Kaoru Fujimura

Natural convection in a hemispherical enclosure heated from below

Natural convection in a hemisphere heated from below was studied experimentally. Correlation between heat transfer rate and Rayleigh number was obtained on a hemispherical surface over wide parameter ranges : 106 ⩽ Ra ⩽ 6 × 1010 and 6 ⩽ Pr ⩽ 13000. The exponent of Nu against Ra changes from 14 to 13 at about Ra ~ 109. Flow patterns in a hemisphere were observed for 2 × 105 ⩽ Ra ⩽ 2 × 109 and 6 ⩽ Pr ⩽ 13000. The patterns were classified into several flow regimes, i.e. steady circulating flow, flow with periodic thermal plumes, flow with irregular thermal plumes and turbulent flow regime. Experimental results show that fully turbulent flow regime occurs at Ra ~ 109.

Takens-Bogdanov bifurcation on the hexagonal lattice for double-layer convection

Abstract In the Benard problem for two-fluid layers, Takens–Bogdanov bifurcations can arise when the stability thresholds for both layers are close to each other. In this paper, we provide an analysis of bifurcating solutions near such a Takens–Bogdanov point, under the assumption that solutions are doubly periodic with respect to a hexagonal lattice. Our analysis focusses on periodic solutions, secondary bifurcations from steady to periodic solutions and heteroclinic solutions arising as limits of periodic solutions. We compute the coefficients of the amplitude equations for a number of physical situations. Numerical integration of the amplitude equations reveals quasiperiodic and chaotic regimes, in addition to parameter regions where steady or periodic solutions are observed.

INTERACTION BETWEEN LONGITUDINAL CONVECTION ROLLS AND TRANSVERSE WAVES IN UNSTABLY STRATIFIED PLANE POISEUILLE FLOW

Nonlinear interaction between a Tollmien–Schlichting wave and longitudinal rolls resulting from Rayleigh–Benard instability has been investigated in an unstably stratified plane Poiseuille flow of infinite extent. Cubic order amplitude equations for the interacting modes are derived on a weakly nonlinear basis in the neighborhood of a crossover point at which both modes become unstable on a linear basis simultaneously. A bifurcation analysis based on use of the actual numerical coefficients obtained from the governing equations and evaluated for various values of the Prandtl number is performed, and the results, such as the effect of longitudinal rolls on the subcritical instability characteristic of a Tollmien–Schlichting wave, are discussed.

Centre manifold reduction and the Stuart‐Landau equation for fluid motions

The centre manifold reduction to derive the Stuart-Landau equation is examined. A double expansion in terms of the Fourier series and linear eigenfunctions is introduced in hydrodynamic equations. A centre manifold reduction scheme is then applied to reduce the resultant system of ordinary differential equations to the Stuart–Landau equation. Through a formal expansion in linear eigenfunctions, the latter equation is shown to be equivalent with the one derived by the method of multiple scales. Numerical coefficients involved in the quintic Stuart–Landau equation are evaluated for plane Poiseuille flow, convection in a vertical slot, and Rayleigh-Bénard convection. In all the cases, the coefficients converge as a dimension of the ODE system increases and approach the numerical values obtained by the method of multiple scales.

The 2:1 steady/Hopf mode interaction in the two-layer Be´nard problem

Abstract The interaction of Hopf and steady modes with wavenumber ratio 2:1 is investigated for a critical situation. Two different immiscible liquids lie in layers between horizontal walls and are heated from below. A situation with a pair of complex conjugate eigenvalues at wavenumber α and a real eigenvalue at wavenumber 2α is at criticality. Weakly nonlinear amplitude equations are derived for the interaction of these oscillatory and steady modes. The two modes generate a two-parameter bifurcation. The coefficients involved in the equations are determined numerically, based on the physical parameters of the system at criticality. Three obvious equilibrium solutions of the amplitude equations are the steady solution, the traveling waves and the mixed standing waves. The eigenvalues governing the stability of these solutions are found explicitly. Numerical results and bifurcation diagrams are given for the critical situation. The steady solution and the traveling wave solution are unstable. There is a region of stability for the standing wave solution. A new equilibrium solution, the asymmetric mixed mode, is found to be stable in a parameter range. Bifurcations from the standing wave solution and the asymmetric mixed mode are described.

Higher harmonic resonance of two-dimensional disturbances in Rayleigh-Bénard convection

A higher harmonic resonance with wavenumber ratio of 1:3 is found to take place in Rayleigh-Benard convection under rigid-rigid boundary conditions. Bifurcation diagrams for two-dimensional motion are obtained for various values of the Prandtl number P. It is found that a pure mode and mixed mode solutions exist as nonlinear equilibrium states of primary roll solutions for relatively high-Prandtl-number fluids ( P ≥ 0.13) while the pure mode, mixed modes, travelling wave and modulated wave solutions exist for relatively low-Prandtl-number fluids ( P ≤ 0.12).

Mixed mode convection in an inclined slot

Nonlinear interaction between transverse disturbances and longitudinal rolls has been investigated for flow in an inclined slot with a heated lower wall when both modes of instability occur at nearly the same value of the control parameter. This condition is shown to be possible for a fluid with Prandtl number greater than 0.263 897, For slightly supercritical values of the Rayleigh number ( R ) when the critical Rayleigh number for longitudinal rolls R L C is somewhat less than that for transverse stationary rolls, R S C , and for transverse travelling waves, R T C , longitudinal rolls occur first and then remain stable as R is increased beyond R S C or R T C ; no mixed mode state occurs. In contrast, if R S c or R T c is slightly below R L C , pure transverse modes exist for only a relatively small range of R beyond R S C or R T C . Thereafter, a three-dimensional mixed mode state occurs well before R L C is reached, i.e. three-dimensionality sets in on a subcritical basis. As R approaches R L C the contribution of the transverse mode decreases continuously until a pure longitudinal roll state emerges for R slightly greater than R L C . Mixed mode convection is also investigated for a special choice of parameters when three modes, namely transverse stationary rolls, transverse travelling waves and longitudinal rolls, become unstable simultaneously. Longitudinal rolls again emerge as the stable supercritical state.

Modulational instability of plane waves in a two-dimensional jet and wake

The modulational instability problem is reformulated by the amplitude expansion method. A generalized version of the Stewartson-Stuart equation is derived, which is applicable to any supercritical state as long as the amplitude of the disturbance is small. The modulational instability of plane waves in a two-dimensional jet and wake is investigated on the basis of this equation, and it is found that the plane waves are stable to side-band disturbances in supercritical states except in the close neighbourhood of the critical state.

A helium-cooled solid breeder concept for the tritium-producing blanket of the international thermonuclear experimental reactor

The usefulness of the tritium-producing blanket in the International Thermonuclear Experimentall Reactor (ITER) to the fusion research and development program can be maximized by selecting design parameters, features, and options that are reactor relevant without significantly increasing the risk in key areas such as device safety and operational reliability. For that reason, a helium-cooled solid breeder (SB) blanket is proposed since it combines the operation of the SB at high reactor-relevant temperatures with the operation of helium at moderate temperature and pressure to minimize risk. Results of the analysis done for this blanket concept indicate that it is very attractive. It can achieve a high tritium breeding ratio without breeding in the space-limited inboard region, It offers important safety features, including the use of inert gas with no chemical reaction or corrosion, low activation SB, and multiple containment of tritium. the concept provides great operational flexibility to accommodate changes in ITER operating parameters, such as power level, and to optimize the operating temperature of the structure. A novel and practical concept is proposed for the thermal resistance gap between the coolant and SB to allow their operating temperatures to be optimized.

Range of Validity of Weakly Nonlinear Theory in the Rayleigh–Bénard Problem

In this paper we examine the equilibrium states of finite amplitude flow in a horizontal fluid layer with differential heating between the two rigid boundaries. The solutions to the Navier-Stokes equations are obtained by means of a perturbation method for evaluating the Landau constants and through a Newton-Raphson iterative method that results from the Fourier expansion of the solutions that bifurcate above the linear stability threshold of infinitesimal disturbances. The results obtained from these two different methods of evaluating the convective flow are compared in the neighborhood of the critical Rayleigh number. We find that for small Prandtl numbers the discrepancy of the two methods is noticeable.