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NASA NTRS1994

The effects of gravity darkening on the ultraviolet continuum polarization produced by circumstellar disks

We investigate the effects of gravity darkening on the UV continuum polarization produced by an axisymmetric disk that surrounds a rapidly rotating star. Although the model is a single scattering approximation, we do include the effects of attenuation (electron scattering plus hydrogen bound-free absorption) by the disk, using an approach similar to that of Sobolev (1963). Because of the gravity darkening of the star and the attenuation within the disk, the radiation field is not axially symmetric about the radius vector. This implies that the polarization source functions are no longer provided by the finite disk depolarization factors of Cassinelli, Nordsieck, & Murison (1987), which are functions of the intensity moments in a spherically symmetric atmosphere. We reformulate the polarization source functions using generalized intensity moment tensors (J, H(sub i), K(sub ij)) that are valid for an arbitrary radiation field and envelope geometry. We find that the polarization source functions are simplest when using intensity moments in the observer's reference frame. On the other hand, the intensity moments are most easily evaluated in the stellar reference frame. Using the rotation transformation properties of the generalized intensity moments, we relate the observer's moments to those evaluated in the stellar reference frame. Our procedure for determining the polarization source functions thus merely involves choosing a set of Euler angles for the coordinate rotations, and then evaluating the associated rotation matrix. The geometrical complications of polarization transfer are thus reduced to obtaining a coordinate rotation matrix.

Bjorkman, J. E.↗
NASA NTRS1995

Critical radiation fluxes and luminosities of black holes and relativistic stars

The critial luminosity at which the outward force of radiation balances the inward force of gravity plays an important role in many astrophysical systems. We present expressions for the radiation force on particles with arbitrary cross sections and analyze the radiation field produced by radiating matter, such as a disk, ring, boundary layer, or stellar surface, that rotates slowly around a slowly rotating gravitating mass. We then use these results to investigate the critical radiation flux and, where possible, the critical luminosity of such a system in genral relativity. We demonstrate that if the radiation source is axisymmetric and emission is back-front symmetric with repect to the local direction of motion of the radiating matter, as seen in the comoving frame, then the radial component of the radiation flux and the diagonal components of the radiation stress-energy tensor outside the source are the same, to first order in the rotation rates, as they would be if the radiation source and gravitating mass were not rotating. We argue that the critical radiation flux for matter at rest in the locally nonrotating frame is often satisfactory as an astrophysical benchmark flux and show that if this benchmark is adopted, many of the complications potentially introduced by rotation of the radiation source and the gravitating mass are avoided. We show that if the radiation field in the absence of rotation would be spherically symmetric and the opacity is independent of frequency and direction, one can define a critical luminosity for the system that is independent of frequency and direction, one can define a critical luminosity for the system that is independent of the spectrum and angular size of the radiation source and is unaffected by rotation of the source and mass and orbital motion of the matter, to first order. Finally, we analyze the conditions under which the maximum possible luminosity of a star or black hole powered by steady spherically symmetric radial accretion is the same in general relativity as in the Newtonian limit.

Lamb, Frederick K.↗
NASA NTRS1995

A study of weak anisotropy in electron pressure in the tail current sheet

We adopt a magnetotail model with stretched field lines where ion motions are generally nonadiabatic and where it is assumed that the pressure anisotropy resides only in the electron pressure tensor. We show that the magnetic field lines with p(perpendicular) greater than p(parallel) are less stretched than the corresponding field lines in the isotropic model. For p(parallel) greater than p(perpendicular), the magnetic field lines become more and more stretched as the anisotropy approaches the marginal firehose limit, p(parallel) = p(perpendicular) + B(exp 2)/mu(sub 0). We also show that the tail current density is highly enhanced at the firehose limit, a situation that might be subject to a microscopic instability. However, we emphasize that the enhancement in the current density is notable only near the center of the tail current sheet (z = 0). Thus it remains unclear whether any microscopic instability can significantly alter the global magnetic field configuration of the tail. By comparing the radius of the field-line curvature at z = 0 with the particle's gyroradius, we suspect that even the conventional adiabatic description of electrons may become questionable very close to the marginal firehose limit.

Lee, D.-Y.↗
NASA NTRS1994

Consequences of magnetotail ion dynamics

The trajectories of a large ensemble of particles are calculated in a modified Tsyganenko magnetic field model with a uniform cross-tail electric field. The model magnetotail can be divided into several distinct dynamical regimes of ion motion. Near Earth, where the field lines are dipolar the adiabatic formalism is adequate. In the mid-tail and distant tail, guiding-center theory breaks down and must be replaced by a quasi-adiabatic formalism. There is an important transition region between the adiabatic and quasi-adiabatic regions where ion trajectories become more complicated and no simple analytical description holds. This wall region is characterized by rapid ion acceleration and a major loss of particles to the dusk flank. The moments of the ion distribution function are constructued from the ion trajectories, including density, temperature, and pressure in the x-z and x-y planes. In the noon-midnight meridian plane, parameters are relatively constant except near the Earth, while the x-y plots show strong gradients across the magnetotail. Magnetotail plasma convects earthward, drifts toward dusk, and is squeezed out of the tail in the near-Earth region. A thin current sheet forms in the quasi-adiabatic region, and the pressure tensor has significant off-diagonal terms at its edges. These terms are the result of quasi-adiabatic ion trajectories which lead to azimuthally asymmetric distribution functions capable of maintaining approximate stress balance across the current sheet. Simplified analytical descriptions provide further physical insight into ion dynamics that are observed.

Ashour-Abdalla, Maha↗
NASA NTRS1994

Experimental study of the flow field inside a whirling annular seal

The flow field inside a whirling annular seal has been measured using a 3-D Laser Doppler Anemometer (LDA) system. The seal investigated has a clearance of 1.27 mm, a length of 37.3 mm and is mounted on a drive shaft with a 50 percent eccentricity ratio. This results in the rotor whirling at the same speed as the shaft rotation (whirl ratio = 1.0). The seal is operated at a Reynolds number of 12000 and a Taylor number of 6300 (3600 rpm). The 3-D LDA system is equipped with a rotary encoding system which is used to produce phase averaged measurements of the entire mean velocity vector field and Reynolds stress tensor field from 0.13 mm upstream to 0.13 mm downstream of the seal. The mean velocity field reveals a highly three-dimensional flow field with large radial velocities near the inlet of the seal as well as a recirculation zone on the rotor surface. The location of maximum mean axial velocity migrates from the pressure side of the rotor at the inlet to the suction side at the exit. Turbulence production is a maximum near the seal inlet as indicated by the rapid increase of the turbulence kinetic energy (kappa). However, turbulence production and dissipation attain equilibrium fairly quickly with kappa remaining relatively constant over the last half of the seal.

Morrison, Gerald L.↗
NASA NTRS1995

A new criterion for bar-forming instability in rapidly rotating gaseous and stellar systems. 1: Axisymmetric form

We analyze previous results on the stability of uniformly and differentialy rotating, self-gravitating, gaseous and stellar, axisymmetric systems to derive a new stability criterion for the appearance of torodial, m = 2 intermediate or I-modes and bar modes. In the process, we demonstrate that the bar modes in stellar systems and the m = 2 I-modes in gaseous systems have many common physical characteristics and only one substantial difference: because of the anisotropy of the stress tensor, dynamical instability sets in at lower rotation in stellar systems. This difference is reflected also in the new stability criterion. The new stability parameter alpha equals (T(sub J))/(absolute value of W) is formulated first for uniformly rotating systems and is based on the angular momentum content rather than on the energy content of a system. (T(sub J) is defined as ((L)(Omega(sub J)))/2; L is the total angular momentum; Omega(sub J) is the Jeans frequency introduced by self-gravity; and W is the total gravitational potential energy.) For stability of stellar systems alpha less than or equal to 0.254-0.258 while alpha less than or equal to 0.341-0.354 for stability of gaseous systems. For uniform rotation, one can write alpha = ((ft)/2)(exp 1/2), where t is defined as T/(absolute value of W), T is the total kinetic energy due to rotation, and f is a function characteristic of the topology/connectedness and the geometric shape of a system. Equivalently, alpha equals t/(chi), where chi is defined as Omega/Omega(sub J) and Omega is the rotation frequency. Using these forms, alpha can be extended to and calculated for a variety of differentially rotating, gaseous and stellar, axisymmetric disk and spheroidal models whose equilibrium structures and stability characteristics are known. In this paper, we also estimate alpha for gaseous torodial models and for stellar disk systems embedded in an inert or responsive 'halo.' We find that the new stability criterion holds equally well for all these previously published axisymmetric models.

Christodoulou, Dimitris M.↗
NASA NTRS1995

Uncertainties for two-dimensional models of solar rotation from helioseismic eigenfrequency splitting

Observed solar p-mode frequency splittings can be used to estimate angular velocity as a function of position in the solar interior. Formal uncertainties of such estimates depend on the method of estimation (e.g., least-squares), the distribution of errors in the observations, and the parameterization imposed on the angular velocity. We obtain lower bounds on the uncertainties that do not depend on the method of estimation; the bounds depend on an assumed parameterization, but the fact that they are lower bounds for the 'true' uncertainty does not. Ninety-five percent confidence intervals for estimates of the angular velocity from 1986 Big Bear Solar Observatory (BBSO) data, based on a 3659 element tensor-product cubic-spline parameterization, are everywhere wider than 120 nHz, and exceed 60,000 nHz near the core. When compared with estimates of the solar rotation, these bounds reveal that useful inferences based on pointwise estimates of the angular velocity using 1986 BBSO splitting data are not feasible over most of the Sun's volume. The discouraging size of the uncertainties is due principally to the fact that helioseismic measurements are insensitive to changes in the angular velocity at individual points, so estimates of point values based on splittings are extremely uncertain. Functionals that measure distributed 'smooth' properties are, in general, better constrained than estimates of the rotation at a point. For example, the uncertainties in estimated differences of average rotation between adjacent blocks of about 0.001 solar volumes across the base of the convective zone are much smaller, and one of several estimated differences we compute appears significant at the 95% level.

Genovese, Christopher R.↗
NASA NTRS1995

An experimental study of a turbulent boundary layer in the trailing edge region of a circulation-control airfoil

This report describes the development of a new near-wall LDV technique and its use to measure the full turbulent stress tensor in a highly three-dimensional separated boundary layer flow. Laser Doppler velocimetry, with small diameter laser beams allowing small beam-crossing angles (i.e., larger fringe spacings), was used, resulting in measurements close to solid surfaces. Dennis Johnson of NASA/Ames combined the advantages offered by FFT signal processors with a beam-steering probe inserted in the flow to devise a new self-aligning, near-wall LDV approach capable of making direct measurements of the transverse velocity component. This new technique was applied to a 2D, zero-pressure-gradient, flat-plate boundary layer in the High Reynolds Number Pilot Channel at Ames. Accurate and repeatable measurements of U and uu to within 30 mm of the wall were obtained. Next, the near-wall LDV technique was employed in a 3D separated boundary layer flow, consisting of a 2-dimensional wall boundary layer approaching a right-angle wedge. The resulting lateral and adverse streamwise pressure gradients generated significant spanwise velocities and turbulent stresses, and a large reverse-flow region. The near-wall LDV was modified further to facilitate measurements in the 3D Wedge Flow.

Brown, Jeffrey↗
NASA NTRS1995

Relaxation approximation in the theory of shear turbulence

Leslie's perturbative treatment of the direct interaction approximation for shear turbulence (Modern Developments in the Theory of Turbulence, 1972) is applied to derive a time dependent model for the Reynolds stresses. The stresses are decomposed into tensor components which satisfy coupled linear relaxation equations; the present theory therefore differs from phenomenological Reynolds stress closures in which the time derivatives of the stresses are expressed in terms of the stresses themselves. The theory accounts naturally for the time dependence of the Reynolds normal stress ratios in simple shear flow. The distortion of wavenumber space by the mean shear plays a crucial role in this theory.

Rubinstein, Robert↗
NASA NTRS1994

Ion velocity distributions in the vicinity of the current sheet in Earth's distant magnetotail

Observations of the three-dimensional velocity distributions of positive ions and electrons have been recently gained for the first time in Earth's distant magnetotail with the Galileo and Geotail spacecraft. For this brief discussion of these exciting results the focus is on the overall character of the ion velocity distributions during substorm activity. The ion velocity distributions within and near the magnetotail current sheet are not accurately described as convecting, isotropic Maxwellians. The observed velocity distributions are characterized by at least two robust types. The first type is similar to the 'lima bean'-shaped velocity distributions that are expected from the nonadiabatic acceleration of ions which execute Speiser-type trajectories in the current sheet. The second distribution is associated with the presence of cold ion beams that presumably also arise from the acceleration of plasma mantle ions in the electric and weak magnetic fields in the current sheet. The ion velocity distributions in a magnetic field structure that is similar to that for plasmoids are also examined. Again the velocity distributions are not Maxwellian but are indicative of nonadiabatic acceleration. An example of the pressure tensor within the plasmoid-like event is also presented because it is anticipated that the off-diagonal elements are important in a description of magnetotail dynamics. Thus our concept of magnetotail dynamics must advance from the present assumption of co-moving electron and ion Maxwellian distributions into reformulations in terms of global kinematical models and nonadiabatic particle motion.

Frank, L. A.↗
NASA NTRS1995

Modeling and simulation of a turbulent far wake

Work continued on two projects which had been started during previous years. Both projects involve calculations of the subsonic, turbulent far wake of a two-dimensional object at a Reynolds number of 1000 (based on wake momentum thickness). This flow was used as a test case for direct comparison of various turbulence models and a direct numerical simulations (DNS) of this flow were undertaken. In the turbulence model comparison studies, for any particular model tested, a unique self-similar solution was obtained far enough downstream, regardless of inlet conditions. Furthermore, different turbulence models led to different far-wake equilibrium solutions. No turbulence model could correctly predict all features of the turbulent far wake. For example, the spreading rate and turbulent shear stresses were underpredicted by all the standard models (both two-equation and full Reynolds stress models). In cases where a more correct spreading rate was achieved, it was at the expense of the turbulent kinetic energy, which was overpredicted. In general, the Algebraic Dissipation Rate Model of Gatski and Speziale, 1992, when added to any of the standard models, improved the results dramatically. Also, full Reynolds stress closure models did a much better job at predicting the shapes of both the mean and turbulence profiles, but the spreading rate was not significantly improved over that predicted by the simpler two-equation models. There are two main conclusions from these studies: First, in a comparison such as this, it is not enough to compare just one parameter, like the spreading rate. A good prediction for one parameter does not necessarily imply good predictions for all parameters in a flow. Second, since no turbulence model could correctly predict the turbulent far wake, much of the important physics of turbulent free shear flows is apparently lost by the assumptions inherent in today's methods of turbulence modeling; turbulence models must be improved. Direct simulations of this flow were begun last year in order to provide a data base through which some of the deficiencies of the existing turbulence models could be identified. Quantities such as the pressure-strain correlation, turbulent diffusion, and the dissipation rate tensor can be easily calculated from the DNS results, whereas these quantities are nearly impossible to measure experimentally. Improvements to existing turbulence models (and development of new models) require knowledge about flow quantities such as these. During this summer, diagnostics codes were written which will calculate the parameters mentioned above, along with other single-point and multi-point statistics. The DNS calculations are still in progress at the time of this writing. When these calculations are complete, the diagnostics codes will be applied so that the results can aid turbulence modelers. In addition, the results will show whether or not there exists a universal equilibrium turbulent far wake, independent of initial conditions.

Cimbala, John M.↗
NASA NTRS1995

Large eddy simulations of compressible turbulent flows

An evaluation of existing models for Large Eddy Simulations (LES) of incompressible turbulent flows has been completed. LES is a computation in which the large, energy-carrying structures to momentum and energy transfer is computed exactly, and only the effect of the smallest scales of turbulence is modeled. That is, the large eddies are computed and the smaller eddies are modeled. The dynamics of the largest eddies are believed to account for most of sound generation and transport properties in a turbulent flow. LES analysis is based on an observation that pressure, velocity, temperature, and other variables are the sum of their large-scale and small-scale parts. For instance, u(i) (velocity) can be written as the sum of bar-u(i) and u(i)-prime, where bar-u(i) is the large-scale and u(i)-prime is the subgrid-scale (SGS). The governing equations for large eddies in compressible flows are obtained after filtering the continuity, momentum, and energy equations, and recasting in terms of Favre averages. The filtering operation maintains only large scales. The effects of the small-scales are present in the governing equations through the SGS stress tensor tau(ij) and SGS heat flux q(i). The mathematical formulation of the Favre-averaged equations of motion for LES is complete.

Porter-Locklear, Freda↗
NASA NTRS1995

Nearly sonic and transsonic convective motions in the solar atmosphere related to the solar wind origin

MHD equations are considered for the solar atmosphere. 15 different simplest MHD regimes are indicated for the momentum transport equation depending on the mutual binary interplay between 6 possible and locally dominant terms: non-stationarity and inhomogeneity of the flow, gas pressure, magnetic tensions, viscous and gravity forces. These regimes are delimited by five physically independent dimensionless parameters, for example, Strouhal, sonic Mach, alfvenic Mach, Reynolds and Froude numbers or their combinations. Another partially overlapping classification of the simplest regimes may be introduced based on the energy conservation equation. There are also 15 independent binary combinations between nonstationary and inhomogeneous convective terms, dissipative energy sinks and sources (viscous, heat-conductive, Joule and radiative ones) in the energy equation. More complicated regimes are considered with multiple dominated terms. All these MHD regimes play their important role somewhere in the solar atmosphere complicated by the tensor transport coefficients in the magnetically dominated regions of the upper atmosphere. Nearly sonic and transsonic nonstationary convective motions with ascending and descending flows are observed in the solar chromosphere. the transition region and the lower pans of the solar corona together with related horizontal velocity components. This convection represents a kind of the 'cocoonery' manufacturing nonstationary vortices generated here and partially connected to the photosphere and to the solar wind. The solar wind originates from this powerful transsonic muddle in the solar atmosphere as a tiny fraction of the streamlines which are temporarily getting detached from the 'cocoons; and going to the infinity. The topologically complicated instantaneous 'runaway surface' around the Sun, i.e., the surface which separates outgoing to the infinity streams from other finite flows in the solar atmosphere was not described in the literature and needs additional investigation. We conclude that a simple one-connected smooth and quasistationary 'critical surface' which is often supposed to be placed somewhere in the solar corona (say, at 4 solar radii) in reality does not exist. Observational and theoretical arguments favor instead of this a highly structured, disordered, patched and time-dependent sonic transition surface in the solar atmosphere permanently fluctuating at positional dependent heights starting sometimes from photospheric (and maybe subphotospheric) levels and more frequently from the chromosphenc network up to several solar radii in the solar corona.

Veselovsky, I. S.↗
NASA NTRS1995

An extended structure-based model based on a stochastic eddy-axis evolution equation

We have proposed and implemented an extension of the structure-based model for weak deformations. It was shown that the extended model will correctly reduce to the form of standard k-e models for the case of equilibrium under weak mean strain. The realizability of the extended model is guaranteed by the method of its construction. The predictions of the proposed model were very good for rotating homogeneous shear flows and for irrotational axisymmetric contraction, but were seriously deficient in the case of plane strain and axisymmetric expansion. We have concluded that the problem behind these difficulties lies in the algebraic constitutive equation relating the Reynolds stresses to the structure parameters rather than in the slow model developed here. In its present form, this equation assumes that under irrotational strain the principal axes of the Reynolds stresses remain locked onto those of the eddy-axis tensor. This is correct in the RDT limit, but inappropriate under weaker mean strains, when the non-linear eddy-eddy interactions tend to misalign the two sets of principal axes and create some non-zero theta and gamma.

Kassinos, S. C.↗
NASA NTRS1995

The radiated noise from isotropic turbulence and heated jets

Our understanding of aerodynamic noise has its foundations in the work of Sir James Lighthill (1952), which was the first major advance in acoustics since the pioneering work of Lord Rayleigh in the last century. The combination of Lighthill's theory of aerodynamic noise as applied to turbulent flows and the experimental growing database from the early 1950's was quickly exploited by various jet propulsion engine designers in reducing the noise of jet engines at takeoff and landing to levels marginally acceptable to communities living in the neighborhoods of airports. The success in this noise containment led to the rapid growth of fast economical subsonic civil transport aircraft worldwide throughout the 1960's and has continued to the present day. One important factor in this success story has been the improvements in the engine cycle that have led to both reductions in specific fuel consumption and noise. The second is the introduction of Noise Certification, which specifies the maximum noise levels at takeoff and landing that all aircraft must meet before they can be entered on the Civil Aircraft Register. The growing interest in the development of a new supersonic civil transport to replace 'Concorde' in the early years of the next century has led to a resurgence of interest in the more challenging problem of predicting the noise of hot supersonic jets and developing means of aircraft noise reduction at takeoff and landing to meet the standards now accepted for subsonic Noise Certification. The prediction of aircraft noise to the accuracy required to meet Noise Certification requirements has necessitated reliance upon experimental measurements and empirically derived laws based on the available experimental data bases. These laws have their foundation in the results from Lighthill's theory, but in the case of jet noise, where the noise is generated in the turbulent mixing region with the external ambient fluid, the complexity of the turbulent motion has prevented the full deployment of Lighthill's theory from being achieved. However, the growth of the supercomputer and its applications in the study of the structure of turbulent shear flows in both unbounded and wall bounded flows, which complements and in certain cases extends the work of the few dedicated experimental groups working in this field for the past forty years, provides an opportunity and challenge to accurately predict the noise from jets. Moreover a combination of numerical and laboratory experiments offers the hope that in the not too distant future the physics of noise generation and flow interaction will be better understood and it will then be possible to not only improve the accuracy of noise prediction but also to explore and optimize schemes for noise reduction. The present challenge is to provide time and space accurate numerical databases for heated subsonic and supersonic jets to provide information on the fourth-order space-time covariance of Lighthill's equivalent stress tensor, T(ij), which governs the characteristics of the farfield radiated noise and the total acoustic power. Validation with available experimental databases will establish how close Lighthill's theory is to the accurate prediction of the directivity and spectrum of jet noise and the total acoustic power, and the need, in the applications of the theory, to include the effects of flow-acoustic interaction.

Lilley, G. M.↗
NASA NTRS1996

Numerical solutions of the complete Navier-Strokes equations

This report describes the development of an enstrophy model capable of predicting turbulence separation and its application to two airfoils at various angles of attack and Mach numbers. In addition, a two equation kappa-xi model with a tensor eddy viscosity was developed. Plans call for this model to be used in calculating three dimensional turbulent flows.

Hassan, H. A.↗
NASA NTRS1996

Investigation of Thermal Stress Convection in Nonisothermal Gases Under Microgravity Conditions

Microgravity conditions offer an environment in which convection in a nonisothermal gas could be driven primarily by thermal stress. A direct examination of thermal stress flows would be invaluable in assessing the accuracy of the Burnett terms in the fluid stress tensor. We present a preliminary numerical investigation of the competing effects of thermal stress, thermal creep at the side walls, and buoyancy on gas convection in nonuniformly heated containers under normal and reduced gravity levels. Conditions in which thermal stress convection becomes dominant are identified, and issues regarding the experimental measurement of the flows are discussed.

Mackowski, Daniel W.↗
NASA NTRS1995

Stress Tuning of Laser Crystals

The topic of stress tunable laser crystals is addressed in this study with the purpose of determining the piezo-optic coefficients of a new laser material. This data was collected using a quadruple pass birefringence technique because of its high degree of sensitivity relative to the other methods examined including fringe shift analysis using a Mach-Zender interferometer. A green He-Ne laser was passed through a light chopper and Glan-Thompson prism before entering a crystal of Erbium doped Yttrium Aluminum Garnet (Er:YAG) (used in order to validate the experimental technique). The Er:YAG crystal is mounted in a press mechanism and the laser is quadruple passed through test specimen before being returned through the prism and the orthogonally polarized portion of the beam measured with a optical sensor. At a later stage, the Er:YAG crystal was replaced with a new crystal in order to determine the piezo-optic coefficients of this uncharacterized material. The applied load was monitored with the use of a 50 lb. load cell placed in line with the press. Light transmission readings were taken using a lock-in amplifier while load cell measurements were taken with a voltmeter from a 5 volt, 0.5 amp power supply. Despite the fact that an effective crystal press damping system was developed, size limitations precluded the use of the complete system. For this reason, data points were taken only once per full turn so as to minimize the effect of non uniform load application on the collected data. Good correlation was found in the transmission data between the experimentally determined Er:YAG and the previously known peizo-optic constants of non-doped crystal with which it was compared. The variation which was found between the two could be accounted for by the aforementioned presence of Erbium in the experimental sample (for which exact empirical data was not known). The same test procedure was then carried out on a Yttrium Gallium Aluminum garnet (YGAG) for the purpose of establishing values of its unknown piezo-optic constant tensor using experimentally collected transmission data. Significant variation between the piezo-optic constants of YAG and YGAG crystals was found however, the excellent data correlation of separate experimental runs carried out on the YGAG sample demonstrates the validity of these results. The data collected during the stressing of the YGAG was of high quality, however the amount of data collected was somewhat limited by a fracture of YGAG specimen which undoubted altered the crystalline lattice structure and hence precluded any further testing.

Carty, Atherton A.↗