We outline the formulation of a novel algorithm which can be used for the solution of both compressible and incompressible Navier-Stokes or Euler equations. Full incompressibility can be dealt with if the algorithm is used in its semi-explicit corm and its structure permits arbitrary interpolation cunctions to be used avoiding the Babuska-Brezzi restriction. In a fully explicit version it introduces a rational form of balancing dissipation avoiding the use of arbitrary parameters and forms for this

A general algorithm for compressible and incompressible flow—Part I. the split, characteristic‐based scheme

SUMMARY An approximate projection scheme based on the pressure correction method is proposed to solve the NavierStokes equations for incompressible flow. The algorithm is applied to the continuous equations; however, there are no problems concerning the choice of boundary conditions of the pressure step. The resulting velocity and pressure are consistent with the original system. For the spatial discretization a high-order spectral element method is chosen. The high-order accuracy allows the use of a diagonal mass matrix, resulting in a very efficient algorithm. The properties of the scheme are extensively tested by means of an analytical test example. The scheme is Mer validated by simulating the laminar flow over a backward-facing step. The solution of the Navier-Stokes equations for unsteady incompressible fluid flow is still a major challenge in the field of computational fluid dynamics. An overview of the most important aspects with respect to the solution of the incompressible Navier-Stokes equations can be found in References 1-5. The Navier-Stokes equations form a set of coupled equations for both velocity and pressure (or, better, the gradient of the pressure). One of the main problems related to the numerical solution of these equations is the imposition of the incompressibility constraint and consequently the calculation of the pressure. The pressure is not a thermodynamic variable, as there is no equation of state for an incompressible fluid. It is an implicit variable which instantaneously 'adjusts itself' in such a way that the velocity remains divergence-free. The gradient of the pressure, on the other hand, is a relevant physical quantity: a force per unit volume. The mathematical importance of the pressure in an incompressible flow lies in the theory of saddle-point problems (of which the steady Stokes equations are an example), where it acts as a Lagrangian multiplier that constrains the velocity to remain divergence-free. There are numerous approaches to solve the Navier-Stokes equations. For the solution of unsteady Navier-Stokes flow perhaps one of the most successful approaches to date is provided by the class of projection methods.&' Projection methods have been developed as a useful way of obtaining an efficient solution algorithm for unsteady incompressible flow. In this paper, projection methods are considered that are applied to the set of continuous equations, yielding very efficient and simple-toimplement algorithms. By decoupling the treatment of velocity and pressure terms, a set of easier-tosolve equations arises: a convectiondiffusion problem for the velocity, yielding an intermediate velocity which is not divergence-free; and a Poisson equation for the pressure (or a related quantity).

An approximate projection scheme for incompressible flow using spectral elements

http://eportfolio.lib.ksu.edu.tw/user/4/9/4990H066/repository/11040421432820342.pdf

Numerical simulation of viscous flow past two circular cylinders of different diameters

An Eulerian description of the streaming process in the lattice Boltzmann equation

The linear stability of incompressible flows is investigated on the basis of the finite element method. The two-dimensional base flows computed numerically over a range of Reynolds numbers are perturbed with three-dimensional disturbances. The three-dimensionality in the flow associated with the secondary instability is identified precisely. First, by using linear stability theory and normal mode analysis, the partial differential equations governing the evolution of perturbation are derived from the linearized Navier–Stokes equation with slight compressibility. In terms of the mixed finite element discretization, in which six-node quadratic Lagrange triangular elements with quadratic interpolation for velocities (P2) and three-node linear Lagrange triangular elements for pressure (P1) are employed, a non-singular generalized eigenproblem is formulated from these equations, whose solution gives the dispersion relation between complex growth rate and wave number. Then, the stabilities of two cases, i.e. the lid-driven cavity flow and flow past a circular cylinder, are examined. These studies determine accurately stability curves to identify the critical Reynolds number and the critical wavelength of the neutral mode by means of the Krylov subspace method and discuss the mechanism of instability. For the cavity flow, the estimated critical results are Rec=920.277±0.010 for the Reynolds number and kc=7.40±0.02 for the wave number. These results are in good agreement with the observation of Aidun et al. and are more accurate than those by the finite difference method. This instability in the cavity is associated with absolute instability [Huerre and Monkewitz, Annu. Rev. Fluid Mech., 22, 473–537 (1990)]. The Taylor–Goertler-like vortices in the cavity are verified by means of the reconstruction of three-dimensional flows. As for the flow past a circular cylinder, the primary instability result shows that the flow has only two-dimensional characteristics at the onset of the von Karman vortex street, when Re<49. The estimated critical values of primary instability are Rec=46.389±0.010 and Stc=0.126 for the Strouhal number. These values are very close to the observation data [Williamson, J. Fluid Mech., 206, 579–627 (1989)] and other stability results [Morzynski and Thiele, Z. Agnew. Math. Mech., 71, T424–T428 (1991); Jackson, J. Fluid Mech., 182, 23–45 (1987)]. This onset of vortex shedding is associated with the symmetry-breaking bifurcation at the Hopf point. Copyright © 1999 John Wiley & Sons, Ltd.

Three-dimensional linear stability analysis of incompressible viscous flows using the finite element method

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. The stability analysis of the one-dimensional purely convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax-Wendroff finite element method. The method is cost effective for incompressible flows, because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present three-step finite element method does not contain any new higher-order derivatives, and is suitable for solving non-linear multi-dimensional problems and flows with complicated outlet boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows, such as the vortex pairing in mixing layer. The properties of the flow fields are displayed by the marker and cell technique. The obtained numerical results are in good agreement with the literature.

A three-step finite element method for unsteady incompressible flows

This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. Stability analysis of the one-dimensional pure convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax--Wendroff finite element method. The method is cost-effective for incompressible flows because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present method does not contain any new higher-order derivatives, which makes it suitable for solving non-linear multidimensional problems and flows with complicated boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows. The numerical results obtained are in good agreement with those in the literature.

The analysis of unsteady incompressible flows by a three-step finite element method

https://iopscience.iop.org/article/10.1016/0169-5983(92)90003-F/pdf

Chun Bo Jiang

Papers

A three-step finite element method for unsteady incompressible flows

The analysis of unsteady incompressible flows by a three-step finite element method

A Taylor–Galerkin-based finite element method for turbulent flows