Knot theory and matrix integrals

The large size limit of matrix integrals with quartic potential may be used to count alternating links and tangles. The removal of redundancies amounts to renormalizations of the potential. This extends into two directions: higher genus and the counting of "virtual" links and tangles; and the counting of "coloured" alternating links and tangles. We discuss the asymptotic behavior of the number of tangles as the number of crossings goes to infinity.Comment: chapter of the book Random Matrix Theory, Eds Akemann, Baik and Di Francesc

Computer program to determine the irrotational nozzle admittance

Irrotational nozzle admittance is the boundary condition that must be satisfied by combustor flow oscillations at nozzle entrance. Defined as the ratio of axial velocity perturbation to the pressure perturbation at nozzle entrance, nozzle admittance can also be used to determine whether wave motion in nozzle under consideration adds or removes energy from combustor oscillations

Spectral analysis of the free orthogonal matrix

We compute the spectral measure of the standard generators $u_{ij}$ of the Wang algebra $A_o(n)$. We show in particular that this measure has support $[-2/\sqrt{n+2},2/\sqrt{n+2}]$, and that it has no atoms. The computation is done by using various techniques, involving the general Wang algebra $A_o(F)$, a representation of $SU^q_2$ due to Woronowicz, and several calculations with orthogonal polynomials.Comment: 22 pages, 4 figure

Measurements of admittances and characteristic combustion times of reactive gaseous propellant coaxial injectors

The results of an experimental investigation that was concerned with the quantitative determination of the capabilities of combustion processes associated with coaxial injectors to amplify and sustain combustor oscillations was described. The driving provided by the combustion process was determined by employing the modified standing-wave method utilizing coaxial injectors and air-acetylene mixtures. Analyses of the measured data indicate that the investigated injectors are capable of initiating and amplifying combustion instabilities under favorable conditions of injector-combustion coupling and over certain frequency ranges. These frequency ranges and the frequency at which an injector's driving capacity is maximum are observed to depend upon the equivalence ratio, the pressure drop across the injector orifices and the number of injector elements. The characteristic combustion times of coaxial injectors were determined from steady state temperature measurements

Characteristics of response factors of coaxial gaseous rocket injectors

The results of an experimental investigation undertaken to determine the frequency dependence of the response factors of various gaseous propellant rocket injectors subject to axial instabilities are presented. The injector response factors were determined, using the modified impedance-tube technique, under cold-flow conditions simulating those observed in unstable rocket motors. The tested injectors included a gaseous-fuel injector element, a gaseous-oxidizer injector element and a coaxial injector with both fuel and oxidizer elements. Emphasis was given to the determination of the dependence of the injector response factor upon the open-area ratio of the injector, the length of the injector orifice, and the pressure drop across the injector orifices. The measured data are shown to be in reasonable agreement with the corresponding injector response factor data predicted by the Feiler and Heidmann model

The prediction of the nonlinear behavior of unstable liquid rockets

Analytical technique for solving nonlinear combustion problems associated with liquid propellant rocket engine

Development of an analytical technique for the optimization of jet engine and duct acoustic liners

A special integral representation of the external solutions of the Helmholtz equation is described. The analytical technique developed for the generation of the optimum acoustic admittance for an arbitrary axisymmetric body is also presented along with some numerical procedures and some preliminary results for a straight duct

The prediction of nonlinear three dimensional combustion instability in liquid rockets with conventional nozzles

An analytical technique is developed to solve nonlinear three-dimensional, transverse and axial combustion instability problems associated with liquid-propellant rocket motors. The Method of Weighted Residuals is used to determine the nonlinear stability characteristics of a cylindrical combustor with uniform injection of propellants at one end and a conventional DeLaval nozzle at the other end. Crocco's pressure sensitive time-lag model is used to describe the unsteady combustion process. The developed model predicts the transient behavior and nonlinear wave shapes as well as limit-cycle amplitudes and frequencies typical of unstable motor operation. The limit-cycle amplitude increases with increasing sensitivity of the combustion process to pressure oscillations. For transverse instabilities, calculated pressure waveforms exhibit sharp peaks and shallow minima, and the frequency of oscillation is within a few percent of the pure acoustic mode frequency. For axial instabilities, the theory predicts a steep-fronted wave moving back and forth along the combustor