Flow Of Inviscid Incompressible Fluid Research Articles

Application of the peridynamic differential operator to the solution of sloshing problems in tanks

PurposeThe purpose of this paper is to apply the Peridynamic differential operator (PDDO) to incompressible inviscid fluid flow with moving boundaries. Based on the potential flow theory, a Lagrangian formulation is used to cope with non-linear free-surface waves of sloshing water in 2D and 3D rectangular and square tanks.Design/methodology/approachIn fact, PDDO recasts the local differentiation operator through a nonlocal integration scheme. This makes the method capable of determining the derivatives of a field variable, more precisely than direct differentiation, when jump discontinuities or gradient singularities come into the picture. The issue of gradient singularity can be found in tanks containing vertical/horizontal baffles.FindingsThe application of PDDO helps to obtain the velocity field with a high accuracy at each time step that leads to a suitable geometry updating for the procedure. Domain/boundary nodes are updated by using a second-order finite difference time algorithm. The method is applied to the solution of different examples including tanks with baffles. The accuracy of the method is scrutinized by comparing the numerical results with analytical, numerical and experimental results available in the literature.Originality/valueBased on the investigations, PDDO can be considered a reliable and suitable approach to cope with sloshing problems in tanks. The paper paves the way to apply the method for a wider range of problems such as compressible fluid flow.

Free Streamline Hydrodynamic Analogy for a Linear Elastic Antiplane Slot Problem with Perfectly Plastic Ligaments at Its Ends

It is shown that a two-dimensional analogy exists between an inviscid incompressible fluid flow around two finite-length flat plates (Riabouchinsky problem) and that of a straight slot cut in a linear elastic plate having rounded ends that are on the verge of yielding plastically. For the plates, a uniform flow at infinity is assumed in a direction perpendicular to the side-by-side parallel lines that constitute the two-dimensional geometry of the plates. Correspondingly, the slot has a boundary condition at infinity of a uniform shear traction in the antiplane direction. In the fluid flow problem, two free streamlines form at both ends of the flat plates, while in the slotted plate problem, the rounded ends of the traction-free slot surfaces constitute ligaments of perfectly plastic material. The specific shape of the rounded ends of the slot problem can be inferred by analogy with the fluid flow problem.

Steady-state flows of inviscid incompressible fluid and related particle dynamics in rectangular channels

In this paper, we investigate inviscid incompressible fluid flow in a 2D channel of finite length described by Euler’s equation with Yudovich’s boundary conditions. A border of the channel includes inlet with prescribed vorticity and normal velocity, an outlet with only normal velocity defined and two horizontal impermeable walls. Using analytic and numerical methods we obtained a set of steady-state flows contains both flow-through and recirculation zones, characterized by various distributions of vorticity and stream function. The properties and structure of flows and the effect on it of parameters such as channel length and velocity of fluids in inlet and outlet of the channel is studied. Steady-state flows linear instability and perturbation response was evaluated using spectral approach and long-time numerical simulation by the vortex-in-cell method for a non-steady problem. The second part of the paper is devoted to the investigation of Lagrangian dynamics of a fluid particle in steady-state and periodically driven flows. The factors affecting on stay time of the particle in the flow-through zone of the channel were investigated.

Solving inverse boundary-value problem of aerohydrodynamics in a new statement

We consider a modified inverse boundary-value problem of aerohydrodynamics, in which it is required to find the shape of a wing profile, streamlined by a potential flow of incompressible inviscid fluid, when the distribution of the velocity potential on one section of the profile is given as a function of abscissa, and on the rest of the profile as a function of the ordinate of the profile point.

On the use of Laplace's equation for pressure and a mesh-free method for 3D simulation of nonlinear sloshing in tanks; Reply to the discussion

In this reply, we re-emphasize on the fact that in the simulation of inviscid incompressible fluid flows using Lagrangian description, under some conditions, the divergence operator and the material derivative operator commute and thus the Laplace of the pressure is applicable in practical engineering problems as it has already been used and verified in numerous studies by many researchers. We show that the incompressibility condition suggested by the author of the discussion (discusser), as the replacement for divergence of the velocity field in using finite time increments, is insufficient and wrong. It will be shown that the concept has totally been misinterpreted by the discusser.

Local well‐posedness and zero‐α limit for the Euler‐α equations

In this paper, we prove the local‐in‐time existence and a blow‐up criterion of solutions in the Besov spaces for the Euler‐α equations of inviscid incompressible fluid flows in . We also establish the convergence rate of the solutions of the Euler‐α equations to the corresponding solutions of the Euler equations as the regularization parameter α approaches 0 in . Copyright © 2016 John Wiley & Sons, Ltd.

Generalized Ertel’s theorem and infinite hierarchies of conserved quantities for three-dimensional time-dependent Euler and Navier–Stokes equations

AbstractLocal conservation laws are systematically constructed for three-dimensional time-dependent viscous and inviscid incompressible fluid flows, in primitive variables and vorticity formulation, using the direct construction method. Complete sets of local conservation laws in primitive variables are derived for the case of conservation law multipliers depending on derivatives up to the second order. In the vorticity formulation, there exists an infinite family of vorticity-dependent conservation laws involving an arbitrary differentiable function of space and time, holding for both viscous and inviscid cases. The infinite conservation law family is used to generate further independent hierarchies of conservation laws that essentially involve vorticity and arbitrary flow parameters, which are determined by known evolution equations such as those for momentum, energy or helicity, though not necessarily in the form of a conservation law. The new conservation laws are not restricted to any reduced flow geometry such as planar or axisymmetric limits. Examples are considered.

Finite-time Euler singularities: A Lagrangian perspective

We address the question of whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows as promising candidates for finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high-resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometric properties of vortex line segments. We then verify the assumptions made in the analytical non-blowup criteria introduced by Deng et al. [Commun. PDE 31 (2006)] connecting vortex line geometry (curvature, spreading) to velocity increase, to rule out singular behavior.

On a Long-Wave Instability of Steady-State Plane-Parallel Flows of an Ideal Fluid with Free Boundary in the Gravity Field

We consider the problem of linear stability of steady-state plane-parallel flows of inviscid incompressible fluid of uniform density with free surface in the gravity field. Using the method of coupling the integrals of motion, we prove that sufficient conditions for the stability of these flows to small plane long-wave perturbations are absent. We construct an analytic example of a steady-state plane-parallel flow with small plane long-wave perturbations superimposed as normal modes

A fast BIE iteration method for an arbitrary body in a flow of incompressible inviscid fluid

The paper is concerned with the new iteration algorithm to solve boundary integral equations arising in boundary value problems of mathematical physics. The stability of the algorithm is demonstrated on the problem of a flow around bodies placed in the incompressible inviscid fluid. With a discrete numerical treatment, we approximate the exact matrix by a certain Töeplitz one and then apply a fast algorithm for this matrix, on each iteration step. We illustrate the convergence of this iteration scheme by a number of numerical examples, both for hard and soft boundary conditions. It appears that the method is highly efficient for hard boundaries, being much less efficient for soft boundaries.

Added mass of a pair of discs

Irrotational flow of inviscid incompressible fluid around a pair of coaxial discs of zero thickness, moving along their common axis, is considered. The added mass of an accelerating disc is modified by the presence of a second, coaxial disc. It is reduced if both discs accelerate equally in the same direction along their common axis and increased if their accelerations are equal in magnitude but in opposite directions. Other accelerations can be treated as linear combinations of these two cases. Analytic predictions are obtained when the disc separation 2z0 is large compared with the disc radius a and also when the discs are close (z0≪a). Results for intermediate separations are obtained as numerical solutions of a pair of dual integral equations.

Exponential basis functions in solution of incompressible fluid problems with moving free surfaces

In this paper, a new simple meshless method is presented for the solution of incompressible inviscid fluid flow problems with moving boundaries. A Lagrangian formulation established on pressure, as a potential equation, is employed. In this method, the approximate solution is expressed by a linear combination of exponential basis functions (EBFs), with complex-valued exponents, satisfying the governing equation. Constant coefficients of the solution series are evaluated through point collocation on the domain boundaries via a complex discrete transformation technique. The numerical solution is performed in a time marching approach using an implicit algorithm. In each time step, the governing equation is solved at the beginning and the end of the step, with the aid of an intermediate geometry. The use of EBFs helps to find boundary velocities with high accuracy leading to a precise geometry updating. The developed Lagrangian meshless algorithm is applied to variety of linear and nonlinear benchmark problems. Non-linear sloshing fluids in rigid rectangular two-dimensional basins are particularly addressed.

Non-linear asymptotic stability for the through-passing flows of inviscid incompressible fluid

The paper addresses the dynamics of inviscid incompressible fluid confined within bounded domain with the inflow and outflow of fluid through certain parts of the boundary. This system is non-conservative essentially since the fluxes of energy and vorticity through the flow boundary are not equal to zero. Therefore, the dynamics of such flows should demonstrate the generic non-conservative phenomena such as the asymptotic stability of the equilibria, the onset of instability or the excitation of the self-oscillations, etc. These phenomena are studied extensively for the flows of the viscous fluids but not for the inviscid ones. In this paper, we prove a sufficient condition for the non-linear asymptotic stability of the inviscid steady flows. In this paper, we consider the flows of inviscid incompressible fluid through a given domain thereby assuming that the fluid is allowed to come in and out the flow domain through the prescribed parts of its boundary. Such sort of boundaries and the related fluid flows are referred to as open ones. The relevant examples are the flows through the finite ducts or pipes with the inflow of fluid at one end and outflow on the other end. In the case of open boundaries, the inviscid fluid represent a non-conservative system. The violation of the conservation laws takes place since the inflow and outflow of fluid supplies to and withdraws from the flow both energy and vorticity. The withdrawal can be considered as some sort of dissipation. Therefore, the dynamics of open flows should demonstrate the generic non-conservative phenomena such as the asymptotic stability of the equilibria, the onset of the instability or excitation of the self-oscillations, etc. These features are quite new for the dynamics of inviscid fluid though they are investigated widely for the Navier-Stokes and allied equations. Generally, the result of the withdrawal-supplying competition depends on boundary conditions at the flow inlet and outlet and the withdrawal dominates sometimes. For example, Morgulis and Yudovich (11) have pointed out wide classes of 2D open flows admitting the decreasing Liapunov functionals. Their construction employs the Arnold stability approach. Later on, Gallaire and Chomaz (7) examined the effect of the boundary conditions on the inviscid swirling flows through the finite pipes. Among other

Variational principle in dynamics of a vortex filament

A variational principle governing the dynamics of a vortex filament in unbounded incompressible inviscid fluid flow was suggested by Rasetti and Regge [Physica A 80, 217 (1975)]. This variational principle holds in the approximation taking into account the logarithmically large terms, on the order of ln(La) , L and a being the length and the cross-section radius of the filament, respectively. In this approximation, the Hamiltonian is a function of L . Accordingly, the filament length L is constant in the course of motion. In this paper, a variational principle is obtained that takes into account also the terms on the order of unity. A characteristic feature of the more precise theory is the evolution of the filament length. The variational principle of the vortex filament dynamics is derived from the variational principle for the arbitrary vortex motion of an incompressible inviscid fluid found recently.

Ideal incompressible flow around a wedge tip

A planar analog of conical flows is considered: an inviscid incompressible fluid flow around a wedge tip. A class of conical flows is found where vorticity is transported along streamlines by the potential component of velocity. Problems of a wave “locked” in the corner region and of a flow accelerating along the rib of a dihedral angle are considered. By analogy with an axisymmetric quasi-conical flow, a planar quasi-conical flow of the fluid is determined, namely, the flow inside and outside the region bounded by tangent curves described by a power law. Conditions are found where vorticity and swirl produce a significant effect. An approximate solution of the problem of the fluid flow inside a “zero” angle is obtained.

Hamiltonian moving-particle semi-implicit (HMPS) method for incompressible fluid flows

Particle methods are meshless simulation techniques in which motion of continua is approximated by discrete dynamics of a finite number of particles, and thus have a great degree of flexibility, for instance, in dealing with the complex motion of surfaces or boundaries. In this paper, a particle method is developed from a direct discretization of the Lagrangian for inviscid incompressible fluid flows. The discretized Lagrangian is transformed into a Hamiltonian with holonomic constraints that stem from the incompressibility condition. The RATTLE algorithm that is a symplectic scheme for holonomically constrained Hamiltonian systems is adopted so that the resulting particle method inherits structures possessed by the partial differential equations that govern inviscid incompressible flows. The algorithm of the particle method is semi-implicit, so we call it the Hamiltonian moving-particle semi-implicit (HMPS) method. Some numerical tests indicate the excellence of the HMPS method in conservation of mechanical energy (as well as linear and angular momenta).

Calculation of two-dimensional flows of inviscid incompressible fluid through a rectilinear duct

Calculation of two-dimensional flows of inviscid incompressible fluid through a rectilinear duct

Remarks on the Blow-up of the Euler Equations and the Related Equations

We consider the Euler system for inviscid incompressible fluid flows, and its perturbations in ℝn, n≥2. We prove global well-posedness of this perturbed Euler system in the Triebel-Lizorkin spaces for initial vorticity which is small in the critical Besov norms. Comparison type theorems about the blow-up of solutions are proved between the Euler system and its perturbations. We also study the possiblity of the self-similar type of blow-up of solutions to the equations.

Capillary and Gravity Waves on Bounded Region of Flow Free Surface

AbstractThe problems of two‐dimensional flow of inviscid incompressible fluid are considered. The flow is bounded on one side with rectilinear wall possessing a gap. The fluid is assumed to be weightless in the first problem and surface tension is taken into account. The fluid is to be gravity in the second problem. The problems are solved at linear and nonlinear statement.

On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

AbstractWe prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ℝn,n≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.