Inviscid Fluid Dynamics Research Articles

Two-Dimensional Modeling of Nonlinear Dynamics of Forcespinning Jet Formation

Abstract Forcespinning is a novel method that makes use of centrifugal forces to produce nanofibers rapidly and at high yields. To improve and enhance the forcespinning production method, a 2D computational forcespinning inviscid fluid dynamics model is developed. Two models, namely, time-independent and time-dependent, are obtained in order to investigate the effects of various parameters on fiber forcespinning formation (trajectory, jet diameter, tangential velocity). The fluid dynamics equations are solved using the method of multiple scales along with the finite difference method and including slender-jet theory assumptions. It is important to produce jets with small diameters in the micro- and nanorange. Both the Weber (We) and Rossby (Rb) numbers were found to expand the jet trajectory as they increased. Increasing We and/or decreasing Rb was found to decrease the jet diameter. Also, by varying forcespinning parameters, it has been found that the jet radius can be decreased by increasing the jet exit angle in the direction of rotation, reducing the spinneret fluid level, increasing the angular velocity of the spinneret, reducing spinneret length, and/or reducing the orifice diameter. Knowing that jet trajectories are important for designing and positioning of the fiber collector, it has been found that the trajectories expand out with the increase in the jet exit angle in the direction of rotation, increase in the fluid level, increase in the angular velocity, and/or increase in the spinneret length. Production rates and jet radii for any predetermined radial collector distance were also determined.

Mechanisms of Lagrangian Analyticity in Fluids

Certain systems of inviscid fluid dynamics have the property that for solutions that are only slightly better than differentiable in Eulerian variables, the corresponding Lagrangian trajectories are analytic in time. We elucidate the mechanisms in fluid dynamics systems that give rise to this automatic Lagrangian analyticity, as well as mechanisms in some particular fluids systems which prevent it from occurring. We give a conceptual argument for a general fluids model which shows that the fulfillment of a basic set of criteria results in the analyticity of the trajectory maps in time. We then apply this to the incompressible Euler equations to prove analyticity of trajectories for vortex patch solutions. We also use the method to prove that the Lagrangian trajectories are analytic for solutions to the pressureless Euler–Poisson equations, for initial data with moderate regularity. We then examine the compressible Euler equations, and find that the finite speed of propagation in the system is incompatible with the Lagrangian analyticity property. By taking advantage of this finite speed we construct smooth initial data with the property that some corresponding Lagrangian trajectory is not analytic in time. We also study the Vlasov–Poisson system, uncovering another mechanism that deters the analyticity of trajectories. In this case, a key nonlocal operator does not preserve analytic dependence in time. For this system we also construct smooth initial data for which the corresponding solution has some non-analytic Lagrangian trajectory, providing a counterexample to Lagrangian analyticity for a system in which there is an infinite speed of propagation.

Numerical Simulations of Two-Layer Flow past Topography. Part I: The Leeside Hydraulic Jump

Abstract Laboratory observations of the leeside hydraulic jump indicate it consists of a statistically stationary turbulent motion in an overturning wave. From the point of view of the shallow-water equations (SWE), the hydraulic jump is a discontinuity in fluid-layer depth and velocity at which kinetic energy is dissipated. To provide a deeper understanding of the leeside hydraulic jump, three-dimensional numerical solutions of the Navier–Stokes equations (NSE) are carried out alongside SWE solutions for nearly identical physical initial-value problems. Starting from a constant-height layer flowing over a two-dimensional obstacle at constant speed, it is demonstrated that the SWE solutions form a leeside discontinuity owing to the collision of upstream-moving characteristic curves launched from the obstacle. Consistent with the SWE solution, the NSE solution indicates the leeside hydraulic jump begins as a steepening of the initially horizontal density interface. Subsequently, the NSE solution indicates overturning of the density interface and a transition to turbulence. Analysis of the initial-value problem in these solutions shows that the tendency to form either the leeside height–velocity discontinuity in the SWE or the overturning density interface in the exact NSE is a feature of the inviscid, nonturbulent fluid dynamics. Dissipative turbulent processes associated with the leeside hydraulic jump are a consequence of the inviscid fluid dynamics that initiate and maintain the locally unstable conditions.

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.

Symmetries and conservation laws of the Euler equations in Lagrangian coordinates

We consider the Euler equations of incompressible inviscid fluid dynamics. We discuss a variational formulation of the governing equations in Lagrangian coordinates. We compute variational symmetries of the action functional and generate corresponding conservation laws in Lagrangian coordinates. We clarify and demonstrate relationships between symmetries and the classical balance laws of energy, linear momentum, center of mass, angular momentum, and the statement of vorticity advection. Using a newly obtained scaling symmetry, we obtain a new conservation law for the Euler equations in Lagrangian coordinates in n-dimensional space. The resulting integral balance relates the total kinetic energy to a new integral quantity defined in Lagrangian coordinates. This relationship implies an inequality which describes the radial deformation of the fluid, and shows the non-existence of time-periodic solutions with nonzero, finite energy.

Two-Layer Stratified Flow past a Valley

Abstract Observations and models of nocturnal katabatic winds indicate strong low-level stability with much weaker stability aloft. When such winds encounter an embedded depression in an otherwise smooth sloping plane, the flow responds in a manner that is largely describable by the inviscid fluid dynamics of stratified flow. Building on earlier work, the present study presents a series of numerical simulations based on the simplest nontrivial idealization relevant to the observations: the height-independent flow of a two-layer stratified fluid past a two-dimensional valley. Stratified flow past a valley has received much less attention than the related problem of stratified flow past a hill. Hence, the present paper gives a detailed review of existing theory and fills a few gaps along the way. The theory is used as an interpretive guide to an extensive set of numerical simulations. The solutions exhibit a variety of behaviors that depend on the nondimensional input parameters. These behaviors range from complete flow through the valley to valley-flow stagnation to situations involving internal wave breaking, lee waves, and quasi-stationary waves in the valley. A diagram is presented that organizes the solutions into flow regimes as a function of the nondimensional input parameters.

On the computation of measure-valued solutions

A standard paradigm for the existence of solutions in fluid dynamics is based on the construction of sequences of approximate solutions or approximate minimizers. This approach faces serious obstacles, most notably in multi-dimensional problems, where the persistence of oscillations at ever finer scales prevents compactness. Indeed, these oscillations are an indication, consistent with recent theoretical results, of the possible lack of existence/uniqueness of solutions within the standard framework of integrable functions. It is in this context that Young measures – parametrized probability measures which can describe the limits of such oscillatory sequences – offer the more general paradigm ofmeasure-valued solutionsfor these problems.We present viable numerical algorithms to compute approximate measure-valued solutions, based on the realization of approximate measures as laws of Monte Carlo sampled random fields. We prove convergence of these algorithms to measure-valued solutions for the equations of compressible and incompressible inviscid fluid dynamics, and present a large number of numerical experiments which provide convincing evidence for the viability of the new paradigm. We also discuss the use of these algorithms, and their extensions, in uncertainty quantification and contexts other than fluid dynamics, such as non-convex variational problems in materials science.

The fate of random initial vorticity distributions for two-dimensional Euler equations on a sphere

The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

An approach to the Riemann problem in the light of a reformulation of the state equation for SPH inviscid ideal flows: a highlight on spiral hydrodynamics in accretion discs

In physically inviscid fluid dynamics, "shock capturing" methods adopt either an artificial viscosity contribution or an appropriate Riemann solver algorithm. These techniques are necessary to solve the strictly hyperbolic Euler equations if flow discontinuities (the Riemann problem) are to be solved. A necessary dissipation is normally used in such cases. An explicit artificial viscosity contribution is normally adopted to smooth out spurious heating and to treat transport phenomena. Such a treatment of inviscid flows is also widely adopted in the Smooth Particle Hydrodynamics (SPH) finite volume free Lagrangian scheme. In other cases, the intrinsic dissipation of Godunov-type methods is implicitly useful. Instead "shock tracking" methods normally use the Rankine-Hugoniot jump conditions to solve such problems. A simple, effective solution of the Riemann problem in inviscid ideal gases is here proposed, based on an empirical reformulation of the equation of state (EoS) in the Euler equations in fluid dynamics, whose limit for a motionless gas coincides with the classical EoS of ideal gases. The application of such an effective solution to the Riemann problem excludes any dependence, in the transport phenomena, on particle smoothing resolution length $h$ in non viscous SPH flows. Results on 1D shock tube tests, as well as examples of application for 2D turbulence and 2D shear flows are here shown. As an astrophysical application, a much better identification of spiral structures in accretion discs in a close binary (CB), as a result of this reformulation is also shown here.

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

From weak discontinuities to nondissipative shock waves

An analysis is presented of the effect of weak dispersion on transitions from weak to strong discontinuities in inviscid fluid dynamics. In the neighborhoods of transition points, this effect is described by simultaneous solutions to the Korteweg—de Vries equation ut′ + uux″ + uxxx‴ = 0 and fifth-order nonautonomous ordinary differential equations. As x2 + t2 →∞, the asymptotic behavior of these simultaneous solutions in the zone of undamped oscillations is given by quasi-simple wave solutions to Whitham equations of the form ri(t, x) = tli x/t2.

Second order front tracking for the Euler equations

A second order front tracking method is developed for solving the hyperbolic system of Euler equations of inviscid fluid dynamics numerically. Meshless front tracking methods are usually limited to first order accuracy, since they are based on a piecewise constant approximation of the solution. Here second order convergence is achieved by deriving a piecewise linear reconstruction of the piecewise constant front tracking solution. The linearization is performed by decomposing the front tracking solution into its wave components and by linearizing the wave solutions separately. In order to construct a physically correct linearization, the physical phenomena of the front are taken into account in terms of the front types of the previously developed improved front interaction model. This front interaction model is also extended to include front numbers used in the wave decomposition. It is illustrated numerically for Sod's Riemann problem, the two interacting blast waves problem, and a two-dimensional supersonic airfoil flow validation study that the proposed front tracking method achieves second order convergence also in the presence of strong discontinuities and their interactions.

Eshbach's handbook of engineering fundamentals

Eshbach's handbook of engineering fundamentals

Some numerical issues on simulation of detonation cell structures

The present study examines several numerical issues on simulation of detonation cell structures. Various stability regimes ranging from weakly to highly unstable detonations are considered. The analysis treats two-dimensional inviscid fluid-dynamics equations and a one-step reaction model. A series of investigations is carried out to identify numerical requirements for high-fidelity simulations of detonation cell structures. Emphasis is placed on the wave-front dynamics and evolution of cellular patterns. The effects of the preexponential factor, grid size, time step, domain length, and exit boundary condition on the cellular structure and cell size are examined systematically. The required numerical grid size is determined and compared with various length scales associated with a steady Zel’dovich-von Neumann-Doring detonation wave. A general rule for the grid-resolution requirement is proposed for the first time: a minimum of 5 grid points should be included in the heat-release zone of the corresponding steady Zel’dovich-von Neumann-Doring detonation wave, in order to achieve an accurate simulation of detonation cell structures.

$L^1$ continuous dependence for the Euler equations of compressible fluids dynamics

We prove the $L^1$ continuous dependence of entropy solutions for the $2 \times 2$ (isentropic) and the $3\times 3$ (non-isentropic) systems of inviscid fluid dynamics in one-space dimension. We follow the approach developed by the second author for solutions with small total variation to general systems of conservation laws in [11, 14]. For the systems of fluid dynamics under consideration here, our estimates are more precise and we cover entropy solutions with large total variation.

Transport phenomena

Transport phenomena

Rotationally Invariant and Partially Invariant Flows of a Viscous Incompressible Fluid and a Viscous Gas

In this manuscript we study the Navier–Stokes equations and viscous gasdynamics equations. These equations play a central role in much of the research withinapplied mathematics, physics and engineering. One of the questions that we study here isthe existence of solutions of special vortex type for the Navier–Stokes equations and viscous gas dynamics equations. This type of solution for the inviscid gas and fluid dynamics equations was introduced by Ovsiannikov [1]. Note that this solution is partiallyinvariant with respect to group of rotations $O$(3). Another part of our study is devotedto the group classification of spherically symmetric viscous gas dynamics equations. Theapproach used is classical group analysis. We use the notions of invariant and partiallyinvariant solutions.

Hamilton-based Numerical Methods for a Fluid-Membrane Interaction in Two and Three Dimensions

The Hamiltonian formulation of incompressible, inviscid fluid dynamics is recalled. Numerical methods based on this formulation have several conservation properties that make them attractive. This formulation is applied to study the interaction of a fluid with a membrane in two and three dimensions. Linear analysis is performed in two and three dimensions for axisymmetric flow. The result in two dimensions agrees with linear analysis obtained by the vorticity formulation. Numerical schemes based on this formulation are derived and applied to the evolution of membranes in two and three dimensions. The numerical results are in agreement with linear analysis.

Hamiltonian description of the ideal fluid

The Hamiltonian viewpoint of fluid mechanical systems with few and infinite number of degrees of freedom is described. Rudimentary concepts of finite-degree-of-freedom Hamiltonian dynamics are reviewed, in the context of the passive advection of a scalar or tracer field by a fluid. The notions of integrability, invariant-tori, chaos, overlap criteria, and invariant-tori breakup are described in this context. Preparatory to the introduction of field theories, systems with an infinite number of degrees of freedom, elements of functional calculus and action principles of mechanics are reviewed. The action principle for the ideal compressible fluid is described in terms of Lagrangian or material variables. Hamiltonian systems in terms of noncanonical variables are presented, including several examples of Eulerian or inviscid fluid dynamics. Lie group theory sufficient for the treatment of reduction is reviewed. The reduction from Lagrangian to Eulerian variables is treated along with Clebsch variable decompositions. Stability in the canonical and noncanonical Hamiltonian contexts is described. Sufficient conditions for stability, such as Rayleigh-like criteria, are seen to be only sufficient in the general case because of the existence of negative-energy modes, which are possessed by interesting fluid equilibria. Linearly stable equilibria with negative energy modes are argued to be unstable whenmore » nonlinearity or dissipation is added. The energy-Casimir method is discussed and a variant of it that depends upon the notion of dynamical accessibility is described. The energy content of a perturbation about a general fluid equilibrium is calculated using three methods. {copyright} {ital 1998} {ital The American Physical Society}« less

Vortices and boundaries

This paper develops a framework in which vortex/boundary interactions can be analysed for the cases of inviscid fluid dynamics and type II superconductors. Comments are made concerning the possibility of reconnection, defined as an interaction in which an initially single-component vortex eventually emerges with two components.