Here we discuss the statistical properties of the surface elevation for long crested waves characterized by Jonswap spectra with random phases. Experiments are performed in deep water conditions in one of the largest wave tank facilities in the world. We show that for long-crested waves and for large values of the Benjamin–Feir index, the second order theory is not adequate to describe the tails of the probability density function of wave crests and wave heights. We show that the probability of finding an extreme wave can be underestimated by more than one order of magnitude if second order theory is considered. We explain these observed deviations in terms of the modulational instability mechanism that for large BFI can take place in random wave spectra.
Numerical simulation of evolution of nonlinear gravity waves is presented. Simulation is done using two-dimensional code, based on conformal mapping of the fluid to the lower half-plane. We have considered two problems: (i) modulation instability of wave train and (ii) evolution of solitons with different steepness of carrier wave. In both cases we have observed formation of freak waves.
Isothermal, incompressible round turbulent free jets of air, issuing from a sharp-edged orifice and from a contoured nozzle into still air surroundings, have been used to study the effects of upstream nozzle shaping on near field jet evolution experimentally. The Reynolds number, based on the diameter of the orifice or the nozzle, was in both jets. Hot-wire anemometry and a pitot-static tube were used to obtain the measured quantities which included the mean streamwise velocity, the turbulent Reynolds normal and shear stresses, the autocorrelation coefficients and one-dimensional energy spectra of the fluctuating streamwise velocity and the mean static pressure. The mean streamwise velocity decay on the jet centerline and the jet half-velocity widths were obtained from the mean streamwise velocity data. To the extent that the results showed that mixing in the sharp-edged orifice round jet was higher than in the contoured nozzle round jet, upstream nozzle shaping was found to affect jet evolution in the near flow field. The distribution of the autocorrelation coefficients of the streamwise fluctuating velocity showed a marked difference in the evolution of the two jets, one of which had a uniform, and the other a non-uniform, exit plane mean streamwise velocity profile. The one-dimensional energy spectra results and also those of the distribution of the autocorrelation coefficients indicated the presence of coherent structures in the near field of the jets and the sharp-edged orifice jet was found to be more “energetic” than the contoured nozzle jet.
This paper concerns long time interaction of envelope solitary gravity waves propagating at the surface of a two-dimensional deep fluid in potential flow. Fully nonlinear numerical simulations show how an initially long wave group slowly splits into a number of solitary wave groups. In the example presented, three large wave events are formed during the evolution. They occur during a time scale that is beyond the time range of validity of simplified equations like the nonlinear Schrödinger (NLS) equation or modifications of this equation. A Fourier analysis shows that these large wave events are caused by significant transfer to side-band modes of the carrier waves. Temporary downshiftings of the dominant wavenumber of the spectrum coincide with the formation large wave events. The wave slope at maximal amplifications is about three times higher than the initial wave slope. The results show how interacting solitary wave groups that emerge from a long wave packet can produce freak wave events. Our reference numerical simulation are performed with the fully nonlinear model of Clamond and Grue [D. Clamond, J. Grue, A fast method for fully nonlinear water wave computations, J. Fluid Mech. 447 (2001) 337–355]. The results of this model are compared with that of two weakly nonlinear models, the NLS equation and its higher-order extension derived by Trulsen et al. [K. Trulsen, I. Kliakhandler, K.B. Dysthe, M.G. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids 12 (10) (2000) 2432–2437]. They are also compared with the results obtained with a high-order spectral method (HOSM) based on the formulation of West et al. [B.J. West, K.A. Brueckner, R.S. Janda, A method of studying nonlinear random field of surface gravity waves by direct numerical simulation, J. Geophys. Res. 92 (C11) (1987) 11 803–11 824]. An important issue concerning the representation and the treatment of the vertical velocity in the HOSM formulation is highlighted here for the study of long-time evolutions.
The aim of this paper is to present a continuum model for thermo-bioconvection of oxytactic bacteria in a porous medium and investigate the combined effects of microorganisms' upswimming and heating from below on the stability of bioconvection in a horizontal layer filled with a fluid saturated porous medium. Different from traditional bioconvection, thermo-bioconvection has two destabilizing mechanisms that contribute to creating the unstable density stratification. This problem may be relevant to a number of geophysical applications, such as the investigation of the dynamics of oxytactic species of thermophiles (heat loving microorganisms) living in hot springs, microbial-enhanced oil recovery, and modeling oil- and gas-bearing sedimentary basins. The utilization of the Galerkin method to solve a linear stability problem leads to a correlation between the critical value of the bioconvection Rayleigh number and the traditional “thermal” Rayleigh number.
The freak wave formation due to the dispersive focusing mechanism is investigated experimentally without wind and in presence of wind. An asymmetric behaviour between the focusing and defocusing stages is found when the wind is blowing over the mechanically generated gravity wave group. This feature corresponds physically to the sustain of the freak wave mechanism on longer periods of time. Furthermore, a weak amplification of the freak wave and a shift in the downstream direction of the point where the waves merge are observed. The experimental results suggest that the Jeffreys' sheltering mechanism could play a key role in the coherence of the group of the freak wave. Hence, the Jeffreys' sheltering theory is introduced in a fully nonlinear model. The results of the numerical simulations confirm that the duration of the freak wave event increases with the wind velocity.
Three reasons for freak wave generation due to interaction of wave with spatially non-uniform current are considered in the paper. They are as follows: wave energy amplification due to wave-current interaction; wave height amplification around caustic due to refraction wave in non-uniform current; non-linear wave interaction in shallow water due to their intersection described by the Kadomtsev–Petviashvili equation. These mechanisms can generate a large wave amplification producing a dangerous natural phenomenon.
The intention of the “von Kármán sodium” (VKS) experiment is to study the hydromagnetic dynamo effect in a highly turbulent and unconstrained flow. Much effort has been devoted to the optimization of the mean flow and the lateral boundary conditions in order to minimize the critical magnetic Reynolds number and hence the necessary motor power. The main focus of this paper lies on the role of “lid layers”, i.e. layers of liquid sodium between the impellers and the end walls of the cylinder. First, we study an analytical test flow to show that lid layers can have an ambivalent effect on the efficiency of the dynamo. The critical magnetic Reynolds number shows a flat minimum for a small lid layer thickness, but increases for thicker layers. For the actual VKS geometry it is shown that static lid layers yield a moderate increase of by approximately 12 per cent. A more dramatic increase by 100 till 150 per cent can occur when some rotational flow is taken into account in those layers. Possible solutions of this problem are discussed for the real dynamo facility.
A second-order surface wave model is used to investigate the effects of the spectral distribution on the statistical properties of the surface elevation. To this end single and double peaked directional wave spectra are considered at different water depths. For unimodal seas (i.e. single peaked), the addition of directional components reduces the effects of the second-order interactions in deep water and increases them in shallower depths. For a bimodal sea (i.e. double peaked), on the other hand, a large angle between the wave trains decreases systematically the vertical asymmetry of the wave profile. However, the nonlinear interactions seem to reach their maximum strength when the two wave spectra are slightly separated in direction. This produces an evident deviation of the wave crest distribution at low probability levels if compared with the unimodal condition.
Results for flow simulations and experiments of different models of the human nasal cavity with and without turbinates and/or spurs are presented. The flow is investigated for normal inspiration and expiration at Reynolds numbers based on the throat diameter of and . The numerical method is second-order accurate on a multi-block structured grid. Flow measurements are based on the method of Digital Particle-Image Velocimetry (DPIV) in transparent nose models. The experimental results corroborate the numerical flow structure thereby evidencing that the nose flow can be considered laminar in the Reynolds number range investigated. Moreover, the analysis of the flow field indicates overall, the higher susceptibility to geometric changes at inspiration and in particular, the lower turbinate to have the major impact on the flow structure especially when air is inhaled.
The problem of uniform potential flow past a circular cylinder is a basic one in fluid dynamics and the solution is well-known. In this paper, an analytical construction is presented to generalize this fundamental result to find solutions for steady irrotational uniform flow past a multi-cylinder configuration in a planar flow in the case when the circulations around the obstacles is taken to vanish. More generally, if a conformal mapping from a canonical multiply connected circular region to the unbounded fluid region exterior to a finite collection of non-cylindrical obstacles of more general shape is known, the formulation also provides solutions for the uniform flow past those obstacles.
A spectral – spectral-element code is used to investigate the hydrodynamic forces acting on a fixed sphere placed in a uniform flow in the Reynolds number interval [10–320] covering the early stages of transition, i.e. the steady axisymmetric regime with detached flow, the steady non-axisymmetric and the unsteady periodic regimes of the sphere wake. The mentioned changes of regimes, shown by several authors to be related to a regular and a Hopf bifurcations in the wake, result in significant changes of hydrodynamic action of the flow on the sphere. In the present paper, we show that the loss of axisymmetry is accompanied not only by an onset of lift but also of a torque and we give accurate values of drag, lift and torque in the whole interval of investigated Reynolds numbers. Among other results show, moreover, that each bifurcation is accompanied also by a change of the trend of the drag versus Reynolds number dependence, the overall qualitative effect of instabilities being an increase of drag.
The analysis of surface waves time series is performed to understand the nature of freak waves. Contributions of quasi-linear dispersive focusing effects and nonlinear self-modulation (Benjamin–Feir) effects are estimated with the help of kinematical description, nonlinear spectral analysis and numerical simulations. The nonlinear dynamics of an envelope soliton over a background wave is investigated and a possible extreme wave appearance is predicted.
We propose an asymptotic model for quite general liquid microchannel flows in the presence of electrical double layers (EDLs). The model provides an “inner” solution for the wall layer, which reflects the dominant balance between electrical forces and viscous forces (tangentially), respectively between electrical forces and pressure and viscous forces (normally). The electrically-neutral core of the flow is governed by the standard Navier–Stokes equations, providing the “outer” solution. The asymptotic matching of both solutions provides a method for the simplified numerical treatment of such EDLs. The superposition of the solutions in both regions then allows to infer an approximate solution, valid within the entire domain. Based on this model, we apply external oscillatory electrical fields to excite secondary flows (i) in microchannels with an internal obstacle or (ii) in folded (meander) microchannels. These secondary flows are demonstrated to greatly enhance the mixing of two liquids flowing in a layered fashion through these microchannels. Thus, electrical excitation has considerable potential if micromixers for ionic liquids are designed within electrically-insulating (e.g. plastics, glass) substrates.
Lorentz forces originating from surface-mounted actuators of permanent magnets and electrodes in weakly conducting fluids like seawater can be used to control flow separation at hydrofoils. The numerical results presented here are based on direct numerical simulation in the laminar flow regime, limited to Reynolds numbers of . Control by steady forcing at the suction side and by oscillatory forcing near the leading edge of the foil is investigated in the post-stall regime. By applying a strong enough steady control, separation can be completely suppressed. Oscillatory forcing always has to compete with the natural shedding process, lock-in behavior may occur. Lift-optimum control for strong amplitudes is found in a frequency band around the natural shedding frequency. In terms of the momentum coefficient describing the control effort, appropriate excitation frequencies in relation to the natural vortex shedding frequency allow for a more effective lift control than steady forcing.
The evolution of the initially random wave field with a Gaussian spectrum shape is studied numerically within the Korteweg–de Vries (KdV) equation. The properties of the KdV random wave field are analyzed: transition to a steady state, equilibrium spectra, statistical moments of a random wave field, and the distribution functions of the wave amplitudes. Numerical simulations are performed for different Ursell parameters and spectrum width. It is shown that the wave field relaxes to the stationary state (in statistical sense) with the almost uniform energy distribution in low frequency range (Rayleigh–Jeans spectrum). The wave field statistics differs from the Gaussian one. The growing of the positive skewness and non-monotonic behavior of the kurtosis with increase of the Ursell parameter are obtained. The probability of a large amplitude wave formation differs from the Rayleigh distribution.
Dean instability for Newtonian fluids in laminar secondary flow in 180° curved channels was studied experimentally and numerically. The numerical study used Fluent CFD code to solve the Navier–Stokes equations, focusing on flow development conditions and the parameters influencing Dean instability. An accurate criterion based on the radial gradient of the axial velocity was defined that allows detection of the instability threshold, and this criterion is used to optimize the grid geometry. The effects on Dean instability of the curvature ratio (from 5.5 to 20) and aspect ratio (from 0.5 to 12) are studied. In particular, we show that the critical value of the Dean number decreases with the increasing duct curvature ratio. The variation of the critical Dean number with duct aspect ratio is less regular. In the experimental study, flows were visualized in several tangential positions of a 180° curved channel with aspect ratio 8 and curvature ratio 10. The flow is hydrodynamically developed at the entrance to the curved channel. The critical Dean number is detected and the development of secondary flow vortices by additional counter-rotating vortex pairs is observed. A diagram of different critical Dean numbers is established.
A simple and accurate boundary-type meshless method of fundamental solutions (MFS) is applied to solve both 2D and 3D Stokes flows based on the dual-potential formulation of velocity potential and stream function vector. Using the dual-potential concept, the solutions of both 2D and 3D Stokes flows are obtained by combining the much simpler fundamental solutions of Laplace (potential) and bi-harmonic equations without using the complicated singular fundamental solutions such as Stokeslets and their derivatives as well as source doublet hypersingularity. The developed algorithm is used to test five numerical experiments for 2D flows: (1) circular cavity, (2) wave-shaped bottom cavity and (3) circular cavity with eccentric rotating cylinder; and for 3D flows: (4) a uniform flow passing a sphere and (5) a uniform flow passing a pair of spheres. Good results are obtained as comparing with solutions of analytical and numerical methods such as FEM, BEM and other meshfree schemes.
The effects of wall corrugation on the stability of wall-bounded shear flows have been examined experimentally in plane channel flows. One of the channel walls has been modified by introduction of the wavy wall model with the amplitude of 4% of the channel half height and the wave number of 1.02. The experiment is focused on the two-dimensional travelling wave instability and the results are compared with the theory [J.M. Floryan, Two-dimensional instability of flow in a rough channel, Phys. Fluids 17 (2005) 044101 (also: Rept. ESFD-1/2003, Dept. of Mechanical and Materials Engineering, The University of Western Ontario, London, Ontario, Canada, 2003)]. It is shown that the flow is destabilized by the wall corrugation at subcritical Reynolds numbers below 5772, as predicted by the theory. For the present corrugation geometry, the critical Reynolds number is decreased down to about 4000. The spatial growth rates, the disturbance wave numbers and the distribution of disturbance amplitude measured over such wavy wall also agree well with the theoretical results.
Two original algorithms are proposed for the computation of bifurcation points in fluid mechanics. These algorithms consist of finding the zero values of a specific indicator. To compute this indicator a perturbation method is used which leads to an analytical expression of this indicator. Two kinds of instability are considered: stationary and Hopf bifurcations. To prove the efficiency and advantages of such numerical methods several numerical tests are discussed.