Experimental and Numerical Study of the Turbulent Flow and Heat Transfer in a Wedge-shaped Channel with Guiding Pin Fin Arrays under Rotating Conditions

Experiments and numerical simulations under stationary and rotating conditions have been conducted to investigate turbulent flow and heat transfer characteristics of innovative guiding pin fin arrays in a wedge-shaped channel, which models the internal cooling passages for gas turbine blade trailing edge. The Reynolds number range is 10,000-80,000, and the inlet rotation number range is 0-0.46. With the increase of Reynolds numbers, the enhancement of heat transfer performance with guiding pin fin arrays is significantly higher than that with conventional circular pin fin arrays. At the highest Reynolds number of Re=80,000, the overall Nusselt number of the channel with guiding pin fin arrays is about 33.7% higher than that of the channel with circular pin fin arrays under the stationary condition, and is about 23.0% higher than the latter under the rotating conditions. At the highest inlet rotation number of Ro=0.46, the heat transfer difference between the trailing side and leading side of the channel is significantly lower with the guiding pin fin arrays. Both the experiments and numerical simulations indicate that the heat transfer uniformity and enhancement of the channel endwall is significantly improved by the guiding pin fin arrays under stationary and rotating conditions, which provide more reasonable flow distribution in the wedge-shaped channel, and can further produce obviously improved heat transfer in the tip region for the trailing edge internal cooling channel.

Topological conjugacy of 𝑛-multiple Cartesian products of circle rough transformations

It is known from the 1939 work of A. G. Mayer that rough transformations of the circle are limited to the diffeomorphisms of Morse – Smale. A topological conjugacy class of orientation-preserving diffeomorphism is entirely determined by its rotation number and the number of its periodic orbits, while for orientation-changing diffeomorphism the topological invariant will be only the number of periodic orbits. Thus, the purpose of this study is to find topological invariants of n-fold Cartesian products of diffeomorphisms of a circle. Methods. This paper explores the rough Morse – Smale diffeomorphisms on the n-torus surface. To prove the main result, additional constructions and formation of subsets of considered sets were used. Results. In this paper, a numerical topological invariant is introduced for n-fold Cartesian products of rough circle transformations. Conclusion.The criterion of topological conjugacy of n-fold Cartesian products of rough transformations of a circle is formulated.

The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

In this paper we study global properties of the Wigner caustic of parameterized closed planar curves. We find new results on its geometry and singular points. In particular, we consider the Wigner caustic of rosettes, i.e. regular closed parameterized curves with non-vanishing curvature. We present a decomposition of a curve into parallel arcs to describe smooth branches of the Wigner caustic. By this construction we can find the number of smooth branches, the rotation number, the number of inflexion points and the parity of the number of cusp singularities of each branch. We also study the global properties of the Wigner caustic on shell (the branch of the Wigner caustic connecting two inflexion points of a curve). We apply our results to whorls—the important object to study the dynamics of a quantum particle in the optical lattice potential.

Quasisymmetric orbit-flexibility of multicritical circle maps

Two given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$ , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems 1.6 and 1.8).

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.

A Note on the Conjugacy Between Two Critical Circle Maps

We study a conjugacy between two critical circle homeomorphisms with irrational rotation number. Let fi, i = 1, 2 be a C3 circle homeomorphisms with critical point x(i) cr of the order 2mi + 1. We prove that if 2m1 + 1 ̸= 2m2 + 1, then conjugating between f1 and f2 is a singular function. Keywords: circle homeomorphism, critical point, conjugating map, rotation number, singular function

Heat Transfer in Rotating, Trailing-Edge, Converging Channels With Smooth Walls and Pin-Fins

The heat transfer and pressure drop characteristics of a rotating cooling channel that has an angled trapezoidal cross section and converges from the hub to the tip in both the streamwise and spanwise directions are experimentally investigated. The channel is oriented 120 deg with respect to the direction of rotation to model the geometry of an internal, trailing-edge cooling passage. Both the leading and trailing sides of the channel are divided into three and six regions in the spanwise and streamwise directions, respectively. The copper plate method is used to obtain regionally averaged heat transfer coefficients. The pressure drop is measured using pressure taps placed at the inlet and outlet of the channel. Experiments were conducted with the inlet Reynolds number ranging from 10,000 to 40,000. The rotational speed varies from 0 rpm to 300 rpm, resulting in the highest rotation number of 0.21. The effects of full pin-fins on the heat transfer and pressure drop characteristics are obtained and compared to the smooth surface converging channel results. The impact of the convergence, which causes variations of flow and geometric parameters through the passage, such as aspect ratio, Reynolds number, and rotation number, on the heat transfer coefficients and pressure drop are addressed. Results show that due to the 120 deg channel orientation, the rotation has a positive impact on the leading and trailing surface heat transfer. Furthermore, the convergence decreases the aspect ratio while increasing the Reynolds number. The convergence significantly enhances heat transfer on both the leading and trailing surfaces along the streamwise and spanwise directions. The convergence also reduces the rotation effect in the streamwise direction for a given mass flow rate.

Experimental Study of Outflow Effects in a Rotating Turbulated Channel

The three-pass turbulated serpentine channel has many applications in internal turbine blade systems. However, the studies on the effects of the outflow ratio are lacking, which decreases the thermal analysis accuracy in such a model. To fill this gap, outflow-ratio experiments are conducted on the Nusselt number distributions of a three-pass turbulated channel. The current experimental results can guide and optimize the turbine blade internal cooling system. The results show with the mass flow of the lateral outlet increasing, the low heat transfer region on the lateral-outflow-passage gradually expands. Increasing the mass flow of the lateral outlet heightens the spanwise-averaged-Nusselt-number of the lateral-outflow-passage, especially under the static condition. In the lateral-outflow-passage, the rotation significantly improves the Nusselt number uniformity, particularly at the high mass-flow-rate of the lateral holes; the rotation shows slight effects on the spanwise-averaged-Nusselt-number of the lateral-outflow-passage at low rotation-numbers, whereas, the profound influence is observed for the spanwise-averaged-Nusselt-number under high rotation-number condition. The rotation can profoundly increase the pressure coefficient leading to a reduced pressure loss with the rotation-number increasing from 0.03 to 0.06.