Steady Incompressible Navier-Stokes Equations Research Articles

Approximate inner solvers for block preconditioning of the incompressible Navier–Stokes problems discretized by isogeometric analysis

AbstractWe deal with efficient numerical solution of the steady incompressible Navier–Stokes equations (NSE) using our in‐house solver based on the isogeometric analysis (IgA) approach. We are interested in the solution of the arising saddle‐point linear systems using preconditioned Krylov subspace methods. Based on our comparison of ideal versions of several state‐of‐the‐art block preconditioners for linear systems arising from the IgA discretization of the incompressible NSE, suitable candidates have been selected. In the present paper, we focus on selecting efficient approximate solvers for solving subsystems within these preconditioning methods. We investigate the impact on the convergence of the outer solver and aim to identify an effective combination. For this purpose, we compare convergence properties of the selected solution approaches for problems with different viscosity values, mesh refinement levels and discretization bases.

A two-level finite element method with grad-div stabilizations for the incompressible Navier–Stokes equations

This article presents and studies a two-level grad-div stabilized finite element discretization method for solving numerically the steady incompressible Navier–Stokes equations. The method consists of two steps. In the first step, we compute a rough solution by solving a nonlinear Navier–Stokes system on a coarse grid. And then, in the second step, we pass the coarse grid solution to a fine grid to linearize the nonlinear term, update the solution by solving a linearized problem based on Newton iterations. In both steps, a grad-div stabilization term is incorporated into the system to reduce the influence of pressure on the approximate velocity. We analyze stability and asymptotic convergence of the approximate solutions, derive explicit dependence of the solution errors on the grad-div stabilization parameter and viscosity. We perform also some numerical tests to validate the theoretical analysis and illustrate the efficiency of the proposed method. Compared with the standard two-level method without stabilizations, the grad-div stabilization term added in present method improves the accuracy of the approximate velocity, accelerates the convergence of the nonlinear iterations for the coarse mesh nonlinear system, and reduces the computational time.

On Leray’s problem in an infinitely long pipe with the Navier-slip boundary condition

The original Leray's problem concerns the well-posedness of weak solutions to the steady incompressible Navier-Stokes equations in a distorted pipe, which approach to the Poiseuille flow subject to the no-slip boundary condition at spacial infinity. In this paper, the same problem with the Navier-slip boundary condition instead of the no-slip boundary condition, is addressed. Due to the complexity of the boundary condition, some new ideas, presented as follows, are introduced to handle the extra difficulties caused by boundary terms. First, the Poiseuille flow in the semi-infinite straight pipe with the Navier-slip boundary condition will be introduced, which will be served as the asymptotic profile of the solution to the generalized Leray's problem at spacial infinity. Second, a solenoidal vector function defined in the whole pipe, satisfying the Navier-slip boundary condition, having the designated flux and equalling to the Poiseuille flow at large distance, will be carefully constructed. This plays an important role in reformulating our problem. Third, the energy estimates depend on a combined $L^2$ estimate of the gradient and the stress tensor of the velocity.

On the multi-objective perspective of discrete topology optimization in fluid-structure interaction problems

Fluid-structure interaction is a challenging topic that addresses fluid and solid physics, as well as the stress coupling between them. Traditional topology optimization methods are performed with coupling load interpolation schemes in order to have some information on fluid flow sensitivity analysis and improve compliance minimization into design-dependent problems. In this paper, we propose a strategy to design fluid-structure systems by combining solid and fluid objectives without the need for the coupling load interpolation scheme. Therefore, we investigate topology optimization applied to fluid-structure interaction problems via a multi-objective formulation. We combine structural compliance with some fluid flow objective functions (energy dissipation, downforce, and drag) subject to a volume constraint. We assume linear elasticity for the solid and the steady incompressible Navier-Stokes equations for the fluid. The optimization problem is solved by using sequential integer linear programming via the TOBS (Topology Optimization of Binary Structures) method with a geometry trimming (GT) technique. It is a gradient-based method that produces explicit (discrete) boundaries that are convenient for coupled physics problems. Numerical results are presented for two and three-dimensional problems considering low to moderate Reynolds numbers. We demonstrate that the proposed multi-objective approach yields physically meaningful designs with improved performance, highlighting that the incorporation of coupling load interpolation into fluid-structure interaction optimization is redundant.

Radial Basis Function Finite Difference Method Based on Oseen Iteration for Solving Two-Dimensional Navier-Stokes Equations.

In this paper, the radial basis function finite difference method is used to solve two-dimensional steady incompressible Navier-Stokes equations. First, the radial basis function finite difference method with polynomial is used to discretize the spatial operator. Then, the Oseen iterative scheme is used to deal with the nonlinear term, constructing the discrete scheme for Navier-Stokes equation based on the finite difference method of the radial basis function. This method does not require complete matrix reorganization in each nonlinear iteration, which simplifies the calculation process and obtains high-precision numerical solutions. Finally, several numerical examples are obtained to verify the convergence and effectiveness of the radial basis function finite difference method based on Oseen Iteration.

A theoretical model for calculating particle trajectory in a vane-type separator

A vane-type separator has been successfully applied to gas-liquid and liquid-liquid separation. In order to evaluate the separation performance of the separator, a theoretical model for calculating the particle motion trajectory in the vane-type separator was developed. Firstly, the simplified steady incompressible Navier-Stokes equations were derived to acquire the equations of tangential velocity and axial velocity of continuous phase flow field. Then, the coefficients were determined by fitting the CFD results and it can be used to obtain the velocity at different swirler structure conditions. Further, the force analysis on the particle was conducted to develop the motion governing equations to predict the trajectory. The predicted separation lengths at different geometric swirl number conditions show a good agreement with the experimental results and its relative error is smaller than 3%. Besides, the effects of the key parameters including density difference, particle size and viscosity of the continuous phase on the separation length were investigated.

A pressure‐robust weak Galerkin finite element method for Navier–Stokes equations

AbstractIn this article, we develop and analyze a novel numerical scheme for the steady incompressible Navier–Stokes equations by the weak Galerkin methods. By employing the divergence‐preserving velocity reconstruction operator, our algorithm can achieve pressure‐robustness, which means, the velocity error is independent of the pressure and the irrotational body force. Error analysis has been established to show the rate of convergence. Numerical experiments are presented to validate the theoretical conclusions.

A three‐step Oseen‐linearized finite element method for incompressible flows with damping

AbstractThis paper introduces a three‐step Oseen‐linearized finite element method for the 2D/3D steady incompressible Navier–Stokes equations with nonlinear damping term. Within this method, we solve a nonlinear problem over a coarse grid followed by solving two Oseen‐linearized problems over a fine grid, which possess the same stiffness matrices with only various right‐hand sides. We theoretically analyze the stability of the present method, and derive optimal error estimates of the finite element solutions. We conduct a series of numerical experiments which support the theoretical analysis and test the effectiveness of the proposed method. We demonstrate numerically that there is a significant improvement in the accuracy of the approximate solutions over those for the standard two‐level method.

Non‐intrusive reduced‐order modeling using convolutional autoencoders

AbstractThe use of reduced‐order models (ROMs) in physics‐based modeling and simulation almost always involves the use of linear reduced basis (RB) methods such as the proper orthogonal decomposition (POD). For some nonlinear problems, linear RB methods perform poorly, failing to provide an efficient subspace for the solution space. The use of nonlinear manifolds for ROMs has gained traction in recent years, showing increased performance for certain nonlinear problems over linear methods. Deep learning has been popular to this end through the use of autoencoders for providing a nonlinear trial manifold for the solution space. In this work, we present a non‐intrusive ROM framework for steady‐state parameterized partial differential equations that uses convolutional autoencoders to provide a nonlinear solution manifold and is augmented by Gaussian process regression (GPR) to approximate the expansion coefficients of the reduced model. When applied to a numerical example involving the steady incompressible Navier–Stokes equations solving a lid‐driven cavity problem, it is shown that the proposed ROM offers greater performance in prediction of full‐order states when compared to a popular method employing POD and GPR over a number of ROM dimensions.

A parallel stabilized quadratic equal-order finite element algorithm for the steady Navier–Stokes equations

This paper is concerned with a parallel stabilized quadratic equal-order finite element algorithm for the steady incompressible Navier–Stokes equations where a fully overlapping domain decomposition is used for parallelization. The basic idea of the studied algorithm is that each processor independently calculates a local stabilized solution in an interesting subdomain on a multiscale mesh that is fine around the subdomain and coarse on the rest of subdomain. The present algorithm can be solved simultaneously by the existing Navier–Stokes codes without substantial recoding. In the light of local a priori estimate for stabilized finite element solution, we analyse the error bound of the stabilized solution. Moreover, several parallel iterative stabilized quadratic equal-order finite element algorithms are presented, and three numerical examples which confirm the high efficiency of the parallel stabilized algorithms are provided.

Local and parallel finite element algorithms for the incompressible Navier-Stokes equations with damping

<p style='text-indent:20px;'>Using two-grid discretizations strategy, we present some local and parallel finite element algorithms for simulating the steady incompressible Navier-Stokes equations with a nonlinear damping term. In these algorithms, we compute a solution of the Navier-Stokes system with a nonlinear damping term on a coarse grid, and then adjust the solution by some local and parallel procedures on overlapped fine grid subomains. With the use of theoretical tool of local a priori estimate of the finite element solution, we estimate the error bounds of the approximate solutions, and derive the algorithmic parameter scalings. Finally, we give some numerical results to verify the theoretical predictions and demonstrate the efficiency of the proposed algorithms.</p>

A novel stabilized Galerkin meshless method for steady incompressible Navier–Stokes equations

In the paper, a novel stabilized meshless method is presented for solving steady incompressible fluid flow problems. For this method, the standard Galerkin discretization is used to momentum equations, where the variational multiscale method is applied to mass conservation equation. Thus, the novel stabilized method can be regarded as a simplification of the variational multiscale element free Galerkin method, but it still retains the advantages of the variational multiscale element free Galerkin method. The present method allows equal linear basis approximation of both velocity and pressure and avoids the Ladyzhenskaya–Babuška–Breezi(LBB) condition. Meanwhile, it can automatically obtain the stabilization tensor. Three Stokes flow and two Navier–Stokes flow problems are applied to validate the accuracy and feasibility of the present method. It is shown that the present stabilized meshless method can guarantee the numerical stability and accuracy for incompressible fluid flow problems. Moreover, it can save computational cost evidently compared with variational multiscale element free Galerkin method.

A Comparative Study on the Hydrodynamic-Energy Loss Characteristics between a Ducted Turbine and a Shaftless Ducted Turbine

The shaftless ducted turbine (abbreviated as SDT), as an extraordinary innovation in tidal current power generation applications, has many advantages, and a wide application prospect. The structure of an SDT resembles a ducted turbine (abbreviated as DT), as both contain blades and a duct. However, there are some structural differences between a DT and a SDT, which can cause significant discrepancy in the hydrodynamic characteristics and flow features. The present work compares the detailed hydrodynamic-energy loss characteristics of a DT and a SDT by means of computational fluid dynamics (CFD), performed by solving the 3D steady incompressible Reynolds-averaged Navier-Stokes (RANS) equations in combination with the Menter’s Shear Stress Transport (SST k−ω) turbulence model and entropy production model. The results show the SDT features a higher power level at low tip speed ratio (TSR) and a potential reduction in potential flow resistance and disturbance with respect to the DT. Moreover, a detail entropy production analysis shows the energy loss is closely related to the flow separation and the reverse flow, and other negative flow factors. The entropy production of the SDT is lessened than that of the DT at different TSR. Unlike the DT, the SDT allows a large mass flow of water to leak through the open-center structure, which plays an important role in improving the wake structure and avoiding the negative flow along the central axis.

Artificial Compressibility Methods for the Incompressible Navier–Stokes Equations Using Lowest-Order Face-Based Schemes on Polytopal Meshes

Abstract We investigate artificial compressibility (AC) techniques for the time discretization of the incompressible Navier–Stokes equations. The space discretization is based on a lowest-order face-based scheme supporting polytopal meshes, namely discrete velocities are attached to the mesh faces and cells, whereas discrete pressures are attached to the mesh cells. This face-based scheme can be embedded into the framework of hybrid mixed mimetic schemes and gradient schemes, and has close links to the lowest-order version of hybrid high-order methods devised for the steady incompressible Navier–Stokes equations. The AC time-stepping uncouples at each time step the velocity update from the pressure update. The performances of this approach are compared against those of the more traditional monolithic approach which maintains the velocity-pressure coupling at each time step. We consider both first-order and second-order time schemes and either an implicit or an explicit treatment of the nonlinear convection term. We investigate numerically the CFL stability restriction resulting from an explicit treatment, both on Cartesian and polytopal meshes. Finally, numerical tests on large 3D polytopal meshes highlight the efficiency of the AC approach and the benefits of using second-order schemes whenever accurate discrete solutions are to be attained.

基于Picard迭代的PN×PN-2谱元法求解定常不可压缩Navier-Stokes方程

A <em>P_N</em>×<em>P</em>_<em>N</em>-2 spectral element method based on the Picard linearized iteration was presented for the solution of 2D steady incompressible Navier-Stokes equations. Through the Picard iteration, the Navier-Stokes equations were converted to a series of Stokes-type equations to be solved with the <em>P_N</em>×<em>P</em>_<em>N</em>-2 spectral element method on the non-staggered grid in each iteration step. In order to eliminate the pseudo pressure mode, the pressure discretization is 2 orders lower than the velocity discretization, and the application of non-staggered grids makes the discretization of the equation convenient and avoids the interpolation error. The Stokes flow, the Kovasznay flow and the lid-driven cavity flow were simulated with the present method. The numerical results show that, the error converges with the spectral accuracy. In addition, avoidance of the pressure oscillation phenomenon indicates the accuracy and reliability of the present method.

A simplified two‐level subgrid stabilized method with backtracking technique for incompressible flows at high Reynolds numbers

AbstractBased on finite element discretization, a simplified two‐level subgrid stabilized method with backtracking technique is proposed for the steady incompressible Navier–Stokes equations at high Reynolds numbers. The method combines the best algorithmic characteristics of the standard two‐level method with backtracking technique and subgrid stabilized method. In this method, we first solve a fully nonlinear Navier–Stokes equations with a subgrid stabilized term on a coarse grid, then solve a simplified subgrid stabilized linear problem on a fine grid, and finally solve a linear correction problem on a coarse grid, where the stabilized term is based on an elliptic projection. The theoretical results show that, with suitable scalings of algorithmic parameters, the method can yield an optimal convergence rate of second‐order. Two numerical results are given to demonstrate the effectiveness of the method.

A posteriori error estimation and adaptivity based on VMS for the incompressible Navier–Stokes equations

In this work an explicit a posteriori error estimator for the steady incompressible Navier–Stokes equations is investigated. The error estimator is based on the variational multiscale theory, where the numerical solution is decomposed in resolved scales (FEM solution) and unresolved scales (FEM error). The error is estimated locally considering the residuals that emerge from the numerical solution and the error inverse-velocity scales, τ’s, associated with each type of residual. These error scales are provided in this paper, which have been computed a-priori solving a set of local problems with unit residuals. Therefore, the computational effort to predict the error is small and its implementation in any FEM code is simple. As an application, a strategy to develop adaptive meshes with the aim of optimizing the computational effort is shown. Numerical examples are presented to test the behavior of the error estimator.

Cavity flow characteristics and applications to kidney stone removal

Abstract

Sobolev regular solutions for the incompressible Navier\u2013Stokes equations in higher dimensions: asymptotics and representation formulae

In this paper, we consider the steady incompressible Navier–Stokes equations in a smooth bounded domain Omega subset mathbb R^n with the dimension nge 3. We first establish asymptotic expansion formulae of Sobolev regular finite energy solutions in Omega. In the second part of this paper, explicit representation formulae of Sobolev regular solutions are showed in the regular polyhedron Omega :=[0,T]^n.

A reduced-order shifted boundary method for parametrized incompressible Navier–Stokes equations

We investigate a projection-based reduced order model of the steady incompressible Navier–Stokes equations for moderate Reynolds numbers. In particular, we construct an “embedded” reduced basis space, by applying proper orthogonal decomposition to the Shifted Boundary Method, a high-fidelity embedded method recently developed. We focus on the geometrical parametrization through level-set geometries, using a fixed Cartesian background geometry and the associated mesh. This approach avoids both remeshing and the development of a reference domain formulation, as typically done in fitted mesh finite element formulations. Two-dimensional computational examples for one and three parameter dimensions are presented to validate the convergence and the efficacy of the proposed approach.