| 研究実績の概要 |
The 2D Tollmien-Schlichting (TS) flow has been computed. It is found to bifurcate directly from the basic state only for values of the streamwise wavenumber smaller than 1.09732, with solution branches for larger values found by homotopy.The lowest value of the Reynolds number (R) that such 2D flows can exist was found to be R = R_2Dmin = 2939, in agreement with previous studies.
Two types of 3D tertiary flows that bifurcate from TS flow have been found, which we label here TW_a and TW_b. TW_a inherits the shift-reflect symmetry possessed by the TS flow while this symmetry is lost in the TW_b flow.
Discrepancies with the results and conclusions of the previous study by Ehrenstein & Koch (1991) have been uncovered. In particular, in contrast to the earlier study, we find that none of the 3D tertiary flows can exist for R < R_2Dmin.
This project also intends to investigate whether new flows found in this study of Poiseuille flows can be extended to other shear flows. As preparation, a corresponding symmetry analysis of Couette flow has been undertaken and implemented in the computational code. As a test of this code, calculations of Ribbon Couette flow, which corresponds to the symmetry of a flow that can arise as a secondary, tertiary and quarternary flow in Poiseuille configuration, have been undertaken. These calculations, performed in collaboration with B. Song and M. Nagata, have revealed that the 2D flow that was thought to explain the first transition in rotating Couette flow is unstable from its first appearance, and so cannot explain such a transition.
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| 現在までの達成度 |
現在までの達成度
2: おおむね順調に進展している
理由
Most of the goals of the research plan for FY2018 have been achieved. In particular, we have successfully computed all of the tertiary travelling wave states that we intended to compute. However, some of these solutions have required a relatively high truncation level in order to guarantee numerical convergence, which has slowed down the calculations exploring the parameter space of these solutions. In order to overcome this problem, the computer code has been revised to use the MPI parallelisation library, which will allow the program to b run more efficiently on the Kyushu University supercomputer cluster. These modifications took some time, so there are still some remaining calculations of the tertiary flow that must be undertaken before investigation of the quarternary flows can commence.
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| 今後の研究の推進方策 |
After completing the final exploration of the parameter space for the tertiary flows, we will then proceed to seek the quarternary flows that bifurcate from these flows. In most cases this will be initiated by performing stability analysis of the tertiary flows, and then using this analysis to construct initial guesses for the nonlinear solutions, guided by symmetry analysis of the quarternary flows (already complete).
Moving into FY2020, the significance of the flows found in the present study will then be investigated. We also intend to explore the relevance of these solutions to other shear flow configurations, with a particular focus on Couette flows.
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