MEETINGS FOR NOVEMBER

 November 5, Friday 2:00pm, Room  BAL 209 
      Dr. F. Hu, Old Dominion University

    "Results of Some Benchmark Problems of Computational Aeroacustics"

    
Aeroacoustics is a branch of fluid dynamics that studies the generation
and propagation of sound. Computational Aeroacoustics (CAA) is a recently
emerged field within Computational Fluid Dynamics (CFD) in which the acoustic
waves are to be computed dircetly from the governing equations of the
compressible flows, namely, the Euler or Navier-Stokes equations.
The traditional CFD algorithms, which are mostly designed for steady state
calculations, are generally considered to be too dissipative and inefficient
for aeroacoutics problems as the latter require the numerical solutions
to be time accurate. NASA has been organizing workshops on Benchmark Problems
of Computational Aeroacoustics to assess the needs and progresses in
this field. The third such workshop, sponsored by NASA Glenn Research Center,
will be held in November at Cleveland, Ohio. In this talk, results of four of
the benchmark problems proposed for the workshop will be presented.
They include the radiation of an acoustic point source inside a
two-dimensional supersionic jet, sound field generated by a rotor inside
a semi-infinite duct, internal propagation of sound waves in a transonic
nozzle and one-dimensional shock-sound interaction in a supersonic nozzle.
All solutions have been obtained by a high order finite difference scheme
optimized for aeroacoutic problems. The methodology of applying the finite
difference method to a given phisical problem will be briefly discussed.
The emphasis will be on the treatment of numerical boundary conditions.
Physical interpretations of the computed results will also be offered.

  Friday, November 12, 1999,   2:00pm BAL 209 

     Dr. John A. Adam Professor
   "The Mathematical Physics of Rainbows"

A detailed qualitative summary of the optical rainbow is provided
at several complementary levels of description, including geometrical
optics (ray theory), the Airy approximation, Mie scattering theory, the
complex angular momentum (CAM) method, and catastrophe theory.  The phenomenon
known commonly as the glory is also discussed from both physical and
mathematical points of view: backward glories, rainbow-glories and forward
glories.  While both rainbows and glories result from scattering of the incident
radiation, the primary rainbow arises from scattering at about 138 degrees
from the forward direction, whereas the backward glory is associated with 
scattering very close to the backward direction.  In fact, it is a more complex
phenomenon physically than the rainbow, involving a variety of different effects
(including surface waves) associated with the scattering droplet. Both sets
of optical phenomena-rainbows and glories- have their counterparts in atomic,
molecular and nuclear scattering and these are addressed also.  The conceptual
foundations for understanding rainbows, glories and their associated features
range from classical geometrical optics, through quantum mechanics
(in particular scattering from a square well potential; the associated Regge
poles and scattering amplitude functions) to diffraction catastrophes. 
Both the scalar and the electromagnetic scattering problems are reviewed, 
the latter providing details about the polarization of the rainbow that the
scalar problem cannot address.  The basis for the complex angular
momentum (CAM)theory (used in both types of scatteringproblem) is a 
modification of the Watson transform, developed by Watson in  the early
part of this century in the study of radio wave diffraction around the
earth.  This modified Watson transform enables a valuable and accurate
approximation to be made to the Mie Partial-wave series, which while exact, 
converges very slowly at high frequencies.  The theory and many applications
of the CAM method were developed in a fundamental series of papers by
Nussenzveig and Khare, but many other contributions have been made to 
the understanding these beautiful phenomena, including descriptions in
terms of so-called diffraction catastrophes.  The rainbow is a fine example
of an observable event which may be described at many levels of mathematical
sophistication using distinct mathematical approaches, and in so doing 
the connections between several seemingly unrelated areas within physics
become evident. 

OLD DOMINION UNIVERSITY COLLEGE OF SCIENCES DEPARTMENT OF MATHEMATICS AND STATISTICS