Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number

1985 ◽  
Vol 16 (1) ◽  
pp. 45-83 ◽  
Author(s):  
Michael R. Herman
Keyword(s):  
Rotation Number ◽  
Local Conjugacy

Effects of Rotation on Heat Transfer for a Single Row Jet Impingement Array With Crossflow

2012 ◽  
Vol 134 (8) ◽  
Author(s):  
Justin A. Lamont ◽  
Srinath V. Ekkad ◽  
Mary Anne Alvin
Keyword(s):  
Heat Transfer ◽  
Liquid Crystal ◽  
Rotation Number ◽  
Coriolis Force ◽  
Room Temperature ◽  
Turbine Blades ◽  
Target Surface ◽  
Channel Height ◽  
Potential Core ◽  
Single Row

The effects of the Coriolis force are investigated in rotating internal serpentine coolant channels in turbine blades. For complex flow in rotating channels, detailed measurements of the heat transfer over the channel surface will greatly enhance the blade designers’ ability to predict hot spots so coolant may be distributed more effectively. The present study uses a novel transient liquid crystal technique to measure heat transfer in a rotating, radially outward channel with impingement jets. A simple case with a single row of constant pitch impinging jets with the crossflow effect is presented to demonstrate the novel liquid crystal technique and document the baseline effects for this type of geometry. The present study examines the differences in heat transfer distributions due to variations in jet Rotation number, Roj, and jet orifice-to-target surface distance (H/dj = 1,2, and 3). Colder air, below room temperature, is passed through a room temperature test section to cause a color change in the liquid crystals. This ensures that buoyancy is acting in a similar direction as in actual turbine blades where walls are hotter than the coolant fluid. Three parameters were controlled in the testing: jet coolant-to-wall temperature ratio, average jet Reynolds number, Rej, and average jet Rotation number, Roj. Results show, such as serpentine channels, the trailing side experiences an increase in heat transfer and the leading side experiences a decrease for all jet channel height-to-jet diameter ratios (H/dj). At a jet channel height-to-jet diameter ratio of 1, the crossflow from upstream spent jets greatly affects impingement heat transfer behavior in the channel. For H/dj = 2 and 3, the effects of the crossflow are not as prevalent as H/dj = 1: however, it still plays a detrimental role. The stationary case shows that heat transfer increases with higher H/dj values, so that H/dj = 3 has the highest results of the three examined. However, during rotation the H/dj = 2 case shows the highest heat transfer values for both the leading and trailing sides. The Coriolis force may have a considerable effect on the developing length of the potential core, affecting the resulting heat transfer on the target surface.

scholarly journals Computation of derivatives of the rotation number for parametric families of circle diffeomorphisms

2008 ◽  
Vol 237 (20) ◽  
pp. 2599-2615 ◽  
Author(s):  
Alejandro Luque ◽  
Jordi Villanueva
Keyword(s):  
Rotation Number ◽  
Parametric Families ◽  
Circle Diffeomorphisms ◽  
Derivatives Of

Influence of Radial Rotation on Heat Transfer in a Rectangular Channel With Two Opposite Walls Roughened by Hemispherical Protrusions at High Rotation Numbers

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
Shyy Woei Chang ◽  
Tong-Miin. Liou ◽  
Wei-Chun Chen
Keyword(s):  
Heat Transfer ◽  
Infrared Thermography ◽  
Rotation Number ◽  
Rectangular Channel ◽  
Flow Region ◽  
Transfer Data ◽  
Nusselt Numbers ◽  
Rotation Numbers ◽  
Heat Transfer Correlations ◽  
Rotating Channel

Detailed heat transfer distributions over two opposite leading and trailing walls roughened by hemispherical protrusions were measured from a rotating rectangular channel at rotation number up to 0.6 to examine the effects of Reynolds (Re), rotation (Ro), and buoyancy (Bu) numbers on local and area-averaged Nusselt numbers (Nu and Nu¯) using the infrared thermography. A set of selected heat transfer data illustrates the Coriolis and rotating buoyancy effects on the detailed Nu distributions and the area-averaged heat transfer performances of the rotating channel. The Nu¯ for the developed flow region on the leading and trailing walls are parametrically analyzed to devise the empirical heat transfer correlations that permit the evaluation of the interdependent and individual Re, Ro, and Bu effect on Nu¯.

Heat Transfer in a Radially Rotating Square-Sectioned Duct With Two Opposite Walls Roughened by 45Deg Staggered Ribs at High Rotation Numbers

2006 ◽  
Vol 129 (2) ◽  
pp. 188-199 ◽  
Author(s):  
Shyy Woei Chang ◽  
Tong-Minn Liou ◽  
Jui-Hung Hung ◽  
Wen-Hsien Yeh
Keyword(s):  
Heat Transfer ◽  
Rotation Number ◽  
Coriolis Force ◽  
Density Ratio ◽  
Inertial Force ◽  
Local Heat Transfer ◽  
Local Heat ◽  
Secondary Flows ◽  
Test Channel ◽  
Rib Roughened

This paper describes an experimental study of heat transfer in a radially rotating square duct with two opposite walls roughened by 45deg staggered ribs. Air coolant flows radially outward in the test channel with experiments to be undertaken that match the actual engine conditions. Laboratory-scale heat transfer measurements along centerlines of two rib-roughened surfaces are performed with Reynolds number (Re), rotation number (Ro), and density ratio (Δρ∕ρ) in the ranges of 7500–15,000, 0–1.8, and 0.076–0.294. The experimental rig permits the heat transfer study with the rotation number considerably higher than those studied in other researches to date. The rotational influences on cooling performance of the rib-roughened channel due to Coriolis forces and rotating buoyancy are studied. A selection of experimental data illustrates the individual and interactive impacts of Re, Ro, and buoyancy number on local heat transfer. A number of experimental-based observations reveal that the Coriolis force and rotating buoyancy interact to modify heat transfer even if the rib induced secondary flows persist in the rotating channel. Local heat transfer ratios between rotating and static channels along the centerlines of stable and unstable rib-roughened surfaces with Ro varying from 0.1 to 1.8 are in the ranges of 0.6–1.6 and 1–2.2, respectively. Empirical correlations for periodic flow regions are developed to permit the evaluation of interactive and individual effects of ribflows, convective inertial force, Coriolis force, and rotating buoyancy on heat transfer.

Local Heat Transfer in Rotating Smooth and Ribbed Two-Pass Square Channels With Three Channel Orientations

1996 ◽  
Vol 118 (3) ◽  
pp. 578-584 ◽  
Author(s):  
S. Dutta ◽  
J.-C. Han
Keyword(s):  
Heat Transfer ◽  
Rotation Number ◽  
Local Heat Transfer ◽  
Local Heat ◽  
Transfer Coefficients ◽  
Channel Changes ◽  
Square Channel ◽  
Heat Transfer Coefficients ◽  
Experimental Heat ◽  
Channel Orientation

This paper presents experimental heat transfer results in a two-pass square channel with smooth and ribbed surfaces. The ribs are placed in a staggered half-V fashion with the rotation orthogonal to the channel axis. The channel orientation varies with respect to the rotation plane. A change in the channel orientation about the rotating frame causes a change in the secondary flow structure and associated flow and turbulence distribution. Consequently, the heat transfer coefficient from the individual surfaces of the two-pass square channel changes. The effects of rotation number on local Nusselt number ratio distributions are presented. Heat transfer coefficients with ribbed surfaces show different characteristics in rotation number dependency from those with smooth surfaces. Results show that staggered half-V ribs mostly have higher heat transfer coefficients than those with 90 and 60 deg continuous ribs.

scholarly journals The Hess–Appelrot case and quantization of the rotation number

2017 ◽  
Vol 13 (3) ◽  
pp. 433-452
Author(s):  
И.А. Бизяев ◽  
◽  
А.В. Борисов ◽  
И.С. Мамаев ◽  
◽  
...  
Keyword(s):  
Rotation Number

scholarly journals Twist Periodic Orbits for Continuous Maps of the Eight Space

2013 ◽  
Vol 6 ◽  
pp. 4-8
Author(s):  
Iftichar Mudhar Talb Al-Shraa
Keyword(s):  
Fixed Point ◽  
Periodic Orbit ◽  
Periodic Orbits ◽  
Rotation Number ◽  
Continuous Map ◽  
Continuous Maps

Let g be a continuous map from 8 to itself has a fixed point at (0,0), we prove that g has a twist periodic orbit if there is a rational rotation number.

scholarly journals Experimental Heat Transfer Investigation of Stationary and Orthogonally Rotating Asymmetric and Symmetric Heated Smooth and Turbulated Channels

1992 ◽  
Author(s):  
H. A. El-Husayni ◽  
M. E. Taslim ◽  
D. M. Kercher
Keyword(s):  
Heat Transfer ◽  
Heat Flux ◽  
Rotation Number ◽  
Symmetric Case ◽  
Wall Heat Flux ◽  
Heated Wall ◽  
Wall Heat Transfer ◽  
Transfer Coefficients ◽  
Heat Transfer Coefficients ◽  
Smooth Channel

An experimental investigation was conducted to determine the effects of variations in wall thermal boundary conditions on local heat transfer coefficients in stationary and orthogonally rotating smooth wall and two opposite-wall turbulated square channels. Results were obtained for three distributions of uniform wall heat flux: asymmetric, applied to the primary wall only; symmetric, applied to two opposite walls only; and fully-symmetric, applied to all four channel walls. Measured stationary and rotating smooth channel average heat transfer coefficients at channel location L/Dh = 9.53 were not significantly sensitive to wall heat flux distributions. Trailing side heat transfer generally increased with Rotation number whereas the leading wall results showed a decreasing trend at low Rotation numbers to a minimum and then an increasing trend with further increase in Rotation number. The stationary turbulated wall heat transfer coefficients did not vary markedly with the variations in wall heat flux distributions. Rotating leading wall heat transfer decreased with Rotation number and showed little sensitivity to heat flux distributions except for the fully-symmetric heated wall case at the highest Reynolds number tested. Trailing wall heat transfer coefficients were sensitive to the thermal wall distributions generally at all Reynolds numbers tested and particularly with increasing Rotation number. While the asymmetric case showed a slight deficit in trailing wall heat transfer coefficients due to rotation, the symmetric case indicated little change whereas the fully-symmetric case exhibited an enhancement.

Effect of Rib Spacing on Heat Transfer in a Two-Pass Rectangular Channel (AR=1:4) With a Sharp Entrance at High Rotation Numbers

2008 ◽  
Author(s):  
Michael Huh ◽  
Yao-Hsien Liu ◽  
Je-Chin Han ◽  
Sanjay Chopra
Keyword(s):  
Heat Transfer ◽  
Aspect Ratio ◽  
Heat Transfer Enhancement ◽  
Rotation Number ◽  
Rectangular Channel ◽  
Transfer Enhancement ◽  
Sharp Bend ◽  
Buoyancy Parameter ◽  
First Pass ◽  
Rib Turbulators

The focus of the current study was to determine the effects of rib spacing on heat transfer in rotating 1:4 AR channels. In the current study, heat transfer experiments were performed in a two-pass, 1:4 aspect ratio channel, with a sharp bend entrance. The channel leading and trailing walls in the first pass and second pass utilized angled rib turbulators (45° to the mainstream flow). The rib height-to-hydraulic diameter ratio (e/Dh) was held constant at 0.078. The channel was oriented 90° to the direction of rotation. Three rib pitch-to-rib height ratios (P/e) were studied: P/e = 2.5, 5, and 10. Each ratio was tested at five Reynolds numbers: 10K, 15K, 20K, 30K and 40K. For each Reynolds number, experiments were conducted at five rotational speeds: 0, 100, 200, 300, and 400 rpm. Results showed that the sharp bend entrance has a significant effect on the first pass heat transfer enhancement. In the second pass, the rib spacing and rotation effect are reduced. The P/e = 10 case had the highest heat transfer enhancement based on total area, whereas the P/e = 2.5 had the highest heat transfer enhancement based on the projected area. The current study has extended the range of the rotation number (Ro) and local buoyancy parameter (Box) for a ribbed 1:4 aspect ratio channel up to 0.65 and 1.5, respectively. Correlations for predicting heat transfer enhancement, due to rotation, in the ribbed (P/e = 2.5, 5, and 10) 1:4 aspect ratio channel, based on the extended range of the rotation number and buoyancy parameter, are presented in the paper.

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