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Fluid flows are omnipresent in nature and engineering disciplines. The reliable computation of fluids has been a long-lasting challenge due to nonlinear interactions over multiple spatio-temporal scales. The compressible Navier-Stokes…

Fluid Dynamics · Physics 2021-12-10 Deniz A. Bezgin , Aaron B. Buhendwa , Nikolaus A. Adams

High-order time-stepping schemes are crucial for simulating incompressible fluid flows due to their ability to capture complex turbulent behavior and unsteady motion. In this work, we propose a third-order accurate numerical scheme for the…

Numerical Analysis · Mathematics 2025-12-22 Kelong Cheng , Jingwei Sun , Hong Zhang

A few basic, intuitive, properties of the Navier-Stokes system of equations for incompressible fluid flows are discussed in this paper. We present a rephrased interpretation of the Navier-Stokes equation in a space having an arbitrary…

General Mathematics · Mathematics 2023-06-28 R. K. Michael Thambynayagam

In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier-Stokes equations based on a high order (Hybrid) Discontinuous Galerkin formulation. The crucial component for the efficiency…

Numerical Analysis · Mathematics 2016-06-29 Christoph Lehrenfeld , Joachim Schöberl

We propose a novel collocated projection method for solving the incompressible Navier-Stokes equations with arbitrary boundaries. Our approach employs non-graded octree grids, where all variables are stored at the nodes. To discretize the…

Numerical Analysis · Mathematics 2025-06-04 Matthew Blomquist , Scott R. West , Adam L. Binswanger , Maxime Theillard

Incompressible flow solvers based on strong-form meshfree methods represent arbitrary geometries without the need for a global mesh system. However, their local evaluations make it difficult to satisfy incompressibility at the discrete…

Numerical Analysis · Mathematics 2026-05-05 Takeharu Matsuda , Satoshi Ii

An immersed-boundary method for the incompressible Navier--Stokes equations is presented. It employs discrete forcing for a sharp discrimination of the solid-fluid interface, and achieves second-order accuracy, demonstrated in examples with…

This paper presents a robust, adaptive numerical scheme for simulating high density ratio and high shear multiphase flows on locally refined Cartesian grids that adapt to the evolving interfaces and track regions of high vorticity. The…

Computational Physics · Physics 2019-09-04 Nishant Nangia , Boyce E. Griffith , Neelesh A. Patankar , Amneet Pal Singh Bhalla

This article investigates matrix-free higher-order discontinuous Galerkin discretizations of the Navier--Stokes equations for incompressible flows with variable viscosity. The viscosity field may be prescribed analytically or governed by a…

Computational Engineering, Finance, and Science · Computer Science 2025-11-04 Richard Schussnig , Niklas Fehn , Douglas Ramalho Queiroz Pacheco , Martin Kronbichler

We present a strong form, meshless point collocation explicit solver for the numerical solution of the transient, incompressible, viscous Navier-Stokes (N-S) equations in two dimensions. We numerically solve the governing flow equations in…

In this study we revisit the problem of computing steady Navier-Stokes flows in two-dimensional unbounded domains. Precise quantitative characterization of such flows in the high-Reynolds number limit remains an open problem of theoretical…

Fluid Dynamics · Physics 2015-01-26 Jonathan Gustafsson , Bartosz Protas

This study proposes an algorithm for modeling compressible flows in spherical shells in nearly incompressible and weakly compressible regimes based on an implicit direction splitting approach. The method retains theoretically expected…

Numerical Analysis · Mathematics 2021-04-21 Roman Frolov , Peter Minev , Aziz Takhirov

For the incompressible Navier--Stokes system with variable density and viscosity, we propose and analyse an IMEX framework treating the convective and diffusive terms semi-implicitly. This extends to variable density and second order in…

Numerical Analysis · Mathematics 2024-10-16 Nicolás Espinoza-Contreras , Gabriel Barrenechea , Ernesto Castillo , Douglas Pacheco

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small…

Fluid Dynamics · Physics 2018-12-26 Niklas Fehn , Wolfgang A. Wall , Martin Kronbichler

The paper develops a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method builds on a quasi-Lagrangian formulation of the problem. The paper provides stability and…

Numerical Analysis · Mathematics 2018-05-15 Alexander Lozovskiy , Maxim A. Olshanskii , Yuri V. Vassilevski

We advance the computation of physical modal expansions for unsteady incompressible flows. Point of departure is a linearization of the Navier-Stokes equations around its fixed point in a frequency domain formulation. While the most…

Fluid Dynamics · Physics 2018-04-24 Marek Morzynski , Wojciech Szeliga , Bernd R. Noack

We propose a fourth order Navier-Stokes solver based on the immersed interface method (IIM), for flow problems with stationary and one-way coupled moving boundaries and interfaces. Our algorithm employs a Runge-Kutta-based projection method…

Fluid Dynamics · Physics 2025-08-22 Xinjie Ji , Changxiao Nigel Shen , Wim M. van Rees

We present a second-order monolithic method for solving incompressible Navier--Stokes equations on irregular domains with quadtree grids. A semi-collocated grid layout is adopted, where velocity variables are located at cell vertices, and…

Numerical Analysis · Mathematics 2022-06-01 Hyuntae Cho , Yesom Park , Myungjoo Kang

We present an immersed interface method for the vorticity-velocity form of the 2D Navier Stokes equations that directly addresses challenges posed by multiply connected domains, nonconvex obstacles, and the calculation of force…

Fluid Dynamics · Physics 2022-07-13 James Gabbard , Thomas Gillis , Philippe Chatelain , Wim M. van Rees

In this note, we consider a viscous incompressible fluid in a finite domain in both two and three dimensions, and examine the question of determining degrees of freedom (projections, functionals, and nodes). Our particular interest is the…

Analysis of PDEs · Mathematics 2021-02-10 Benjamin Faktor , Michael Holst