Axisymmetric Obstacle Research Articles

Stability of an hypersonic magnetohydrodynamic shock wave

The shape of the shock wave formed by an hypersonic conducting flow around a plane or axisymmetric obstacle is determined by the velocity at infinity and the Lorentz force generated by a magnetic field generated within the body. This shape is described by a nonlinear differential equation. When linearized near the trivial shock wave (the boundary of the obstacle), it becomes amenable to a stability analysis. The study is conveniently separated in the regions where the slope of the body is not very small and the flatter sections. In both cases the main parameter is the sign of the Lorentz force; stability generally occurs when it presses the fluid inwards, but the precise criteria depend also on several other parameters.

On the shape of the cavity created by a thin magnetized airfoil

The shape of the cavity created in a conducting fluid by a magnetic field within a thin airfoil is studied by means of an asymptotic expansion in powers of the airfoil width. In the two dimensional case, for a supersonic flow the interface between the cavity and the external flow starts at the tip of the airfoil and to order zero satisfies a simple formula given by the balance between the inside and outside pressures. This may be generalized to three dimensional conical axisymmetric obstacles. For subsonic flows the appropriate formula is more complex, involving the Hilbert transform of the total pressure. In this case cavities do not form unless the mean of the perturbation is zero, and when they do they start far before the airfoil.

Vortex impingement onto an axisymmetric obstacle – subcritical bifurcation to vortex breakdown

Abstract

Drop squeezing between arbitrary smooth obstacles

Abstract

Drag associated with 3D trapped lee waves over an axisymmetric obstacle in two‐layer atmospheres

Mountain‐wave drag is evaluated explicitly using linear theory and verified against numerical simulations for the flow of idealized two‐layer atmospheres with piecewise‐constant stratification over an axisymmetric mountain. Static stability is either higher in the bottom layer and lower in the top layer (Scorer's atmosphere) or neutral in the bottom layer and positive in the top layer, separated by a sharp temperature inversion (Vosper's atmosphere). The drag receives contributions from long mountain waves propagating vertically in the upper layer and from short trapped lee waves propagating downstream, either in the lower layer or at the inversion. This trapped lee‐wave drag, which is typically not represented in parametrizations, acts on the atmosphere at low levels. As in flow over a 2D ridge, this drag has several maxima as a function of the height of the interface between the two layers for Scorer's atmosphere and is maximized by a marked Scorer parameter contrast between those layers. In Vosper's atmosphere, there is a single trapped lee‐wave drag maximum for Froude numbers near one, when the wind speed matches the phase speed of the dominant interfacial waves, and this drag is maximized for relatively low interface elevations, for which waves at the inversion have higher amplitude. The 3D flow geometry allows resonant wave modes to have various horizontal orientations and a continuous spectrum, forming a dispersive ‘Kelvin ship wave’ pattern, and expanding the regions in parameter space where the drag is non‐zero relative to 2D flow, but it also decreases the drag magnitude dispersively. Nevertheless, the trapped lee‐wave drag on an axisymmetric obstacle can still equal or exceed the drag associated with vertically propagating waves and the reference hydrostatic drag valid for a uniformly stratified atmosphere.

Drag Produced by Waves Trapped at a Density Interface in Nonhydrostatic Flow over an Axisymmetric Hill

Abstract Linear nonhydrostatic theory is used to evaluate the drag produced by 3D trapped lee waves forced by an axisymmetric hill at a density interface. These waves occur at atmospheric temperature inversions, for example, at the top of the boundary layer, and contribute to low-level drag possibly misrepresented as turbulent form drag in large-scale numerical models. Unlike in 2D waves, the drag has contributions from a continuous range of wavenumbers forced by the topography, because the waves can vary their angle of incidence to match the resonance condition. This leads to nonzero drag for Froude numbers (Fr) both <1 and >1 and a drag maximum typically for Fr slightly below 1, with lower magnitude than in hydrostatic conditions owing to wave dispersion. These features are in good agreement with laboratory experiments using two axisymmetric obstacles, particularly for the lower obstacle, if the effects of a rigid lid above the upper layer and friction are taken into account. Quantitative agreement is less satisfactory for the higher obstacle, as flow nonlinearity increases. However, even in that case the model still largely outperforms both 3D hydrostatic and 2D nonhydrostatic theories, emphasizing the importance of both 3D and nonhydrostatic effects. The associated wave signatures are dominated by transverse waves for Fr lower than at the drag maximum, a dispersive “Kelvin ship-wave” pattern near the maximum, and divergent waves for Fr beyond the maximum. The minimum elevation at the density-interface depression existing immediately downstream of the obstacle is significantly correlated with the drag magnitude.

Radiation torque, streaming, and viscous absorption due to orthogonal waves in slightly viscous fluids

Torque generated by orthogonal waves in slightly viscous fluids is analyzed to explore its connection with the near-boundary viscous absorption and streaming [Zhang and Marston, J. Acoust. Soc. Am. 131, 2917–2921 (2014)]. The analysis is based on a viscous correction and beam superposition of vortex fields. The analysis reports an approximation for the torque on a compressible small solid sphere and the viscous power absorption [Eqs. (15) and (21) in Zhang and Marston]. The torque expression in the dense sphere limit recovers prior expressions using near-boundary flow and streaming analyses: [Busse and Wang, J. Acoust. Soc. Am. 69, 1634–1638 (1981)] and [Lee and Wang, J. Acoust. Soc. Am. 85, 1081–1088 (1989)]. Our analysis extends prior results to give the torque on small compressible solid spheres of finite mass. The torque does not depend explicitly on the compressibility of the sphere, but does depend on the density of the sphere. Our analysis also gives the proportional relationship between the torque and the near-boundary viscous absorption. The results are generalized to cases of standing orthogonal waves and acoustic vortex fields, and other small axisymmetric obstacles such as circular cylinders and disks. [Work supported by the 2013-14 F. V. Hunt Postdoctoral Fellowship (Zhang) and by ONR (Marston).]

Acoustic radiation torque on small objects in viscous fluids and connection with viscous dissipation

This analysis reports a formula for torque and viscous power dissipation in scattering of orthogonal waves and vortex beams by a small compressible solid sphere in a slightly viscous fluid. The analysis is based on a viscous correction to far-field scattering, together with beam superposition. The analysis revels the relation between the torque and dissipation. The torque in a heavy sphere limit agrees with a prior analysis by Busse and Wang using boundary flow analysis. The results are applicable to arbitrary sound fields with proper phase distribution, and are extended to other small axisymmetric obstacles such as circular disks and cylinders.

Disorganization of a hole tone feedback loop by an axisymmetric obstacle on a downstream end plate

AbstractThis study investigates the suppression of the sound produced when a jet, issued from a circular nozzle or hole in a plate, goes through a similar hole in a second plate. The sound, known as a hole tone, is encountered in many practical engineering situations. The mean velocity of the air jet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}u_0$ was $6\text {--}12\ \mathrm{m}\ {\mathrm{s}}^{-1}$. The nozzle and the end plate hole both had a diameter of 51 mm, and the impingement length $L_{im}$ between the nozzle and the end plate was 50–90 mm. We propose a novel passive control method of suppressing the tone with an axisymmetric obstacle on the end plate. We find that the effect of the obstacle is well described by the combination ($W/L_{im}$, $h$) where $W$ is the distance from the edge of the end plate hole to the inner wall of the obstacle, and $h$ is the obstacle height. The tone is suppressed when backflows from the obstacle affect the jet shear layers near the nozzle exit. We do a direct sound computation for a typical case where the tone is successfully suppressed. Axisymmetric uniformity observed in the uncontrolled case is broken almost completely in the controlled case. The destruction is maintained by the process in which three-dimensional vortices in the jet shear layers convect downstream, interact with the obstacle and recursively disturb the jet flow from the nozzle exit. While regions near the edge of the end plate hole are responsible for producing the sound in the controlled case as well as in the uncontrolled case, acoustic power in the controlled case is much lower than in the uncontrolled case because of the disorganized state.

Wave patterns generated by an axisymmetric obstacle in a two-layer flow

Gravity waves generated by a moving obstacle in a two-layer stratified fluid are investigated. The experimental configuration is three-dimensional with an axisymmetric obstacle which is towed in one of the two layers. The experimental method used in the present study is based on a stereoscopic technique allowing the 3D reconstruction of the interface between the two layers. Investigation into the wave pattern as a function of the Froude number, Fr, based on the relative density of the fluid layers and the velocity of the towed obstacle is presented. Specific attention is paid to the transcritical regime for which Fr is close to one. Potential energy trapped in the wave field patterns is also extracted from the experimental results and is analyzed as a function of both the Froude number, Fr, and the transcritical similarity parameter \(\varGamma\). In particular, a remarkable increase in the potential energy around Fr = 1 is observed and a scaling allowing to assemble data resulting from different experimental parameters is proposed.

Momentum Fluxes of Gravity Waves Generated by Variable Froude Number Flow over Three-Dimensional Obstacles

Abstract Fully nonlinear mesoscale model simulations are used to investigate the momentum fluxes of gravity waves that emerge at a “far-field” height of 6 km from steady unsheared flow over both an axisymmetric and elliptical obstacle for nondimensional mountain heights ĥm = Fr−1 in the range 0.1–5, where Fr is the surface Froude number. Fourier- and Hilbert-transform diagnostics of model output yield local estimates of phase-averaged momentum flux, while area integrals of momentum flux quantify the amount of surface pressure drag that translates into far-field gravity waves, referred to here as the “wave drag” component. Estimates of surface and wave drag are compared to parameterization predictions and theory. Surface dynamics transition from linear to high-drag (wave breaking) states at critical inverse Froude numbers Frc−1 predicted to within 10% by transform relations. Wave drag peaks at Frc−1 < ĥm ≲ 2, where for the elliptical obstacle both surface and wave drag vacillate owing to cyclical buildup and breakdown of waves. For the axisymmetric obstacle, this occurs only at ĥm = 1.2. At ĥm ≳ 2–3 vacillation abates and normalized pressure drag assumes a common normalized form for both obstacles that varies approximately as ĥm−1.3. Wave drag in this range asymptotes to a constant absolute value that, despite its theoretical shortcomings, is predicted to within 10%–40% by an analytical relation based on linear clipped-obstacle drag for a Sheppard-based prediction of dividing streamline height. Constant wave drag at ĥm ∼ 2–5 arises despite large variations with ĥm in the three-dimensional morphology of the local wave momentum fluxes. Specific implications of these results for the parameterization of subgrid-scale orographic drag in global climate and weather models are discussed.

Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation

Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Γ) M5/3, where Γ = (F-1)M−2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the ‘equivalent aerofoil’ specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M≲0.4 and |Γ| ≲ 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev–Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.For Γ ≳ 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be $c_d M^2/(F\sqrt{F^2-1})$, where cd is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Korteweg–de Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of ${\cal A}= 3M/(2\delta^2 \sqrt{F^2-1})$, where δ is the ratio of the undisturbed layer depth to the radial scale of the obstacle.

Two-dimensional leaps in near-critical flow over isolated orography

This paper describes steady two-dimensional disturbances forced on the interface of a two-layer fluid by flow over an isolated obstacle. The oncoming flow speed is close to the linear longwave speed and the layer densities, layer depths and obstacle height are chosen so that the equations of motion reduce to the forced two-dimensional Korteweg–de Vries equation with cubic nonlinearity, i.e. the forced extended Kadomtsev–Petviashvili equation. The distinctive feature noted here is the appearance in the far lee-wave wake behind obstacles in subcritical flow of a ‘table-top’ wave extending almost one-dimensionally for many obstacles widths across the flow. Numerical integrations show that the most important parameter determining whether this wave appears is the departure from criticality, with the wave appearing in slightly subcritical flows but being destroyed by two-dimensional effects behind even quite long ridges in even moderately subcritical flow. The wave appears after the flow has passed through a transition from subcritical to supercritical over the obstacle and its leading and trailing edges resemble dissipationless leaps standing in supercritical flow. Two-dimensional steady supercritical flows are related to one-dimensional unsteady flows with time in the unsteady flow associated with a slow cross-stream variable in the two-dimensional flows. Thus the wide cross-stream extent of the table-top wave appears to derive from the combination of its occurrence in a supercritical region embedded in the subcritical flow and the propagation without change of form of table-top waves in one-dimensional unsteady flow. The table-top wave here is associated with a resonant steepening of the transition above the obstacle and a consequent twelve-fold increase in drag. Remarkably, the table-top wave is generated equally strongly and extends laterally equally as far behind an axisymmetric obstacle as behind a ridge and so leads to subcritical flows differing significantly from linear predictions.

Trapped modes in cylindrical waveguides

We prove the existence of trapped modes in the presence of two classes of obstacles in cylindrical acoustic waveguides. First we prove that trapped modes exist whenever the obstacle is thin and has a normal which is everywhere perpendicular to the generators of the cylinder. Secondly we prove that for the case of a circular cylindrical guide containing an axisymmetric obstacle, an infinite sequence of trapped modes exists, the frequency of the modes tending to infinity. In each case we consider an example where the trapped mode frequencies can be calculated numerically using the residue calculus method.

A Small Axisymmetric Obstacle in the Presence of an Underwater Point-Source Field

A small, acoustically hard and axisymmetric object is placed in a deep homogeneous sea environment with a hard plane bottom. The free surface of the sea is assumed to be soft. The source and the receiver are placed on the same vertical line, far away from the object. Given the positions of the source and the receiver, two problems are solved: the determination of the pressure field at the receiver from the position and the shape of the object, and the determination of the position and the shape of the object from the pressure field at the receiver. The special case of smooth objects generated by the rotation of differentiable curves is studied. We provide results for the case of a floating object and for the case of an object or a boss at the bottom of the sea.

Impedances of azimuthally symmetric irises and cavities with semielliptical profile in a beam pipe

The beam coupling impedances of small axisymmetric obstacles having a semielliptical cross section along the beam in the vacuum chamber of an accelerator are calculated at frequencies for which the wavelength is large compared to a typical size of the obstacle. Analytical results are obtained for both the irises and the cavities with such a shape, which allows simple estimates of their broadband impedances.

Effects on the Jovian magnetosheath arising from solar wind flow around nonaxisymmetric bodies

We investigate the MHD structure of the Jovian magnetosheath along the Sun‐Jupiter line and, particularly, the region where the interplanetary magnetic field (IMF) exerts a large influence on the magnetosheath flow (the “magnetic barrier”). We do this by integrating numerically the dissipationless MHD equations in their “magnetic string” formulation. The lack of axisymmetry of the magnetospheric obstacle introduces corresponding asymmetries in the Jovian magnetosheath. The dominant effect on the flow is produced by the IMF component orthogonal to Jupiter's rotational equator. The thicknesses of the magnetosheath and magnetic barrier depend sensitively on the orientation of the IMF, decreasing monotonically as the inclination of the IMF to the rotational equator decreases. The magnetic barrier practically disappears when the IMF vector lies in the equator. For an arbitrary orientation of the IMF the magnetosheath magnetic field along the stagnation streamline is not only compressed as the magnetopause is approached but also rotates smoothly toward the direction of the Jovian rotation axis. This effect is absent in the case of flow around axisymmetric obstacles, such as the terrestrial magnetosphere.

Electromagnetic Scattering from Dielectrically Coated Axisymmetric Objects Using the Generalized Point-Matching Technique (GPMT)

In Part I, the theoretical formulation of a numerical solution to electromagnetic scattering problems encompassing arbitrarily oriented, three-dimensional axisymmetric perfectly conducting or dielectric objects embedded in an axisymmetric dielectric obstacle has been developed in a form suitable for an efficient computer solution. Here, numerical results for a wide variety of scatterer configurations are presented in order to demonstrate the validity and accuracy of our generalized point-matching technique formulation. Whenever possible, the Mie theory, the extended Mie theory, the homogeneous extended boundary-condition method (EBCM), the multilayered EBCM, and the Rayleigh-Gans-Debye approximation have been used in the process of examining the applicability of the suggested method. In all of the cases considered, excellent agreement has been achieved.

Resonant generation of finite‐amplitude waves by the uniform flow of a uniformly rotating fluid past an obstacle

In a previous paper (Grimshaw, 1990a) we showed that the resonant, or critical, flow of a rotating fluid past an axisymmetric obstacle placed on the axis of a cylindrical tube is described by a forced Korteweg-de Vries equation for the amplitude of the dominant resonant mode. Here we show that in the anomalous but important case when the oncoming flow is uniform with uniform angular velocity a different theory is required which leads to an evolution equation describing finite-amplitude waves. Some numerical solutions of this equation are described.

ETUDE DU BRUIT ENGENDRÉ PAR L'IMPACT D'UN JET SUPERSONIQUE CHAUD SUR UN OBSTACLE

This document presents experimental works on supersonic hot jet impingement noise which have been done in an anechoic wind tunnel. Far field noise directivity measurements were done for a supersonic jet impingement on an axisymmetric obstacle. Source localizations with focused microphone arrays were realized simultaneously. Introduction of a board in a jet increases the sound pressure level upstream of the board. The localizations with focused microphone arrays show that the predominant sources of noise are located near the impingement region. However, this apparition of sound sources does not amount to a large increase of the total sound power level