3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler–Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=?angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler–Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis $r=0$.     

The Evolution of Linearized Perturbations of Parallel Flows

The equations of an incompressible fluid are linearized for small perturbations of a basic parallel flow. The initial-value problem is then posed by use of Fourier transforms in space. Previous results are systematized in a general framework and used to solve a series of problems for prototypical examples of basic shear flow and of initial disturbance. The prototypes of shear flow are (a) plane Couette flow bounded by rigid parallel walls, (b) plane Couette flow bounded by rigid walls at constant pressure, (c) unbounded two-layer flow with linear velocity profile in each layer, (d) a piecewise linear profile of a boundary layer on a rigid wall. The prototypes of initial perturbation are the fundamental ones: (i) a point source of the field of the transverse velocity (represented by delta functions), (ii) an unbounded sinusoidal field of the transverse velocity, (iii) a point source of the lateral component of vorticity, (iv) a sinusoidal field of the lateral vorticity. Detailed solutions for an inviscid fluid are presented, but the problem for a viscous fluid is only broached.     

Study of free convective boundary layer of isothermal lateral surface of axisymmetrical horizontal body

Approximate analytical solution of simplified Navier–Stokes and Fourier–Kirchhoff equations describing free convective heat transfer from isothermal surface has been presented. It is supposed that the surface has the horizontal axis of symmetry and its axial cross-section lateral boundary is a concave function. The equation for the boundary layer thickness is derived for typical for natural convection assumptions. The most important are that the convective fluid flow is stationary and the normal to the surface component of velocity is negligibly small in comparison with the tangential one. The theoretical results are verified by two characteristic cases of the revolution surfaces namely for horizontal conic and vertical round plate. Both limits of presented solution coincide with known formulas.     

The application of the solid-angle theorem in the special theory of relativity

Ishlinskii's theorem, well known in classical mechanics, asserts that if an axis, selected in a rigid body, having zero projection of the angular velocity onto this axis, described a closed conical surface during the motion of the body, then, after the axis has returned to its initial position the body will have described an angle around it numerically equal to solid angle of the described cone. It is shown that the same relation also exists in the Special Theory of Relativity—the angle of rotation described by a rigid body during motion along a curvilinear trajectory due to the Thomas precession effect, is numerically equal to the solid angle observed in a fixed frame of reference described by an axis connected with the body due to a change in the rotation of the image of the rigid body. The latter phenomenon is due to the Lorentz contraction of the length and the retardation of light radiated by different parts of the body [10–13].     

Flow around an ellipsoid of revolution in a harmonic coaxial vector field

We give an integral equation defining a coaxial magnetic field near the surface of a superconductive axisymmetric body and the velocity of the liquid near the surface of an axisymmetric body situated coaxially to the flow of an ideal liquid. Using this equation in the case when the axisymmetric magnetic field before the placement of an ellipsoid of revolution coaxially to the field changed along the axis by a polynomial law, we analytically define the densities of the surface current and the force with which the magnetic field acts on the ellipsoid. Also the velocity of the liquid is determined near the surface of the ellipsoid of revolution and the force acting on the ellipsoid placed coaxially in the flow of an ideal liquid when the velocity of the liquid before the placement of the ellipsoid changed along the axis of symmetry by a polynomial law.     

On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

Latitudinally deforming rotating sphere

In this study, we consider a sphere with a surface that is fully covered by a stretchable elastic material. The radius of the sphere is fixed and it is also rotating about its radial axis. We investigate how the axisymmetric motion of a triggered fluid flow around the sphere is affected by the presence of both sphere rotation and latitudinal stretching. Considering that the deformation over the sphere commences at the pole, the problem is formulated such that the fluid flow near the pole is similar to the induced flow due to a linearly stretchable rotating disk, which has been described well in previous studies. When the rotation is omitted, the flow develops two-dimensionally under the action of pure stretching; otherwise, a three-dimensional axisymmetric fluid flow occurs, which is computed at each latitudinal angle both numerically and using a perturbation approach. The solution with wall deformation is different from the traditional character of the solution due to a solely rotating sphere. This solution is then used to compute the surface shears due to the physical drag and torque acting over the sphere. The contribution of wall stretching reduces the drag, whereas high rotation suppresses the effects of stretching to enhance the drag. More torque is required to rotate the sphere when both stretching and rotation mechanisms are in action.     

Stability of a flame front in a divergent flow

We study the evolution of perturbations on the surface of a stationary plane flame front in a divergent flow of a combustible mixture incident on a plane wall perpendicular to the flow. The flow and its perturbations are assumed to be two-dimensional; i.e., the velocity has two Cartesian components. It is also assumed that the front velocity relative to the gas is small; therefore, the fluid can be considered incompressible on both sides of the front; in addition, it is assumed that in the presence of perturbations the front velocity relative to the gas ahead of it is a linear function of the front curvature. It is shown that due to the dependence (in the unperturbed flow) of the tangential component of the gas velocity on the combustion front on the coordinate along the front, the amplitude of the flame front perturbation does not increase infinitely with time, but the initial growth of perturbations stops and then begins to decline. We evaluate the coefficient of the maximum growth of perturbations, which may be large, depending on the problem parameters. It is taken into account that the characteristic spatial scale of the initial perturbations may be much greater than the wavelengths of the most rapidly growing perturbations, whose length is comparable with the flame front thickness. The maximum growth of perturbations is estimated as a function of the characteristic spatial scale of the initial perturbations.     

Large Eddy Simulations of Swirling Nozzle-Jet Flow Undergoing Vortex Breakdown

We developed a numerical setup to simulate swirling jet flow undergoing vortex breakdown. Our simulation code CONCYL solves the compressible Navier-Stokes equations in cylindrical coordinates using high-order numerical schemes. A nozzle is included in the computational domain to account for more realistic inflow boundary conditions. Preliminary results of a Re = 5000 compressible swirling jet at Mach number M a = 0.6 with an azimuthal velocity as high as the maximum axial velocity (swirl number S = 1.0 ) capture the fundamental characteristics of this flow type: At a certain point in time the jet spreads and develops into a conical vortex breakdown. A stagnation point-flow in the vicinity of the jet axis is clearly visible with the stagnation point located close to the nozzle exit. The stagnation point precesses in time around the jet axis, moving up- and downstream. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical modeling of hydrodynamic and hydrothermal characteristics in subtropical alpine lake

A three-dimensional, time-dependent hydrodynamic and hydrothermal model was performed and applied to the subtropical alpine Yuan-Yang Lake (YYL) in northeastern region of Taiwan. The model was driven with discharge inflow, heat, and wind stress to simulate the hydrodynamic and hydrothermal in the lake. The model was validated with measured water surface elevation, current, and temperature in 2008. The overall model simulation results are in quantitative agreement with the available field data. The validated model was then used to investigate wind-driven current, mean circulation, and residence time in the YYL. The modeling results reveal that the velocity field along the wind axis present the variations over depth with return current where the velocity at the surface layer is along the wind direction while it is opposite near 1 m below water surface. The simulated mean current indicates that the surface currents flow towards the southwest direction and form a clock-wise rotation. The calculated residence time is strongly dependent on the inflows and wind effects. Regression analysis of model results reveals that an exponential regression equation can be employed to correlate the residence time to change of discharge input. The residence time without wind stress is higher than that with wind effect, indicating that wind plays an important role in lake mixing. The calculated residence time is approximately 2-2.5 days under low inflow with wind effect.     

Analytical solutions of laminar swirl decay in a straight pipe

In this work, the laminar swirl flow in a straight pipe is revisited and solved analytically by using prescribed axial flow velocity profiles. Based on two axial velocity profiles, namely a slug flow and a developed parabolic velocity profiles, the swirl velocity equation is solved by the separation of variable technique for a rather general inlet swirl velocity distribution, which includes a forced vortex in the core and a free vortex near the wall. The solutions are expressed by the Bessel function for the slug flow and by the generalized Laguerre function for the developed parabolic velocity. Numerical examples are calculated and plotted for different combinations of influential parameters. The effects of the Reynolds number, the pipe axial distance, and the inlet swirl profiles on the swirl velocity distribution and the swirl decay are analyzed. The current results offer analytical equations to estimate the decay rate and the outlet swirl intensity and velocity distribution for the design of swirl flow devices.     

Normal Feedback Stabilization of Periodic Flows in a Two-Dimensional Channel

We consider a two-dimensional incompressible channel flow with periodic condition along one axis. We stabilize the linearized system by a boundary feedback controller with vertical velocity observation, which acts on the normal component of the velocity only. The stability is achieved without any a priori condition on the viscosity coefficient, that is, on the Reynolds number.     

First stages of drop impact on a dry surface: asymptotic model

After impact of a viscous liquid drop on a dry wall surrounded by a gas, the drop surface is highly deformed, leading to the formation of an axisymmetrical lateral lamella along the wall. A local asymptotic model for the potential flow and unsteady boundary layer flow is developed to describe the lamella dynamics at early stages after impact. The second-order potential flow displaced by the unsteady boundary layer is taken into account. The lamella shape, its velocity and pressure are calculated with this model in parametrical forms. The three model parameters are evaluated here by fitting with recent experimental findings.     

UNSTEADY FREE CONVECTIVE MHD FLOW AND MASS TRANSFER THROUGH POROUS MEDIUM IN A ROTATING SYSTEM WITH FLUCTUATING HEAT SOURCE/SINK AND CHEMICAL REACTION

An investigation of unsteady MHD free convective flow and mass transfer during the motion of a viscous incompressible fluid through a porous medium in the presence of heat source, bounded by an infinite vertical porous surface, in a rotating system is presented. The porous plane surface and the porous medium are assumed to rotate in a solid body rotation. The vertical surface is subject to uniform constant suction perpendicular to it and the temperature at this surface fluctuates in time about a non-zero constant mean. Analytical expressions for the velocity, temperature and concentration fields are obtained using perturbation technique. Normal 0 false false false EN-US X-NONE X-NONE     

Intersection area of a stationary circle and a circle under a coupled axial rotation and translation

A circle is placed concentrically in a circle of equal or larger size. The circle is then rotated along a vertical axis, creating an ellipse, and translated along the horizontal axis. The intersection area of the circle and circle/ellipse is determined as function of the rotation angle and the relative size of the initial circles. This configuration corresponds to the closing of a ball valve used to control the flow of fluids through pipes.     

Continuous and discontinuous solutions for optimum thrust nozzles of given length

In its first sections, the paper deals with optimum thrust nozzles of given length and exit radius for flows with swirl. The computation is based on a modification of methods familiar for flows without swirl. Rather extensive numerical results show that the swirl does not impair the specific impulse attainable at a given nozzle length. The analysis suggests that the assumption of isentropic continuous flows, on which this approach is based, may sometimes be too restrictive. A survey of plane nozzles shows, on the other hand, that discontinuities need to be admitted only if, besides the length, a rather large radius of the nozzle is prescribed. Discontinuous solutions have been thoroughly investigated by Shmyglevskiy. At least in principle, we use the same line of thought, but considerable simplifications are possible if one starts with the variational formulation of Rao. In its numerical discussion and also in some analytical details, the present paper goes beyond Shmyglevskiy's results. The problem is conveniently discussed in astate plane, which has the local state of the flow (flow direction and speed or Mach number) as independent variables. By taking into account second variations, one can determine the boundary of the region for which continuous solutions give the (local) maximum. This boundary coincides with the locus of points at which the solution in the physical plane would fold back into itself. Another limitation of the original approach emerges if one asks under which conditions the thrust can be increased by admitting along the control surface values of the entropy that are higher than those of the oncoming flow. The conditions for isentropic and nonisentropic jumps are formulated and evaluated next, and a survey of the discontinuities which satisfy conditions for isentropic and also for selected nonisentropic jumps is given. Up to this point, the analysis is concerned only with the state distribution along the control surface. Jumps of the state in the interior require the occurrence of centered compression waves. Sample computations show that, in most cases, flow fields of this character can be generated by the choice of the nozzle shape. In some cases, no nozzle contours exist which generate the optimizing state distribution along the control surface as determined by the present analysis. It would then be necessary to include from the very beginning conditions for the realizability of the flow field.     

The symmetry of rotating fluid bodies

Consider an incompressible fluid body (in outer space) rotating about an axis with a given angular velocity , and which is in equilibrium relative to the potential energy of its own gravitational field and the surface energy due to surface tension. We show that such a body possesses a plane of symmetry perpendicular to the axis of rotation such that any line parallel to the axis and meeting the body cuts it in a line segment whose center lies on the plane of symmetry. This extends an earlier result of L. Lichtenstein [4].This work was done with the support of S.F.B. 72 while the author was a visitor at the University of Bonn     

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

It is proved that, in Minkowski 3-space, a CSM-helicoidal surface, i.e., a helicoidal surface under cubic screw motion is isometric to a rotation surface so that helices on the helicoidal surface correspond to parallel circles on the rotation surface. By distinguishing a CSM-helicoidal surface as three cases, that is, the case of type I, the case of type II with negative and positive pitch, the relations are discussed between the mean curvatures or Gauss maps of a pair of isometric helicoidal and rotation surface. A CSM-helicoidal surface of Case 1 or 2 and its isometric rotation surface with null axis have same mean curvatures (resp. Gauss maps) if and only if they are minimal. But each pair of isometric CSM-helicoidal surface of Case 3 and rotation surface with spacelike axis have different Gauss maps.     

The vibrations of a rotating thin piezoceramic cylindrical shell

G. H. Bryan [8] discovered and studied the dynamic phenomena that arise during the vibrations of thinwalled shells in natural modes while rotating uniformly about an axis of symmetry. In the present article a more detailed mathematical analysis has been conducted for the solution of the problem of the vibrations of a ring rotating with constant angular velocity under feasible conditions of perturbation of the vibrations and realistic mechanical properties of ceramic materials. It is established that two slowly precessing waves arise during the vibration of the ring, one of which (the one discovered by Bryan) is an inverse precession wave having an angular velocity of precession less than the angular velocity of rotation of the ring, while the second is a direct precession wave having angular velocity larger than the angular velocity of rotation of the ring. Under the actual working conditions for piezoceramic resonators the amplitudes of the direct and inverse wave become comparable. Consequently the vibrations present as a modulated standing wave, that is, the phenomenon of beats in the mode of steady-state normal vibration occurs. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 7–18.     

Statistical characteristics and time autocorrelation coefficients of the turbulent swirl flow in pipe

In this paper is presented experimental investigation of the turbulent swirl flow in pipe generated by three axial fans of various geometries. One of these fans (model ZP) generates Rankine swirl, while other two, industrial fans, produce mainly solid body circumferential velocity profile. One-component laser Doppler anemometry (LDA) is employed in this research. Downstream transformations of the non-dimensional time-averaged velocity profiles are analyzed. Distributions of turbulence levels are discussed for all three fans in both measuring sections, as well as statistical moments of higher orders for fluctuations in the axial, radial and circumferential directions. Applied correlation theory revealed turbulence structure and its statistical nature. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)