Inviscid Motion Research Articles

Nonlinear dynamics of two helical vortices: A dynamical system approach

Two interwoven helical vortices of the same circulation and pitch display a rich variety of dynamics. Using simplified models of straight vortices, ring vortices, and helical filaments, we introduce dynamical systems of a few degrees of freedom which we analyze in terms of orbits in phase spaces structured by hyperbolic and elliptic points. Possible inviscid motions, namely leapfrogging, overtaking and fluttering, are systematically described as a function of vortex pitch, core size, and initial configuration.

Kinematics of spherical shock waves in an interstellar ideal gas clouds with dust particles

In this article, a system of partial differential equations governing the one‐dimensional motion of inviscid, self‐gravitating, spherical dusty gas cloud is considered. The evolutionary behavior of spherical shock waves of arbitrary strength in an interstellar dusty gas cloud is examined. By utilizing the method based on the kinematics of the one‐dimensional motion of shock waves, we derived an infinite set of transport equations governing the strength of shock waves and induced discontinuity behind it. By applying the truncation procedure to the infinite set of transport equations, we get an efficient system of finite number ordinary differential equations describing shock propagation, which can be regarded as a good approximation of the infinite hierarchy of the system. The truncated equations describing the shock strength and the induced discontinuity are used to analyze the growth and decay behavior of shock waves of arbitrary strength in the dusty gas medium. We considered the first two truncation approximations and the obtained results for the exponent from the successive approximation and compared our results with Guderley's exact similarity solution and the characteristic rule.

Entry flow in curved pipes: turbulent boundary layer approach

ABSTRACTThe development of secondary boundary layer for a steady entry of fluid flow in a horizontal curved pipe is analysed using the turbulent boundary layer approach. Outside the boundary layer (within the central core), the inviscid fluid motion is treated by using the Euler and the continuity equations, whereas inside the boundary layer, the Pohlhausen method is applied, assuming a one-seventh power law of velocity for solving the integrals of the Navier–Stokes equations. The growth of the secondary boundary layer is slow with an increase in azimuthal angle, while it is gradual along the outer pipe-wall with an increase in radial angle and becomes abruptly faster near the separation point. The wall shear stress decreases with azimuthal angle, whereas it increases with radial angle attaining a peak and then decreases to zero at the separation point along the outer pipe-wall.

Suction–shear–Coriolis instability in a flow between parallel plates

AbstractA rotating fluid flow between differentially translating parallel plates, which induce uniform suction and injection, is studied as a canonical model of swirling flow where suction, shear and Coriolis effects compete. This relatively simple modelling yields several reduced equations that are valid for asymptotically large suction, shear and/or rotation rates. The linear stability problems derived from the full Navier–Stokes and reduced problems are numerically solved and compared. In addition to Taylor-vortex modes, transverse-roll-type instabilities are found in Rayleigh-stable and -unstable parameter regions when weak suction is applied. These instabilities, separated by the so-called Rayleigh line, are characterised by vortices attached to the suction wall. Another type of instability, which exists beyond the Rayleigh line and shows inviscid motion in the fluid core, is found when suction is sufficiently strong. The relation of this instability to the stability results by Gallet, Doering & Spiegel (Phys. Fluids, vol. 22, 2010, 034105) is discussed. Our nonlinear analyses indicate subcritical and supercritical bifurcations of finite-amplitude solutions for the near-wall and fluid-core instabilities, respectively.

Dynamics of sessile drops. Part 1. Inviscid theory

AbstractA sessile droplet partially wets a planar solid support. We study the linear stability of this spherical-cap base state to disturbances whose three-phase contact line is (i) pinned, (ii) moves with fixed contact angle and (iii) moves with a contact angle that is a smooth function of the contact-line speed. The governing hydrodynamic equations for inviscid motions are reduced to a functional eigenvalue problem on linear operators, which are parameterized by the base-state volume through the static contact angle and contact-line mobility via a spreading parameter. A solution is facilitated using inverse operators for disturbances (i) and (ii) to report frequencies and modal shapes identified by a polar $k$ and azimuthal $l$ wavenumber. For the dynamic contact-line condition (iii), we show that the disturbance energy balance takes the form of a damped-harmonic oscillator with ‘Davis dissipation’ that encompasses the dynamic effects associated with (iii). The effect of the contact-line motion on the dissipation mechanism is illustrated. We report an instability of the super-hemispherical base states with mobile contact lines (ii) that correlates with horizontal motion of the centre-of-mass, called the ‘walking’ instability. Davis dissipation from the dynamic contact-line condition (iii) can suppress the instability. The remainder of the spectrum exhibits oscillatory behaviour. For the hemispherical base state with mobile contact line (ii), the spectrum is degenerate with respect to the azimuthal wavenumber. We show that varying either the base-state volume or contact-line mobility lifts this degeneracy. For most values of these symmetry-breaking parameters, a certain spectral ordering of frequencies is maintained. However, because certain modes are more strongly influenced by the support than others, there are instances of additional modal degeneracies. We explain the physical reason for these and show how to locate them.

Motion induced between parallel plates with offset centers of radial stretching and shrinking

The flow between parallel plates separated by distance $$h$$ is investigated where the upper plate radially stretches at a rate $$a$$ , the lower plate radially shrinks at the same rate, and the centers of stretching and shrinking are horizontally separated by a distance $${2\,l}$$ . A reduction of the Navier–Stokes equation yields two ordinary differential equations dependent on a Reynolds number $$R = ah^2/\nu $$ . In addition, a free parameter $$\gamma $$ appears that corresponds to a uniform pressure gradient acting along the line connecting the stretching/shrinking centers. We consider three cases: $$\gamma = 0$$ , $$\gamma = O(1)$$ , and $$\gamma = O(R)$$ . The flow is described by two functions of the plate-normal coordinate $$\eta = z/h$$ : the first $$f(\eta )$$ has an analytical solution, whereas the second $$g(\eta )$$ must be solved numerically. Small- $$R$$ solutions are found, and large- $$R$$ asymptotic behaviors of the wall shear stresses and the centerline velocities are obtained by matching the viscous boundary-layer flows to the interior inviscid motion.

Analytical studies of sloshing in half-cylindrical tanks

The technically important problem of two-dimensional inviscid standing-wave motion of a fluid in a semi-cylindrical vessel is examined using a variety of the Boussinesq wave equation as well as the directed fluid sheet technique. The associated eigenvalue problems are dealt with by employing perturbative procedures and series expansions. The predictions were compared with the outcome of laboratory tank experiments and also with ‘exact' results. The Boussinesq equation perturbative scheme based on an approximate representation of the vessel geometry not only gave the best results, but was also the most straightforward to apply. Furthermore, it yielded practically useful explicit formulae for the frequencies at which sloshing may be initiated. A limitation is, however, that this formalism is only applicable to systems with comparatively small fill ratios.

Biaxial extensional motion of an inertially driven radially expanding liquid sheet

We consider the inertially driven, time-dependent biaxial extensional motion of inviscid and viscous thinning liquid sheets. We present an analytic solution describing the base flow and examine its linear stability to varicose (symmetric) perturbations within the framework of a long-wave model where transient growth and long-time asymptotic stability are considered. The stability of the system is characterized in terms of the perturbation wavenumber, Weber number, and Reynolds number. We find that the isotropic nature of the base flow yields stability results that are identical for axisymmetric and general two-dimensional perturbations. Transient growth of short-wave perturbations at early to moderate times can have significant and lasting influence on the long-time sheet thickness. For finite Reynolds numbers, a radially expanding sheet is weakly unstable with bounded growth of all perturbations, whereas in the inviscid and Stokes flow limits sheets are unstable to perturbations in the short-wave limit.

Coupled oscillations of deformable spherical-cap droplets. Part 1. Inviscid motions

AbstractA spherical drop is constrained by a solid support arranged as a latitudinal belt. This belt support splits the drop into two deformable spherical caps. The edges of the support are given by lower and upper latitudes yielding a ‘spherical belt’ of prescribed extent and position: a two-parameter family of constraints. This is a belt-constrained Rayleigh drop. In this paper we study the linear oscillations of the two coupled spherical-cap surfaces in the inviscid case, and the viscous case is studied in Part 2 (Bostwick & Steen, J. Fluid Mech., vol. 714, 2013, pp. 336–360), restricting to deformations symmetric about the axis of constraint symmetry. The integro-differential boundary-value problem governing the interface deformation is formulated as a functional eigenvalue problem on linear operators and reduced to a truncated set of algebraic equations using a Rayleigh–Ritz procedure on a constrained function space. This formalism allows mode shapes with different contact angles at the edges of the solid support, as observed in experiment, and readily generalizes to accommodate viscous motions (Part 2). Eigenvalues are mapped in the plane of constraints to reveal where near-multiplicities occur. The full problem is then approximated as two coupled harmonic oscillators by introducing a volume-exchange constraint. The approximation yields eigenvalue crossings and allows post-identification of mass and spring constants for the oscillators.

The initial value problem for the equation of motion of irrotational inviscid and heat conductive fluids

The Cauchy problem of the equation of motion of irrotational inviscid and heat conductive fluids is considered. It is proved that the heat diffusion prevents the development of singularities in small amplitude classical solutions, using an equivalent reformulation of the Cauchy problem to obtain effective energy estimates. The full solution converges to their equilibrium state at the rate (1+t)−34 in the L2-norm as the non-isentropic compressible Navier–Stokes system.

Finite-amplitude dynamics of coupled cylindrical menisci

The cylindrical meniscus is a liquid/gas interface of circular-cap cross-section constrained along its axis and bounded by end-planes. The inviscid motions of coupled cylindrical menisci are studied here. Motions result from the competition between inertia and surface tension forces. Restriction to shapes that are of circular-cap cross-section leads to an ordinary differential equation (ode) model, with the advantage that finite-amplitude stability can be examined. The second-order nonlinear ode model has a Hamiltonian structure, showing dynamical behavior like the Duffing-oscillator. The energy landscape has either a single- or double-welled potential depending on the extent of volume overfill. Total liquid volume is a bifurcation parameter, as in the corresponding problem for coupled spherical-cap droplets [1]. Unlike the spherical-cap problem, however, axial disturbances can also destabilize, depending on overfill. For large volumes, previously known axial stability results are applied to find the limit at which axial symmetry is lost and comparison is made to the Plateau–Rayleigh limit.

Impulsive displacement of a liquid in a pipe at high Reynolds numbers

We consider the problem of an impulsive displacement of a liquid, originally at rest in a circular pipe, which is displaced by another liquid. The purpose of this analysis is to show that at a sufficiently high inertia the initial essentially inviscid motion can be extended to cover the entire displacement process, thus creating an inviscid window to which an inviscid analysis can be applied. We simplify the problem first, by considering a 1-liquid problem where the displacing liquid and displaced liquid are the same. We identify two characteristic times in this problem: the time it takes an inviscid liquid to be displaced, and the time it takes a viscous liquid to attain a steady state. Taking the ratio of the two defines the Reynolds number for the problem and we show that the motion becomes essentially inviscid once the Reynolds number is sufficiently high. We obtain the general solution of the 1-liquid problem which determines the nondimensional viscous displacement time as a function of the Reynolds number. We derive from the general solution: a critical Reynolds number above which the motion remains unsteady throughout the entire displacement process, and a formula which determines quantitatively whether applying an inviscid analysis to the 1-liquid viscous problem at a given Reynolds number is admissible within an acceptable error tolerance. We also show that at the limit the Reynolds number approaches infinity the viscous displacement time approaches the inviscid displacement time and that the velocity profile and the shape of the material surface separating the displacing from the displaced liquid approach their counterpart in the inviscid solution. Second, based on these results we propose that an inviscid solution is applicable to the 2-liquid viscous problem once the condition of a high Reynolds number is independently met by the two participating liquids. We obtain the solution to the inviscid 2-liquid displacement problem and calculate various examples. Finally, we present a stability analysis of the flat interface between the two inviscid liquids, which shows which of the examples is stable, neutrally stable, or unstable. The paucity of data for an impulsive displacement in the high Reynolds number range makes quantitative comparisons difficult. However, the excellent agreement obtained between the critical Reynolds number derived in this analysis and the result obtained in a numerical analysis of the viscous 2-liquid problem elsewhere constitutes at least a partial validation of the theory. Additional confirmation is obviously recommended.

On the motion of thin vortex tubes

The motion of a tube of vorticity with a cross sectional radius that is everywhere small compared to local radius of curvature of the tube is considered. In particular, we determine the inviscid motion of the 3D space curve that traces the centerline of the tube for an arbitrary distribution of axial vorticity within the core.

2D Slightly Compressible Ideal Flow in an Exterior Domain

We consider the Euler equations of barotropic inviscid compressible fluids in the exterior domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension 2 such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small.

Boussinesq and Anelastic Approximations Revisited: Potential Energy Release during Thermobaric Instability

Abstract Expressions are derived for the potential energy of a fluid whose density depends on three variables: temperature, pressure, and salinity. The thermal expansion coefficient is a function of depth, and the application is to thermobaric convection in the oceans. Energy conservation, with conversion between kinetic and potential energies during adiabatic, inviscid motion, exists for the Boussinesq and anelastic approximations but not for all approximate systems of equations. In the Boussinesq/anelastic system, which is a linearization of the thermodynamic variables, the expressions for potential energy involve thermodynamic potentials for salinity and potential temperature. Thermobaric instability can occur with warm salty water either above or below cold freshwater. In both cases the fluid may be unstable to large perturbations even though it is stable to small perturbations. The energy per mass of this finite-amplitude instability varies as the square of the layer thickness. With a 4-K temperature difference and a 0.6-psu salinity difference across a layer that is 4000 m thick, the stored potential energy is ∼0.3 m2 s−2, which is comparable to the kinetic energy of the major ocean currents. This potential could be released as kinetic energy in a single large event. Thermobaric effects cause parcels moving adiabatically to follow different neutral trajectories. A cold fresh parcel that is less dense than a warm salty parcel near the surface may be more dense at depth. Examples are given in which two isopycnal trajectories cross at one place and differ in depth by 1000 m or more at another.

Thermohaline circulation induced by bottom friction in sloping-boundary basins

We show that a velocity field in geostrophic and hydrostatic balance on the f-plane can be diagnosed from an arbitrarily prescribed distribution of buoyancy in a basin with closed depth contours. We emphasize the steady-state circulation associated with a large-scale horizontal buoyancy gradient, attained in the absence of wind forcing. For inviscid motion, the diagnosed field contains a free barotropic along-isobath flow which can be chosen arbitrarily, e.g. in such a way that the buoyant “southern” pool of surface water essentially recirculates. Including bottom friction, we show that steady motion requires that the net Ekman transport across closed depth contours must vanish. This constraint determines the free barotropic motion and thereby the entire velocity field, which proves to be independent of the strength of the bottom friction. The barotropic flow component serves to create a “thermohaline” circulation, i.e. a circulation which tends to spread the buoyant water horizontally. Analytical solutions and results from a numerical experiment are presented to illustrate the steady flow resulting in a basin where the upper-ocean density increases across the basin.

The contact angle in inviscid fluid mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived; however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many ‘solutions’ in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense ‘weak-type solutions’; they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.

The formation and dynamics of a blob on free and wall sheets induced by a drop impact on surfaces

A physical model for describing an inviscid motion of free and wall liquid sheets induced by drop impact is presented. This model takes into account the formation of thick borders at the edge of a spreading drop, and the influence of advancing and receding contact angles on the dynamics of blob formation and motion. It has been shown that the blob motion on a free liquid sheet is described by a universal relationship (independent of the Weber number) in terms of dimensionless blob coordinates at both advancing and receding stages. For the case of blob formation and motion on a wall sheet at high Weber numbers, only the advancing stage is described by a universal relationship. The receding stage depends on the ratio of advancing and receding contact angles, but not on the Weber number. At the instant when the drop is at its maximum extension, the blob speed becomes equal to the liquid sheet velocity and the total kinetic energy of the drop is greater than zero. It is shown that the ratio of the instantaneous capillary wavelength to the sheet thickness is a constant 2.619 for a free sheet, when the capillary wave propagates away from the blob.

Life span of 2‐D irrotational compressible fluids in the halfplane

AbstractWe consider the Euler equations of barotropic inviscid compressible fluids in the half plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In dimension two such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial velocity. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small.We consider smooth irrotational solutions. First, we study the life span, i.e. the largest time interval T(ε) of existence of classical solutions, when the initial data are a small perturbation of size εfrom a constant state. For the proof of this result we use a combination of energy and decay estimates. Then, the estimate of the life span allows to show, by a suitable scaling of variables, the existence of irrotational solutions on any arbitrary time interval, for any small enough Mach number. Copyright © 2002 John Wiley & Sons, Ltd.

On Slightly Compressible Ideal Flow¶in the Half-Plane

We consider the Euler equations of barotropic inviscid compressible fluids in the half-plane. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In 2D (two dimensions) such limit solution exists on any arbitrary time interval, with no restriction on the size of the initial data. It is then natural to expect the same for the compressible solution, if the Mach number is sufficiently small.