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Volume 15, Issue 4, April 1972

Numerical Solutions for Time‐Dependent Flow Past an Impulsively Started Sphere
View Description Hide DescriptionA numerical method is given for determining the transient flow past a sphere which is impulsively started from rest with constant velocity in a viscous fluid. One of the features of the method is that the calculation of the flow at early times is performed using boundary‐layer variables, which leads to very accurate solutions. The problem is formulated in terms of the stream function and the vorticity. The method used is the semi‐analytical one of series truncation in which the stream function and vorticity are expanded in a series of Legendre functions with argument , where is the polar angle. The governing equations are thus reduced to sets of time‐dependent differential equations in the radial variable. In theory the number of these equations is infinite but in practice only a finite number can be solved to give an approximation to the flow. They are solved numerically by the Crank‐Nicolson procedure. Numerical solutions are given for the cases = 20, 40, 100, 200, 500, 1000, and , where is the Reynolds number based on the diameter of the sphere. For = 20, 40, and 100 the equations are integrated to times at which the solutions may be considered close to steady motion. The developments of various physical properties of the flow are calculated and compared, where possible, with the results of other calculations.

Viscous Stability Theory for Thermally Stratified Flow: Discontinuous Jets and Shear Layers
View Description Hide DescriptionThe method used by Drazin to investigate the stability of unbounded, viscous, homogeneous, parallel shear flow to small wavenumber disturbances is extended to study the effect of thermal stratification on the stability of unbounded jets and shear layers. By this method the stability characteristics of continuous profiles are inferred from the stability characteristics of discontinuous profiles. The characteristic value problem for discontinuous jet and shear layers is posed by the requirement that the solutions of the governing differential equation satisfy the matching conditions and boundedness conditions for layers that extend to infinity. The analysis leads to a characteristic determinant which is required to vanish for the characteristic values of the parameters: the Reynolds number, the wavenumber, and the wave speed. The stabilizing effect of the thermal stratification as parameterized by the Richardson number was found to be most stabilizing for small wavenumber (large‐scale) disturbances.

Convective Stability of a General Viscoelastic Fluid Heated from Below
View Description Hide DescriptionThe stability problem for a plane layer of a general viscoelastic (simple) fluid heated from below is investigated. The nature of the problem suggests that linear viscoelasticity assumptions are sufficient to fully describe the phenomena. It is shown that under certain conditions the fluid is overstable; namely, an oscillating cell structure will be created before the classical (Bénard) steady secondary‐flow instability appears. Stability criteria for the oscillatory modes have been found as well as wavenumber and oscillation periods for both rigid and free boundaries. The theoretical results have been applied to a Maxwell fluid and to some real viscoelastic solutions. The numerical results for the latter suggest that although oscillation of Bénard cells is theoretically possible, very high‐temperature gradients or high gravitational fields would be required before the oscillating cells could be observed in common polymer solutions of moderate viscosity.

Burgers' Turbulence Model with Two Velocity Components
View Description Hide DescriptionBurgers' turbulence model with two velocity components has a periodic motion as the equilibrium turbulent state. Numerical experiments have shown that this equilibrium state is a stable limit cycle and hence attainable from the arbitrary initial conditions except for some singular ones. Since the equilibrium state is continuous with respect to the Reynolds number, no evidence has been found of the finite‐jump and turbulence‐turbulence transitions of the sort that Case and Chiu have suggested.

Laser Anemometer Measurements of Turbulence in Non‐Newtonian Pipe Flows
View Description Hide DescriptionThe structure of turbulence for pipe flows of dilute polyethylene oxide and polyacrylamide solutions is investigated using a laser anemometer technique. The instrumentation of the optical system and signal processing needed to satisfy the optical requirements and the signal‐to‐noise ratio are presented. The spatial requirement of the beam signal is described with regard to the lens, aperture, wavelength, and the angular alignment of the scattered and reference beams. A gravity‐flow system is used to minimize the degradation of polymer solutions. The measurements of wall pressure drops indicate that the polymer additives used give a consistent delay of transition from laminar flow to turbulence, as compared with Newtonian fluids. The axial turbulence intensities for polymer solutions are found to be substantially reduced, compared with results obtained for water.

Spin Down of Radiation‐Penetrated, Opaque, Compressible Fluid in a Circular Cylinder
View Description Hide DescriptionThe linear impulsive spin down from a rotational equilibrium of rigid body rotation of an opaque, compressible fluid in a circular cylinder under the effect of radiation is investigated. The opacity of the fluid is assumed to result from the electron scattering, and the radiative Prandtl number is assumed to be proportional to the square root of the Ekman number where , and is the kinematic viscosity, is the radiative thermal diffusivity, is the radius of the cylinder, and is the unperturbed angular velocity, respectively. The treatment is also restricted to the case for which the ratio of the thermal energy to the rotational kinetic energy ( , where and are the temperature and the universal gas constant, respectively) is very large corresponding to typical conditions in stellar spin down. As a new aspect of the linear spin down of the non‐Boussinesq compressible fluid, the quasisteady asymptotic state is shown to be rotating nonuniformly despite the fact that such an asymptotic state in Boussinesq fluid is characterized by rigid body rotation. This nonuniform asymptotic state is one in which each horizontal fluid layer rotates like a rigid body with a perturbation angular velocity proportional to the unperturbed temperature. It is also shown that the spin‐down time is of the order of , where is the thermal diffusion time. Because can be rephrased to be equal to the square ratio of the Brunt‐Väisälä frequency to , we may say that our spin‐down process is similar to that of the Boussinesq fluid case except for the above quasistatic asymptotic state.

Experimental Investigation of Normal Shock Wave Velocity Distribution Functions in Mixtures of Argon and Helium
View Description Hide DescriptionThe electron beam fluorescence technique for measuring simple one‐dimensional moments of helium velocity distribution functions has been extended for use with argon. Measurements made in argon‐helium gas mixture shock waves at low Mach numbers are presented. Parallel and perpendicular temperature profiles are shown for each gas. Macroscopic velocities for each gas were measured by determining the blue shifts in the line profiles when the line of sight was directly upstream. The results are compared with existing theories and other experimental results.

Development of the Distribution Function on the Centerline of a Free Jet Expansion
View Description Hide DescriptionSpherical source flow which closely approximates the flow on the centerline of an axisymmetric free jet expansion, is used as a theoretical model. The ellipsoidal statistical model is used to describe the intermolecular collisions. Three partial moments of the distribution function, and the value of the distribution function for a zero perpendicular component of molecular velocity, are calculated as functions of distance and molecular velocity for two values of the Prandtl number , and for two gross collision rates such that the viscosity would be proportional to and , where is the temperature. The quantities calculated correspond to idealized experimental measurements. It is found that the distribution function is, to a good approximation, ellipsoidal in velocity space for , where and are the parallel and perpendicular temperatures, respectively. It is also found that the deduction of and from the experimental quantities by fitting with an assumed Gaussian velocity dependence can lead to serious errors as the flow becomes frozen. There is a significant effect of the Prandtl number upon some of the results.

Prediction of Molecular Scattering in Free‐Jet Expansions
View Description Hide DescriptionA theoretical model of the background interaction with a free‐jet expansion is developed. The background is shown to be separable into two components: one originating from jet particles reflected from the chamber walls and the other from residual particles unremoved by the pumping system. Comparisons of the data of several investigators are made with theoretically predicted values of intensity as a function of either separation distance or source pressure.

Kinetic Theory of Parallel Plate Heat Transfer in a Polyatomic Gas
View Description Hide DescriptionA kinetic model equation for a polyatomic gas is used to investigate heat transfer between two parallel plates. In particular, a general kinetic boundary‐value problem is studied for the case of arbitrary accommodation coefficients on the plates. This problem is solved for all ranges of Knudsen number using a variational principle with a simple trial function. The dependence of heat transfer on Knudsen number, the internal energy, the Eucken factor and the ratio of translational collision time to internal collision time for a polyatomic gas has been obtained. Moreover, density and temperature profiles have been calculated. Comparison is made with measurements on nitrogen, and it is found that the solution is in very good agreement with experiment for heat transfer, and in qualitative agreement with experimental density profiles.

Magnetohydrodynamic Stability of the Developing Laminar Flow in a Parallel‐Plate Channel
View Description Hide DescriptionLinear stability of the developing laminar flow of an electrically conducting, incompressible fluid in a parallel‐plate channel under a transverse magnetic field is investigated. The case of small magnetic Reynolds numbers is treated. The developing flows whose stability characteristics are studied correspond to those induced by a uniform and a parabolic velocity distribution at the channel inlet. The stability of the fully developed Hartmann flow is also reexamined. Neutral stability curves and axial variations of the critical Reynolds number are presented for a range of Hartmann numbers between 0 and 4. It is found that for Hartmann numbers larger than one, the developing flow induced by a uniform inlet velocity distribution is the least stable, while the flow induced by a parabolic inlet velocity distribution could be the most stable in the entrance region of the channel. A comparison with the available experimental data shows that for a given Hartmann number the transition Reynolds number is about two orders of magnitude lower than the critical Reynolds number.

Nonequilibrium Viscous Shock Layer in a Partially Ionized Gas
View Description Hide DescriptionThe structure of a nonequilibrium, partially ionized viscous shock layer of a blunt body which is at floating potential is described. Electron temperature and electron‐ion density profiles across the shock layer have been theoretically investigated for the case where electrons are in thermal nonequilibrium with heavy particles in a free stream. Immediately behind the shock, the electron temperature profile across the shock layer has been matched with that in a free stream, using the relation for the jump of the electron temperature gradient across the shock. The method of matching is described. Dependence of the electron temperature profile across the shock layer upon the degree of thermal nonequilibrium in the free stream has been investigated.

Flow Velocity Measurements in a T Tube
View Description Hide DescriptionThe flow velocity of a T‐tube plasma has been measured by recording the attenuation of a known shock wave propagating through the T‐tube flow. This technique yields results with typical accuracies of 0.1 km/sec which is about 10% of the velocity of the probing shock. The interpretation of the results invites a comparison of the T‐tube flow field with an overexpanded detonation or breakdown wave in the high‐velocity regime and with a radiation driven shock front in the low‐velocity regime.

Revision and Test of the Quasilinear Theory
View Description Hide DescriptionThe quasilinear theory is revised analytically and its consequences are tested by a numerical simulation for a one‐dimensional periodic electron plasma with discrete electrostatic modes. The theory is formulated for unstable and damped modes and a series of constants of motion derived. The numerical model is a Vlasov‐Poisson system where the mode‐mode coupling terms have been omitted. The result is that the predictions of our analytical theory are confirmed by simulation. For all cases considered there are always several modes which are initially unstable, but become damped at a later time so that an inclusion of damped modes in the theory is absolutely necessary.

Electrostatic Stability Theory of Tokamaks from Two‐Component Fluid Equations
View Description Hide DescriptionThe electrostatic stability theory of tokamak plasmas is developed from the two‐component fluid equations. The consistent treatment of the collisional equations with the transport processes from Coulomb scattering shows the relative importance of the transport processes of resistivity, viscosity, the thermoelectric effect, and thermal conduction in the several plasma modes. The plasma modes derived from the electrostatic equations are the sound waves, the drift wave, and thermal modes. The growth rates are given as functions of the driving forces of the tokamak current, the density gradient, and the temperature gradients. At finite aspect ratio, the equations describing the current driven drift‐sound waves are solved for the toroidal shift in the growth rate. In addition to the gradient‐ drifts previously modeled by a periodic gravity, it is shown that consistency requires inclusion of the plasma compression due to the modulation of the parallel and perpendicular fluxes in the nonuniform magnetic field.

Stability of Bernstein‐Greene‐Kruskal Wave with Small Fraction of Trapped Electrons
View Description Hide DescriptionFor , the normal modes are essentially oscillations of the background plasma and these can be made unstable due to coupling to the periodic motion of the trapped electrons with bound frequency , where and is an integer. However, the most unstable waves have frequency , and in this regime, plasma oscillations with wave number and , where , are coupled; instability is associated with coupling due to resonant and nonresonant interaction with the trapped electrons. If the trapped electron distribution function is monotonically increasing (decreasing) in energy, then the lower (upper) frequency side band with wavenumber ) is dominant.

Anomalous Reflection and Transmission of Cyclotron Waves
View Description Hide DescriptionA formalism is developed to describe the propagation along a slowly varying magnetic field of purely transverse cyclotron waves. By an expansion of the coupled set of Vlasov‐Maxwell equations, a one‐dimensional wave equation is derived, which is applied to the case where the field configuration is a magnetic mirror with small field variation which includes points of cyclotron resonance where the local gyrofrequency is equal to the wave frequency. An investigation is made of the processes whereby particles trapped in the mirror field can cause anomalous transmission and reflection of the wave by a nonlocal mechanism. It is found that the nonlocal processes are most significant when the function which gives the total phase difference observed by a trapped particle on its orbit between the point of damping of the incident wave and that of excitation of the transmitted or reflected wave is stationary with respect to both the energy and magnetic moment of the particle.

Effect of Finite on the Drift Cyclotron Instability
View Description Hide DescriptionThe effect of finite (particle pressure/magnetic pressure) on the drift‐cyclotron instability is investigated. It is shown that for the instability range is shifted but not reduced, whereas, for the growth of the drift‐cyclotron mode is much reduced.

Theory and Simulation of the Beam Cyclotron Instability
View Description Hide DescriptionA detailed theory in conjunction with the results of computer simulation experiments is presented for the beam cyclotron instability. The main results are (1) After a period of exponential quasilinear development, turbulent wave‐particle interactions cause cross‐field diffusion of the electrons which smears out the electron gyroresonances. This occurs at a level of turbulence which scales as , where and are the electron cyclotron and plasma frequencies, and results in a transition to ordinary ion sound modes that would occur in an unmagnetized plasma. The magnetic field serves to reduce the effects of electron trapping. (2) This level of turbulence appears to have virtually no effect on long wavelength fluid modes. (3) At this level the instability stabilizes if ordinary ion sound is stable due to ion Landau damping. For cold ions it continues to develop at the slower ion acoustic growth rate until the fields become strong enough to trap the ions. After the fields saturate, further plasma heating is much slower than exponential.

Electron Cyclotron Heating as Resonant Diffusion
View Description Hide DescriptionThe heating of electrons by externally applied microwave fields is treated as a resonant diffusion in velocity space. The diffusion is calculated directly from electron orbits with the microwave fields considered to be small perturbations. The motion of an electron is found to be adiabatic outside resonant zones where the microwave frequency matches the cyclotron frequency. Heating rates are found which are independent of collisional time scales if the correlation length of the external fields is long enough; several wavelengths are required. Finite cyclotron radius effects are included to find the dominant contribution to scattering into the loss cone of a plasma in a mirror field and spatial diffusion rates. The theory is found to agree qualitatively with experiment, but relativistic dynamics must be used for a detailed comparison.