Hideshi Hanazaki

A numerical study of three-dimensional stratified flow past a sphere

On suppose la vitesse uniforme et la stratification lineaire. Les resultats numeriques montrent des changements de regime decoulement avec le nombre de Froude en accord avec de precedents resultats theoriques et experimentaux

Upstream advancing columnar disturbances in two‐dimensional stratified flow of finite depth

A numerical study of the two‐dimensional flow of linearly stratified Boussinesq fluid past a vertical flat plate in a channel of finite depth is described. It is found that there are time‐dependent oscillations in each vertical mode of the upstream advancing columnar disturbances which correspond to the unsteadiness in the drag coefficient found in previous experiments. The long‐time behavior of the upstream columnar disturbances shows that the time‐averaged strength of each mode approaches some constant value that is not zero. This determines the drag coefficient in the long‐time limit. In many points the numerical solutions of the Navier–Stokes equation agree with the solutions of the forced Korteweg–de Vries (KdV) equation with a cubic nonlinear term or the forced KdV–Burgers equation. It is also suggested that the strong downstream columnar disturbances predicted by linear theory for steady flow do not exist.

A numerical study of nonlinear waves in a transcritical flow of stratified fluid past an obstacle

A numerical study of the flow of stratified fluid past an obstacle in a horizontal channel is described. Upstream advancing of waves near criticality (resonance) appears in the case of ordinary two‐layer flow, in which case the flow is described well by the solution of the forced extended Korteweg–de Vries (fEKdV) equation which has a cubic nonlinear term. It is shown theoretically that the upstream waves in the general two‐layer flow cannot be well described by the forced KdV equation except when the wave amplitude is very small. The critical‐level flow is also governed by the fEKdV equation. However, because of the smallness of the coefficient of the quadratic nonlinear term, the bore cannot propagate upstream at exact resonance. The results for the linearly stratified Boussinesq flow show good agreement with the solution of the Grimshaw and Yi [J. Fluid Mech. 229, 603 (1991)] equation at least for exact resonance. The cases of various strengths of forcing and the negative forcing are investigated to se...

Drag coefficient and upstream influence in three-dimensional stratified flow of finite depth

A numerical study of the three-dimensional stratified flow past a vertical square flat plate in a channel of finite depth is described. Particular attention is paid to the anomalous dependence of the drag coefficient CD on parameter K( = ND/-πU), where N is the Brunt-Vaisala frequency, D is the half depth of the channel and U is the upstream velocity. It is shown that CD generally increases with K, while it decreases locally at integral values of K. Time development of the upstream columnar disturbance and the corresponding variation of CD reveals that the periodic variation of CD with time for K > 1 comes from the successive upstream radiation of the columnar disturbances of the first internal wave mode. Although the propagation speed of the columnar disturbance is consistent with the prediction of linear theory, its time-dependent structure is different from the weakly nonlinear theory as has been shown by laboratory experiments.

On the three-dimensional internal waves excited by topography in the flow of a stratified fluid

A numerical study of the three-dimensional internal waves excited by topography in the flow of a stratified fluid is described. In the resonant flow of a nearly two-layer fluid, it is found that the time-development of the nonlinearly excited waves agrees qualitatively with the solution of the forced KP equation or the forced extended KP equation. In this case, the upstream-advancing solitary waves become asymptotically straight crested because of abnormal reflection at the sidewall similar to Mach reflection. The same phenomenon also occurs in the subcritical flow of a nearly two-layer fluid. However, in the subcritical flow of a linearly stratified Boussinesq fluid, the two-dimensionalization of the upstream waves can be interpreted as the separation of the lateral modes due to the differences in the group velocity of the linear wave, although this does not mean in general that the generation of upstream waves is describable by the linearized equation.

On the nonlinear internal waves excited in the flow of a linearly stratified Boussinesq fluid

A numerical study of the internal gravity waves excited by an obstacle in the flow of a linearly stratified Boussinesq fluid is described. Solutions of the Navier–Stokes equations agree quantitatively well with the solutions of Grimshaw and Yi’s equation [J. Fluid Mech. 229, 603 (1991)] near resonance. The equation of Grimshaw and Yi can describe well the upstream waves even when the Froude number F is fairly small (F=0.5). Therefore it is a good model of the nonlinear internal waves excited in a linearly stratified Boussinesq fluid. When the wave amplitude is large, the first disappearance of the horizontal velocity is also well predicted by the solution of the equation of Grimshaw and Yi, although the equation cannot predict the subsequent time development.

Upstream‐advancing nonlinear waves in an axisymmetric resonant flow of rotating fluid past an obstacle

A numerical study of rotating flow past an axisymmetric obstacle in a cylindrical tube is described. Long nonlinear inertial waves propagate upstream of the obstacle even when the theoretical speed of long waves as given by the linearized equations is equal to the incoming axial flow speed. This occurs when the upstream swirling velocity field is of the Brugers vortex type and the quadratic nonlinear term in the forced KdV equation is not zero. If the flow has a rigid‐body rotation, so that the quadratic nonlinear term vanishes, the nonlinear correction of the wave speed is small and cannot be identified even after large times of evolution.

On the wave excitation and the formation of recirculation eddies in an axisymmetric flow of uniformly rotating fluids

The inertial waves excited in a uniformly rotating fluid passing through a long circular tube are studied numerically. The waves are excited either by a local deformation of the tube wall or by an obstacle located on the tube axis. When the flow is subcritical, i.e. when the phase and group velocity of the fastest wave mode in their long-wave limit are larger than the incoming axial flow velocity, the excited waves propagate upstream of the excited position. The non-resonant waves have many linear aspects, including the upstream-advancing speed of the wave and the coexisting lee wavelength. When the flow is critical (resonant), i.e. when the long-wave velocity is nearly equal to the axial flow velocity, the large-amplitude waves are resonantly excited. The time development of these waves is described well by the equation derived by Grimshaw & Yi (1993). The integro-differential equation, which describes the strongly nonlinear waves until the axial flow reversal occurs, can predict the onset time and position of the recirculation eddies observed in the solutions of the Navier-Stokes equations.

Upstream‐advancing nonlinear waves excited in an axisymmetric transcritical flow of rotating fluid

A numerical study of the inertial waves in an axisymmetric transcritical flow of rotating fluid passing through a long circular tube is described. The waves are excited either by an obstacle on the axis of the tube or by a local undulation of the tube wall. In all the calculations, the flow has a circulation of the Burgers‐vortex type. The solutions of the Navier–Stokes equations are compared with the solutions of the forced KdV (Korteweg–de Vries) equation and it is found that the waves that appear in the solutions of the Navier–Stokes equations are described, at least qualitatively, by the solutions of the forced KdV equation. In this study, the forced KdV equation that has also a cubic nonlinear term is derived. However, the effect of the cubic nonlinearity is found to be small, at least for the upstream circulation distribution used in this study.

The quasi-three-dimensional instability of an elliptical vortex subject to a strain field in a rotating stratified fluid

The linear instability of a steady elliptical vortex in a stably stratified rotating fluid is investigated, using the quasi-geostrophic, f-plane approximation. The vortex is embedded in a uniform background straining field e with uniform vorticity 2γ (the Moore-Saffman vortices). An elliptical vortex in an irrotational strain field (γ = 0) is shown to be unstable to long wave quasi-three-dimensional disturbances of azimuthal wavenumber m = 1 (bending wave). The long wave instability has a two-dimensional origin. An elliptical vortex in a simple shear, whose major axis is parallel to the shear streamlines (e = γ), is stable against any disturbance. In contrast, an ellipse in a simple shear, whose major axis is perpendicular to the shear streamlines (e = −γ), is unstable to quasi-three-dimensional bending modes irrespective of a/b. Short wave quasi-three-dimensional disturbances grow faster than two-dimensional instability modes in the parameter range − 0.5 < γ < 0. The origin of short wave instability is attributed to resonance between inertial waves and the imposed strain field e.