Beltrami states for compressible barotropic flows

Beltrami states for compressible barotropic flows are deduced by minimizing the total kinetic energy while keeping the total helicity constant. A Hamiltonian basis for these Beltrami states is sketched. An interesting physical application of the compressible barotropic Beltrami state arises with the Kuzmin-Oseledets formulation of compressible Euler equations. Further, Ertel's invariant is shown to become degenerate in the compressible barotropic Beltrami state.     

Concentration-cancellation for the velocity fields in two dimensional incompressible fluid flows

We show that the weak-L ² limit of a sequence of solutions of the two dimensional incompressible Euler equation is still a solution, provided that a (strong) concentration set for the reduced defect measure has locally finite one dimensional Hausdorff measure in space and time.     

Stability of axisymmetric swirl flows of viscous incompressible fluid

A new method of solution to the problem of stability of the swirl flow of viscous incompressible fluid is developed. The method based on expansion of the required function into power series of radial coordinate allows an avoidance of difficulties related to numerical integration of the system of differential equations with a singular point. Stability of the Poiseuille flow in a rotating pipe is considered as an example.     

Asymmetric self-similar flows of a viscous incompressible fluid along a right-angle corner

Symmetric and asymmetric self-similar flows of a viscous incompressible fluid along a semi-infinite right-angle dihedral corner with a preset streamwise pressure gradient have been considered. Equations describing such flows in the framework of boundary layer approximation have been derived. The asymptotic behavior of solutions of the derived equations far from the corner edge has been theoretically investigated. A new method of computation of these solutions has been developed. Solutions for two types of asymptotic behavior have been obtained.     

On the characteristics-based ACM for incompressible flows

In this paper, the revised characteristics-based (CB) method for incompressible flows recently derived by Neofytou [P. Neofytou, Revision of the characteristic-based scheme for incompressible flows, J. Comput. Phys. 222 (2007) 475–484] has been further investigated. We have derived all the formulas for pressure and velocities from this revised CB method, which is based on the artificial compressibility method (ACM) [A.J. Chorin, A numerical solution for solving incompressible viscous flow problems, J. Comput. Phys. 2 (1967) 12]. Then we analyze the formulations of the original CB method [D. Drikakis, P.A. Govatsos, D.E. Papatonis, A characteristic based method for incompressible flows, Int. J. Numer. Meth. Fluids 19 (1994) 667–685; E. Shapiro, D. Drikakis, Non-conservative and conservative formulations of characteristics numerical reconstructions for incompressible flows, Int. J. Numer. Meth. Eng. 66 (2006) 1466–1482; D. Drikakis, P.K. Smolarkiewicz, On spurious vortical structures, J. Comput. Phys. 172 (2001) 309–325; F. Mallinger, D. Drikakis, Instability in three-dimensional, unsteady stenotic flows, Int. J. Heat Fluid Flow 23 (2002) 657–663; E. Shapiro, D. Drikakis, Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Parts I. Derivation of different formulations and constant density limit, J. Comput. Phys. 210 (2005) 584–607; Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756] to investigate their consistency with the governing flow equations after convergence has been achieved. Furthermore we have implemented both formulations in an unstructured-grid finite volume solver [Y. Zhao, B. Zhang, A high-order characteristics upwind FV method for incompressible flow and heat transfer simulation on unstructured grids, Comput. Meth. Appl. Mech. Eng. 190 (5–7) (2000) 733–756]. Detailed numerical experiments show that both methods give almost identical solutions and convergence rates. Both can generate solutions which agree well with published results and experimental measurements. We thus conclude that both methods, being upwind schemes designed for the ACM, have the same performances in terms of accuracy and convergence speed, even though the revised method is more complex with less stringent assumptions made, while the original CB method is simpler due to the use of extra simplifying assumptions.     

A hybrid scheme for computing incompressible two-phase flows

We propose a hybrid scheme for computations of incompressible two-phase flows. The incompressible constraint has been replaced by a pressure Poisson-like equation and then the pressure is updated by the modified marker and cell method. Meanwhile, the moment equations in the incompressible Navier-Stokes equations are solved by our semidiscrete Hermite central-upwind scheme, and the interface between the two fluids is considered to be continuous and is described implicitly as the 0.5 level set of a smooth function being a smeared out Heaviside function. It is here named the hybrid scheme. Some numerical experiments are successfully carried out, which verify the desired efficiency and accuracy of our hybrid scheme.     

A numerical algorithm for viscous incompressible interfacial flows

We present a new algorithm to numerically simulate two-dimensional viscous incompressible flows with moving interfaces. The motion is updated in time by using the backward difference formula through an iterative procedure. At each iteration, the pseudo-spectral technique is applied in the horizontal direction. The resulting semi-discretized equations constitute a boundary value problem in the vertical coordinate which is solved by decoupling growing and decaying solutions. Numerical tests justify that this method achieves fully second-order accuracy in both the temporal variable and vertical coordinate. As an application of this algorithm, we study the motion of Stokes waves in the presence of viscosity. Our numerical results are consistent with the recently published asymptotic solution for Stokes waves in slightly viscous fluids.     

Viktor Trkal,Beltrami fields,and Trkalian flows

A 1919 contribution of Viktor Trkal on Beltrami fields is contextualized and shown to be of significance in fluid mechanics, time-harmonic electromagnetism and astrophysics.This paper introduces the English translation of an article by V. Trkal from 1919 — see next paper in this issue, p. 97.I gratefully acknowledge the assistance of Drs. Per Lindstrom (Bergen, Norway), Avadh B. Saxena (Los Alamos, New Mexico), Chandra S. Vikrarn (Huntsville, Alabama) and Miloslav Znojil (Prague) in locating old publications. I also thank the editors of this journal for asking me to write this introductory piece as well as for readily agreeing to publish an English translation of Trkal's paper.     

An efficient numerical method for the equations of steady and unsteady flows of homogeneous incompressible Newtonian fluid

In the present paper, we present some numerical methods to solve the equations of steady and unsteady flows, such as those in the microcirculatory bed and large blood vessels (arteries and veins), respectively. In the case of steady flows, the method does not need neither any boundary conditions on pressure nor any small parameter, and the main computation consists of solving some Poisson equations. In the case of unsteady flows, the scheme uses a consistent Neumann boundary condition for the pressure Poisson equation. At each time step, a Poisson and heat equation are solved for the pressure and each velocity component, respectively. The accuracy and efficiency of scheme are checked by a set of numerical tests.     

The Hamiltonian description of incompressible fluid ellipsoids

We construct the noncanonical Poisson bracket associated with the phase space of first order moments of the velocity field and quadratic moments of the density of a fluid with a free-boundary, constrained by the condition of incompressibility. Two methods are used to obtain the bracket, both based on Dirac’s procedure for incorporating constraints. First, the Poisson bracket of moments of the unconstrained Euler equations is used to construct a Dirac bracket, with Casimir invariants corresponding to volume preservation and incompressibility. Second, the Dirac procedure is applied directly to the continuum, noncanonical Poisson bracket that describes the compressible Euler equations, and the moment reduction is applied to this bracket. When the Hamiltonian can be expressed exactly in terms of these moments, a closure is achieved and the resulting finite-dimensional Hamiltonian system provides exact solutions of Euler’s equations. This is shown to be the case for the classical, incompressible Riemann ellipsoids, which have velocities that vary linearly with position and have constant density within an ellipsoidal boundary. The incompressible, noncanonical Poisson bracket differs from its counterpart for the compressible case in that it is not of Lie-Poisson form.     

Turbulent crossed fluxes in incompressible flows

We show in the framework of the stochastic calculus the existence of turbulent crossed fluxes in incompressible flows. Physically, these fluxes are related to the dependence of the phenomenological coefficients on the temperature and concentration variables.     

A massively parallel fractional step solver for incompressible flows

This paper presents a parallel implementation of fractional solvers for the incompressible Navier–Stokes equations using an algebraic approach. Under this framework, predictor–corrector and incremental projection schemes are seen as sub-classes of the same class, making apparent its differences and similarities. An additional advantage of this approach is to set a common basis for a parallelization strategy, which can be extended to other split techniques or to compressible flows.     

Pressure boundary conditions for computing incompressible flows with SPH

In Smoothed Particle Hydrodynamics (SPH) methods for fluid flow, incompressibility may be imposed by a projection method with an artificial homogeneous Neumann boundary condition for the pressure Poisson equation. This is often inconsistent with physical conditions at solid walls and inflow and outflow boundaries. For this reason open-boundary flows have rarely been computed using SPH. In this work, we demonstrate that the artificial pressure boundary condition produces a numerical boundary layer that compromises the solution near boundaries. We resolve this problem by utilizing a “rotational pressure-correction scheme” with a consistent pressure boundary condition that relates the normal pressure gradient to the local vorticity. We show that this scheme computes the pressure and velocity accurately near open boundaries and solid objects, and extends the scope of SPH simulation beyond the usual periodic boundary conditions.     

Two-dimensional MRT LB model for compressible and incompressible flows

In the paper we extend the Multiple-Relaxation-Time （MRT） Lattice Boltzmann （LB） model pro- posed in [Europhys. Lctt., 2010, 90： 54003] so that it is suitable also for incompressible flows. To decrease tile artificial oscillations, the convection term is discretized by the flux linfiter scheme with splitting technique. A new model is validated by some well-known benchmark tests, including Rie- mann problem and Couette flow, and satisfying agreements are obtained between the sinmlation results and ana.lytical ones. In order to show the merit of LB model over traditional methods, the non-equilibrium characteristics of system are solved. The simulation results are consistent with the physical analysis.