Time step restrictions using semi-explicit methods for the incompressible Navier–Stokes equations

2006 ◽  
Vol 195 (33-36) ◽  
pp. 4433-4447 ◽  
Author(s):  
Wendy Kress ◽  
Per Lötstedt
Keyword(s):  
Stokes Equations ◽  
Explicit Methods ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations

A finite element method for the Navier-Stokes equations in moving domain with application to hemodynamics of the left ventricle

2017 ◽  
Vol 32 (4) ◽  
Author(s):  
Alexander Danilov ◽  
Alexander Lozovskiy ◽  
Maxim Olshanskii ◽  
Yuri Vassilevski
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Left Ventricle ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Computed Tomography Images ◽  
Time Dependent Domain ◽  
Element Method

AbstractThe paper introduces a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method is based on a quasi-Lagrangian formulation of the problem and handling the geometry in a time-explicit way. We prove that numerical solution satisfies a discrete analogue of the fundamental energy estimate. This stability estimate does not require a CFL time-step restriction. The method is further applied to simulation of a flow in a model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

NUMERICAL SOLUTIONS OF 2D AND 3D SLAMMING PROBLEMS

2021 ◽  
Vol 153 (A2) ◽  
Author(s):  
Q Yang ◽  
W Qiu
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
The Body ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
2D And 3D

Slamming forces on 2D and 3D bodies have been computed based on a CIP method. The highly nonlinear water entry problem governed by the Navier-Stokes equations was solved by a CIP based finite difference method on a fixed Cartesian grid. In the computation, a compact upwind scheme was employed for the advection calculations and a pressure-based algorithm was applied to treat the multiple phases. The free surface and the body boundaries were captured using density functions. For the pressure calculation, a Poisson-type equation was solved at each time step by the conjugate gradient iterative method. Validation studies were carried out for 2D wedges with various deadrise angles ranging from 0 to 60 degrees at constant vertical velocity. In the cases of wedges with small deadrise angles, the compressibility of air between the bottom of the wedge and the free surface was modelled. Studies were also extended to 3D bodies, such as a sphere, a cylinder and a catamaran, entering calm water. Computed pressures, free surface elevations and hydrodynamic forces were compared with experimental data and the numerical solutions by other methods.

scholarly journals Spectral Direction Splitting Schemes for the Incompressible Navier-Stokes Equations

2011 ◽  
Vol 1 (3) ◽  
pp. 215-234 ◽  
Author(s):  
Lizhen Chen ◽  
Jie Shen ◽  
Chuanju Xu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Splitting Schemes ◽  
Time Step ◽  
Order Of Accuracy ◽  
Navier Stokes Equations ◽  
Pressure Stabilization ◽  
Legendre Spectral Method ◽  
Direction Splitting ◽  
The Stability

AbstractWe propose and analyze spectral direction splitting schemes for the incompressible Navier-Stokes equations. The schemes combine a Legendre-spectral method for the spatial discretization and a pressure-stabilization/direction splitting scheme for the temporal discretization, leading to a sequence of one-dimensional elliptic equations at each time step while preserving the same order of accuracy as the usual pressure-stabilization schemes. We prove that these schemes are unconditionally stable, and present numerical results which demonstrate the stability, accuracy, and efficiency of the proposed methods.

scholarly journals Numerical Investigation on the Water Entry of Several Different Bow-Flared Sections

2020 ◽  
Vol 10 (22) ◽  
pp. 7952
Author(s):  
Qiang Wang ◽  
Boran Zhang ◽  
Pengyao Yu ◽  
Guangzhao Li ◽  
Zhijiang Yuan
Keyword(s):  
Stokes Equations ◽  
Water Entry ◽  
Navier Stokes ◽  
Hydrodynamic Characteristics ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Surface Evolution ◽  
Mesh Method ◽  
The Difference ◽  
Dynamics Method

The bow-flared section may be simplified in the prediction of slamming loads and whipping responses of ships. However, the difference of hydrodynamic characteristics between the water entry of the simplified sections and that of the original section has not been well documented. In this study, the water entry of several different bow-flared sections was numerically investigated using the computational fluid dynamics method based on Reynolds-averaged Navier–Stokes equations. The motion of the grid around the section was realized using the overset mesh method. Reasonable grid size and time step were determined through convergence studies. The application of the numerical method in the water entry of bow-flared sections was validated by comparing the present predictions with previous numerical and experimental results. Through a comparative study on the water entry of one original section and three simplified sections, the influences of simplification of the bow-flared section on hydrodynamic characteristics, free surface evolution, pressure field, and impact force were investigated and are discussed here.

Numerical integration of the three-dimensional Navier-Stokes equations for incompressible flow

1969 ◽  
Vol 37 (4) ◽  
pp. 727-750 ◽  
Author(s):  
Gareth P. Williams
Keyword(s):  
Poisson Equation ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Wave Flow ◽  
Trigonometric Interpolation ◽  
Time Step ◽  
Navier Stokes Equations ◽  
The Poisson Equation

A method of numerically integrating the Navier-Stokes equations for certain three-dimensional incompressible flows is described. The technique is presented through application to the particular problem of describing thermal convection in a rotating annulus. The equations, in cylindrical polar co-ordinate form, are integrated with respect to time by a marching process, together with the solving of a Poisson equation for the pressure. A suitable form of the finite difference equations gives a computationally-stable long-term integration with reasonably faithful representation of the spatial and temporal characteristics of the flow.Trigonometric interpolation techniques provide accurate (discretely exact) solutions to the Poisson equation. By using an auxiliary algorithm for rapid evaluation of trigonometric transforms, the proportion of computation needed to solve the Poisson equation can be reduced to less than 25% of the total time needed to’ advance one time step. Computing on a UNIVAC 1108 machine, the flow can be advanced one time-step in 2 sec for a 14 × 14 × 14 grid upward to 96 sec for a 60 × 34 × 34 grid.As an example of the method, some features of a solution for steady wave flow in annulus convection are presented. The resemblance of this flow to the classical Eady wave is noted.

Numerical Study of Wave Slamming on an Oscillating Flap

2014 ◽  
Author(s):  
Yanji Wei ◽  
Alan Henry ◽  
Olivier Kimmoun ◽  
Frederic Dias
Keyword(s):  
Stokes Equations ◽  
Numerical Study ◽  
Flow Characteristics ◽  
Ocean Waves ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Numerical Studies ◽  
Wave Slamming ◽  
Volume Method

Bottom hinged Oscillating Wave Surge Converters (OWSCs) are efficient devices for extracting power from ocean waves. There is limited knowledge about wave slamming on such devices. This paper deals with numerical studies of wave slamming on an oscillating flap to investigate the mechanism of slamming events. In our model, the Navier–Stokes equations are discretized using the Finite Volume method with the Volume of Fluid (VOF) approach for interface capturing. Waves are generated by a flap-type wave maker in the numerical wave tank, and the dynamic mesh method is applied to model the motion of the oscillating flap. Basic mesh and time step refinement studies are performed. The flow characteristics in a slamming event are analysed based on numerical results. Various simulations with different flap densities, water depths and wave amplitudes are performed for a better understanding of the slamming.

Numerical Solution of Incompressible Unsteady Flows in Turbomachinery

2000 ◽  
Vol 123 (3) ◽  
pp. 680-685 ◽  
Author(s):  
L. He ◽  
K. Sato
Keyword(s):  
Stokes Equations ◽  
Three Dimensional ◽  
Unsteady Flows ◽  
Navier Stokes ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Flow Solver ◽  
Time Stepping ◽  
Incompressible Viscous Flow ◽  
Dual Time Stepping

A three-dimensional incompressible viscous flow solver of the thin-layer Navier-Stokes equations was developed for the unsteady turbomachinery flow computations. The solution algorithm for the unsteady flows combines the dual time stepping technique with the artificial compressibility approach for solving the incompressible unsteady flow governing equations. For time accurate calculations, subiterations are introduced by marching the equations in the pseudo-time to fully recover the incompressible continuity equation at each real time step, accelerated with a multi-grid technique. Computations of test cases show satisfactory agreements with corresponding theoretical and experimental results, demonstrating the validity and applicability of the present method to unsteady incompressible turbomachinery flows.

Numerical Solution of 3-D Water Entry Problems With a Constrained Interpolation Profile Method

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Qingyong Yang ◽  
Wei Qiu
Keyword(s):  
Numerical Solutions ◽  
Stokes Equations ◽  
Type Equation ◽  
Water Entry ◽  
Navier Stokes ◽  
Time Step ◽  
Constrained Interpolation ◽  
Navier Stokes Equations ◽  
Highly Nonlinear ◽  
Calm Water

This paper presents the numerical solutions of slamming problems for 3D bodies entering calm water with vertical and oblique velocities. The highly nonlinear water entry problems are governed by the Navier-Stokes equations and were solved by a constrained interpolation profile (CIP)-based finite difference method on a fixed Cartesian grid. In the computation, the 3D CIP method was employed for the advection calculations and a pressure-based algorithm was applied for the nonadvection calculations. The solid body and the free surface interfaces were captured by density functions. For the pressure computation, a Poisson-type equation was solved at each time step by using the conjugate gradient iterative method. Validation studies were carried out for a 3D wedge, a cusped body vertically entering calm water, and the oblique entry of a sphere into calm water. The predicted hydrodynamic forces on the wedge, the cusped body, and the sphere were compared with experimental data.

A parallel partition of unity method for the nonstationary Navier-Stokes equations

2017 ◽  
Vol 27 (8) ◽  
pp. 1675-1686 ◽  
Author(s):  
Guangzhi Du ◽  
Liyun Zuo
Keyword(s):  
Stokes Equations ◽  
Partition Of Unity ◽  
Navier Stokes ◽  
Computational Time ◽  
The Finite Element Method ◽  
Partition Of Unity Method ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Content Type ◽  
Standard Galerkin Method

Purpose The purpose of this paper is to propose a parallel partition of unity method (PPUM) to solve the nonstationary Navier-Stokes equations. Design/methodology/approach This paper opted for the nonstationary Navier-Stokes equations by using the finite element method and the partition of unity method. Findings This paper provides one efficient parallel algorithm which reaches the same accuracy as the standard Galerkin method but saves a lot of computational time. Originality/value In this paper, a PPUM is proposed for nonstationary Navier-Stokes. At each time step, the authors only need to solve a series of independent local sub-problems in parallel instead of one global problem.

Navier-Stokes Simulation of the Flow Around an Airfoil in Darrieus Motion

1994 ◽  
Vol 116 (4) ◽  
pp. 870-876 ◽  
Author(s):  
Ko-Foa Tchon ◽  
Ion Paraschivoiu
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
External Boundary ◽  
Time Step ◽  
Navier Stokes Equations ◽  
Infinite Elements ◽  
Quadrilateral Elements ◽  
Minimum Residual ◽  
Naca 0015 ◽  
First Time

In order to study the dynamic stall phenomenon on a Darrieus wind turbine, the incompressible flow field around a moving airfoil is simulated using a noninertial stream function-vorticity formulation of the two-dimensional unsteady Navier-Stokes equations. Spatial discretization is achieved by the streamline upwind Petrov-Galerkin finite element method on a hybrid mesh composed of a structured region of quadrilateral elements in the vicinity of solid boundaries, an unstructured region of triangular elements elsewhere, and a layer of infinite elements surrounding the domain and projecting the external boundary to infinity. Temporal discretization is achieved by an implicit second order finite difference scheme. At each time step, a nonlinear algebraic system is solved by a Newton method. To accelerate computations, the generalized minimum residual method with an incomplete triangular factorization preconditioning is used to solve the linearized Newton systems. The solver is applied to simulate the flow around a NACA 0015 airfoil in Darrieus motion and the results are compared to experimental observations. To the authors’ knowledge, it is the first time that the simulation of such a motion has been performed using the Navier-Stokes equations.

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