Statistical properties of MHD turbulence and the mechanism of turbulent dynamo are investigated by direct numerical simulations of three-dimensional MHD equations. It is assumed that the turbulent field has a high symmetry and that the fluid has hyperviscosity and hypermagnetic diffusivity. An external force is exerted on the fluid as kinetic energy and helicity sources. The main concern of the present study is whether magnetic fields of scales comparable to the dominant fluid motions can be generated or not. It is shown that the turbulent dynamo is effective if hypermagnetic diffusivity is smaller than a critical value. The total energy spectrum is close to the k−5/3 power law in the inertial range. The energy transfer between kinetic and magnetic fields is discussed.

The instability and transition of flow past two circular cylinders arranged in tandem are investigated numerically. A steady symmetric flow is realized at small Reynolds numbers, but the flow becomes unstable above a critical Reynolds number and makes a transition to an oscillatory flow. We obtained the symmetric flow numerically and analyze its stability by applying linear stability theory. The nonlinear oscillatory flow arising from the instability is obtained not only by numerical simulation but also by direct numerical calculation of the equilibrium solution, and the bifurcation diagram for the nonlinear equilibrium solution is depicted. We focused our attention on the effect of the gap spacing between the two cylinders on the stability and transition of the flow. The transition of the flow from a steady state to an oscillatory state is clarified to occur due to a supercritical or subcritical Hopf bifurcation depending upon the gap spacing. We found that there is a certain range of the gap spacing where physical quantities such as the drag and lift coefficients and the Strouhal number show an abrupt change when the gap spacing is continuously changed. We identified the origin of the abrupt change as the existence of multiple stable solutions for the flow.

The stability of a two-dimensional flow in a symmetric channel with a suddenly expanded part is investigated numerically and analyzed by using the method of the nonlinear stability theory. From results of the numerical simulation, it is shown that the flow is steady, symmetric and unique at very low Reynolds numbers, while the symmetric flow loses its stability at a critical Reynolds number resulting in an appearance of asymmetric flow. The transition from the steady symmetric flow to the steady asymmetric one is found to occur due to the symmetry breaking pitchfork bifurcation when the aspect ratio, the ratio of the length of the expanded part to its width, is large. It is also found that the bifurcated flow becomes symmetric again when the Reynolds number is increased and the resultant symmetric flow loses its stability becoming periodic in time as the Reynolds number is further increased. On the other hand, when the aspect ratio is small there occurs no pitchfork bifurcation and the direct transition from the steady symmetric flow to a periodic flow occurs due to a Hopf bifurcation. The critical aspect ratio is found to be about 2.3. The critical Reynolds numbers for these bifurcations are evaluated.

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The stability and transition of flow past a pair of circular cylinders in a side-by-side arrangement are investigated by numerical simulations and linear stability analyses. Various flow patterns around the cylinders have been reported to appear due to an instability of the steady symmetric flow that is realized at small Reynolds numbers, among which deflected oscillatory flow is particularly noticeable. The physical origin of the flow is explored by bifurcation analyses of the numerical data. We found that the deflected oscillatory flow arises from the steady symmetric flow through sequential instabilities due to stationary and oscillatory unstable modes. Steady asymmetric flow with respect to the streamwise centreline between the two cylinders was also found to be induced by the instability due to a stationary mode in a very narrow range of the gap width between the two cylinders. We classify the instability modes of the steady symmetric flow into four groups in the parameter space of the gap width, and evaluate the critical Reynolds number for each mode of instability.

Transitions and instabilities of two-dimensional flow in a symmetric channel with a suddenly expanded and contracted part are investigated numerically by three different methods, i.e. the time marching method for dynamical equations, the SOR iterative method and the finite-element method for steady-state equations. Linear and weakly nonlinear stability theories are applied to the flow. The transitions are confirmed experimentally by flow visualizations. It is known that the flow is steady and symmetric at low Reynolds numbers, becomes asymmetric at a critical Reynolds number, regains the symmetry at another critical Reynolds number and becomes oscillatory at very large Reynolds numbers. Multiple stable steady-state solutions are found in some cases, which lead to a hysteresis. The critical conditions for the existence of the multiple stable steady-state solutions are determined numerically and compared with the results of the linear and weakly nonlinear stability analyses. An exchange of modes for oscillatory instabilities is found to occur in the flow as the aspect ratio, the ratio of the length of the expanded part to its width, is varied, and its relation with the impinging free-shear-layer instability (IFLSI) is discussed.

A method of multiple-scale expansion is applied to the theory of incompressible isotropic turbulence in order to close the infinite system of cumulant equations. The dynamical equation for the energy spectrum derived from this method is found to give positive-definite solutions at all Reynolds numbers. At large Reynolds numbers the spectrum takes the form of Kolmogorov's$-\frac{5}{3}$power spectrum in the inertial subrange, whose extent increases indefinitely with Reynolds number. The spectrum in the energy-containing range satisfies an inviscid similarity law, so that the rate of energy decay or of viscous dissipation is also independent of the viscosity. In the higher wavenumber region beyond the inertial subrange the spectrum takes a universal form which is independent of its structure at lower wavenumbers. The universal spectrum is composed of three different subspectra, which are, in order of increasing wavenumber, the$k^{-\frac{5}{3}}$spectrum, thek−1spectrum and the exp [−σk1·5] spectrum, σ being a constant. Various statistical quantities such as the energy, the skewness of the velocity derivative, the microscale and the microscale Reynolds number are calculated from the numerical data for the energy spectrum. Theoretical results are discussed in detail in comparison with experimental results.

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