Wavelength Of Disturbance Research Articles

Linear Temporal Instability of a Two-Layer Axisymmetric Cylindrical Liquid Sheet

The instability of an axial uniform flow of a two-layer axisymmetric cylindrical sheet composed of two inviscid fluids is examined analytically and numerically on the basis of thin-layer approximation. As the eigenmodes for disturbances of large wavelength, we obtain an internal mode in which the displacement of interface between these fluids is larger than the displacements of two free surfaces of the sheet, as well as the modes similar to varicose and sinuous modes in a single-layer case. All the modes are unstable only for disturbances of large wavelength, and the growth rate of sinuous-like mode is much larger than that of two other modes. Furthermore, unlike a single-layer case, the wavenumber of disturbances of maximum growth rate is non-zero if the density ratio is sufficiently far from one. In such cases, the ratio of amplitudes of these disturbances is much different from that in a typical sinuous mode. The dependences of maximum growth rate on the ratios of surface tension coefficients, fluid densities, and sheet thicknesses are also examined.

Stability of wavy downflow of films calculated by the Navier-Stokes equations

The process of downflow of viscous films on a smooth surface is analyzed theoretically with the use of the full Navier-Stokes equations. The limits of applicability of the asymptotic and integral approaches to the description of waves on falling films are determined. Various nonlinear wavy downflow regimes are calculated in a wide range of the Reynolds and Kapitsa numbers, and stability of these regimes is studied. For low values of the Kapitsa number, the results of the asymptotic approach are demonstrated to be inapplicable almost for all Reynolds numbers. For high values of the Kapitsa number, the solution obtained by the asymptotic method starts to differ significantly from the result obtained by solving the Navier-Stokes equations beginning from moderate Reynolds numbers. For high Reynolds numbers, the wavelength of neutral disturbances is independent of the flow rate of the liquid, and the phase velocity of neutral disturbances is close to the velocity of the free surface. Calculations of nonlinear wavy regimes with moderate Reynolds numbers predict the existence of internal vortex zones. It is shown that there are only a few families of steady traveling solutions (a countable set of different families of such solutions was obtained in calculations by the integral model).

Breakup of compound liquid jets under periodic excitation at small core‐to‐shell mass ratios

A periodic excitation was applied to a compound liquid jet, which was composed of a core liquid, water, surrounded by a shell liquid, diesel oil, for an experimental investigation on its breakup. The core‐to‐shell mass ratio, γ, and the ratio of disturbance wavelength to undisturbed diameter of the compound jet, λ/D, were taken into consideration as parameters to describe the breakup processes of the compound jet. For γ < 0.36, the compound jet breakup was classified into six patterns. By comparing the breakup of a single diesel jet to that of a water‐in‐diesel compound jet, two conclusions were drawn. First, the uniform drop regime of the compound jet was wider than that of the single jet. Second, both the single jet and water‐in‐diesel compound jet broke randomly in the range of λ/D ≤ 3.

Numerical simulations of the Proudman resonance

The paper documents numerical modelling exercises designed to reproduce the Proudman resonance on a shelf. Various sinusoidal air pressure disturbances were simulated to travel over the homogeneous waters on a flat or slope bathymetry, qualifying and quantifying the strength of resonance and near-resonance processes in the sea. The results were discussed in light of the worldwide documented destructive meteotsunami events, putting the attention particularly on the effects of the disturbance wavelength, on the weakening of the resonance in near-resonant conditions and on the effect of a slope to the resonance strength.

Unsteady phenomena in the edge tone

Despite its geometrical simplicity, the edge tone displays a remarkably complex behaviour. A plane jet oscillates around the wedge-shaped object with a relatively stable frequency and under certain circumstances emits an audible tone. This configuration plays a central role in the sound production of several wind instruments but occurs in industrial situations too. The flow exhibits various interesting nonlinear phenomena reported in the literature which are not entirely explained. In this paper, detailed high precision numerical simulations of the flow are reported under various conditions. Several phenomena are reproduced in agreement with the literature such as the existence of “stages”, the dependence of oscillation frequency on the outflow velocity and the orifice–edge distance within one stage, the pressure distribution on the edge surface, etc. A criterion for the appropriate time step for constant accuracy has been derived. The location of force action is surprisingly stable; it remains in a very narrow region of the wedge surface independently of the Reynolds number and the orifice–edge distance but it is much further behind the edge tip than reported in the literature. The various stages can coexist in different ways: jumping back and forth between stages or being superposed on each other. Regardless of the form, the first stage continues to be dominant even when the second and third stage appears. The question of disturbance propagation velocity and disturbance wavelength is also investigated. The development of higher harmonics of a single stage along the orifice–edge tip distance is presented.

Fixed contact line helical interfaces in zero gravity

Fluid interfaces supported in microgravity by a helical structure are shown to have a more robust stability than more common structures such as liquid bridges. In particular, helical interfaces can take the form of infinite right circular cylinders over a broad range of configurations. In the case of a single fixed contact line support, the infinite cylinder is stable for all cases in which the pitch to diameter ratio is less than π∕3 (more tightly coiled interfaces). When there are two or more equally spaced fixed contact line supports, the infinite cylinder is stable for all configurations. Furthermore, in the two support case (the double helix), stability persists for all volumes from the cylinder to zero volume, when the pitch to diameter ratio is greater than 2.082 (more loosely coiled interfaces). The equivalent to the axisymmetric Young-Laplace equation is derived for helical interfaces. Interfacial stability is determined from equilibrium branch structure following the application of Maddocks’ method by Lowry and Steen [Proc. R. Soc. London, Ser. A 449, 411 (1995)]. Perturbations to finite wavelength disturbances are considered for the case of a single helical support. Overall stability envelopes are presented for single and multiple support cases. Limited experimental results verify the infinite length stability limit for the single helical support case.

Experiment of the Two-Dimensional Roughness Effect on Hypersonic Boundary-Layer Transition

An experimental investigation of the effects of two-dimensional roughness on hypersonic boundary layers was carried out at the JAXA 0.5 m hypersonic wind tunnel using a 5 deg half-angle sharp cone at a freestream Mach number of 7.1 and a wide range of stagnation conditions. Aerodynamic heating distributions and surface pressure fluctuations were measured with and without roughness elements applied to the cone. A wavy wall roughness with a wavelength of 2� located well upstream of the breakdown region had the effect of delaying transition. Surface pressure spectrum densities indicate that disturbances of the roughness wavelength are amplified compared to the smooth wall case. At a lower stagnation temperature, however, wavy wall roughness configurations had the same effect as regular roughness, and no discernible differences between wavy wall roughness and spherical roughness were observed, either in the transition point or in the pressure fluctuation spectrum.

Mechanical stresses in an irradiated target with a disturbed surface

Mechanical stresses generated in a disturbed-surface solid target irradiated by intense fluxes of accelerated charged particles are studied numerically and analytically. The disturbed surface is needed to analyze the influence of microirregularities present on real surfaces on the stress pattern. Three basic components of the stress field are revealed: a shock wave; stresses localized in the energy liberation region; and disturbed stresses, which are due to the disturbed surface. The disturbed stresses are localized in a surface layer of width comparable to the disturbance wavelength and account for about 30% of the field stress energy. It is concluded that the disturbance of the surface and disturbance-induced stresses should be taken into account in analysis of structural transformations under irradiation.

Suppression of instability in liquid flow down an inclined plane by a deformable solid layer

The linear stability of a liquid layer flowing down an inclined plane lined with a deformable, viscoelastic solid layer is analyzed in order to determine the effect of the elastohydrodynamic coupling between the liquid flow and solid deformation on the free-surface instability in the liquid layer. The stability of this two-layer system is characterized by two qualitatively different interfacial instability modes: In the absence of the deformable solid layer, the free surface of the liquid film undergoes a long-wave instability due to fluid inertia. With the presence of the deformable solid layer, the interface between the fluid and the solid undergoes a finite-wavelength instability when the deformable solid becomes sufficiently soft. The effect of the solid layer deformability on the free-surface instability of the liquid film flow is analyzed using a long-wave asymptotic analysis. The asymptotic results show that for a fixed Reynolds number and inclination angle, the free-surface instability is completely suppressed in the long-wave limit when the nondimensional (inverse) solid elasticity parameter Gamma=Va eta/(GR)increases beyond a critical value. Here, Va is the average velocity of the liquid film flow, eta is the viscosity of the liquid, G is the shear modulus of the solid layer, and is R the thickness of the liquid layer. The predictions of the asymptotic analysis are verified and extended to finite wavelengths using a numerical solution, and this indicates that the suppression of the free-surface instability indeed continues to finite wavelength disturbances. Further increase of Gamma is found to have two consequences: first, the interface between the liquid and the deformable solid layer could become unstable at finite wavelengths; second, the free-surface interfacial mode could also become unstable at finite wavelengths due to an increase in solid layer deformability. However, our numerical results demonstrate that, for a given average velocity, there exists a sufficient window in the value of shear modulus G where both the unstable modes are absent at all wavelengths. Our study therefore suggests that soft solid layer coatings could potentially provide a passive method of suppressing free-surface instabilities in liquid film flows.

On undercompressive shocks in constrained two-layer flows

We consider the countercurrent flow of two incompressible immiscible viscous fluids in an inclined channel. The lower fluid is denser than the upper fluid, making this configuration relevant to air-water systems. Flow is driven by an imposed pressure gradient and gravity. From a lubrication approximation based on the ratio of the channel height to the downstream disturbance wavelength, we derive a nonlinear system of evolution equations that govern the interfacial shape separating the two fluids and the leading-order pressure. This system includes the physical effects of advection, capillarity, inertia and hydrostatic pressure. Our interest is to compare the dynamics of the solutions under different flow constraints. In particular, we compare the dynamics when the liquid volumetric flow rate and the downstream pressure drop are held fixed to the case when the gas volumetric flow rate and the interfacial height at ends of the channel are held fixed. In both of these systems, Lax shocks, undercompressive shocks and rarefaction waves are investigated. Through a numerical verification, we find that the dynamics of both scenarios are different, resulting in unsteady interfacial profiles for flows driven by fixed liquid flow rate and pressure drop.

On undercompressive shocks and flooding in countercurrent two-layer flows

We consider the countercurrent flow of two incompressible immiscible viscous fluids in an inclined channel. This configuration may lead to the phenomena of ‘flooding’, i.e. the transition from a countercurrent regime to a cocurrent regime. This transition is marked by a variety of transient behaviour, such as the development of large-amplitude waves that impede the flow of one of the fluids to the reversal of the flow of the denser fluid. From a lubrication approximation based on the ratio of the channel height to the downstream disturbance wavelength, we derive a nonlinear system of evolution equations that govern the interfacial shape separating the two fluids and the leading-order pressure. This system, which assumes fluids with disparate density and dynamic viscosity ratios, includes the effects of viscosity stratification, inertia, shear and capillarity. Since the experimental constraints for this effective system are unclear, we consider two ways to drive the flow: either by fixing the volumetric flow rate of the gas phase or by fixing the total pressure drop over a downstream length of the channel. The latter forcing results in a single evolution equation whose dynamics depends non-locally on the interfacial shape. From both of these driven systems, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. These criteria, through a numerical verification, do not depend significantly on the inertial effects within the more dense layer. The choice of the local/non-local constraints appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding.

Linear stability of cylindrical Couette flow in the convection regime

The instability of steady circular Couette flow with radial heating across a vertically oriented annulus with a rotating inner cylinder and a stationary outer cylinder is investigated using a linear stability analysis. The convection regime base flow is developed for an infinite aspect ratio geometry and constant fluid properties with buoyancy included through the Boussinesq approximation. The base flow is characterized by a dimensionless stratification parameter γ that is proportional to the vertical temperature gradient. Critical stability boundaries are calculated for this assumed base flow with respect to both toroidal and helical disturbances. The numerical investigation is primarily restricted to a radius ratio of 0.6 at a Prandtl number of 100. Critical stability boundaries in Taylor–Grashof number space are presented for two values of the stratification parameter γ (4 and 13). The results follow the development of critical stability from Taylor cells at small Grashof numbers up to a maximum Grashof number used in this calculation of 20 000 and 80 000 for γ=4 and 13, respectively. Results show that increasing the stratification parameter stabilizes the isothermal Taylor vortices, followed by a destabilization at higher azimuthal mode numbers (n&amp;gt;0). The results also show that for γ=4 (close to the conduction regime), two modes are obtained: one is axisymmetric and the other is nonaxisymmetric. However, for the convection regime (large γ) six asymmetric modes are obtained. Finally, the disturbance wavelength, phase speed, and spiral inclination angle are presented as a function of the critical Grashof number for the stratification parameters considered in this work.

Factors of Three-Dimensional Laminar Boundary Layers Affecting the Characteristics of Cross-Flow Instability

The correlations between the characteristic values of the three dimensional boundary layer and the characteristics of the cross-flow instability have studied by a linear stability analysis of the Falkner-Skan-Cooke boundary layers. Several parameters to predict the critical point of the cross-flow instability and prediction functions of the critical Reynolds number are suggested. It has been found that new parameter defined in this study is available to predict the critical point of the cross-flow instability. It is also shown that the shape factor obtained from the velocity profile of the cross-flow velocity component and the non-dimensional cross-flow velocity at the inflection point of the cross-flow velocity profile are available to predict the critical Reynolds number. The wave length of the critical disturbance in the cross-flow direction has been found to be about 3-4 times as long as the 1% boundary layer thickness of the cross-flow velocity profile.

The partition of unity finite element method for short wave acoustic propagation on non-uniform potential flows

A novel numerical method is proposed for modelling time-harmonic acoustic propagation of short wavelength disturbances on non-uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non-uniform. Many wavelengths can be included within a single element leading to ultra-sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid-based schemes. Numerical results for lined uniform two-dimensional ducts and for non-uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.

Regime diagram of the development of long-wave near-wall disturbances in a hypersonic boundary layer

A regime diagram of the development of slow near-wall disturbances induced by an unsteady self-induced pressure perturbation in a hypersonic boundary layer is constructed for a disturbance wavelength greater than the boundary layer thickness. It is shown that the main factors shaping the perturbed flow are the gas enthalpy near the body surface, the intensity of the viscous-inviscid interaction, and the nature (sub- or supersonic) of the main part of the boundary layer. Nonlinear boundary-value problems are formulated for regimes in which the near-wall boundary layer region plays a decisive role. Numerical and analytical solutions are obtained in the linear approximation. It is shown that intensification of the viscous-inviscid interaction or an increase in the role of the supersonic main region of the boundary layer impart generally “supersonic” properties to the main part of the boundary layer, i.e. the upstream propagation of the disturbances is damped and the disturbance growth downstream becomes more intense. Damping of the viscous-inviscid interaction and an increase in the role of the subsonic main part of the boundary layer have the opposite effect. Surface cooling increases the effect of the main part of the boundary layer on the formation of pressure disturbances and surface heating leads to an increase in the effect of the near-wall boundary layer region. It is also shown that for the regimes considered disturbances propagate in a direction opposite to that of the free stream from the turbulent flow region located downstream of the local disturbance development region.

On the stability of shoaling Stokes waves over a slowly varying bottom

The third order cubic Schrödinger equation for weakly non-linear water waves over slowly varying bottom is extended to include the next higher order variable depth effects beyond shoaling. This involved a re-ordering of the dispersion relative to non-linearity that further allowed a wider frequency bandwidth. The effect of the depth variation on the stability of shoaling Stokes waves is analyzed. It is shown that the higher order depth terms introduced a higher order instability for all small wavelength disturbances beyond the Benjamin–Feir type.

The Helmholtz equation for convection in two-dimensional porous cavities with conducting boundaries

It is well-known that every two-dimensional porous cavity with a conducting and impermeable boundary is degenerate, as it has two different eigensolutions at the onset of convection. In this paper it is demonstrated that the eigenvalue problem obtained from a linear stability analysis may be reduced to a second-order problem governed by the Helmholtz equation, after separating out a Fourier component. This separated Fourier component implies a constant wavelength of disturbance at the onset of convection, although the phase remains arbitrary. The Helmholtz equation governs the critical Rayleigh number, and makes it independent of the orientation of the porous cavity. Finite-difference solutions of the eigenvalue problem for the onset of convection are presented for various geometries. Comparisons are made with the known solutions for a rectangle and a circle, and analytical solutions of the Helmholtz equation are given for many different domains.

Laser-induced break-up of water jet waveguide

In this article, an optical method to control the break-up of high-speed liquid jets is proposed. The method consists of focusing the light of a pulsed laser source into the jet behaving as a waveguide. Experiments were performed with the help of a Q-switched frequency doubled Nd:Yag laser (λ=532 nm). The jet diameter was 48 µm and jet velocities from 100 to 200 m/s. To study the laser-induced water jet break-up, observations of the jet coupled with the high power laser were performed for variable coupling and jet velocity conditions. Experimentally determined wavelength and growth rate of the laser-generated disturbance were also compared with the ones predicted by linear stability theory of free jets.

Theory of Mach Cones in Magnetized Dusty Plasmas

Theoretical models for Mach cones in magnetized dusty plasmas of Saturn’s rings are presented. Dispersion properties of different types of longitudinal and transverse waves (viz. dust–acoustic waves, fast and slow dust–magnetoacoustic waves) in magnetized dusty plasmas are presented. Numerical results are presented for Vd/Vp (> 1), which determines the existence of Mach cones as well as their opening angles [θ = sin-1(Vp/Vd)], where Vd is the dust boulder speed and Vp ( = ω/k) is the phase speed of short or long wavelength disturbances in magnetized dusty plasmas. It is found that short wavelength (in comparison with the ion gyroradius) modified dust–acoustic waves and long wavelength slow dust–magnetoacoustic waves (DMW) are viable candidates for the formation of Mach cones in the equatorial plane of Saturn’s rings. They should be observed by on board instruments of the CASSINI spacecraft that is supposed to enter Saturn’s orbit in July 2004.

Three‐dimensional simulation of the ASTEX Lagrangian 1 field experiment with a regional numerical weather prediction model

AbstractThe Atlantic Stratocumulus Transition Experiment (ASTEX) first Lagrangian experiment (Lagrangian 1) is here simulated with a modified version of the regional forecast model HIRLAM (High Resolution Limited Area Model). The main modification is that moist turbulent fluxes are accounted for in the model. Trajectory calculations show good agreement with earlier estimations. The initially rather shallow stratocumulus topped marine boundary layer is deepening along the trajectory, and in the end cumulus clouds are formed that penetrate the boundary‐layer top. The model predicts this change in cloudiness, but the boundary layer is too shallow in the model. A simulation with modified initial conditions shows improved results, but is still too slow in increasing the boundary‐layer depth. Additional factors that influence the boundary‐layer growth are: the increase in sea surface temperatures, lower modelled wind speeds, low entrainment rates due to coarse vertical resolution, and synoptic‐scale subsidence.An anticyclone at the surface moved slightly northward during the simulation. The anticyclone was accompanied at 500 hPa by a deepening cyclone and, therefore, one would expect synoptic subsidence in the area of the Lagrangian 1. The modelled negative vertical wind component at the boundary‐layer top oscillates, and this is examined using spectral analysis. The results show that the vertical velocity is influenced by cumulus clouds on time‐scales up to 15 h with a peak at 9 h. The horizontal and vertical wavelengths of the vertical velocity disturbances are estimated from model output to be 400–500 km and 6–10 km, respectively. Using the estimated vertical wavelength and linear theory for hydrostatic inertia–gravity waves, a horizontal wavelength of 350–550 km was calculated for a frequency of 9 h. The model results thus indicate that these types of waves are responsible for the undulating vertical velocity.Finally, an estimation of the synoptic‐scale vertical velocity is calculated by filtering out all scales smaller than 15 h from the vertical velocity signal. This results in subsidence both at the beginning and the end of the Lagrangian with vertical velocities between −0.1 and −0.4 cm s−1. Copyright © 2004 Royal Meteorological Society