Calculation of some determinants using the s-shifted factorial

Several determinants with gamma functions as elements are evaluated. This kind of determinants are encountered in the computation of the probability density of the determinant of random matrices. The s-shifted factorial is defined as a generalization for non-negative integers of the power function, the rising factorial (or Pochammer's symbol) and the falling factorial. It is a special case of polynomial sequence of the binomial type studied in combinatorics theory. In terms of the gamma function, an extension is defined for negative integers and even complex values. Properties, mainly composition laws and binomial formulae, are given. They are used to evaluate families of generalized Vandermonde determinants with s-shifted factorials as elements, instead of power functions.Comment: 25 pages; added section 5 for some examples of application

Asymptotic behavior of a flat plate wake

An experimental study has been conducted to investigate the far-field, self-similar properties of a flat plate wake. A plane turbulent wake was generated at the trailing edge of a smooth splitter plate separating two legs of a Mixing Layer Wind Tunnel, with both initial boundary layers tripped. For the present study, both legs were operated at a free-steam velocity in the test section of 15 m/s, giving a Reynolds number based on wake momentum thickness of about 1750. Single profile measurements were obtained at five streamwise locations using a Pitot probe for the mean velocity measurements and a single cross-wire probe for the turbulence data, which included statistics up to third order. The mean flow data indicated a self-similar behavior beyond a streamwise distance equivalent to about 350 wake momentum thicknesses. However, the turbulence data show better collapse beyond a distance equivalent to about 500 momentum thicknesses, with all the measured peak Reynolds stresses achieving constant, asymptotic levels. The asymptotic mean flow behavior and peak primary stress levels agree well with theoretical predictions based on a constant eddy viscosity model. The present data also agree reasonably well with previous measurements, of which only one set extends into the self-similar region. Detailed comparisons with previous data are presented and discussed in this report

Contraction design for small low-speed wind tunnels

An iterative design procedure was developed for 2- or 3-dimensional contractions installed on small, low speed wind tunnels. The procedure consists of first computing the potential flow field and hence the pressure distributions along the walls of a contraction of given size and shape using a 3-dimensional numerical panel method. The pressure or velocity distributions are then fed into 2-dimensional boundary layer codes to predict the behavior of the boundary layers along the walls. For small, low speed contractions, it is shown that the assumption of a laminar boundary layer originating from stagnation conditions at the contraction entry and remaining laminar throughout passage through the successful designs is justified. This hypothesis was confirmed by comparing the predicted boundary layer data at the contraction exit with measured data in existing wind tunnels. The measured boundary layer momentum thicknesses at the exit of four existing contractions, two of which were 3-D, were found to lie within 10 percent of the predicted values, with the predicted values generally lower. From the contraction wall shapes investigated, the one based on a 5th order polynomial was selected for newly designed mixing wind tunnel installation

Design and calibration of the mixing layer and wind tunnel

A detailed account of the design, assembly and calibration of a wind tunnel specifically designed for free-shear layer research is contained. The construction of this new facility was motivated by a strong interest in the study of plane mixing layers with varying initial and operating conditions. The Mixing Layer Wind tunnel is located in the Fluid Mechanics Laboratory at NASA Ames Research Center. The tunnel consists of two separate legs which are driven independently by centrifugal blowers connected to variable speed motors. The blower/motor combinations are sized such that one is smaller than the other, giving maximum flow speeds of about 20 and 40 m/s, respectively. The blower speeds can either be set manually or via the Microvax II computer. The two streams are allowed to merge in the test section at the sharp trailing edge of a slowly tapering splitter plate. The test section is 36 cm in the cross-stream direction, 91 cm in the spanwise direction and 366 cm in length. One test section side-wall is slotted for probe access and adjustable so that the streamwise pressure gradient may be controlled. The wind tunnel is also equipped with a computer controlled, three-dimensional traversing system which is used to investigate the flow fields with pressure and hot-wire instrumentation. The wind tunnel calibration results show that the mean flow in the test section is uniform to within plus or minus 0.25 pct and the flow angularity is less than 0.25 deg. The total streamwise free-stream turbulence intensity level is approximately 0.15 pct. Currently the wind tunnel is being used in experiments designed to study the three-dimensional structure of plane mixing layers and wakes

Detailed studies of aviation fuel flowability

Six Jet A fuels, with varying compositions, were tested for low temperature flowability in a 190-liter simulator tank that modeled a section of a wing tank of a wide-body commercial airplane. The insulated tank was chilled by circulating coolant through the upper and lower surfaces. Flow-ability was determined as a function of fuel temperature by holdup, the fraction of unflowable fuel remaining in the tank after otherwise complete withdrawal. In static tests with subfreezing tank conditions, hold up varied with temperature and fuel composition. However, a general correlation of two or three classes of fuel type was obtained by plotting holdup as a function of the difference between freezing point and boundary-layer temperature, measured 0.6 cm above the bottom tank surface. Dynamic conditions of vibrations and slosh or rate of fuel withdrawal had very minor effects on holdup. Tests with cooling schedules to represent extreme, cold-day flights showed, at most, slight holdup for any combination of fuel type or dynamic conditions. Tests that superimposed external fuel heating and recirculation during the cooldown period indicates reduced hold up by modification of the low-temperature boundary layer. Fuel heating was just as effective when initiated during the later times of the tests as when applied continuously

Green's Functions and the Adiabatic Hyperspherical Method

We address the few-body problem using the adiabatic hyperspherical representation. A general form for the hyperangular Green's function in $d$-dimensions is derived. The resulting Lippmann-Schwinger equation is solved for the case of three-particles with s-wave zero-range interactions. Identical particle symmetry is incorporated in a general and intuitive way. Complete semi-analytic expressions for the nonadiabatic channel couplings are derived. Finally, a model to describe the atom-loss due to three-body recombination for a three-component fermi-gas of $^{6}$Li atoms is presented.Comment: 14 pages, 8 figures, 2 table

Three-Body Recombination in One Dimension

We study the three-body problem in one dimension for both zero and finite range interactions using the adiabatic hyperspherical approach. Particular emphasis is placed on the threshold laws for recombination, which are derived for all combinations of the parity and exchange symmetries. For bosons, we provide a numerical demonstration of several universal features that appear in the three-body system, and discuss how certain universal features in three dimensions are different in one dimension. We show that the probability for inelastic processes vanishes as the range of the pair-wise interaction is taken to zero and demonstrate numerically that the recombination threshold law manifests itself for large scattering length.Comment: 15 pages 7 figures Submitted to Physical Review