Incompressible Flow Problems Research Articles

An adaptive iterative linearised finite element method for implicitly constituted incompressible fluid flow problems and its application to Bingham fluids

In this work, we further extend the theory in [C. Kreuzer, E. Süli, Adaptive finite element approximation of steady flows of incompressible fluids with implicit power-law-like rheology, ESAIM: Math. Model. Numer. Anal. 50 (5) (2016) 1333–1369] on the adaptive finite element analysis of implicitly constituted incompressible fluid flow problems by taking into account the approximation of the nonlinear finite element solutions by an iterative solver. Thereby we show that the computable sequence generated by the modified algorithm still has a weak limit point, which is a solution of the given problem. Our abstract theory can be applied to Bingham fluids, both with and without the convective term. The performance of the adaptive iterative linearised finite element algorithm in this context is illustrated by two numerical experiments.

A phase-change model for diffusion-driven mass transfer problems in incompressible two-phase flows

We present a VOF-based numerical method for incompressible Direct Navier–Stokes (DNS) equations for diffusion-driven phase-change flows. A special emphasis is placed on the treatment of velocity discontinuities across the interface. A novel algorithm is presented to smoothly extend the liquid velocity field across the interface in a way that the interface can be transported by a divergence-free velocity field. The transport of species is treated with a two-scalar approach and special attention is paid to the advection and diffusion steps in order to prevent artificial mass transfer. The methodology is implemented in the open-source code Basilisk and is validated against analytical and semi-analytical models. The relative errors on the relevant quantities are generally below 1% for the finest grids. The method is finally applied to study the growth of electrochemically generated bubbles on planar electrodes and the effect of contact angles and number of nucleation sites is investigated.

Mesh‐free model for Hopf's bifurcation points in incompressible fluid flows problems

AbstractIn this article, a new numerical model is proposed to detect numerically the Hopf bifurcation point of incompressible fluid flows problems. The concept of this model consists to propose a Hopf bifurcation indicator which is solved by a high order mesh free algorithm (HO‐MFA) using the moving least squares (MLS) approximation. This numerical model is based on a strong formulation of Navier–Stokes equations. This procedure allowed us to recover the results of the literature in using only a modelization for low Reynolds number. This new model is tested on the classical flow around a cylindrical obstacle and the backward‐facing step flow to show the advantage and ability of the proposed model to detect the bifurcation points and to determinate the flow periodicity.

Difference between smoothed particle hydrodynamics and moving particle semi-implicit operators

Smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods are representative meshfree particle methods used to compute Lagrangian mechanics. The approximations of differential operators in the SPH and MPS methods have several similarities, but the theoretical discussion of the difference between them is limited. This study mathematically describes the difference via a comprehensive derivation of the first- and second-order derivative operators for each method. The comprehensive derivation indicates that the SPH and MPS operators are consistent with the pressure Poisson equation and moving least-squares approximation, respectively. The variation in consistency can explain the difference in the schemes of the incompressible flow problem. Additionally, the comprehensive derivation of the MPS operators can result in novel second-order and anisotropic operators. This study strengthens the theoretical understanding of the SPH and MPS methods and facilitates the selection of the appropriate method by users. Furthermore, the proposed MPS operators contribute to developing methods with adaptive or multiscale particle distributions.

ISI TAŞINIMI İÇİN YAPISAL OLMAYAN AĞLARDA BİR HIZLANDIRILMIŞ SÜREKSİZ GALERKİN METDU: FORMÜLASYONU VE DOĞRULANMASI

We present a GPU-accelerated method for large scale, coupled incompressible fluid flow and heat transfer problems. A high-order, nodal discontinuous Galerkin method is utilized to discretize governing equations on unstructured triangular meshes. A semi-implicit scheme with explicit treatment of the advective terms and implicit treatment of the split Stokes operators are used for time discretization. The pressure system is solved with a conjugate gradient method together with a fully GPU-accelerated multigrid preconditioner. The code is built on scalable libParanumal solver which is a library of high-performance kernels for high-order discretizations. Performance portability is achieved by using the open concurrent compute abstraction, OCCA. A set of numerical experiments including free and mixed convection problems indicate that our approach experimentally reaches design order of accuracy.

A fourth-order accurate adaptive solver for incompressible flow problems

We present a numerical solver for the incompressible Navier–Stokes equations that combines fourth-order-accurate discrete approximations and an adaptive tree grid (i.e. h-refinement). The scheme employs a novel compact-upwind advection scheme and a 4th-order accurate projection algorithm whereby the numerical solution exactly satisfies the incompressibility constraint. Further, we introduce a new refinement indicator that is tailored to this solver. We show tests and examples to illustrate the consistency, convergence rate and the application for the adaptive solver. The combination of the solver scheme and the proposed grid adaptation algorithm result in fourth-order convergence rates whilst only tuning a single grid-refinement parameter. The speed performance is benchmarked against a well-established second-order accurate adaptive solver alternative. We conclude that the present 4th order solver is an efficient design for problems with strong localization in the spatial-temporal domain and where a high degree of convergence for the solution statistics is desired.

Development of a high-order continuous Galerkin sharp-interface immersed boundary method and its application to incompressible flow problems

The sharp-interface immersed boundary method is a strategy to impose boundary conditions on complex geometries while simplifying the meshing process. This method can offer a high-order of accuracy. Most sharp-interface methods have been developed in the context of the finite volume or the finite difference methods. In this paper, we introduce, verify and validate a novel high-order sharp-interface immersed boundary method in the context of the finite element method. We apply this method to the incompressible Navier–Stokes equations using a pressure-stabilizing/Petrov–Galerkin (PSPG) stabilization. We verify that we obtain a high order of convergence using a Taylor–Couette flow. We validate the results obtained for the drag, lift, and Strouhal number of the flow behind a cylinder at Re=200. We investigate the flow around a sphere at Re=100 and compare the drag force and the characteristics of the recirculating zones with experimental and numerical results obtained in the literature. Finally, the sharp-interface method is used to study a packing of 10 spheres at Re=50, and the results are compared to those obtained with a conformal mesh. It is shown that the sharp-interface immersed boundary preserves the high-order of the finite element scheme and accurately predicts the steady-state and transient flow around particles including the evaluation of the particle-fluid forces.

Robust output-feedback stabilization for incompressible flows using low-dimensional mathcal {H}_{infty }-controllers

Output-based controllers are known to be fragile with respect to model uncertainties. The standard mathcal {H}_{infty }-control theory provides a general approach to robust controller design based on the solution of the mathcal {H}_{infty }-Riccati equations. In view of stabilizing incompressible flows in simulations, two major challenges have to be addressed: the high-dimensional nature of the spatially discretized model and the differential-algebraic structure that comes with the incompressibility constraint. This work demonstrates the synthesis of low-dimensional robust controllers with guaranteed robustness margins for the stabilization of incompressible flow problems. The performance and the robustness of the reduced-order controller with respect to linearization and model reduction errors are investigated and illustrated in numerical examples.

Enriched Nonconforming Multiscale Finite Element Method for Stokes Flows in Heterogeneous Media Based on High-order Weighting Functions

This paper addresses an enriched nonconforming multiscale finite element method (MsFEM) to solve viscous incompressible flow problems in genuine heterogeneous or porous media. In the work of [B. P. Muljadi, et al., Multiscale Model. Simul., 13 (2015), pp. 1146--1172] and [G. Jankowiak and A. Lozinski, arXiv:1802.04389, 2018], a nonconforming MsFEM has been first developed for Stokes problems in such media. Based on these works, we propose an innovative enriched nonconforming MsFEM where the approximation space of both velocity and pressure are enriched by weighting functions which are defined by polynomials of higher-degree. Numerical experiments show that this enriched nonconforming MsFEM improves significantly the accuracy of the nonconforming MsFEMs. Theoretically, this method provides a general framework which allows one to find a good compromise between the accuracy of the method and the computing costs, by varying the degrees of polynomials.

On the initial higher-order pressure convergence in equal-order finite element discretizations of the Stokes system

In incompressible flow problems, the finite element discretization of pressure and velocity can be done through either stable spaces or stabilized pairs. For equal-order stabilized methods with piecewise linear discretization, the classical theory guarantees only linear convergence for the pressure approximation. However, a higher order is often observed, yet seldom discussed, in numerical practice. Such experimental observations may, in the absence of a sound a priori error analysis, mislead the selection of finite element spaces in practical applications. Therefore, we present here a numerical analysis demonstrating that an initial higher-order pressure convergence may in fact occur under certain conditions, for equal-order elements of any degree. Moreover, our numerical experiments clearly indicate that whether and for how long this behavior holds is a problem-dependent matter. These findings confirm that an optimal pressure convergence can in general not be expected when using unbalanced velocity-pressure pairs.

A novel interpolation-free sharp-interface immersed boundary method

This paper describes a novel 2nd order direct forcing immersed boundary method designed for simulation of 2D and 3D incompressible flow problems with complex immersed boundaries. In this formulation, each cell cut by the immersed boundary (IB) is reshaped to conform to the shape of the IB. IBs are modeled as a series of 2D planes in 3D space that connect seamlessly at the edges of the cut cells, in a way that mimics conformal grid. IBs are represented in a continuous and consistent fashion from one cell to another, thus eliminating spatial pressure oscillations originating from inconsistent description of the IB as well as the traditional stair-step problem, leading to a more accurate resolution of the boundary layer. Boundary conditions are enforced at the exact location of the IB devoid of interpolation, which guarantees sound simulations even on grids with high aspect ratio, and enables simulations of flow packed with multiple IBs in close proximity. Boundary conditions for each phase across the IB are enforced independently, yielding a unique capability to solve flows with zero-thickness IBs. Simulations of a large number of 2D and 3D test cases confirm the prowess of the devised immersed boundary method in solving flows over multiple loosely/closely-packed IBs; stationary, moving and highly morphing IBs; as well as IBs with zero-thickness. Results show that predictions from the proposed scheme agree remarkably well with theoretical models and experimental data for both integral variables and local flow fields and they are often with less than 1% deviation from solutions obtained by conformal grid of similar resolution.

Modelling and Simulation of Fluid Flow through a Circular Cylinder with High Reynolds Number: A COMSOL Multiphysics Study

In this study, we intend to investigate the steady‐state and laminar flow of a viscous fluid through a circular cylinder fixed between two parallel plates keeping the aspect ratio of 1 : 5 from cylinder radius to height of the channel. The two‐dimensional, incompressible fluid flow problem has been simulated using COMSOL Multiphysics 5.4 which implements finite element’s procedure. The flow pattern will be investigated by using the Reynolds number from 100 to 1000. The reattachment length formed at the back of the cylinder and drag force when the fluid comes to strike with the front surface of the cylinder is expressed in terms of Reynolds numbers. We propose to calculate the velocity and the pressure before and after the cylinder. For this purpose, two‐line graphs before and after the cylinder will be drawn to check the impact of cylinder on both velocity and pressure. It was found that the percentage change in the velocity as well as pressure before to after the cylinder is changing their behaviours at Re = 700. The study is important because the empirical equations between the vortex’s lengths formed along the cylinder using the linear regression process obtained in this study may be used for future implementation.

A discrete adjoint approach based on finite differences applied to topology optimization of flow problems

Topology optimization methods have been vastly applied to fluid problems with different methods. In this work, the discrete adjoint (DA) approach is used in combination with a finite differences scheme to calculate the sensitivity of incompressible and compressible flow problems. The proposed methodology is especially useful for complex physical problems, where the derivation of the adjoint system can be a challenging task. The algorithm considers the finite volume model to solve the state equations, and uses an adaptive mesh refinement (AMR) scheme during the optimization to enhance the definition of the solid–fluid interface while trying to maintain a competitive computational cost. The implementation is done by using the OpenFOAM platform, which can be modified to address algorithm needs (DA and AMR). The results show an evaluation of the proposed methodology by considering traditional 2D cases of incompressible flow and exploring compressible flow cases with meshes using different cell types and 3D models.

Boundary variation diminished conservative semi-Lagrangian method for both compressible and incompressible flows

Motivated by the enlightenment that diminishing the jump at the cell boundary can effectively reduce numerical dissipation near the critical region, and a novel constrained interpolation profile conservative semi-Lagrangian method is proposed based on a newly designed boundary variation diminishing algorithm. First, a constrained interpolation profile conservative semi-Lagrangian scheme with the piecewise tangent of hyperbola for interface capturing scheme is proposed as one candidate to represent jump-like discontinuities. Second, the constrained interpolation profile conservative semi-Lagrangian scheme with a fourth-order weighted essentially non-oscillatory limiter is used as another candidate to keep the high-order and non-oscillatory reconstruction for smooth solutions. The selection criterion of these two candidates is designed by minimizing the total variations of the first derivative at cell boundaries. A unified pressure-based projection formulation with a fractional step procedure is implemented with the proposed scheme to simulate both compressible and incompressible flows. A variety of numerical tests are studied, including linear and nonlinear scalar wave transport problems and compressible and incompressible flow problems. Results show that the proposed method can effectively eliminate numerical oscillation and diffusion, suggesting it has great potential to be applied to various types of engineering problems including both compressible and incompressible flows.

Separated response surfaces for flows in parametrised domains: Comparison of a priori and a posteriori PGD algorithms

Reduced order models (ROM) are commonly employed to solve parametric problems and to devise inexpensive response surfaces to evaluate quantities of interest in real-time. There are many families of ROMs in the literature and choosing among them is not always a trivial task. This work presents a comparison of the performance of a priori and a posteriori proper generalised decomposition (PGD) algorithms for an incompressible Stokes flow problem in a geometrically parametrised domain. This problem is particularly challenging as the geometric parameters affect both the solution manifold and the computational spatial domain. The difficulty is further increased because multiple geometric parameters are considered and extended ranges of values are analysed for the parameters and this leads to significant variations in the flow features. Using a set of numerical experiments involving geometrically parametrised microswimmers, the two PGD algorithms are extensively compared in terms of their accuracy and their computational cost, expressed as a function of the number of full-order solves required.

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 high-order accurate meshless method for solution of incompressible fluid flow problems

Meshless solution to differential equations using radial basis functions (RBF) is an alternative to grid based methods commonly used. Since the meshless method does not need an underlying connectivity in the form of control volumes or elements, issues such as grid skewness that adversely impact accuracy are eliminated. Gaussian, Multiquadrics and inverse Multiquadrics are some of the most popular RBFs used for the solutions of fluid flow and heat transfer problems. But they have additional shape parameters that have to be fine tuned for accuracy and stability. Moreover, they also face stagnation error when the point density is increased for accuracy. Recently, Polyharmonic splines (PHS) with appended polynomials have been shown to solve the above issues and give rapid convergence of discretization errors with the degree of appended polynomials. In this research, we extend the PHS-RBF method for the solution of incompressible Navier-Stokes equations. A fractional step method with explicit convection and explicit diffusion terms is combined with a pressure Poisson equation to satisfy the momentum and continuity equations. Systematic convergence tests have been performed for five model problems with two of them having analytical solutions. We demonstrate fast convergence both with refinement of number of points and degree of appended polynomials. The method is further applied to solve problems such as lid-driven cavity and vortex shedding over circular cylinder. We have also analyzed the performance of this approach for solution of Euler equations. The proposed method shows promise to solve fluid flow and heat transfer problems in complex domains with high accuracy.

On the Problem of Resonant Incompressible Flow in Ventilated Double Glazing

On the Problem of Resonant Incompressible Flow in Ventilated Double Glazing

Development of artificial compressibility method with elliptic relaxation approach

An artificial compressibility (AC) method resorting to the pressure-based algorithm is developed on a non-orthogonal non-staggered grid using a cell-centered finite-volume Δ-approximation for incompressible fluid flow problems. A pressure time-derivative is used to perturb the equation of continuity. This artifact provokes a density preconditioning which transforms conservative variables to primitive ones; the physical relevance of density preconditioning signifies to convoke compatible linearizations of residuals in collaboration with an AC parameter. The avoidance of pressure-velocity decoupling is enhanced by a consistently formulated cell-face dissipation approach, preserving an increased accuracy with a greater flexibility analogous to the MUSCL (monotone upstream-centered schemes for conservation laws) approach. An elliptic-relaxation scheme having accountability to promoting anisotropic diffusion coefficients is applied to smooth out the pressure residual. Numerical experiments dictate that pertaining to the pseudo-time integration method, the overall contrivance is benefited with an enhanced robustness having sensible oscillation damping characteristics.

Adaptive level set method applied to drag minimization problems constrained by Stokes equations

Abstract We present a level set based adaptive mesh method for solving the drag minimization problem of incompressible flow governed by Stokes equations. Shape sensitivity analysis of the cost functional is derived. During the optimization process, two levels of meshes are adopted. A uniform coarse mesh for evolving the level set function is defined over the whole computational domain. The level set function also serves as the refinement indicator with an easy strategy. The coarse mesh that contains the interfaces then further divided into a uniform fine mesh and the computation is mainly near the interfaces. Therefore, the computational cost is significantly reduced compared with the uniform mesh over the whole domain that achieves the same resolution. Above all, the value of the shape derivative on the boundary can be obtained implicitly, which is a very difficult task in classical optimal shape design problems. The overall optimization procedure is presented and verified with benchmark examples.