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A Transformation Theory of the Partial Differential Equations of Gas Dynamics

Description: Note presenting a transformation theory of systems of partial differential equations which allows the construction of classes of pressure-density relations depending on parameters for which the equations governing the flow can be transformed into an essentially simpler form, into the system corresponding to the wave equation in the subsonic region, and finally into the form corresponding to the Tricomi equation in the transonic region.
Date: April 1950
Creator: Loewner, Charles
Partner: UNT Libraries Government Documents Department

Error analysis of the TAI equation

Description: There is a certain tolerance of error implicit in any calculation based upon measured physical data. Occasionally, the error is explicitly recognized by attaching a plus or minus variation to the calculated quantity. More often, however, the error is simply neglected. The present analysis was performed to quantify the error implicit in a typical TAI (Trip-After-Instability) limit calculation. This error results from the precision of measurement of each of the parameters used in the calculation of the TAI limit. Though not specifically considered, the results of this work could be used to establish the required minimum level of accuracy of the tube outlet temperature monitoring equipment. Further, some of the generalized equations could probably be programmed into the ROL (Reactor Operating Limits) to indicate how far each Panellit gauge could be ``rolled`` without exceeding a fixed minimum TAI limit. Since the examination of these other possibilities is beyond the scope of this report, they are left as suggestions for future work only.
Date: October 23, 1961
Creator: Cremer, B. R.
Partner: UNT Libraries Government Documents Department

Numerical modeling of liquid geothermal systems

Description: A mathematical model describing the physical behavior of hot-water geothermal systems is presented. The model consists of a set of coupled partial differential equations for heat and mass transfer in porous media and an equation of state relating fluid density to temperature and pressure. The equations are solved numerically using an integrated finite difference method which can treat arbitrary nodal configurations in one, two, or three dimensions. The model is used to analyze cellular convection in permeable rock layers heated from below. Results for cases with constant fluid and rock properties are in good agreement with numerical and experimental results from other authors.
Date: January 1, 1978
Creator: Sorey, M.L.
Partner: UNT Libraries Government Documents Department

Beam tilt due to e/sup +/e/sup /minus// crossing at an angle

Description: The problem of beam tilt caused by beam-beam crossing at an angle has been treated by B. Richter and D. Ritson. An electron at a longitudinal position z relative to the center of the bunch experiences on the average a transverse kick proportional to z from the on-coming bunch. This kick produces a closed orbit for the electron being considered. The closed orbit is different for electrons with different values of z. Consequently, the beam will become tilted relative to its direction of motion. On the other hand, there is a longitudinal kick proportional to z of the particle due to both the rf focusing and the beams crossing at an angle. This longitudinal kick will cause the particle's energy to change and the energy change will, in turn, change the particle's z in the next revolution. It is, therefore, clear that a complete treatment of this problem should take the synchrotron oscillation, as well as the transverse betatron oscillation, into account. We will assume in a later calculation that the longitudinal defocusing effect due to beam-beam crossing at an angle is a dominated by the focusing effect provided by the rf cavities. The present note is to recalculate this beam tilt with the synchrotron dimension included. The result shows the structure of a synchrobetatron resonance. The difference between this calculation and that of previous authors is expected to be small away from the synchrobetatron coupling resonances since the synchrotron oscillation is indeed slow compared with the revolution frequency. Numerical estimate shows that this beam tilt is negligible away from synchrobetatron resonances. 4 refs.
Date: May 1, 1978
Creator: Chao, A.W. & Morton, P.L.
Partner: UNT Libraries Government Documents Department

Modeling the effects of a flat wiggler on a storage ring beam

Description: The purpose of the present note is to show how the various effects of the wiggler may be modeled in a simple way suitable for use in machine control. It will be seen that in general a total of about 17 functions are involved. However, in typical designs many of these functions vanish identically because of symmetries, and others are neglibly small. Furthermore, each of the functions may be modeled quite accurately by a single power law in (B/sub o//E)/sup n/ where B is a measure of the field excitation. E is the beam energy, and n is an integer which takes on values of either 0, 2, 3, 4, for 5 for the different functions. Magnet saturation may cause the field distribution to vary with excitation so that the series coefficients would vary slowly with B/sub o/. A computer program has been used to obtain numerical results for typical wiggler designs. In practice, the required functions could be determined either by computer analysis of the measured field data, or by experimental calibration using the stored beam in the ring. 9 refs., 3 figs., 11 tabs.
Date: June 1, 1978
Creator: Helm, R.H.
Partner: UNT Libraries Government Documents Department

Studies of Flow Problems With the Simulator shaft78

Description: In recent years, a number of numerical simulators for geothermal reservoirs have been developed. The general purpose of these is to aid reservoir engineers in (i) determining characteristic parameters of reservoirs (most important among those being the reserves of fluid and heat), and (ii) simulating the performance of reservoirs upon production and injection. The various simulators differ in the approximations made in the underlying physical model (e.g., dependence of rock and fluid properties upon thermodynamic variables), in the geometrical definition of the reservoir (one-, two-, or three-dimensional, regular or irregular shape); in the choice of thermodynamic variables, and in the mathematical techniques used for solving the coupled mass and energy transport equations. Criteria for desirable performance of numerical simulators depend in part upon the particular problems to be investigated. Different problems will often differ in the required level of detail to be resolved, and in the optimum balance of speed and accuracy of computation. Much can be learned about two-phase flow in porous media from model studies for idealized systems. Such studies can be performed with less-than-three-dimensional model and algorithms which are based on regular grid spacings will be perfectly acceptable. For modeling natural geothermal reservoirs, on the other hand, it is important that irregular three-dimensional geometries may be handled easily. In comparison with other two-phase simulators which have been discussed in the literature, the main distinctive feature of SHAFT78 is that it uses an integrated finite difference method (IFD). We solve finite difference equations that are obtained by integrating the basic partial differential equations for mass and energy flow over discrete surface and volume elements. This method is as easily applicable to irregular geometries of actual reservoirs as it is to idealized, regular geometries; yet the relative simplicity of the finite difference method is retained in the theory and ...
Date: November 1, 1978
Creator: Pruess, K.; Schroeder, R. C. & Zerzan, J.
Partner: UNT Libraries Government Documents Department

Heavy quarks and new particles

Description: Some aspects of particles containing c, b, and heavier quarks are reviewed. It is shown how these particles can provide tests of the short-range nature of the strong interactions, and how elementary quantum mechanics can reveal the color and charge of the constituents and the number of states below flavor threshold. Some new results are presented on the inverse problem for confining potentials. Properties of flavored hadrons (masses and lifetimes), of gluonic bound states, and of particles containing very heavy objects are also discussed.
Date: January 1, 1979
Creator: Rosner, J.L.
Partner: UNT Libraries Government Documents Department

Frequencies of small oscillations in the vicinity of a fixed point: Application to sextupole insertions

Description: The stability criterion of A.C. Bountis for the periodic orbits of a dynamical mapping is applied to the horizontal motion of a particle in a storage ring with sextupole magnets. The approach developed takes into account any number of non-linear ''kicks'' in one superperiod of the ring, non-equal spacing between nonlinear elements and a strong focusing magnet system. The stability criterion for a fixed point is given provided such a fixed point is found. The formula for the tune of the small oscillations for any stable fixed point is derived. 2 refs.
Date: January 10, 1979
Creator: Kheifets, S.
Partner: UNT Libraries Government Documents Department

Weak-strong instability as a diffusion process

Description: The phenomenological analysis of the weak-strong instability for an electron storage ring is developed. The vertical size of the weak beam is found to depend on two machine parameters: {radical}{eta}, which is proportional to {Delta}Q, and b, which depends on the aspect ratio of the strong beam. The model also contains one fitting parameter. Experimental consequences of such dependence are discussed. 5 refs., 7 figs.
Date: February 1, 1979
Creator: Kheifets, S.A.
Partner: UNT Libraries Government Documents Department

Single feedback systems for simultaneous damping of horizontal and longitudinal coherent oscillations

Description: To describe the horizontal motion of the bunch, we need four coordinates, x and z are the horizontal and longitudinal displacements of the bunch center relative to the ideal trajectory; x' is the angle between the bunch's direction of motion and the ideal trajectory; and delta=..delta..E/E is relative energy error of the bunch. Among the four variables, x and z are easy to measure by position monitors, while x' and delta are easy to change by electromagnetic devices. In combination, this suggests four possible types of feedback systems. In the following, we will present a complete analysis of the Type (x, delta) feedback system, using a matrix method. The analyses of other types are similar to that of Type (x, delta) and only the results are included. We then include some comparisons of these types of feedback schemes in terms of power consumptions and the effectiveness in damping the horizontal-betatron and synchrotron oscillations. We will also discuss some effects of position measuring errors on the performance of the feedback system. 2 refs., 3 tabs.
Date: March 1, 1979
Creator: Chao, A.W.; Morton, P.L. & Rees, J.R.
Partner: UNT Libraries Government Documents Department

Polarized beams in high energy circular accelerators

Description: In recent years, high energy physicists have become increasingly interested in the possible spin effects at high energies. To study those spin effects, it is desirable to have beams with high energy, high intensity and high polarization. In this talk, we briefly review the present status and the prospects for the near future of high energy polarized beams. 30 refs.
Date: May 1, 1979
Creator: Chao, A.W.
Partner: UNT Libraries Government Documents Department

Stability of the higher order longitudinal modes for PEP

Description: The theory of longitudinal instabilities of bunched beams is well known. Application to PEP has recently been considered by Pellegrini and Sands, who were mainly concerned with the rigid bunch oscillation of three bunches (i.e. the dipole mode with mode number m = 1). In this note, we will look at the stability of modes with arbitrary mode number m, first for a single-bunch beam and then or a beam consisting of three equally spaced bunches. The method of analysis is essentially that of Sacherer's, whose main result is an integral equation for the eigenmodes and eigenfrequencies of the longitudinal bunch oscillations. It turns out that the integral equation can be solved if the unperturbed beam has a uniform distribution inside an ellipse in the longitudinal phase space (the water-bag model). We study the impedance of a parallel resonator circuit representing the rf accelerating cavity and obtain expressions for the damping rates and frequency shifts for mode number m. In deriving these expressions, we have included contributions from the tails of the impedance and not only from its peak. The importance of these contributions has been pointed out by Zotter. For m = 1, our expression for the damping rate is the same as that for Robinson damping. We also find that the damping rate of a higher order mode (m />=/ 2) is small compared with that of the dipole mode (m = 1)--- this is expected since the rf wavelength is much longer than the bunch length. On the other hand, due to the contribution from the impedance tail, the mode frequency shift does not decrease with m rapidly. We have also included an estimate of the damping rates of the quadrupole modes using the PEP parameters. 4 refs., 1 fig.
Date: July 1, 1979
Creator: Chao, A.W.
Partner: UNT Libraries Government Documents Department

Beam excitation and damping with the transverse feedback system

Description: The questions often come up, ''What is the strength if the beam excitation system. How much damping can the transverse feedback provide.'' The design is now advanced enough to answer these questions; also, laboratory tests of some components have been conducted and we know what can be expected of the hardware. This paper discusses these questions.
Date: August 1, 1979
Creator: Pellegrin, J.L. & Rees, J.R.
Partner: UNT Libraries Government Documents Department

Transverse resonances of periodically widened cyclindrical tubes with circular cross section

Description: The transverse resonances of the electromagnetic field in periodic cylindrical cavities connected by concentric side tubes are calculated for arbitrary azimuthal mode numbers by expansion of the Hertz potentials in subregions bounded by coordinate surfaces. The expansion coefficients are determined by matching of the tangential field components across the common surfaces of the subregions. Resonances are characterized by the existence of solutions of the homogeneous wave equation in the absence of source terms, and the resonant frequencies are given by the vanishing of the determinant of an infinite matrix. They can be calculated on a computer by truncating the matrix. Expressions for the stored energy, the loss-factor, and for R/Q are then obtained by integration over the fields at resonance. These results are of interest for the determination of the energy loss and the stability of the beam in high-energy particle accelerators and storage rings when cavities are formed by pairs of cross section variations of the vacuum chamber. 7 refs., 1 fig.
Date: September 1, 1979
Creator: Zotter, B. & Bane, K.
Partner: UNT Libraries Government Documents Department

Application of matrix formalism to problems with memory

Description: On an example of the transverse one-dimensional coherent bunch motion in the presence of feedback, we developed a matrix formalism which allows us to find the decrement (or increment) and the tune shift of the motion due to a feedback system which remembers many previous passages of the bunch. The derived formulae are applied to a particular case of exponential decay of a resonant kicker signal. The possible detuning of the kicker resonator is also considered. The formulae are further applied to the case of a parasitic cavity. 15 refs., 16 figs.
Date: October 1, 1979
Creator: Kheifets, S.
Partner: UNT Libraries Government Documents Department

Damping rates and frequency shifts produced by a feedback system in coherent oscillations of multi-bunched beams

Description: The matrix formalism developed in an earlier PEP Note is generalized here to the case of motion of any number of bunches in each of two counter-rotating beams. The motion of the bunches in both beams is coupled through the feedback memory which arises from the finiteness of the feedback system bandwidth. The damping rates and the frequency shifts of one-dimensional coherent oscillations are calculated. Numerical examples are given for the particular case of bunches uniformly spaced around the orbit. 3 refs., 8 figs.
Date: November 1, 1979
Creator: Kheifets, S. & Rees, J.R.
Partner: UNT Libraries Government Documents Department

Transverse instability excited by rf deflecting modes for PEP

Description: We have looked at the possible transverse instability effects which are caused by the deflecting modes of the rf cavities in PEP. The results are obtained by applying the expression of the instability damping rate. We have assumed that there equal bunches equally spaced in PEP. We have worked out the equivalent for a single bunch beam. The effect of chromaticity xi is included as a frequency shift in the bunch mode spectra. We rewrite this result in terms of the transverse wake field instead of the impedance. We include an application of the Sacherer formalism to the case of resistive wall. The resulting expression of the damping rate contains two terms. The first term corresponds to the effect of the short wake fields; it agrees with the result of the head-tail instability as derived by Sands. A numerical estimate of this resistive-wall head tail case for PEP is given. It re-confirms that the resistive wall instability is not a serious problem for PEP. The second term gives the effect of long wake fields and it agrees with the result of Courant and Sessler. 10 refs., 2 figs.
Date: November 1, 1979
Creator: Chao, A.W. & Yao, C.Y.
Partner: UNT Libraries Government Documents Department

A users guide to Poisson

Description: This writeup describes the POISSON program package in operation at SLAC. The program can be used to solve two-dimensional or cylindrical magnetostatic or electrostatic problems. The program uses an irregular triangular mesh of up to 6600 points in the finite difference approximation to Poisson's equation. Symmetry properties of a problem are taken advantage of so that usually, only part of a given problem has to be set up. For example: only a 45 degree segment of a symmetrical quadrupole is solved in order to obtain a complete solution. Successive over-relaxation (SOR) is used to find values of the vector potential at the mesh points from the boundary conditions and input currents. Non-linear regions (iron) are treated in a quasi-linear manner by holding the permeabilities constant during a relaxation cycle, computing local fields at the end of a cycle, performing a table lookup and under-correcting the permeability to keep the problem stable. 10 figs.
Date: November 1, 1979
Creator: Early, R.A.
Partner: UNT Libraries Government Documents Department

Steady state distribution for unbunched beams colliding in a drift space

Description: A self-consistent time-independent solution of the system of coupled equations for the distribution of two colliding unbunched beams of opposite charges is found. The solution for each beam contains an arbitrary constant which characterizes the lateral size of the beam. On the other hand, the angular divergence is uniquely determined by the line charge density of the opposite beam in case the beams have cylindrical symmetry. The possible implications of this solution for a linac collider are discussed. Further work on the stability of this solution is needed. 6 refs., 9 figs., 2 tabs.
Date: December 1, 1979
Creator: Kheifets, S. & Chao, A.W.
Partner: UNT Libraries Government Documents Department

Wakefields in a pillbox cavity

Description: The wakefields due to a bunch in a pillbox cavity can be found by mode-expansion of the fields given by the wave-equation. Usually one considers first the wakefield due to an infinitely short pulse and then integrates over the current distribution. Since the series for the delta function pulse converge rather poorly, it is preferable to use the wakefield due to a step-function current, which can also be integrated to give the result for an arbitrary current distribution. For the most important case of fields with azimuthal symmetry, the results are expressed as double series over the radial mode number n, and the longitudinal mode number p. For a beam traveling with light-velocity, these series have been summed analytically. However, the results are restricted to observation times before reflections from the cylindrical side-wall reach the axis, i.e., they are actually valid only for a parallel-plate geometry. In order to find the wakefields in cylindrical cavities of finite radius we have to re-examine the infinite series. Although no closed-form expression could be found for the sidewall reflections, they can be determined from a function defined by a single infinite sum with only one variable. Extension to modes with azimuthal variation is straightforward. However, extension to beam velocities smaller than light-velocity is more complicated and again would require the use of functions defined by infinite series. The influence of finite holes in the end-walls has not been investigated in this report, since no analytic solutions are known for this case. However, the character of the solutions for the closed cavity should help in understanding the numerical results obtained for this case. 7 refs., 7 figs.
Date: December 1, 1979
Creator: Zotter, B.
Partner: UNT Libraries Government Documents Department

CSQII: a two-dimensional Eulerian code for computation of material motion in two dimensions

Description: CSQII is a general-purpose code with a wide variety of options: rectangular or cylindrical coordinates, up to ten materials, allowance for void, elestic-plastic or distended material, inclusion of energy sources or gravity. The basic solution scheme integrates the Lagrangian form of the conservation equations through a time step and then explicity rezones back to the Eulerian grid. The following aspects are treated in some detail: equation of state, momentum balance, energy balance, rezoning, and radiation. (RWR)
Date: January 1, 1980
Creator: McGlaun, J.M. & Thompson, S.L.
Partner: UNT Libraries Government Documents Department

Ion optics arithmetic and its implications for the positive ion CTR program

Description: This paper discusses ion extraction optics formulations in which presheath ionization is shown to have a negligible effect on ion optics at optimum perveance; otherwise, the examples shown establish an ionization gradient instability. Infinite slot optics as a function of perveance and potential partitioning is delineated for the TFTR tetrode from 2-D considerations; finite slot optics at optimum perveance is delineated from 3-D considerations. Finally, further 2-D considerations yield an end slot design.
Date: January 1, 1980
Creator: Whealton, J.H.
Partner: UNT Libraries Government Documents Department