Linear Stability of Fluid Flows and its Applications (Topology optimization theory and applications toward wide fields of natural sciences)

"Topology optimization theory and applications toward wide fields of natural sciences". May 7～9, 2014. edited by Takashi Nakazawa. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.First, mathematical formulation of linear stability analysis in fluid mechanics is given. Next, several examples in which the linear stability analysis was successfully applied are presented. Then, bifurcation diagrams are shown, and the basic equations by which fluid motions are calculated for specific boundary conditions are explained. Finally, numerical methods to obtain quasi‐analytic solutions in high accuracy are addressed, on which the perturbation is superimposed and thus the linear stability analysis can be applied, in the case that an exact solution is difficult to obtain

Wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube in a simple shear flow

the mechanism of wrap, tilt and stretch of vorticity lines around a strong thin straight vortex tube of circulation &#915; starting with a vortex filament in a simple shear flow (U=SX2x^1, S being a shear rate) is investigated analytically. an asymptotic expression for the vorticity field is obtained at a large reynolds number &#915;/&#957; » 1, &#957; being the kinematic viscosity of fluid, and during the initial time St « 1 of evolution as well as St « (&#915;/&#957;)1/2. the vortex tube, which is inclined from the streamwise (X1) direction both in the vertical (X2) and spanwise (X3) directions, is tilted, stretched and diffused under the action of the uniform shear and viscosity. the simple shear vorticity is on the other hand, wrapped and stretched around the vortex tube by a swirling motion, induced by it to form double spiral vortex layers of high azimuthal vorticity of alternating sign. the magnitude of the azimuthal vorticity increases up to O((&#915;/&#957;)1/3S) at distance r=O((&#915;/&#957;)1/3 (&#957;t)1/2) from the vortex tube. the spirals induce axial flows of the same spiral shape with alternate sign in adjacent spirals which in turn tilt the simple shear vorticity toward the axial direction. as a result, the vorticity lines wind helically around the vortex tube accompanied by conversion of vorticity of the simple shear to the axial direction. the axial vorticity increases in time as s2t, the direction of which is opposite to that of the vortex tube at r=O((&#915;/&#957;)1/2 (&#957;t)1/2) where the vorticity magnitude is strongest. in the near region r « (&#915;/&#957;)1/3 (&#957;t)1/2, on the other hand, a viscous cancellation takes place in tightly wrapped vorticity of alternate sign, which leads to the disappearance of the vorticity normal to the vortex tube. only the axial component of the simple shear vorticity is left there, which is stretched by the simple shear flow itself. as a consequence, the vortex tube inclined toward the direction of the simple shear vorticity (a cyclonic vortex) is intensified, while the one oriented in the opposite direction (an anticyclonic vortex) is weakened. the growth rate of vorticity due to this effect attains a maximum (or minimum) value of ±S2/33/2 when the vortex tube is oriented in the direction of X^1+X^2[minus-or-plus sign] X^3. the present asymptotic solutions are expected to be closely related to the flow structures around intense vortex tubes observed in various kinds of turbulence such as helical winding of vorticity lines around a vortex tube, the dominance of cyclonic vortex tubes, the appearance of opposite-signed vorticity around streamwise vortices and a zig-zag arrangement of streamwise vortices in homogeneous isotropic turbulence, homogeneous shear turbulence and near-wall turbulence.</p

Extracting dominant turbulent structures in supersonic flow using two-dimensional Fourier transform

A new image process for quantifying both convection velocities (U C) and scales (λ d) of turbulent structures captured in a fast-framing schlieren movie is presented. We obtained 90 time-series schlieren images of a transverse jet into a Mach 2 supersonic flow with 1-MHz sampling. The schlieren images captured not only the shock and expansion waves but also the turbulent structures within the jet and the boundary layer. The image intensities were extracted along the outer edges of the jet and the boundary layer and were remapped as a time–space intensity map. The time–space map exhibited swept stripe patterns, indicating that stable turbulent structures were periodically generated and convected downstream. The angle and interval of the stripe patterns were efficiently extracted using the two-dimensional Fourier transform, which corresponded to U C and λ d of the dominant structures. The zero-padding fast Fourier transform and the sub-pixel estimation of the spectral peak positions in the Fourier domain improved the accuracy for evaluating the angle and interval of the stripes, which resulted in the accurate evaluation of U C and λ d. The proposed method was validated by comparing U C obtained using the proposed method to those obtained via schlieren image velocimetry for both the transverse jet and the supersonic boundary layer

Large-scale vortical structure detection using microphone array in a semiconductor single wafer spin cleaner

Fluctuating pressure (p’ ) of a large-scale vortical structure generated in a semiconductor single wafer spin cleaner was detected by using microphone array. Twelve microphones were installed on the exhaust cover under the rotating disk of the cleaner with their interval of 7.5° or 15°. Power spectrum densities (PSD) of p’ were compared with those of fluctuating velocity measured by PIV for various rotation angular velocities to identify fluctuations due to convection of the large-scale vortical structure. Good agreement of PSDs indicates that the large-scale structure could be detected by using microphone. Cross-correlation of p’ measured at different positions revealed that the large-scale structure convected to the downstream in the rotational direction of the disk. The convection speed was about 12 % of the angular velocity of the rotating disk. Number of the vortex in the large-scale structure was also evaluated from the time-series p’ data. Time-space contour map was made for p’ based on the data measured at the different angular position, and showed periodical swept strip patterns. Presences of the strip patterns indicate the pressure disturbances were stably convected to the downstream. From this time-space map, two-dimensional Fourier transform efficiently extracted the number of vortices in the large-scale structure

Numerical study of air-entraining and submerged vortices in a pump sump

Numerical detection of harmful vortices in pump sumps, such as an air-entraining vortex (AEV) and a submerged vortex (SMV), is crucially important to develop the drain pump machinery. We performed numerical simulations of the benchmark experiments of the pump sump conducted by Matsui et al. (2006 and 2016) using the OpenFOAM and compared the simulation results with the experimental data considering the effects of turbulence model, grid density and detection method of the vortices. We studied the threshold of the gas-liquid volume fraction of the VOF method and the second invariant of velocity gradient tensor to identify AEV and SMV. The methods proposed in the present paper were found to be very effective for the detection of the vortices, and the simulation results by RANS with the SST k-omega model successfully reproduced the experimental data. LES with the Smagorinsky model, however, was sensitive to the grid system and difficult to reproduce the experimental data even for the finest grid system having 3.7 million cells in the present study

Screening of sperm velocity by fluid mechanical characteristics of a cyclo-olefin polymer microfluidic sperm-sorting device

The microfluidic sperm-sorting (MFSS) device is a promising advancement for assisted reproductive technology. Previously, poly(dimethylsiloxiane) and quartz MFSS devices were developed and used for intracytoplasmic sperm injection. However, these disposable devices were not clinically suitable for assisted reproduction, so a cyclo-olefin polymer MFSS (COP-MFSS) device was developed. By micromachining, two microfluidic channels with different heights and widths (chip A: 0.3 x 0.5 mm; chip B: 0.1 x 0.6 mm) were prepared. Sorted sperm concentrations were similar in both microfluidic channels. Linear-velocity distribution using the microfluidic channel of chip B was higher than that of chip A. Using confocal fluorescence microscopy, it was found that the highest number of motile spermatozoa swam across the laminar flow at the bottom of the microfluidic channel. The time required to swim across the laminar flow was longer at the bottom and top of the microfluidic channels than in the middle because of the low fluid velocity. These results experimentally demonstrated that the width of microfluidic channels should be increased in the region of laminar flow from the semen inlet to the outlet for unsorted spermatozoa to selectively recover spermatozoa with high linear velocity

Periodical structure of vortices in a semiconductor single wafer spin cleaner

We experimentally and numerically investigated large-scale structures formed by vortices in a single wafer spin cleaner. The Q-criterion identified the vortices developed in the cleaner as the flow regions with positive second invariant of the velocity gradient tensor obtained by both the PIV and LES. The time-series two-components PIV data shows that small-vortices were clustered near and under the edge of the rotating disk and were periodically emanated from there to the housing wall of the cleaner. The emanation frequency was increased with increasing in the angular velocity of the rotating disk. Three-dimensional LES reveal that six longitudinal vortices were spirally developed from under the edge of the rotating disk to the housing wall. This structure stably rotated slower than the disk speed. Fourier analysis of the LES data agreed with that of the PIV data. This supports that the passages of the stable spiral vortices on the PIV measurement region resulted in the periodical emanation of the clustered small-vortices observed in the PIV. Such a very large-scale spiral structure will induce reattachment of contaminants on the wafer surface, and should be destructed for development of much higher efficient cleaner