimation using BBO DTM-Pade’ Semi-analytical Technique

The study reviewed the use of DTM-Pade’ approximation method of Basset–Boussinesq–Oseen equation (BBO) in predicting particle slip velocity. A hypothetical example is used in the study with three different diameters, 3,5 and 7mm, of spherical particle assumed to represent wide rang of drill cuttings’ sizes. In addition, the drilling fluid is assumed to be incompressible fluid in stationary mode. The Pade’ approximation (K=20, [8 8]) is used, based on literature, in calculating slip velocities through 2 seconds time span. It is concluded that this technique is a practical mean of particle slip velocity prediction at high Pad’ orders using Matlab for a short period of time, around (o.1 to 0.2 seconds). However, BBO equation application range such as, single particle fall in infinite incompressible fluid, small particle size rang (less than 3mm), low particle to fluid density ratio (higher than 0.38), limits its application in oil well drilling operations. Moreover, the necessity of more factors considerations in simulating drilling operations including Non-Newtonian fluids, wall effect, and wide range of irregular shaped drill cuttings densities and sizes urges for the use of generalized BBO equation. In drilling operations, concisely, drill cuttings are usually mixed in the wellbore annulus and for more reliable predictions the application of more sophisticated techniques such as Machine Learning (MA) and Artificial Intelligence (AI) become indispensable.

Coupled clustering strategies for hierarchical matrix preconditioners in saddle point problems

AbstractFluid flow problems can be modeled by the Navier–Stokes or, after linearization, by the Oseen equations. Their discretization results in discrete saddle point problems. These systems of equations are typically very large and need to be solved iteratively. Standard (block‐) preconditioning techniques for saddle point problems rely on an approximation of the Schur complement. Such an approximation can be obtained by a hierarchical (‐) matrix LU‐decomposition, which first approximates the Schur complement explicitly. The computational complexity of this computation depends, among other things, on the hierarchical block structure of the involved matrices. However, widely used techniques do not consider the connection between the discretization grids for the velocity field and the pressure, respectively. Here, we present a hierarchical block structure for the finite element discretization of the gradient operator that is improved by considering the connection between the two involved grids. Numerical results imply that the improved block structure allows for a faster computation of the Schur complement, which is the bottleneck for the set‐up of the ‐matrix LU‐decomposition.

A new class of stabilized virtual element methods for the time-dependent Oseen equations

A new class of stabilized virtual element methods for the time-dependent Oseen equations

An evolving space framework for Oseen equations on a moving domain

This article considers non-stationary incompressible linear fluid equations in a moving domain. We demonstrate the existence and uniqueness of an appropriate weak formulation of the problem by making use of the theory of time-dependent Bochner spaces. It is not possible to directly apply established evolving Hilbert space theory due to the incompressibility constraint. After we have established the well-posedness, we derive and analyse a first order time discretisation of the system.

A generalized weak Galerkin method for Oseen equation

In this work, the authors introduce a generalized weak Galerkin (gWG) finite element method for the evolutionary Oseen equation. The gWG method is based on a new framework for approximating the gradient operator. Both a semi-discrete and a fully-discrete numerical schemes are developed and analyzed for their convergence, stability, and error estimates. The backward Euler discretization is employed in the design of the fully-discrete scheme. Error estimates of optimal order are established mathematically, and they are validated numerically with some benchmark examples.

Continuous interior penalty stabilization for divergence-free finite element methods

Abstract In this paper, we propose, analyze and test numerically a pressure-robust stabilized finite element for a linearized problem in incompressible fluid mechanics, namely, the steady Oseen equation with low viscosity. Stabilization terms are defined by jumps of different combinations of derivatives for the convective term over the element faces of the triangulation of the domain. With the help of these stabilizing terms, and the fact the finite element space is assumed to provide a point-wise divergence-free velocity, an $\mathcal O\big(h^{k+\frac 12}\big)$ error estimate in the $L^2$-norm is proved for the method (in the convection-dominated regime), and optimal order estimates in the remaining norms of the error. Numerical results supporting the theoretical findings are provided.

Modeling virus transport and dynamics in viscous flow medium

ABSTRACT In this paper, we developed a mathematical model to simulate virus transport through a viscous background flow driven by the natural pumping mechanism. Two types of respiratory pathogens viruses (SARS-Cov-2 and Influenza-A) are considered in this model. The Eulerian–Lagrangian approach is adopted to examine the virus spread in axial and transverse directions. The Basset–Boussinesq–Oseen equation is considered to study the effects of gravity, virtual mass, Basset force, and drag forces on the viruses transport velocity. The results indicate that forces acting on the spherical and non-spherical particles during the motion play a significant role in the transmission process of the viruses. It is observed that high viscosity is responsible for slowing the virus transport dynamics. Small sizes of viruses are found to be highly dangerous and propagate rapidly through the blood vessels. Furthermore, the present mathematical model can help to better understand the viruses spread dynamics in a blood flow.

Second-order inertial forces and torques on a sphere in a viscous steady linear flow

We compute the full set of second-order inertial corrections to the instantaneous force and torque acting on a small spherical rigid particle moving unsteadily in a general steady linear flow. This is achieved by using matched asymptotic expansions and formulating the problem in a coordinate system co-moving with the background flow. Effects of unsteadiness and fluid-velocity gradients are assumed to be small, but to dominate in the far field over those of the velocity difference between the body and fluid, making the results essentially relevant to weakly positively or negatively buoyant particles. The outer solution (which at first order is responsible for the Basset–Boussinesq history force at short time and for shear-induced forces such as the Saffman lift force at long time) is expressed via a flow-dependent tensorial kernel. The second-order inner solution brings a number of different contributions to the force and torque. Some are proportional to the relative translational or angular acceleration between the particle and fluid, while others take the form of products of the rotation/strain rate of the background flow and the relative translational or angular velocity between the particle and fluid. Adding the outer and inner contributions, the known added-mass force or the spin-induced lift force are recovered, and new effects involving the velocity gradients of the background flow are revealed. The resulting force and torque equations provide a rational extension of the classical Basset–Boussinesq–Oseen equation incorporating all first- and second-order fluid inertia effects resulting from both unsteadiness and velocity gradients of the carrying flow.

High-order unfitted characteristic finite element methods for moving interface problem of Oseen equations

Moving interface problems have broad applications in multi-phase flows and fluid–structure interactions. The design of high-order numerical methods for such problems becomes challenging, in particular, when the interface is severely deformed. We propose kth-order unfitted characteristic finite element methods (k=2,3,4) for solving two-dimensional Oseen equations with a time-varying interface. In each time step, the interface is not specified with an explicit function, but constructed numerically with a (k+1)th-order interface-tracking algorithm. We obtain optimal convergence orders of both the discrete velocity and the discrete pressure under the H1-norm. Our convergence analysis includes errors of the approximate interface and the time integration and spatial discretization of the governing equations. Numerical experiments are performed on a rotating object and a severely deformed interface and show optimal convergence orders of the method.

BDDC Algorithms for Oseen problems with HDG Discretizations

Abstract The balancing domain decomposition by constraints (BDDC) methods are applied to the linear system arising from the hybridizable discontinuous Galerkin (HDG) discretization of the Oseen equation of incompressible flow. The generalized minimal residual method (GMRES) is used to accelerate the convergence. The original system is first reduced to a subdomain interface problem that is asymmetric indefinite, but can be positive definite in a special subspace. Edge/face average constraints can ensure all BDDC-preconditioned GMRES iterates stay in this special subspace. The convergence of the algorithm is analyzed, and additional edge/face constraints are used to improve the convergence. If the subdomain size is small enough, the number of iterations is independent of the number of subdomains, and depends only slightly on the subdomain problem size when the viscosity is large. When the viscosity is small, the convergence deteriorates. However, the numerical examples can give satisfactory results for the linear HDG discretization even with a small viscosity.

Meshfree-based physics-informed neural networks for the unsteady Oseen equations

We propose the meshfree-based physics-informed neural networks for solving the unsteady Oseen equations. Firstly, based on the ideas of meshfree and small sample learning, we only randomly select a small number of spatiotemporal points to train the neural network instead of forming a mesh. Specifically, we optimize the neural network by minimizing the loss function to satisfy the differential operators, initial condition and boundary condition. Then, we prove the convergence of the loss function and the convergence of the neural network. In addition, the feasibility and effectiveness of the method are verified by the results of numerical experiments, and the theoretical derivation is verified by the relative error between the neural network solution and the analytical solution.

Propagation of H1N1 virus through saliva movement in oesophagus: a mathematical model.

H1N1 (Swine flu) is caused by the influenza A virus which belongs to the Orthomyxoviridae family. Influenza A is very harmful to the elderly, and people with chronic respiratory disease and cardiovascular disease. Therefore, it is essential to analyse the behaviour of virus transmission through the saliva movement in oesophagus. A mathematical paradigm is developed to study the saliva movement under the applications of transverse magnetic field. Jeffrey fluid model is considered for saliva to show the viscoelastic nature. The flow nature is considered creeping and assumptions of long wavelength and low Reynolds number are adopted for analytical solutions. The Basset–Boussinesq–Oseen equation is employed to understand the propagation of H1N1 virus through saliva under the effect of applicable forces such as gravity, virtual mass, basset force, and drag forces. The suitable data for saliva, oesophagus and H1N1 virus are taken from the existing literature for simulation of the results using MATLAB software. From the graphical results, it is observed that the susceptibility to viral infections is less because the magnetic field reduces the motion of the virus particle. Further, the chances of infections in males are more as compared to females and children due to variation in viscosity of saliva. Such findings provide an understanding of the mechanics of the virus floating through the saliva (viscoelastic fluids) in the oesophagus.

Force on inertial particles crossing a two layer stratified turbulent/non-turbulent interface

We study the peculiar motion of inertial solid particles across stratified turbulent/non-turbulent interfaces (STNTI). Previous studies in quiescent stratified layers demonstrated that inertial particles slow down substantially due to an additional force term related to the stratification. Here we report for the first time a similar effect on inertial particles moving across a two-layer STNTI of finite thickness. This problem is addressed both experimentally and numerically: we utilize the three-dimensional particle tracking velocimetry (3D-PTV) in a index-matched STNTI experiment under an oscillating grid, and two direct numerical simulation (DNS) cases of STNTI. The DNSs test the effects of different turbulent forcings on inertial spheres in the turbulent layer and across STNTI and extends the parameter ranges of Reynolds and Froude numbers unfeasible in the experiments. Turbulence is produced in the DNSs using a convective forcing (heat source at the domain boundary) in one case, and a forcing that mimics a vertically oscillating grid in the other. The numerical spheres are tracked, through one-way coupling approach, using a modified Basset–Boussinesq–Oseen equation which includes a stratification-induced term. The stratification force is modelled as an additional buoyancy of a caudal wake with varying density. This algorithm creates Lagrangian trajectories that resemble the motion of inertial particles across stratified interfaces in quiescent and turbulent experiments. Furthermore, numerical results for the STNTI cases help to distinguish the essential features observed in the experiments that are caused by stratification from those that relate to turbulence–particle interactions.