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This work presents an iterative method for solving time-asymmetric partial differential equation systems analytically. In this procedure, two symmetric systems in the spatial variable are constructed, and the associated correction functionals are then generated. Applicability of this iterative method is demonstrated in solving the governing equations for both incompressible fluid flow and incompressible micropolar fluid flow.

Smoothed particle hydrodynamics (SPH) is a Lagrangian, meshfree particle method and has been widely applied to different areas in engineering and science. Since its original extension to modeling free surface flows by Monaghan in 1994, SPH has been gradually developed into an attractive approach for modeling viscous incompressible fluid flows. This paper presents an overview on the recent progresses of SPH in modeling viscous incompressible flows in four major aspects which are closely related to the computational accuracy of SPH simulations. The advantages and disadvantages of different SPH particle approximation schemes, pressure field solution approaches, solid boundary treatment algorithms and particle adapting algorithms are described and analyzed. Some new perspectives and future trends in SPH modeling of viscous incompressible flows are discussed.

This review paper presents an overview of Vortex Methods for flow simulation and their different sub-approaches, from their creation to the present. Particle methods distinguish themselves by their intuitive and natural description of the fluid flow as well as their low numerical dissipation and their stability. Vortex methods belong to Lagrangian approaches and allow us to solve the incompressible Navier-Stokes equations in their velocity-vorticity formulation. In the last three decades, the wide range of research works performed on these methods allowed us to highlight their robustness and accuracy while providing efficient computational algorithms and a solid mathematical framework. On the other hand, many efforts have been devoted to overcoming their main intrinsic difficulties, mostly relying on the treatment of the boundary conditions and the distortion of particle distribution. The present review aims to describe the Vortex methods by following their chronological evolution and provides for each step of their development the mathematical framework, the strengths and limits as well as references to applications and numerical simulations. The paper ends with a presentation of some challenging and very recent works based on Vortex methods and successfully applied to problems such as hydrodynamics, turbulent wake dynamics, sediment or porous flows.

Inf-sup stable FEM applied to time-dependent incompressible Navier–Stokes flows are considered. The focus lies on robust estimates for the kinetic and dissipation energies in a twofold sense. Firstly, pressure–robustness ensures the fulfilment of a fundamental invariance principle and velocity error estimates are not corrupted by the pressure approximability. Secondly, Re-semi-robustness means that constants appearing on the right-hand side of kinetic and dissipation energy error estimates (including Gronwall constants) do not explicitly depend on the Reynolds number. Such estimates rely on the essential regularity assumption ∇u∈L1(0,T;L∞(Ω)) which is discussed in detail. In the sense of best practice, we review and establish pressure- and Re-semi-robust estimates for pointwise divergence-free H1 -conforming FEM (like Scott–Vogelius pairs or certain isogeometric based FEM) and pointwise divergence-free H(div)-conforming discontinuous Galerkin FEM. For convection-dominated problems, the latter naturally includes an upwind stabilisation for the velocity which is not gradient-based.

We study the time evolution of an incompressible fluid with axial symmetry without swirl when the vorticity is sharply concentrated on N annuli of radii of the order of [Formula omitted] and thickness [Formula omitted]. We prove that when [Formula omitted], [Formula omitted], the vorticity field of the fluid converges for [Formula omitted] to the point vortex model, in an interval of time which diverges as [Formula omitted]. This generalizes previous result by Cavallaro and Marchioro in (J Math Phys 62:053102, 2021), that assumed [Formula omitted] and in which the convergence was proved for short times only.

In this paper, a penalty virtual element method (VEM) on polyhedral mesh for solving the 3D incompressible flow is proposed and analyzed. The remarkable feature of VEM is that it does not require an explicit computation of the trial and test space, thereby bypassing the obstacle of standard finite element discretizations on arbitrary mesh. The velocity and pressure are approximated by the practical significative lowest equal-order virtual element space pair (Xh,Qh) which does not satisfy the discrete inf-sup condition. Combined with the penalty method, the error estimation is proved rigorously. Numerical results on the 3D polygonal mesh illustrate the theoretical results and effectiveness of the proposed method.

Purpose The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial compressibility method (ACM), and further to study the performance of the combined method SCM-ACM for three-dimensional incompressible flow and heat transfer. Design/methodology/approach The partial differentials in space are discretized by the SCM with Chebyshev polynomial and Chebyshev–Gauss–Lobbatto collocation points. The unsteady artificial compressibility equations are solved to obtain the steady results by the ACM. Three-dimensional exact solutions with trigonometric function form and exponential function form are constructed to test the accuracy of the combined method. Findings The SCM-ACM is developed successfully for three-dimensional incompressible flow and heat transfer with high accuracy that the minimum value of variance can reach. The accuracy increases exponentially along with time marching steps. The accuracy is also improved exponentially with the increasing of nodes before stable accuracy is achieved, while it keeps stably with the increasing of the time step. The central processing unit time increases exponentially with the increasing of nodes and decreasing of the time step. Research limitations/implications It is difficult for the implementation of the implicit scheme by the developed SCM-ACM. The SCM-ACM can be used for solving unsteady impressible fluid flow and heat transfer. Practical implications The SCM-ACM is applied for two classic cases of lid-driven cavity flow and natural convection in cubic cavities. The present results show good agreement with the published results with much fewer nodes. Originality/value The combined method SCM-ACM is developed, firstly, for solving three-dimensional incompressible fluid flow and heat transfer by the SCM and ACM. The performance of SCM-ACM is investigated. This combined method provides a new choice for solving three-dimensional fluid flow and heat transfer with high accuracy.

We study the time evolution of an incompressible fluid with axial symmetry without swirl, assuming initial data such that the initial vorticity is very concentrated inside N small disjoint rings of thickness ε and vorticity mass of the order of |logε|-1 . When ε→0 , we show that the motion of each vortex ring converges to a simple translation with constant speed (depending on the single ring) along the symmetry axis. We obtain a sharp localization of the vorticity support at time t in the radial direction, whereas we state only a concentration property in the axial direction. This is obtained for arbitrary (but fixed) intervals of time. This study is the completion of a previous paper [5], where a sharp localization of the vorticity support was obtained both along the radial and axial directions, but the convergence for ε→0 worked only for short times.

Output-based controllers are known to be fragile with respect to model uncertainties. The standard H∞-control theory provides a general approach to robust controller design based on the solution of the H∞-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.

In this paper, we consider efficient numerical approximations for the phase field model of two-phase incompressible flows. To develop easy-to-implement time stepping scheme, we introduce two types of nonlocal auxiliary variables to achieve highly efficient and fully-decoupled scheme based on the Crank–Nicolson–Leapfrog (CNLF) formula and artificial compression method. We prove that the scheme is linear and unconditionally energy stable. Ample numerical experiments are performed to demonstrate the accuracy, stability and efficiency.