Transonic flow theory of airfoils and wings

There are plans to use the supercritical wing on the next generation of commercial aircraft so as to economize on fuel consumption by reducing drag. Computer codes have served well in meeting the consequent demand for new wing sections. The possibility of replacing wind tunnel tests by computational fluid dynamics is discussed. Another approach to the supercritical wing is through shockless airfoils. A novel boundary value problem in the hodograph plane is studied that enables one to design a shockless airfoil so that its pressure distribution very nearly takes on data that are prescribed

Numerical design of shockless airfoils

An attempt is made to indicate and briefly discuss only the most significant achievements of the research. The most successful contribution from the contract was the code for two dimensional analysis of airfoils in transonic flow

The NYU inverse swept wing code

An inverse swept wing code is described that is based on the widely used transonic flow program FLO22. The new code incorporates a free boundary algorithm permitting the pressure distribution to be prescribed over a portion of the wing surface. A special routine is included to calculate the wave drag, which can be minimized in its dependence on the pressure distribution. An alternate formulation of the boundary condition at infinity was introduced to enhance the speed and accuracy of the code. A FORTRAN listing of the code and a listing of a sample run are presented. There is also a user's manual as well as glossaries of input and output parameters

Relic Radiation from an Evaporating Black Hole

We present a non-string-theoretic calculation of the microcanonical entropy of relic integer-spin Hawking radiation -- at fixed total energy $E$. The only conserved macroscopic quantity is the total energy $E$ (the total energy of the relic radiation). Data for a boundary-value approach, with massless, integer-spin perturbations, are set on initial and final space-like hypersurfaces. In the resulting 1-dimensional statistical-mechanics problem, the real part of the (complex) time separation at spatial infinity, $T = {\mid}T{\mid}\exp(-i\delta), \delta >0$, is the variable conjugate to the total energy. We count the number of weak-field configurations on the final space-like hypersurface with energy $E$. One recovers the Cardy formula and the Bekenstein-Hawking entropy, if Re(T) is of the order of the black-hole life- time, leading to a statistical interpretation of black-hole entropy. The microcanonical entropy includes a logarithmic correction to the black-hole area law, which is {\it universal} (independent of black-hole parameters). Here, the discreteness of the energy levels is crucial. This approach is compared with that of string theory for the transition to the fundamental-string r\'egime in the final stages of evaporation. The squared coupling, $g^2$, regulating the transition to a highly-excited string state and {\it vice versa}, can be related to the angle, $\delta$, of complex-time rotation above. The strong-coupling r\'egime corresponds to a Euclidean black hole, while the physical limit of a Lorentzian space-time (as $\delta \to 0_+$) corresponds to the weak-coupling r\'egime. This resembles the transition to a highly-excited string-like state which subsequently decays into massless particles, thereby avoiding the naked singularity.Comment: To appear in International Journal of Modern Physics

Coincidence of length spectra does not imply isospectrality

Penrose--Lifshits mushrooms are planar domains coming in nonisometric pairs with the same geodesic length spectrum. Recently S. Zelditch raised the question whether such billiards also have the same eigenvalue spectrum for the Dirichlet Laplacian (conjecturing ``no''). Here we show that generically (in the class of smooth domains) the two members of a mushroom pair have different spectra.Comment: 8 pages, 5 figure

Transonic airfoil codes

Computer codes for the design and analysis of transonic airfoils are considered. The design code relies on the method of complex characteristics in the hodograph plane to construct shockless airfoil. The analysis code uses artificial viscosity to calculate flows with weak shock waves at off-design conditions. Comparisons with experiments show that an excellent simulation of two dimensional wind tunnel tests is obtained. The codes have been widely adopted by the aircraft industry as a tool for the development of supercritical wing technology

Initial Data for General Relativity with Toroidal Conformal Symmetry

A new class of time-symmetric solutions to the initial value constraints of vacuum General Relativity is introduced. These data are globally regular, asymptotically flat (with possibly several asymptotic ends) and in general have no isometries, but a $U(1)\times U(1)$ group of conformal isometries. After decomposing the Lichnerowicz conformal factor in a double Fourier series on the group orbits, the solutions are given in terms of a countable family of uncoupled ODEs on the orbit space.Comment: REVTEX, 9 pages, ESI Preprint 12

Liouville field theory with heavy charges. II. The conformal boundary case

We develop a general technique for computing functional integrals with fixed area and boundary length constraints. The correct quantum dimensions for the vertex functions are recovered by properly regularizing the Green function. Explicit computation is given for the one point function providing the first one loop check of the bootstrap formula.Comment: LaTeX 26 page

Time dependence of Bragg forward scattering and self-seeding of hard x-ray free-electron lasers

Free-electron lasers (FELs) can now generate temporally short, high power x-ray pulses of unprecedented brightness, even though their longitudinal coherence is relatively poor. The longitudinal coherence can be potentially improved by employing narrow bandwidth x-ray crystal optics, in which case one must also understand how the crystal affects the field profile in time and space. We frame the dynamical theory of x-ray diffraction as a set of coupled waves in order to derive analytic expressions for the spatiotemporal response of Bragg scattering from temporally short incident pulses. We compute the profiles of both the reflected and forward scattered x-ray pulses, showing that the time delay of the wave $\tau$ is linked to its transverse spatial shift $\Delta x$ through the simple relationship $\Delta x = c\tau \cot\theta$, where $\theta$ is the grazing angle of incidence to the diffracting planes. Finally, we apply our findings to obtain an analytic description of Bragg forward scattering relevant to monochromatically seed hard x-ray FELs.Comment: 11 pages, 6 figure

Laplacian Growth, Elliptic Growth, and Singularities of the Schwarz Potential

The Schwarz function has played an elegant role in understanding and in generating new examples of exact solutions to the Laplacian growth (or "Hele- Shaw") problem in the plane. The guiding principle in this connection is the fact that "non-physical" singularities in the "oil domain" of the Schwarz function are stationary, and the "physical" singularities obey simple dynamics. We give an elementary proof that the same holds in any number of dimensions for the Schwarz potential, introduced by D. Khavinson and H. S. Shapiro [17] (1989). A generalization is also given for the so-called "elliptic growth" problem by defining a generalized Schwarz potential. New exact solutions are constructed, and we solve inverse problems of describing the driving singularities of a given flow. We demonstrate, by example, how \mathbb{C}^n - techniques can be used to locate the singularity set of the Schwarz potential. One of our methods is to prolong available local extension theorems by constructing "globalizing families". We make three conjectures in potential theory relating to our investigation