Papers
The pulsatile motion of a semi-infinite bubble in a channel: flow fields, and transport of an inactive surface-associated contaminant
- MAXIMILLIAN E. ZIMMER, HARVEY A. R. WILLIAMS, DONALD P. GAVER
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- 04 August 2005, pp. 1-33
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We investigate a theoretical model of the pulsatile motion of a contaminant-doped semi-infinite bubble in a rectangular channel. We examine the fluid mechanical behaviour of the pulsatile bubble, and its influence on the transport of a surface-inactive contaminant (termed surfinactant). This investigation is used to develop a preliminary understanding of surfactant responses during unsteady pulmonary airway reopening. Reopening is modelled as the pulsatile motion of a semi-infinite gas bubble in a horizontal channel of width 2$a$ filled with a Newtonian liquid of viscosity $\mu$ and constant surface tension $\gamma$. A modified Langmuir sorption model is assumed, which allows for the creation and respreading of a surface multilayer. The bubble is forced via a time-dependent volume flux $Q(t)$ with mean and oscillatory components ($Q_{M}$ and $Q_{\omega }$, respectively) at frequency $\omega $. The flow behaviour is governed by the dimensionless parameters: Ca$_{M} \,{=}\,\mu Q_{M}/(2a\gamma $), a steady-state capillary number, which represents the ratio of viscous to surface tension forces; Ca$_{\Omega } \,{=}\,\mu Q_{\omega }/(2a\gamma $), an oscillatory forcing magnitude; $\Omega \,{=}\,\omega \mu a/\gamma $, a dimensionless frequency that represents the ratio of viscous relaxation to oscillatory-forcing timescales; and $A\,{=}\,2\hbox{\it Ca}_{\Omega }/\Omega $, a dimensionless oscillation amplitude. Our simulations indicate that contaminant deposition and retention in the bubble cap region occurs at moderate frequencies if retrograde bubble motion develops during the oscillation cycle. However, if oscillations are too rapid the ensuing large forward tip velocities cause a net loss of contaminant from the bubble tip. Determination of an optimal oscillation range may be important in reducing ventilator-induced lung injury associated with infant and adult respiratory distress syndromes by increasing surfactant transport to regions of collapsed airways.
Precipitate formation in a porous rock through evaporation of saline water
- GEORGE G. TSYPKIN, ANDREW W. WOODS
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- 04 August 2005, pp. 35-53
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We examine the motion of a high-pressure aqueous solution, through a low-permeability fracture, towards a low-pressure well. As the liquid decompresses in the fractures it expands, and for sufficiently high initial temperature the liquid reaches the boiling point. A vaporization front then develops, so that vapour issues from the well. As the fluid evaporates near the well, the salt concentration of the residual fluid increases. If the salt concentration increases beyond the saturation limit, then the evaporation leads to precipitation of salt in the fracture. We find a new family of self-similar solutions to describe the boiling and precipitation in a single idealized fracture, which at long times remains approximately isothermal owing to the cross-fracture heat transfer. The solutions describe the mass of salt that precipitates as a function of the initial salt concentration, the reservoir temperature and pressure, and the well pressure. In fact, this family of self-similar solutions is multi-valued: we identify a liquid-advection-dominated regime, in which the boiling front advances slowly and the fracture porosity decreases significantly, and a boiling-dominated regime, in which the boiling front advances more rapidly, and less precipitate forms at each point in the fracture. As the pressure difference between the well and the far field reservoir increases, these solutions converge, and eventually coincide. Beyond this critical point, there is no similarity solution, since the advective flux of salt from the far-field would produce more precipitate than can be taken up in the fracture adjacent to the boiling front. Instead, the rock will become fully sealed through precipitation, thereby suppressing flow into the well. We extend the model to show that an analogous result also occurs within an extensive porous layer. However in that case, the system is not isothermal; instead, the heat flux is supplied in the direction of flow, while the cross-flow heat flux is small. We discuss the relevance of the work to the natural venting of steam in high-temperature geothermal systems.
Linearized analysis of Richtmyer–Meshkov flow for elastic materials
- JEEYEON N. PLOHR, BRADLEY J. PLOHR
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- 04 August 2005, pp. 55-89
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We present a study of Richtmyer–Meshkov flow for elastic materials. This flow, in which a material interface is struck by a shock wave, was originally investigated for gases, where growth of perturbations of the interface is observed. Here we consider two elastic materials in frictionless contact. The governing system of equations comprises conservation laws supplemented by constitutive equations. To analyse it, we linearize the equations around a one-dimensional background solution under the assumption that the perturbation is small. The background problem defines a Riemann problem that is solved numerically; its solution contains transmitted and reflected shock waves in the longitudinal modes. The linearized Rankine–Hugoniot condition provides the interface conditions at the longitudinal and shear waves; the frictionless material interface conditions are also linearized. The resulting equations, a linear system of partial differential equations, is solved numerically using a finite-difference method supplemented by front tracking. In verifying the numerical code, we reproduce growth of the interface in the gas case. For the elastic case, in contrast, we find that the material interface remains bounded: the non-zero shear stiffness stabilizes the flow. In particular, the linear theory remains valid at late time. Moreover, we identify the principal mechanism for the stability of Richtmyer–Meshkov flow for elastic materials: the vorticity deposited on the material interface during shock passage is propagated away by the shear waves, whereas for gas dynamics it stays on the interface.
Transient growth on boundary layer streaks
- JÉRÔME HŒPFFNER, LUCA BRANDT, DAN S. HENNINGSON
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- 04 August 2005, pp. 91-100
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The linear perturbations evolving on streamwise boundary layer streaks which yield maximum energy growth are computed. The steady and spanwise-periodic streaks arising from the nonlinear saturation of optimally growing streamwise vortices are considered as base flow. It is shown that significant transient growth may occur for both sinuous antisymmetric perturbations and for varicose symmetric modes. The energy growth is observed at amplitudes significantly below the threshold beyond which the streaks become linearly unstable and is largest for sinuous perturbations, to which the base flow considered first become unstable. The optimal initial condition consists of velocity perturbations localized in the regions of highest shear of the streak base flow, tilted upstream from the wall. The optimal response is still localized in the areas of largest shear but it is tilted in the flow direction. The most amplified perturbations closely resemble the unstable eigenfunctions obtained for streaks of higher amplitudes. The results suggest the possibility of a transition scenario characterized by the non-modal growth of primary perturbations, the streaks, followed by the secondary transient growth of higher frequency perturbations. The implication for turbulent flow is also discussed.
The non-Boussinesq lock-exchange problem. Part 1. Theory and experiments
- RYAN J. LOWE, JAMES W. ROTTMAN, P. F. LINDEN
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- 04 August 2005, pp. 101-124
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The results of an experimental study of the non-Boussinesq lock-exchange problem are described. The experiments were performed in a rectangular channel using water and either a sodium iodide solution or a sodium chloride solution as the two fluids. These combinations of fluids have density ratios (light over heavy density) in the range 0.61 to 1. A two-layer hydraulic theory is developed to model the experiments. The theory assumes that a light gravity current propagates in one direction along the top of the channel and a heavy gravity current propagates in the opposite direction along the bottom of the channel. The two currents are assumed to be connected by either a combination of an internal bore and an expansion wave, or just an expansion wave. The present results, previous experimental results and two-dimensional numerical simulations from a companion paper are compared with the theory. The results of the comparison lead to the conclusion that the theory without the internal bore is the most appropriate.
The non-Boussinesq lock-exchange problem. Part 2. High-resolution simulations
- V. K. BIRMAN, J. E. MARTIN, E. MEIBURG
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- 04 August 2005, pp. 125-144
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The present investigation explores the unsteady dynamics of large density contrast non-Boussinesq lock-exchange flows by means of high-resolution two-dimensional simulations of the incompressible variable-density Navier–Stokes equations, employing a combination of spectral and compact finite-difference methods. For small density contrasts, the simulations closely reproduce earlier Boussinesq results for corresponding flows. Across the entire range of density contrasts, good agreement is obtained between the computed front propagation velocities and corresponding experimental observations reported in Part 1 of this investigation and by other authors. The simulations yield the required quantitative information with respect to the light and dense front heights, their propagation velocities, and the spatial structure of the dissipation fields in order to determine conclusively which of the scenarios developed in Part 1 is observed in reality. Simulations are conducted for fluids with the same kinematic viscosity, as well as for fluids with the same dynamic viscosity. For both slip and no-slip boundary conditions, and for all $\hbox{\it Re}$ values, we find that for larger density contrasts, the dense front dissipates an increasing amount of energy. In contrast, the energy dissipated by the light front remains near its Boussinesq level for all values of the density ratio. In addition, for all density ratios, the height of the light front is very close to half the channel height, and it propagates with a non-dimensional velocity close to a half. This provides strong evidence that the dynamics of the light front is indeed approximated by the energy-conserving solution described in an earlier theoretical analysis. In contrast, the height of the dense front is substantially less than half the channel height. In addition, its velocity is close to the value derived in Part 1 for a dissipative gravity current. Together with the above results for the dissipation field, this confirms that the dense front behaves as a dissipative gravity current.
Convection in a narrow annular channel rotating about its axis of symmetry
- F. H. BUSSE
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- 04 August 2005, pp. 145-154
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The onset of convection in a narrow cylindrical annulus heated from below and rotating about its vertical axis of symmetry is considered in the case when the rigid cylindrical walls are thermally insulating. An analytical expression is derived for the Rayleigh number $R$ for onset of convection as a function of rotation rate and azimuthal wavenumber. The critical value $R_c$ for high rotation rates is much lower than the corresponding value in an extended layer. At finite amplitudes the convection flow generates a differential rotation which is antisymmetric with respect to the middle of the layer and is prograde near the outer wall for low rotation rates, but changes sign for higher values of the rotation parameter.
Dynamics of crescent water wave patterns
- D. FRUCTUS, C. KHARIF, M. FRANCIUS, Ø. KRISTIANSEN, D. CLAMOND, J. GRUE
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- 04 August 2005, pp. 155-186
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The nonlinear dynamics of three-dimensional instabilities of uniform gravity-wave trains evolving to crescent wave patterns is investigated numerically. A new mechanism of generation of oscillating horseshoe patterns is proposed and a detailed discussion on their occurrence in a water wave tank is given. It is suggested that these patterns are more likely to be observed naturally in water of finite depth. A critical wave steepness for the onset of three-dimensional wave breaking due to the nonlinear evolution of quintet resonant interactions corresponding to the phase-locked crescent-shaped structures (class II instability) is provided when the quartet resonant interaction (class I instability) is absent. The nonlinear coupling between quartet resonant interactions (class I instability) and quintet resonant interactions (class II instability) leading to three-dimensional breaking waves, as shown experimentally by Su & Green (1984, 1985), is numerically investigated.
Optimal and robust control of streaks in pipe flow
- M. I. GAVARINI, A. BOTTARO, F. T. M. NIEUWSTADT
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- 04 August 2005, pp. 187-219
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Control theory is used to determine optimal disturbances in pipe flow and the forcing, in the form of blowing and suction at the wall, capable of attenuating them. An approach is adopted, based on a parabolic approximation of the linear Navier–Stokes equations, which is appropriate when dealing with asymptotically elongated (in the streamwise direction) flow structures. A cost functional is introduced and maximized in the optimal-perturbation problem or minimized, for a given inflow perturbation, in the optimal-control problem. The extrema of the cost functional are reached by means of an iterative technique, based on the numerical solution of the equations for the state and the adjoint state, coupled via transfer and optimality conditions. Central to the control is the determination of the Green's function expressing the receptivity of the flow to wall forcing. A considerable reduction in output disturbance energy, as compared to the uncontrolled case, is obtained for control laws operating both over a long section of the pipe or over shorter strips. Finally, a robust control is sought, by simultaneously computing the worst inflow condition and the corresponding best control at the wall.
Evolution of a chemically reacting plume in a ventilated room
- D. T. CONROY, STEFAN G. LLEWELLYN SMITH, C. P. CAULFIELD
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- 04 August 2005, pp. 221-253
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The dynamics of a second-order chemical reaction in an enclosed space driven by the mixing produced by a turbulent buoyant plume are studied theoretically, numerically and experimentally. An isolated turbulent buoyant plume source is located in an enclosure with a single external opening. Both the source and the opening are located at the bottom of the enclosure. The enclosure is filled with a fluid of a given density with a fixed initial concentration of a chemical. The source supplies a constant volume flux of fluid of different density containing a different chemical of known and constant concentration. These two chemicals undergo a second-order non-reversible reaction, leading to the creation of a third product chemical. For simplicity, we restrict attention to the situation where the reaction process does not affect the density of the fluids involved. Because of the natural constraint of volume conservation, fluid from the enclosure is continually vented. We study the evolution of the various chemical species as they are advected by the developing ventilated filling box process within the room that is driven by the plume dynamics. In particular, we study both the mean and vertical distributions of the chemical species as a function of time within the room. We compare the results of analogue laboratory experiments with theoretical predictions derived from reduced numerical models, and find excellent agreement. Important parameters for the behaviour of the system are associated with the source volume flux and specific momentum flux relative to the source specific buoyancy flux, the ratio of the initial concentrations of the reacting chemical input in the plume and the reacting chemical in the enclosed space, the reaction rate of the chemicals and the aspect ratio of the room. Although the behaviour of the system depends on all these parameters in a non-trivial way, in general the concentration within the room of the chemical input at the isolated source passes through three distinct phases. Initially, as the source fluid flows into the room, the mean concentration of the input chemical increases due to the inflow, with some loss due to the reaction with the chemical initially within the room. After a finite time, the layer of fluid contaminated by the inflow reaches the opening to the exterior at the base of the room. During an ensuing intermediate phase, the rate of increase in the concentration of the input chemical then drops non-trivially, due to the extra sink for the input chemical of the outflow through the opening. During this intermediate stage, the concentration of the input chemical continues to rise, but at a rate that is reduced due to the reaction with the fluid in the room. Ultimately, all the fluid (and hence the chemical) that was originally within the room is lost, both through reaction and outflow through the opening, and the room approaches its final steady state, being filled completely with source fluid.
Planform selection in Rayleigh–Bénard convection between finite slabs
- B. HOLMEDAL, M. TVEITEREID, E. PALM
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- 04 August 2005, pp. 255-270
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Thermal convection in a thin horizontal fluid layer enclosed between two rigid slabs of arbitrary thicknesses and conductivities has been investigated. We have found a mathematical transformation between this problem and the problem of the upper and lower slabs being interchanged. A weakly nonlinear expansion has been applied to reduce the governing equations to a set of Landau equations. Their extremum principle combined with an analytical solution for the case of insulating slabs has been used to prove that rhombuses and rolls are the only stable solutions. Hexagons, quasi-patterns and any solution involving higher numbers of modes, are proved to be unstable. Stability regions of rolls and rhombuses have been found numerically for a wide range of slab conductivities and thicknesses. The wavenumber selection has been investigated by studying two coupled Ginzburg–Landau equations. Earlier stability analyses of Proctor's equation valid for the limit of poorly conducting slabs has revealed that the wavenumbers of squares, i.e. rhombuses with orthogonal wave vectors, are restricted by a zigzag instability and by a truly three-dimensional instability. We show here that the wavenumber selection for more general cases with finite conductivities and thicknesses of the slabs are always restricted by the same types of instability. In addition, we show how the stability and wavenumber selection of another solution of the Ginzburg–Landau equations, the undulated rolls, is restricted by a cross-roll instability.
Stability of squares and rolls in Rayleigh–Bénard convection in an infinite-Prandtl-number fluid between slabs
- B. HOLMEDAL
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- 04 August 2005, pp. 271-284
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Nonlinear solutions in the form of squares and rolls are investigated for Rayleigh–Bénard convection in an infinite-Prandtl-number fluid enclosed between two symmetric slabs. It is found that the heat transfer depends strongly on the thickness and thermal conductivity of the slabs, but hardly on the planform of convection. Examples of stability regions of rolls are calculated, showing that for certain slab selections, rolls remain stable at even larger Rayleigh numbers than with fixed temperatures at the boundaries. The region of stable squares is restricted by a zigzag and a long-wavelength cross-roll instability in addition to a new three-dimensional instability. As the slab conductivity is increased, the stability region of the squares shrinks onto a point located well above the critical point for the onset of convection. For a small range of slab conductivities, stability regions for squares and rolls both exist for the same set-up. In the present calculations, the regions never overlap. An example, where both patterns are stable at the same Rayleigh number, provides an explanation for the co-existence of rolls and squares where transparent slabs with a low thermal conductivity were applied.
Effects of rotation and sloping terrain on the fronts of density currents
- J. C. R. HUNT, J. R. PACHECO, A. MAHALOV, H. J. S. FERNANDO
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- 04 August 2005, pp. 285-315
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The initial stage of the adjustment of a gravity current to the effects of rotation with angular velocity $f/2$ is analysed using a short time analysis where Coriolis forces are initiated in an inviscid von Kármán–Benjamin gravity current front at $t_F\,{=}\,0$. It is shown how, on a time-scale of order $1/f$, as a result of ageostrophic dynamics, the slope and front speed $U_F$ are much reduced from their initial values, while the transverse anticyclonic velocity parallel to the front increases from zero to $O(N H_0)$, where $N\,{=}\,\sqrt{g'/H_0}$ is the buoyancy frequency, and $g'\,{=}\,g \Delta \rho /\rho_0$ is the reduced acceleration due to gravity. Here $\rho_0$ is the density and $\Delta \rho$ and $H_0$ are the density difference and initial height of the current. Extending the steady-state theory to account for the effect of the slope $\sigma$ on the bottom boundary shows that, without rotation, $U_{F}$ has a maximum value for $\sigma \,{=}\, \upi/6$, while with rotation, $U_{F}$ tends to zero on any slope. For the asymptotic stage when $ft_F \,{\gg}\, 1$, the theory of unsteady waves on the current is reviewed using nonlinear shallow-water equations and the van der Pol averaging method. Their motions naturally split into a ‘balanced’ component satisfying the Margules geostrophic relation and an equally large ‘unbalanced’ component, in which there is horizontal divergence and ageostrophic vorticity. The latter is responsible for nonlinear oscillations in the current on a time scale $f^{-1}$, which have been observed in the atmosphere and field experiments. Their magnitude is mainly determined by the initial potential energy in relation to that of the current and is proportional to the ratio $\sqrt{\hbox{\it Bu}} \,{=}\, L_R/R_0$, where $L_R\,{=}\,N H_0/f$ is the Rossby deformation radius and $R_0$ is the initial radius. The effect of slope friction also prevents the formation of a steady front. From the analysis it is concluded that a weak mean radial flow must be driven by the ageostrophic oscillations, preventing the mean front speed $U_F$ from halting sharply at $f t_F \,{\sim}\, 1$. Depending on the initial value of $L_R/R_0$, physical arguments show that $U_F$ decreases slowly in proportion to $(f t_F)^{-1/2}$, i.e. $U_F/U_{F_0}\,{=}\,F(ft_F,\hbox{\it Bu})$. Thus the front only tends to the geostrophic asymptotic state of zero radial velocity very slowly (i.e. as $f t_F \,{\rightarrow}\, \infty $) for finite values of $L_R/R_0$. However, as $L_R/R_0 \,{\rightarrow}\, 0$, it reaches this state when $f t_F \,{\sim}\, 1$. This analysis of the overall nonlinear behaviour of the gravity current is consistent with two two-dimensional non-hydrostatic (Navier–Stokes) and axisymmetric hydrostatic (shallow-water) Eulerian numerical simulations of the varying form of the rotating gravity current. When the effect of surface friction is considered, it is found that the mean movement of the front is significantly slowed. Furthermore, the oscillations with angular frequency $f$ and the slow growth of the radius, when $ft_F \,{\ge}\, 1$, are consistent with recent experiments.
Analysis of weakly turbulent dilute-spray flames and spray combustion regimes
- JULIEN REVEILLON, LUC VERVISCH
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- 04 August 2005, pp. 317-347
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Spray combustion is analysed using a full simulation of the continuous gaseous carrier phase, while dilute-spray modelling is adopted for the discrete phase. The direct numerical simulation of the flow is performed in an Eulerian context and a Lagrangian description is used for the spray. The numerous physical parameters controlling spray flames are first studied to construct two synthetic model problems of spray combustion: a laminar spray flame that propagates freely over a train of droplets and a weakly turbulent spray-jet with coflowing preheated air. It is observed that the flame structures can be classified with respect to three dimensionless quantities, which characterize the fuel/air equivalence ratio within the core of the spray-jet, the ratio between the mean distance between the droplets and the flame thickness, and the ratio between an evaporation time and a flame time. A large variety of reaction zone topologies is found when varying those parameters, and they are scrutinized by distinguishing between premixed and diffusion combustion regimes. Partially premixed combustion is observed in most of the spray-jet flames and the spray parameters that make the flame transition from non-premixed to premixed combustion are determined. A combustion diagram for dilute-spray combustion is then proposed from the identification of those various regimes.
Investigation of noise sources in high-speed jets via correlation measurements
- J. PANDA, R. G. SEASHOLTZ, K. A. ELAM
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- 04 August 2005, pp. 349-385
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To locate noise sources in high-speed jets, the far-field sound pressure fluctuations $p^\prime $ were correlated with each of density $\rho $, axial velocity $u$, radial velocity $v$, $\rho uu$ and $\rho vv$ fluctuations measured from various points in jet plumes. Detailed surveys were conducted in fully expanded, unheated plumes of Mach 0.95, 1.4 and 1.8. The velocity and density fluctuations were measured simultaneously using a recently developed non-intrusive point measurement technique based on molecular Rayleigh scattering. The technique uses a continuous-wave narrow line-width laser, Fabry–Perot interferometer and photon counting electronics. Laser light scattered by air molecules from a 1.06 mm long region on the narrow beam was collected and spectrally resolved by the interferometer. It was observed that the fluctuation spectra for air density inside the plume were in general similar to those of axial velocity spectra, while the radial velocity spectra were somewhat different. For the correlation study, microphone polar angles were varied from 30$^\circ$ to 90$^\circ$ to the jet axis. The sound pressure fluctuations $p^\prime $ at the shallowest 30$^\circ$ angle provided the highest correlation with turbulent fluctuations. The correlations sharply decreased as the polar angle was increased to 60$^\circ$, beyond which all data mostly fell below the experimental noise floor. Among all turbulent fluctuations $\langle\rho uu; p^\prime\rangle$ correlations showed the highest values. Correlation with the first-order terms $\langle\rho ^\prime \bar{u} \bar{u}; p^\prime \rangle$, $\langle\skew3\bar\rho \bar{u}u^\prime; p^\prime\rangle$ and third-order terms $\langle\rho ^\prime u^\prime u^\prime ;p^\prime\rangle$ was higher than that from the second-order terms $\langle\skew3\bar\rho u^\prime u^\prime ;p^\prime \rangle$ and $\langle\bar u\rho ^\prime u^\prime ; p^\prime \rangle$. Both $\langle v^\prime ; p^\prime \rangle$ and $\langle\rho vv; p^\prime \rangle$ correlations with the 90$^\circ$ microphone signal fell below the experimental noise floor, while that from the shallow 30$^\circ$ microphone showed weaker values. By moving the laser probe to various locations in the jet, it was found that the strongest noise source lay downstream of the end of the potential core and extended many diameters beyond. Correlation measurements from turbulent fluctuations along the lip shear layer showed a Mach-number dependency: significant values were measured in supersonic jets, while correlations fell below the noise floor for subsonic jets. Various additional analyses showed that fluctuations from large coherent structures mostly contributed to the measured correlation, while that from small-scale structures fell below the noise floor.
Experimental and numerical investigation of the dynamics of an underwater explosion bubble near a resilient/rigid structure
- E. KLASEBOER, K. C. HUNG, C. WANG, C. W. WANG, B. C. KHOO, P. BOYCE, S. DEBONO, H. CHARLIER
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- 04 August 2005, pp. 387-413
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This paper deals with an experimental and numerical study of the dynamics of an underwater explosion and its associated fluid–structure interaction. Experimental studies of the complex fluid–structure interaction phenomena were carried out in a specially designed test pond. The pond is equipped with a high-speed camera and pressure and displacement sensors. The high-speed camera was used to capture the expansion and collapse of the gas bubble created by the explosion. Several different structures were used in the experiments, including both rigid and resilient plates of circular shape. The deformation of the plate was measured with a non-contact laser telemetry device. The numerical simulations of the explosion bubble interacting with a submerged resilient structure were performed using a three-dimensional bubble dynamics code in conjunction with a structural code. The bubble code is based on the boundary-element method (BEM) and has been coupled to a structural finite-element code (PAM-CRASH$^{\rm TM})$. The experimental results were compared against the numerical results for different bubble–structure configurations and orientations. Several physical phenomena that have been observed, such as bubble jetting and bubble migration towards the structure are discussed.
The Rayleigh–Taylor instability of two-dimensional high-density vortices
- L. JOLY, J. FONTANE, P. CHASSAING
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- 04 August 2005, pp. 415-431
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We investigate the stability of variable-density two-dimensional isolated vortices in the frame of incompressible mixing under negligible gravity. The focus on a single vortex flow stands as a first step towards vortex interactions and turbulent mixing. From heuristic arguments developed on a perturbed barotropic vortex, we find that high-density vortices are subject to a Rayleigh–Taylor instability. The basic mechanism relies on baroclinic vorticity generation when the density gradient is misaligned with the centripetal acceleration field. For Gaussian radial distributions of vorticity and density, the intensity of the baroclinic torque due to isopycnic deformation is shown to increase with the ratio $\delta/\delta_\rho$ of the vorticity radius to the density radius. Concentration of mass near the vortex core is confirmed to promote the instability by the use of an inviscid linear stability analysis. We measure the amplification rate for the favoured azimuthal wavenumbers $m\,{=}\,2,3$ on the whole range of positive density contrasts between the core and the surroundings. The separate influence of the density-contrast and the radius ratio is detailed for modes up to $m\,{=}\,6$. For growing azimuthal wavenumbers, the two-dimensional structure of the eigenmode concentrates on a ring of narrowing radial extent centred on the radius of maximum density gradient. The instability of the isolated high-density vortex is then explored beyond the linear stage based on high-Reynolds-number numerical simulations for modes $m\,{=}\,2,3$ and a moderate density contrast $C_{\rho}\,{=}\,0.5$. Secondary roll-ups are seen to emerge from the nonlinear evolution of the vorticity and density fields. The transition towards $m$ smaller vortices involves vorticity exchange between initially-rotating dense fluid particles and the irrotational less-dense medium. It is shown that baroclinic enstrophy production is associated with the centrifugal mass ejection away from the vortex centre.
The unstable spectrum of swirling gas flows
- S. LEBLANC, A. LE DUC
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- 04 August 2005, pp. 433-442
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The asymptotic structure of the discrete spectrum of a compressible inviscid swirling flow with arbitrary radial distributions of density, pressure and velocity is described for disturbances with large wavenumbers. It is shown that discrete eigenmodes are unstable when a criterion derived by Eckhoff & Storesletten (1978) is satisfied. In general, these modes are characterized by a length scale of order $|m|^{-3/4}$ where $|m|\,{\gg}\,1$ is the azimuthal wavenumber of the disturbance. They have a spatial structure similar to the incompressible modes obtained by Leibovich & Stewartson (1983). In the particular case of solid-body rotation with a positive gradient of entropy, the unstable discrete spectrum contains modes which scale with $|m|^{-1/2}$. If the modes are localized near a solid boundary, they scale with $|m|^{-2/3}$.
Corrigenda
CORRIGENDUM
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- 04 August 2005, p. 443
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Viscous effects on transient long-wave propagation
BY PHILIP L.-F. LIU AND ALEJANDRO ORFILA
Journal of Fluid Mechanics, vol. 520 (2004), pp. 83–92