Shinya Okino

A new nonlinear vortex state in square-duct flow

A new nonlinear travelling-wave solution for a flow through an isothermal square duct is discovered. The solution is found by a continuation approach in parameter space, starting from a case where the fluid is heated internally. The Reynolds number for which the travelling wave emerges is much lower than that of the solutions discovered recently by an analysis based on the self-sustaining process (Wedin et al., Phys. Rev. E, vol. 79, 2009, p. 065305; Uhlmann et al., Advances in Turbulence XII, 2009, pp. 585-588). Furthermore, the new travelling-wave solution is shown to be unstable from the onset.

Turbulence in a Fluid Stratified by a High Prandtl-Number Scalar

Turbulence in the fluid stratified by a high Prandtl-number (Pr) scalar such as heat (Pr = 7) or salinity (Pr = 700) has been simulated by direct numerical simulations, using 40963 grid points. Computations have been performed using the 1024 nodes of NEC SX-ACE, which have enabled us to resolve the smallest scale of salinity fluctuations \(\sqrt{ 700}(\sim 26)\) times smaller than the smallest eddy size. In our simulations, buoyancy initially affects only the large scale motions, and the k−1 spectrum predicted by Batchelor (J. Fluid Mech. 5:113–133, 1959) for a passive scalar could be observed in the spectrum of potential energy, i.e. the salinity fluctuations. However, as time proceeds, the buoyancy affects the smaller-scale motions, and the salinity fluctuations begin to show a unique spatially localised structure. At the same time, there appears a flat spectrum ( ∝ k0) instead of the k−1 spectrum. The localised structure and the flat spectrum could be observed only for the salinity (Pr = 700) and not for heat (Pr = 7).

Travelling Wave Solutions in Square Duct Flow

Three nonlinear travelling wave solutions for square duct flow are discovered. One of them, the asymmetric solution, which is predicted by the stability analysis, bifurcates from the mirror-symmetric solution found by Okino et al., J. Fluid Mech. 657, 413 (2010). The solution is characterized by asymmetric crosssectional flow patterns with streamwise vortices attached to one of the side walls. The other two solutions, which have rotational symmetry by π and π /2, respectively, are found by a homotopy approach using artificially arranged body forces. Each of the new solutions shows striking similarity to that of pipe flow.