Flow structures in spanwise rotating plane Poiseuille flow based on thermal analogy

The analogy between rotating shear flow and thermal convection suggests the existence of plumes, inertial waves and plume currents in plane Poiseuille flow under spanwise rotation. The existence of these flow structures is examined with the results of three-dimensional and two-dimensional three-component direct numerical simulations. The dynamics of plumes near the unstable side is embodied in a truncated exponential distribution of turbulent fluctuations. For large rotation numbers, inertial waves are identified near the stable side, and these can be used to explain the abnormal flow statistics, such as the large root-mean-square of the streamwise velocity fluctuation and the nearly negligible Reynolds shear stress. For small or medium rotation numbers, plumes generated from the unstable side form large-scale plume currents and the patterns of the plume currents show different capabilities in scalar transport.

Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers

We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J ⁢ z ′ = ∇ ⁡ H ⁢ ( t , z ) {Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) {\frac{\partial H}{\partial x}(t,x,y)} , uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

Lyapunov exponent of random dynamical systems on the circle

We consider products of an independent and identically distributed sequence in a set $\{f_1,\ldots ,f_m\}$ of orientation-preserving diffeomorphisms of the circle. We can naturally associate a Lyapunov exponent $\lambda $ . Under few assumptions, it is known that $\lambda \leq 0$ and that the equality holds if and only if $f_1,\ldots ,f_m$ are simultaneously conjugated to rotations. In this paper, we state a quantitative version of this fact in the case where $f_1,\ldots ,f_m$ are $C^k$ perturbations of rotations with rotation numbers $\rho (f_1),\ldots ,\rho (f_m)$ satisfying a simultaneous diophantine condition in the sense of Moser [On commuting circle mappings and simultaneous diophantine approximations. Math. Z.205(1) (1990), 105–121]: we give a precise estimate of $\lambda $ (Taylor expansion) and we prove that there exist a diffeomorphism g and rotations $r_i$ such that $\mbox {dist}(gf_ig^{-1},r_i)\ll |\lambda |^{{1}/{2}}$ for $i=1,\ldots , m$ . We also state analogous results for random products of $2\times 2$ matrices, without any diophantine condition.

HEAT TRANSFER IN A ROTATING, TWO-PASS, VARIABLE ASPECT RATIO COOLING CHANNEL WITH PROFILED V-SHAPED RIBS

The thermal performance of two V-type rib configurations is measured in a rotating, two-pass cooling channel. The coolant travels radially outward in the rectangular first pass (AR = 4:1), and travels radially inward in the second pass (AR = 2:1). Both the passages are oriented 90° to the direction of rotation. The LS and TS of the channel are roughened with V-type ribs. The first V-shaped configuration has a narrow gap at the apex of the V. The configuration is modified by off-setting one leg of the V to create a staggered discrete, V-shaped configuration. The ribs are oriented 45° relative to the streamwise coolant direction. The heat transfer enhancement and frictional losses are measured with varying Reynolds and rotation numbers. The Reynolds number varies from 10,000 to 45,000 in the AR = 4:1 first pass; this corresponds to 16,000 to 73,500 in the AR = 2:1 second pass. The maximum rotation numbers are 0.39 and 0.16 in the first and second passes, respectively. The heat transfer enhancement on both the leading and trailing surfaces of the first pass of the 45° V-shaped channel is slightly reduced with rotation. In the second pass, the heat transfer increases on the leading surface while it decreases on the trailing surface. The 45° staggered, discrete V-shaped ribs provide increased heat transfer and thermal performance compared to the traditional V-shaped and standard, 45° angled rib turbulators.

Spectral Theory of Schrödinger Operators over Circle Diffeomorphisms

We initiate the study of Schrödinger operators with ergodic potentials defined over circle map dynamics, in particular over circle diffeomorphisms. For analytic circle diffeomorphisms and a set of rotation numbers satisfying Yoccoz’s ${{\mathcal{H}}}$ arithmetic condition, we discuss an extension of Avila’s global theory. We also give an abstract version and a short proof of a sharp Gordon-type theorem on the absence of eigenvalues for general potentials with repetitions. Coupled with the dynamical analysis, we obtain that, for every $C^{1+BV}$ circle diffeomorphism, with a super Liouville rotation number and an invariant measure $\mu $, and for $\mu $-almost all $x\in{{\mathbb{T}}}^1$, the corresponding Schrödinger operator has purely continuous spectrum for every Hölder continuous potential $V$.

Numerical Investigation on Heat Transfer and Oxidation Deposition of Aviation Fuel in a Rotatory U-Channel

Liquid-fuel regenerative cooling is a promising turbine cooling technology. We developed a numerical model of heat transfer coupled with oxidation deposition in a rotatory channel for regenerative cooling applications. Source terms for the centrifugal and Coriolis forces caused by rotation were added to the momentum equations and turbulent transport equations. A kinetic model for the thermal oxidation and deposition of supercritical hydrocarbon fuel was used to predict the oxidation deposition process. Coupled fluid–solid simulations of the heat transfer and oxidation deposition of hydrocarbon fuel in a U-shaped channel at five rotation numbers showed that the rotation improves convective heat transfer in the cooling channel and prevents the occurrence of a heat transfer deterioration zone. The average deposition rate in the channel decreased with increasing rotation number. In the centrifugal section of the rotatory channel, the Coriolis force caused the temperatures of the leading wall to be higher than those of the trailing wall, but the differences became smaller and nearly disappeared in the elbow and centripetal sections. The deposition rate on the leading wall was higher than that on the trailing wall in the straight centrifugal channel. In the bending section, the oxidation deposits were more prone to form on the inner edge than on the outer edge.