7 results
Radiation of short waves from the resonantly excited capillary–gravity waves
- M. Hirata, S. Okino, H. Hanazaki
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- Journal:
- Journal of Fluid Mechanics / Volume 810 / 10 January 2017
- Published online by Cambridge University Press:
- 24 November 2016, pp. 5-24
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Capillary–gravity waves resonantly excited by an obstacle (Froude number: $Fr=1$) are investigated by the numerical solution of the Euler equations. The radiation of short waves from the long nonlinear waves is observed when the capillary effects are weak (Bond number: $Bo<1/3$). The upstream-advancing solitary wave radiates a short linear wave whose phase velocity is equal to the solitary waves and group velocity is faster than the solitary wave (soliton radiation). Therefore, the short wave is observed upstream of the foremost solitary wave. The downstream cnoidal wave also radiates a short wave which propagates upstream in the depression region between the obstacle and the cnoidal wave. The short wave interacts with the long wave above the obstacle, and generates a second short wave which propagates downstream. These generation processes will be repeated, and the number of wavenumber components in the depression region increases with time to generate a complicated wave pattern. The upstream soliton radiation can be predicted qualitatively by the fifth-order forced Korteweg–de Vries equation, but the equation overestimates the wavelength since it is based on a long-wave approximation. At a large Bond number of $Bo=2/3$, the wave pattern has the rotation symmetry against the pattern at $Bo=0$, and the depression solitary waves propagate downstream.
Numerical simulation of jets generated by a sphere moving vertically in a stratified fluid
- H. Hanazaki, S. Nakamura, H. Yoshikawa
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- Journal:
- Journal of Fluid Mechanics / Volume 765 / 25 February 2015
- Published online by Cambridge University Press:
- 22 January 2015, pp. 424-451
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The flow past a sphere moving vertically at constant speeds in a salt-stratified fluid is investigated numerically at moderate Reynolds numbers $\mathit{Re}$. Time development of the flow shows that the violation of density conservation is the key process for the generation of the buoyant jet observed in the experiments. For example, if the sphere moves downward, isopycnal surfaces are simply deformed and dragged down by the sphere while the density is conserved along the flow. (The flow pattern is inverted if the sphere moves upward. Some explanations are given in the introduction.) Then, the flow will never become steady. As density diffusion becomes effective around the sphere surface and generates a horizontal hole in the isopycnal surface, fluid with non-conserved density is detached from the isopycnal surface and moves upward to generate a buoyant jet. These processes will constitute a steady state near the sphere. With lengths scaled by the sphere diameter and velocities by the downward sphere velocity, the duration of density conservation at the rear/upper stagnation point, or the maximum distance that the isopycnal surface is dragged downward, is proportional to the Froude number $\mathit{Fr}$, and estimated well by ${\rm\pi}\mathit{Fr}$ for $\mathit{Fr}\gtrsim 1$ and $\mathit{Re}\gtrsim 200$, corresponding to a constant potential energy. The radius of a jet defined by the density and velocity distributions, which would have correlations with the density and velocity boundary layers on the sphere, is estimated well by $\sqrt{\mathit{Fr }/2\mathit{Re }\mathit{ Sc}}$ and $\sqrt{\mathit{Fr }/2\mathit{Re}}$ respectively for $\mathit{Fr}\lesssim 1$, where $\mathit{Sc}$ is the Schmidt number. Numerical results agree well with the previous experiments, and the origin of the conspicuous bell-shaped structure observed by the shadowgraph method is identified as an internal wave.
Jets generated by a sphere moving vertically in a stratified fluid
- H. HANAZAKI, K. KASHIMOTO, T. OKAMURA
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- Journal:
- Journal of Fluid Mechanics / Volume 638 / 10 November 2009
- Published online by Cambridge University Press:
- 24 September 2009, pp. 173-197
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Experiments are performed on the flow past a sphere moving vertically at constant speeds in a salt-stratified fluid. Shadowgraph method and fluorescent dye are used for the flow visualization, and particle image velocimetry is used for the velocity measurement in the vertical plane. Vertical ‘jets’ or columnar structures are observed in the shadowgraph for all the Froude numbers Fr(0.2 ≲ Fr ≲ 70) investigated, and the wake structures in the whole parameter space of Fr and the Reynolds number Re(30 ≲ Re ≲ 4000) are classified into seven types, five of which are newly found. Those include two types of thin jets, one of which is short with its top disturbed by internal waves to have a peculiar ‘bell-shaped’ structure, while the other has an indefinitely long length. There are two other new types of jet with periodically generated ‘knots’, one of which is straight, while the other has a spiral structure. A simply meandering jet has also been found. These wake structures are significantly different from those in homogeneous fluids except under very weak stratification, showing that the stratification effects on vertical motion are much more significant than those on horizontal motion.
Linear processes in unsteady stably stratified turbulence
- H. Hanazaki, J. C. R. Hunt
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- Journal:
- Journal of Fluid Mechanics / Volume 318 / 10 July 1996
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- 26 April 2006, pp. 303-337
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Unsteady turbulence in uniformly stratified unsheared flow is analysed using rapid distortion theory (RDT). For inviscid flow with no molecular diffusion the theory shows how the initial conditions, such as the initial turbulent kinetic energy KE0 and potential energy PE0, determine the partition of energy between the potential energy associated with density fluctuation and the kinetic energy associated with each of the velocity components during the subsequent development of the turbulence. One parameter is an exception to this sensitivity to initial conditions, namely the limit at large time of the ratio of potential energy to vertical kinetic energy. In the linear theory, this ratio depends neither on the Reynolds number Re, nor the Prandtl number Pr nor the Froude number Fr. This is consistent with turbulence measurements in the atmosphere, wind tunnel and water tank experiments, and with large-eddy simulations, where similar values of the ratio are found. The RDT results are extended to show the effects of viscosity and diffusion where Re is not very large, explaining the sensitivity of the spectra and the fluxes to the value of the Prandtl number Pr. When Pr is larger than 1, the high-wavenumber components of the three-dimensional spectra induce a vertical flux of temperature (density) that is positive (negative), and therefore ‘countergradient.’ On the other hand, when the thermal diffusivity is stronger and Pr is less than 1, lower-wavenumber components become countergradient sooner since the high-wavenumber components are prevented from becoming countergradient. When all the wavenumber components are integrated to derive the total vertical density flux, it becomes countergradient more quickly and more strongly in high-Pr than in low-Pr turbulence. All these theoretically derived differences between high-Pr and low-Pr turbulence are consistent with the experimental measurements in water tank and wind tunnel experiments and numerical simulations. It is shown that the initial kinetic and potential energy spectrum forms E(k) and S(k) near k = 0 determine the long-time limit values of the variances and the covariances, including their decay rate with time. In the special case of Pr = 1, the oscillation time period of the three-dimensional spectrum function is independent of the wavenumber and is the same as that of an inviscid fluid with the effect of viscosity/diffusion being limited to the damping of all the wavenumber components in-phase with each other. Furthermore, the non-dimensional ratios of the covariances, including the normalized vertical density flux and the anisotropy tensor, agree with the inviscid results if S(k) is proportional to E(k), or if either S(k) or E(k) is identically zero. However, even when Pr = 1, in the ‘one-dimensional spectrum’ in the x-direction, there is a transitory countergradient flux for high wavenumbers; only in this case is there a qualitative difference with the three-dimensioanl spectrum. This paper shows that the characteristic differences in the behaviour of stably stratified turbulence reported in previous DNS experiments at moderate Reynolds numbers can largely be explained by linear oscillations and simple molecular or eddy diffusion rather than by any new kinds of nonlinear mixing processes.
Structure of unsteady stably stratified turbulence with mean shear
- H. HANAZAKI, J. C. R. HUNT
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- Journal:
- Journal of Fluid Mechanics / Volume 507 / 25 May 2004
- Published online by Cambridge University Press:
- 12 May 2004, pp. 1-42
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The statistics of unsteady turbulence with uniform stratification $N$ (Brunt–Väisälä frequency) and shear $\alpha ({=}\,{\rm d}U_{1}/{\rm d}x_{3})$ are analysed over the entire time range ($0 \,{<}\, \alpha t \,{<}\, \infty$) using rapid distortion theory (RDT) over a wide range of Richardson number ${\hbox{\it Ri}} ({=}\,N^{2}/\alpha^{2})$, and initial conditions. The solutions are found to be described by the Legendre functions of complex degree with pure-imaginary argument and are compared with previously published results of both direct numerical simulations (DNS) and experiments. In the initial stage of development many of the characteristics are similar to those in stratified flow with no shear, since the turbulence is determined by $\hbox{\it Nt}$ at the leading order, and the effects of vertical shear $\alpha$ generally appear at higher order. It is shown how in developing turbulence for ${\hbox{\it Ri}} \,{>}\, 0$ and ${\hbox{\it Ri}} \,{>}\, 0.25$ respectively, oscillatory momentum and positive and negative density fluxes develop. Above a critical value of ${\hbox{\it Ri}}_{\hbox{\scriptsize\it crit}} ({\sim}\,0.3)$, their average values are persistently countergradient. This structural change in the turbulence is the primary mechanism whereby stable stratification reduces the fluxes and the production of variances. It is quite universal and differs from the energy and stability mechanisms of Richardson (1926) and Taylor (1931). The long-time asymptotics of the energy ratio ER$({=}\,\hbox{\it PE/VKE}$) of the potential energy to the vertical kinetic energy generally decreases with ${\hbox{\it Ri}}({\geq}\,0.25)$, reaching the smallest value of $3/2$ when there is no shear (${\hbox{\it Ri}} \,{\rightarrow}\, \infty$). For strong mean shear (${\hbox{\it Ri}}\,{<}\,0.25$), RDT significantly overestimates ER since (as in unstratified shear flow) it underestimates the vertical kinetic energy VKE. The RDT results show that the asymptotic values of the energy ratio ER and the normalized vertical density flux are independent of the initial value of ER, in agreement with DNS. This independence of the initial condition occurs because the ratios of the contributions from the initial values $P\!E_{0}$ and $K\!E_{0}$ are the same for PE and VKE and can be explained by the linear processes. Stable stratification generates buoyancy oscillations in the direction of the energy propagation of the internal gravity wave and suppresses the generation of turbulence by mean shear. Because the shear distorts the wavenumber fluctuations, the low-wavenumber spectrum of the vertical kinetic energy has the general form $E_{33}(k)\,{\propto}\, (\alpha tk)^{-1}$, where $(L_{X} \alpha t)^{-1} \,{\ll}\,k \,{\ll}\,L_{X}^{-1}$ ($L_{X}$: integral scale). The viscous decay is controlled by the shear, so that the components of larger streamwise wavenumber $k_{1}$ decay faster. Then, combined with the spectrum distortion by the shear, the energy and the flux are increasingly dominated by the small-$k_{1}$ components as time elapses. They oscillate at the buoyancy period $\pi/N$ because even in a shear flow the components as $k_{1} \,{\rightarrow}\, 0$ are weakly affected by the shear. The effects of stratification $N$ and shear $\alpha$ at small scales are to reduce both VKE and PE. Even for the same ${\hbox{\it Ri}}$, larger $N$ and $\alpha$ reduce the high-wavenumber components of VKE and PE. This supports the applicability of the linear assumption for large $N$ and $\alpha$. At large scales, the stratification and shear effects oppose each other, i.e. both VKE and PE decrease due to the stratification but they increase due to the shear. We conclude that certain of these unsteady results can be applied directly to estimate the properties of sheared turbulence in a statistically steady state, but others can only be applied qualitatively.
Linear processes in stably and unstably stratified rotating turbulence
- H. HANAZAKI
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- Journal:
- Journal of Fluid Mechanics / Volume 465 / 25 August 2002
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- 02 September 2002, pp. 157-190
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Unsteady turbulence in stably and unstably stratified flow with system rotation around the vertical axis is analysed using the rapid distortion theory (RDT). Complete linear solutions for the spectra, variances and covariances are obtained analytically, and their characteristics, including the short- and long-time asymptotics and the effect of initial conditions, are examined in detail. It has been found that the rotation modifies the energy partition among the three kinetic energy components and the potential energy, and the ratio of the Coriolis parameter f to the Brunt–Väisälä frequency N, i.e. f/N, determines the final steady values of these components. The ratio also determines the phase of the energy/flux oscillation. Depending on whether f/N > 1 or f/N < 1, there is a phase shift of ±π/4. However, unsteady aspects are largely dominated by stratification. This occurs because the effects of the Coriolis parameter f appear only in the form of fk3, which vanishes for the horizontal wavenumber components (k3 = 0), which contribute most to the energies and the fluxes. For example, the oscillation frequency of the energies and the fluxes asymptotes to 2N over a long time, in agreement with the stratified non-rotating turbulence. The initial time development is also dominanted by the stratification, and the short-time asymptotics (Nt, ft [Lt ] 1) agree with those for non-rotating stratified fluids in the lowest-order approximation. In the special case of f = N, all the wavenumber components oscillate in phase, leading to no inviscid decay of oscillation. This is in contrast to the general case of f ≠ N, in which inviscid decay has been observed. For pure rotation (f ≠ 0, N = 0), analytical solutions showed that any turbulence that is initially axisymmetric around the rotation axis recovers exact three-dimensional isotropy in the kinetic energy components. Comparison with previous DNS and experiments shows that many of the unsteady aspects of the kinetic and potential energies and the vertical density flux can be explained by the linear processes described by RDT. Even the time development of the vertical vorticity, which would represent the small-scale characteristics of turbulence, agrees well with DNS. For unstably stratified turbulence, the initial processes observed in DNS and experiments, such as the initial decay of the kinetic energy due to viscosity and the subsequent rapid growth of the vertical kinetic energy compared to the horizontal kinetic energy, could be explained by RDT.
Flow past a sphere moving vertically in a stratified diffusive fluid
- C. R. TORRES, H. HANAZAKI, J. OCHOA, J. CASTILLO, M. VAN WOERT
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- Journal:
- Journal of Fluid Mechanics / Volume 417 / 25 August 2000
- Published online by Cambridge University Press:
- 25 August 2000, pp. 211-236
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Numerical studies are described of the flows generated by a sphere moving vertically in a uniformly stratified fluid. It is found that the axisymmetric standing vortex usually found in homogeneous fluids at moderate Reynolds numbers (25 [les ] Re [les ] 200) is completely collapsed by stable stratification, generating a strong vertical jet. This is consistent with our experimental visualizations. For Re = 200 the complete collapse of the vortex occurs at Froude number F ≃ 19, and the critical Froude number decreases slowly as Re increases. The Froude number and the Reynolds number are here defined by F = W/Na and Re = 2Wa/v, with W being the descent velocity of the sphere, N the Brunt–Väisälä frequency, a the radius of the sphere and v the kinematic viscosity coefficient. The inviscid processes, including the generation of the vertical jet, have been investigated by Eames & Hunt (1997) in the context of weak stratification without buoyancy effects. They showed the existence of a singularity of vorticity and density gradient on the rear axis of the flow and also the impossibility of realizing a steady state. When there is no density diffusion, all the isopycnal surfaces which existed initially in front of the sphere accumulate very near the front surface because of density conservation and the fluid in those thin layers generates a rear jet when returning to its original position. In the present study, however, the fluid has diffusivity and the buoyancy effects also exist. The density diffusion prevents the extreme piling up of the isopycnal surfaces and allows the existence of a steady solution, preventing the generation of a singularity or a jet. On the other hand, the buoyancy effect works to increase the vertical velocity to the rear of the sphere by converting the potential energy to vertical kinetic energy, leading to the formation of a strong jet. We found that the collapse of the vortex and the generation of the jet occurs at much weaker stratifications than those necessary for the generation of strong lee waves, showing that jet formation is independent of the internal waves. At low Froude numbers (F [les ] 2) the lee wave patterns showed good agreement with the linear wave theory and the previous experiments by Mowbray & Rarity (1967). At very low Froude numbers (F [les ] 1) the drag on a sphere increases rapidly, partly due to the lee wave drag but mainly due to the large velocity of the jet. The jet causes a reduction of the pressure on the rear surface of the sphere, which leads to the increase of pressure drag. High velocity is induced also just outside the boundary layer of the sphere so that the frictional drag increases even more significantly than the pressure drag.