Two-dimensional Irrotational Flow Research Articles

Natural sloshing frequencies and modes in a rectangular tank with a slat-type screen

Being installed in tanks, screens play the role of slosh-suppressing devices which may strongly change resonant sloshing frequencies and yield an extra nonlinear damping due to cross-flow resulting in either flow separation or jet flow. Employing the linear sloshing theory and domain decomposition method, we construct an accurate analytical approximation of the natural sloshing modes in a rectangular tank with a slat-type screen at the tank middle. Two-dimensional irrotational flow of an ideal incompressible liquid is assumed. Because the considered flow model does not account for flow separation and jet flow at the screen, the velocity field is locally singular at the sharp edges. The constructed solution captures this singularity. Analyzing this solution establishes a complex dependence of the natural sloshing frequencies on the solidity ratio, the number of submerged screen gaps, the liquid depth, and the position of perforated openings relative to the mean free surface. Results are compared with experimental data. Natural surface wave profiles are discussed in the context of a jump of the velocity potential at the screen and the local inflow component to the screen.

The dipolar field of rotating bodies in two dimensions

The fluid velocity far from a translating body in two-dimensional irrotational flow is generally dipolar. This is a classical result. Here we ask when the dipolar component vanishes. Lamb (1945, § 126) provides symmetry conditions on the virtual mass tensor for this to be the case. We show that these conditions are not necessary and obtain the sufficient and necessary condition in terms of the shape of the body using conformal maps. Some explicit examples are constructed based on this condition.

Flow in a waterfall with large Froude number

An approximate method is developed to solve the full nonlinear equations governing two-dimensional irrotational flow in a free waterfall, falling under the influence of gravity, at high Froude number based on conditions far upstream. Schwarz—Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique, for large Froude number, are used as a basis for the approximate solution of the problem. A complete solution, up to second-order approximation, for the downstream free-surfaces profiles, for different Froude number, is discussed and illustrated. The obtained approximate solutions are compared with those of other authors. Favourable agreement with other results suggests that this method is effective in dealing with flow problems strongly influenced by gravity and high Froude number. The results obtained by this method are sufficiently accurate for practical purposes.

The effect of heat addition on slightly compressible flow: The example of vortex pair motion

The effect of a point heat source on the two-dimensional irrotational flow of a slightly compressible nonviscous and nondiffusive fluid is studied. Heat is added at a constant rate and with a prescribed spatial distribution. The Mach number M is assumed small and the governing equations are solved to order M2 by a generalization of the Rayleigh–Janzen expansion. The interaction between entropy variations produced by the heat source and the pressure field generates a wake consisting of a vortex sheet and a dipole sheet extending downstream of the heat source, and this in turn perturbs the flow at order M2. A general result is that if heat is added where the gas is compressed by the flow, the impulse directed against the free stream is increased and vice versa. Cross-stream impulse is produced if the heat source is not on the axis of symmetry (if any) of the flow. This would result in a lifting force if a cylinder is providing the perturbation to the free stream. When heat is added on the downstream dividing streamline of a symmetrical flow the order M2 velocity field at infinity is that due to an apparent mass source and it is shown that the required flux of mass and momentum is supplied from infinity via the wake. Detailed calculations are performed for the case of the self-propagation of a symmetrical vortex pair, where heat is added downstream of the rearward stagnation point. The resultant increase in the pair impulse is shown to lead to a slow increase in the vortex pair separation.

Biofluiddynamics of balistiform and gymnotiform locomotion. Part 2. The pressure distribution arising in two-dimensional irrotational flow from a general symmetrical motion of a flexible flat plate normal to itself

When, for an otherwise unbounded fluid, the unique irrotational flow compatible with the instantaneous motion of an immersed body has been calculated, it is straightforward to deduce the pressure field from the unsteady form of Bernoulli's equation if the body is rigid. On the other hand, if the body is flexible, a somewhat subtle analysis is required to determine the time derivative of velocity potential, ∂ϕ/∂t, which occurs in that equation. This is because no simple relationship exists between the instantaneous form of ϕ and its form at a nearby instant.In the case of two-dimensional flow, however, the two forms of ϕ for a flexible body may be related, not in general by a simple translational and/or rotational mapping as for a rigid-body motion, but by a conformal mapping. The example of a flexible flat plate is used here to illustrate this approach to calculating the pressure field.In the analysis of balistiform motion by elongated-body theory (Lighthill & Blake 1990), one part of the propulsive force on the fish has magnitude equal to P, the area integral of the pressure field just described. This area integral is shown in §3 below to take a simple form or for E, so that P itself is also not enhanced. For the relevance of these findings to the efficiency of balistiform motion, see Lighthill & Blake (1990).

On integrals and invariants for inviscid, irrotational flow under gravity

Simplified proofs are given of the eight basic conservation laws for two-dimensional irrotational flow of a perfect fluid. The corresponding relations for rotational flow are also discussed.

On the Irrotational Flow Around a Horizontal Cylinder in Waves

Milne-Thomson’s circle theorem is used to study the characteristics of the two-dimensional irrotational flow around a horizontal cylinder under long-crested waves. Even when the cylinder diameter is small compared with the wavelength, the circumferential velocity and pressure distributions are unsteady and differ markedly from those corresponding to uniform ambient flow with similar velocity and acceleration vectors.

Part 6. Present plate boundary seismic, volcanic and kinematic processes: Subduction-zone vortices

Sinking slabs of lithosphere must push the asthenosphere if they pull the plates. The resulting flow in the asthenosphere then goes rapidly round the ends of a slab, pulling the lithosphere with it. The motion can be analyzed with the help of Hele?Shaw's mathematical analogy between viscous flow in a thin layer of fluid and classical two-dimensional irrotational flow.

Uniform Asymptotic Solutions for the Two-Dimensional Potential Field About a Slender Body

The two-dimensional potential field about an arbitrary slender body, with either a Dirichlet or Neumann boundary condition on the body, is studied. In particular, the two-dimensional irrotational flow of an incompressible inviscid fluid past a slender cylindrical airfoil and the two-dimensional electrostatic field in the exterior of a slender cylindrical conductor are considered. The complete uniform asymptotic expansion of the potential field is obtained with respect to the parameter $\varepsilon $, the slenderness ratio of the profile curve of the cylinder. The camber line of the profile curve is completely arbitrary and hence these results extend the work of Geer and Keller (1968). The potential due to the presence of the body is represented as the superposition of potentials of point sources and point vortices distributed along an arc inside the body. The source and vortex strength distributions satisfy a pair of coupled linear integral equations. The kernels in both of the equations become singular as $\varepsilon $ approaches zero. The complete uniform asymptotic expansion of the solution to these equations is found by modifying and adapting the method of Handelsman and Keller (1967). Complete expansions of the virtual mass, the polarizability, and the lift of the cylinder are obtained.

Irrotational Flow with Irregular Boundary Shapes

An algorithm for the numerical solution of two-dimensional irrotational flows based on an inverse transformation is presented as an efficient means of solving flows with irregular boundary shapes. Using the intersections of a set of streamlines and equipotential lines as computational nodal points, the algorithm determines the coordinates of these intersections. The streamlines and equipotential lines can be arbitrarily selected to concentrate the nodal points in areas of high velocity flow so as to increase the accuracy in these areas. The algorithm is particularly well suited to modern high speed computers, and FORTRAN statements are given. Since the program operates on many different boundary shapes with no internal alterations, it is suitable for use by persons with limited background in numerical procedures.

The Iterated Equation of Generalized Axially Symmetric Potential Theory

Milne-Thomson's well-known circle theorem [1] gives the stream function for steady two-dimensional irrotational flow of a perfect fluid past a circular cylinder when the flow in the absense of the cylinder is known. Butler's sphere theorem [2] gives the corresponding result for axially symmetric irrotational flow of a perfect fluid past a sphere. Collins [3] has obtained a sphere theorem for axially symmetric Stokes flow of a viscous liquid which gives a stream function satisfying the appropriate viscous boundary conditions on the surface of a sphere when the stream function for irrotational flow in the absence of the sphere is known.

Uniform Asymptotic Solutions for Potential Flow Around a Thin Airfoil and the Electrostatic Potential About a Thin Conductor

The two-dimensional irrotational flow of an incompressible inviscid fluid past a thin cylindrical airfoil and the two-dimensional electrostatic field in the exterior of a thin cylindrical conductor are considered. The complete uniform asymptotic expansions of the two potentials are obtained with respect to $\epsilon $, the slenderness ratio of the profile curve C of the cylinder. Only curves C which have axes of symmetry are treated. To find the symmetric part of the potential due to the presence of the body, it is represented as a superposition of potentials of point sources distributed along a segment of the axis inside C. The source strength distribution satisfies a linear integral equation. The uniform asymptotic expansion of the solution of this equation is obtained by adapting the method of Handelsman and Keller [5], [6].The antisymmetric part is obtained by using the harmonic conjugate of an appropriate symmetric potential. Complete expansions of the virtual mass, the polarizability, and the lift o...

Impossibility of Three Confluent Shocks in Two-Dimensional Irrotational Flow

The configuration where three shocks separate three uniform regions is impossible. This has been proved by Courant and Friedrichs for materials having a polytropic equation of state. A proof independent of the form of the equation of state is presented. The conditions of mass and momentum conservation are incompatible for this configuration. For the configuration which occurs in Mach reflection the material flowing across the two shocks attains a greater velocity than that portion flowing across only one. This result holds even if the flow is considered isentropic.

Stokes Flow of an Electrically Conducting Fluid in a Uniform Magnetic Field

The flow of an incompressible, viscous, electrically conducting fluid past an obstacle in a uniform magnetic field is investigated using Stokes approximation. No particular configuration of the flow and the magnetic field is assumed, so that the result applies to the general three-dimensional problems. §§ 2 and 3 deal with the general theory. It is found that the neutrality of the electric charge density does not hold exactly, when the undisturbed magnetic field is not perpendicular to the vorticity vector. It is also found that the vorticity and the electric current density are confined in a paraboloidal region, thus making a `wake' which extends in the direction of the undisturbed magnetic field. Distribution of the electric charge density also shows the same structure. In § 4, the flow past a sphere is investigated as an example. The drag is obtained in a power series of the Hartmann number M . The component of the drag perpendicular to the undisturbed magnetic line of force is found to be larger than its parallel component. It is pointed out as an interesting feature of the three-dimensional cases that the velocity field includes components which express the two-dimensional irrotational flow.

Divided Flow Through a Divergent Inlet Conduit

Determination of characteristics of divided flow phenomena is investigated; one model is conduit with constant cross-section having opening perpendicular to its length; other is conduit with divergent inlet having side opening at end of divergent; analysis is developed by means of free stream line theory, assuming two-dimensional irrotational flow of incompressible fluid.

A note on Stokes's stream function for motion with a spherical boundary

The circle theorem of Milne-Thomson(1) connecting the complex potential in a two-dimensional irrotational flow about a circular cylinder with that of the flow when the cylinder is absent has a three-dimensional counterpart in the result due to Weiss (3) for the perturbed velocity potential in an unlimited irrotational flow when the rigid spherical boundary r = a is inserted.