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The theoretical work reported herein studies the free-surface profile, the flow structure, and the pressure distribution of a finite-amplitude solitary wave on shallow water with uniform vorticity. The kinematic problem for the stream function is formulated employing the vertical coordinate and the free surface as the independent variables of the Poisson equation with variable coefficients that are functions of the Hamiltonian of the rotational solitary wave. The exact solution of the boundary-value kinematic problem for the stream function is derived in the form of a power series complemented by a recurrence relation. The dynamic problems for the Hamiltonian and the free surface are solved globally in the Boussinesq–Rayleigh approximation. To find angles enclosed by the branches of the solution at critical points and points of bifurcation the surface streamline is also treated locally by an exact topological solution. The complete analysis of the four-dimensional Hamiltonian maps presented in §4 specifies critical values of the Froude number and the vorticity for five flow regimes: the emergence of the solitary wave, the flow separation near the bottom, the flow separation near the crest, the critical regime for an instability, and the formation of a limiting configuration. The streamlines of the recirculating flow are obtained as a single-eddy bifurcation that preserves continuity of all derivatives on the boundary streamline. The eddy separated near the crest forms the limiting configuration by blocking the upstream current. The results are compared with weakly nonlinear theory, with numerical simulations and with field observations with satisfactory agreement.
Laboratory experiments were performed in which an intrusive gravity current was observed using shadowgraph and particle tracking methods. The intrusion was generated in a two-layer fluid with a sharp interface by mixing the fluid behind a vertical lock gate and then suddenly withdrawing the gate from the tank. The purpose of the experiments was to determine the structure of the velocity field inside the intrusion and the stability characteristics of the interface. Soon after the removal of the lock gate, the front of the intrusive gravity current travelled at a constant speed close to the value predicted by theory for an energy-conserving gravity current. The observed structure of the flow inside the intrusion can be divided into three regions. At the front of the intrusion there is an energy-conserving head region in which the fluid velocity is nearly uniform with speed equal to the front speed. This is followed by a dissipative wake region in which large billows are present with their associated mixing and in which the fluid velocity is observed to be non-uniform and have a maximum speed approximately 50% greater than the front speed. Behind the wake region is a tail region in which there is very little mixing and the velocity field is nearly uniform with a speed slightly faster than the front speed.
Crossflow-vortex-induced laminar breakdown in a three-dimensional flat-plate boundary-layer flow is investigated in detail by means of spatial direct numerical simulations. The base flow is generic for an infinite swept wing, with decreasing favourable chordwise pressure gradient. First, the downstream growth and nonlinear saturation states initiated by a crossflow-vortex-mode packet as well as by single crossflow-vortex modes with various spanwise wavenumbers are simulated. Second, the secondary instability of the flow induced by the saturated crossflow vortices is scrutinized, clearly indicating the convective nature of the secondary instability and strengthening knowledge of the conditions for its onset. Emphasis is on the effect of crossflow-vortex-mode packets and of the spanwise vortex spacing on the secondary stability properties of the saturation states. Saturated uniform crossflow vortices initiated by single crossflow-vortex modes turn out to be less unstable than vortices initiated by a packet of vortex modes, and closely spaced saturated vortices are even stable. Third, we investigate the transition control strategy of upstream flow deformation by appropriate steady nonlinear vortex modes as applied in wind tunnel experiments at the Arizona State University. A significant transition delay is shown in the base flow considered here, and the underlying mechanisms are specified.
To examine possible links between a global instability and laminar–turbulent breakdown in a three-dimensional boundary layer, the spatio-temporal stability of primary and secondary crossflow vortices has been investigated for the DLR swept-plate experiment. In the absence of any available procedure for the direct verification of pinching for three-dimensional wave packets the alternative saddle-point continuation method has been applied. This procedure is known to give reliable results only in a certain vicinity of the most unstable ray. Therefore, finding no absolute instability by this method does not prove that the flow is absolutely stable. Accordingly, our results obtained this way need to be confirmed experimentally or by numerical simulations. A geometric interpretation of the time-asymptotic saddle-point result explains certain convergence and continuation problems encountered in the numerical wave packet analysis. Similar to previous results, all our three-dimensional wave packets for primary crossflow vortices were found to be convectively unstable.
Due to prohibitive CPU time requirements the existing procedure for the verification of pinching for two-dimensional wave packets of secondary high-frequency instabilities could not be implemented. Again saddle-point continuation was used. Surprisingly, all two-dimensional wave packets of high-frequency secondary instabilities investigated were also found to be convectively unstable. This finding was corroborated by recent spatial direct numerical simulations of Wassermann & Kloker (2001) for a similar problem. This suggests that laminar–turbulent breakdown occurs after the high-frequency secondary instabilities enter the nonlinear stage, and spatial marching techniques, such as the parabolized stability equation method, should be applicable for the computation of these nonlinear states.
A row of fifty identical, truncated vertical cylinders is submitted to regular head waves, with wave periods in a narrow range around the period of the so-called Neumann trapped mode. The free-surface elevation is measured at 14 locations along the array. Response amplitude operators of the free-surface motion are compared with numerical predictions from a potential flow model. Resonance effects, at wave periods equal to or larger than the critical one, are found to be much less than given by the numerical model. It is advocated that these discrepancies are due to dissipative effects taking place in the boundary layers at the cylinder walls. An artificial means is devised to incorporate dissipation in the potential flow model, whereby the cylinder walls are made slightly porous; the inward normal velocity of the flow is related to the dynamic pressure. The coefficient of proportionality is based on existing knowledge for circular cylinders in oscillatory flows. With this modification in the numerical code, excellent agreement is obtained with the experiments. The numerical model is further used for the case of a very long array composed of 1000 cylinders; it is found that with dissipation at the cylinder walls, the wave action steadily decreases along the array, even for wave periods substantially larger than the critical one. On the other hand, at wave periods less than the critical one, dissipation plays a negligible role; the observed decay is solely due to diffraction effects. Implications of these results for very large structures such as column-supported floating airports are discussed. In particular, it is concluded that scale effects may be an important issue in the experimental analysis of such multi-column structures.
An extensive experimental study is carried out to examine the properties of a quasi-two-dimensional MHD turbulent shear flow. Axisymmetric shear of a mercury layer is enforced by the action of a steady vertical magnetic field and a radial horizontal electric current flowing between a ring set of electrodes and a cylindrical wall. This shear layer is unstable, and the properties of the turbulent flow are studied for a wide range of Hartmann (up to 1800) and Reynolds numbers (up to 106). The mean velocity profiles exhibit a turbulent free shear layer, of thickness larger than that predicted by the laminar theory by two orders of magnitude. The profiles yield the expected linear dependence between the total angular momentum and the electric current when the magnetic field is large enough, but demonstrate a systematic deviation when it is moderate (Ha [lsim ] 250). The quasi-two-dimensional turbulence is characterized by an energy transfer towards the large scales, which leads to a relatively small number of large coherent structures. The properties of these structures result from the competition between the energy transfer and the Joule dissipation within the Hartmann layers. In the intermediate range of wavenumbers (k[lscr ] < k < ki, where k[lscr ] is the integral-length-scale wavenumber and ki the injection wavenumber), the energy spectra exhibit a power law close to k−5/3 when the Joule dissipation is weak and close to k−3 when it is significant. The properties of the turbulent flow in this latter regime depend on only one non-dimensional parameter, the ratio (Ha/Re)(l⊥/h)2 (Ha is the Hartmann number, Re the Reynolds number based on the cell radius, l⊥ a typical transverse scale, and h the layer width).
The effects of thermal modulation with time on the thermocapillary instability of a thin horizontal fluid layer with a deformable free surface are investigated on the basis of linear stability theory. First, a sinusoidal heating with a mean component is applied at the lower wall, corresponding to boundary conditions either in the form of prescribed temperature or heat flux. For finite-wavelength convection the thermal modulation exerts a strong effect, giving rise to a family of looped regions of instability corresponding to alternating synchronous or subharmonic responses. In the case of prescribed heat flux, the critical curve consists of significantly fewer loops than in the case of prescribed temperature. Thermal modulation with moderate modulation amplitude tends to stabilize the mean basic state, and optimal values of frequency and amplitude of modulation are determined to yield maximum stabilization. However, large-amplitude modulation can be destabilizing. A basic state with zero mean is then considered and the critical Marangoni number is obtained as a function of frequency. The effects of modulation are also investigated in the long-wavelength limit. For the case of prescribed temperature, the modulation does not affect the onset of the long-wavelength mode associated with the mean basic state and a purely oscillating basic state is always stable with respect to long-wavelength disturbances. For the case of prescribed heat flux both at the wall and free surface, by contrast, thermal modulation exerts a significant effect on the onset of convection from a mean basic state and long-wavelength convection can occur even for a purely oscillating basic state. The modulation can be stabilizing or destabilizing, depending on the frequency.
In decaying two-dimensional Navier–Stokes turbulence, Batchelor's similarity hypothesis fails due to the existence of coherent vortices. However, it is shown that decaying two-dimensional turbulence governed by the Charney–Hasegawa–Mima (CHM) equation
(∂/∂t)(∇2φ−λ2φ) +J(φ, ∇2φ) = D,
where D is a damping, is described well by Batchelor's similarity hypothesis for wave numbers k [Lt ] λ (the so-called AM regime). It is argued that CHM turbulence in the AM regime is a more ‘ideal’ form of two-dimensional turbulence than is Navier–Stokes turbulence itself.
This paper is concerned with convective and absolute instabilities in the boundary-layer flow over the outer surface of a sphere rotating in an otherwise still fluid. Viscous and streamline-curvature effects are included and the analysis is conducted between latitudes of 10° and 80° from the axis of rotation. Both convective and absolute instabilities are found at each latitude within specific parameter spaces. The results of the convective instability analysis show that a crossflow instability mode is the most dangerous below θ = 66°. Above this latitude a streamline-curvature mode is found to be the most dangerous, which coincides with the appearance of reverse flow in the radial component of the mean flow. At low latitudes the disturbances are considered to be stationary, but at higher latitudes they are taken to rotate at 76% of the sphere surface speed, as observed in experimental studies. Our predictions of the Reynolds number and vortex angle at the onset of convective instability are consistent with existing experimental measurements. Results are also presented that suggest that the occurrence of the slowly rotating vortices is associated with the dominance of the streamline-curvature mode at θ = 66°. The local Reynolds number at the predicted onset of absolute instability matches experimental data well for the onset of turbulence at θ = 30°; beyond this latitude the discrepancy increases but remains relatively small below θ = 70°. It is suggested that this absolute instability may cause the onset of transition below θ = 70°. Close to the pole the predictions of each stability analysis are seen to approach those of existing rotating disk investigations.
By considering an idealized model of helically forced flow in an extended domain that allows scale separation, we have investigated the interaction between dynamo action on different spatial scales. The evolution of the magnetic field is studied numerically, from an initial state of weak magnetization, through the kinematic and into the dynamic regime. We show how the choice of initial conditions is a crucial factor in determining the structure of the magnetic field at subsequent times. For a simulation with initial conditions chosen to favour the growth of the small-scale field, the evolution of the large-scale magnetic field can be described in terms of the α-effect of mean field magnetohydrodynamics. We have investigated this feature further by a series of related numerical simulations in smaller domains. Of particular significance is that the results are consistent with the existence of a nonlinearly driven α-effect that becomes saturated at very small amplitudes of the mean magnetic field.
The non-Newtonian rheology is calculated numerically to second order in the volume fraction in steady simple shear flows for Brownian hard spheres in the presence of hydrodynamic and excluded volume interactions. Previous analytical and numerical results for the low-shear structure and rheology are confirmed, demonstrating that the viscosity shear thins proportional to Pe2, where Pe is the dimensionless shear rate or Péclet number, owing to the decreasing contribution of Brownian forces to the viscosity. In the large Pe limit, remnants of Brownian diffusion balance convection in a boundary-layer in the compressive region of the flow. In consequence, the viscosity shear thickens when this boundary-layer coincides with the near-contact lubrication regime of the hydrodynamic interaction. Wakes are formed at large Pe in the extensional zone downstream from the reference particle, leading to broken symmetry in the pair correlation function. As a result of this asymmetry and that in the boundary-layer, finite normal stress differences are obtained as well as positive departures in the generalized osmotic pressure from its equilibrium value. The first normal stress difference changes from positive to negative values as Pe is increased when the hard-sphere limit is approached. This unusual effect is caused by the hydrodynamic lubrication forces that maintain particles in close proximity well into the extensional quadrant of the flow. The study demonstrates that many of the non-Newtonian effects observed in concentrated suspensions by experiments and by Stokesian dynamics simulations are present also in dilute suspensions.
An investigation is made into the trapping of surface gravity waves by totally submerged three-dimensional obstacles and strong numerical evidence of the existence of trapped modes is presented. The specific geometry considered is a submerged elliptical torus. The depth of submergence of the torus and the aspect ratio of its cross-section are held fixed and a search for a trapped mode is made in the parameter space formed by varying the radius of the torus and the frequency. A plane wave approximation to the location of the mode in this space is derived and an integral equation and a side condition for the exact trapped mode are obtained. Each of these conditions is satisfied on a different line in the plane and the point at which the lines cross corresponds to a trapped mode. Although it is not possible to locate this point exactly, because of numerical error, existence of the mode may be inferred with confidence as small changes in the numerical results do not alter the fact that the lines cross.
If the torus makes small vertical oscillations, it is customary to try to express the fluid velocity as the gradient of the so-called heave potential, which is assumed to have the same time dependence as the body oscillations. A necessary condition for the existence of this potential at the trapped mode frequency is derived and numerical evidence is cited which shows that this condition is not satisfied for an elliptical torus. Calculations of the heave potential for such a torus are made over a range of frequencies, and it is shown that the force coefficients behave in a singular fashion in the vicinity of the trapped mode frequency. An analysis of the time domain problem for a torus which is forced to make small vertical oscillations at the trapped mode frequency shows that the potential contains a term which represents a growing oscillation.
The run-up of non-breaking and breaking solitary waves on a uniform plane beach connected to a constant-depth wave tank was investigated experimentally and numerically. If only the general characteristics of the run-up process and the maximum run-up are of interest, for the case of a breaking wave the post-breaking condition can be simplified and represented as a propagating bore. A numerical model using this bore structure to treat the process of wave breaking and subsequent shoreward propagation was developed. The nonlinear shallow water equations (NLSW) were solved using the weighted essentially non-oscillatory (WENO) shock capturing scheme employed in gas dynamics. Wave breaking and post-breaking propagation are handled automatically by this scheme and ad hoc terms are not required. A computational domain mapping technique was used to model the shoreline movement. This numerical scheme was found to provide a relatively simple and reasonably good prediction of various aspects of the run-up process. The energy dissipation associated with wave breaking of solitary wave run-up (excluding the effects of bottom friction) was also estimated using the results from the numerical model.
This paper investigates the steady round laminar jet discharging into a coaxial duct when the jet Reynolds number, Rej, is large and the ratio of the jet radius to the duct radius, ε, is small. The analysis considers the distinguished double limit in which the Reynolds number Rea = Rejε for the final downstream flow is of order unity, when four different regions can be identified in the flow field. Near the entrance, the outer confinement exerts a negligible influence on the incoming jet, which develops as a slender unconfined jet with constant momentum flux. The jet entrains outer fluid, inducing a slow back flow motion of the surrounding fluid near the backstep. Further downstream, the jet grows to fill the duct, exchanging momentum with the surrounding recirculating flow in a slender region where the Reynolds number is still of the order of Rej. The streamsurface bounding the toroidal vortex eventually intersects the outer wall, in a non-slender transition zone to the final downstream region of parallel streamlines. In the region of jet development, and also in the main region of recirculating flow, the boundary-layer approximation can be used to describe the flow, while the full Navier–Stokes equations are needed to describe the outer region surrounding the jet and the final transition region, with Rea = Rejε entering as the relevant parameter to characterize the resulting non-slender flows.
The dynamics of a cyclonic monopolar vortex on a topographic beta-plane are studied experimentally and theoretically. Detailed measurements of the vortex structure are conducted using high-resolution quantitative velocity measurements. The initial velocity profiles were described in terms of a radius Rvm, maximum azimuthal velocity vθm, and a dimensionless parameter α which characterizes the steepness of the velocity profile. The initial direction of motion of the monopolar vortex is critically dependent on α and weakly dependent of the initial strength and size of the vortex: isolated vortices (α ∼ 3) move north, whereas non-isolated vortices characterized by α ∼ 1 move northwest. When the azimuthal velocity decays slowly with radial distance (α < 1.4), Rossby wave generation dominates the vortex dynamics and the translational speed of the vortex correlates with the Rossby wave speed. When the azimuthal velocity decays rapidly with radial distance (α > 1.4) the vortex is isolated and the translational speed is much slower than the Rossby wave speed. To interpret the effect of the vortex structure on the direction of motion, a mechanistic model is developed which includes the Rossby force and a lift force arising from circulation around the vortex, but does not include the effect of Rossby waves. The Rossby force results from the integrated effect of the Coriolis force on the vortex and drives the vortex north; the lift force is determined from the circulation around the vortex and drives the vortex west. Comparison with the experimental data reveals two regimes: α < 1.4, where the vortex dynamics are dominated by Rossby waves whereas for α > 1.4 Rossby waves are weak and favourable agreement is found with the mechanistic model.
This paper considers nonlinear acoustic waves propagating unidirectionally in a gas-filled tube under an axial temperature gradient, and examines whether the energy flux of the waves can be amplified by thermoacoustic effects. An array of Helmholtz resonators is connected to the tube axially to avoid shock formation which would otherwise give rise to nonlinear damping of the energy flux. The amplification is expected to be caused by action of the boundary layer doing reverse work, in the presence of the temperature gradient, on the acoustic main flow outside the boundary layer. By the linear theory, the velocity at the edge of the boundary layer is given in terms of the fractional derivatives of the axial velocity of the gas in the acoustic main flow. It is clearly seen how the temperature gradient controls the velocity at the edge. The velocity is almost in phase with the heat flux into the boundary layer from the wall. With effects of both the boundary layer and the array of resonators taken into account, nonlinear wave equations for unidirectional propagation in the tube are derived. Assuming a constant temperature gradient along the tube, the evolution of compression pulses is solved numerically by imposing the initial profiles of both an acoustic solitary wave and of a square pulse. It is revealed that when a positive gradient is imposed, the excess pressure decreases while the particle velocity increases and that the total energy flux can indeed be amplified if the gradient is suitable.