AIP Publishing

The forces on a small rigid sphere in a nonuniform flow are considered from first prinicples in order to resolve the errors in Tchen’s equation and the subsequent modified versions that have since appeared. Forces from the undisturbed flow and the disturbance flow created by the presence of the sphere are treated separately. Proper account is taken of the effect of spatial variations of the undisturbed flow on both forces. In particular the appropriate Faxen correction for unsteady Stokes flow is derived and included as part of the consistent approximation for the equation of motion.

The stability of a plane current layer is analyzed in the hydromagnetic approximation, allowing for finite isotropic resistivity. The effect of a small layer curvature is simulated by a gravitational field. In an incompressible fluid, there can be three basic types of ``resistive'' instability: a long-wave ``tearing'' mode, corresponding to breakup of the layer along current-flow lines; a short-wave ``rippling'' mode, due to the flow of current across the resistivity gradients of the layer; and a low-g gravitational interchange mode that grows in spite of finite magnetic shear. The time scale is set by the resistive diffusion time τR and the hydromagnetic transit time τH of the layer. For large S = τR/τH, the growth rate of the ``tearing'' and ``rippling'' modes is of order τR−3/5τH−2/5, and that of the gravitational mode is of order τR−1/3τH−2/3. As S → ∞, the gravitational effect dominates and may be used to stabilize the two nongravitational modes. If the zero-order configuration is in equilibrium, there are no overstable modes in the incompressible case. Allowance for plasma compressibility somewhat modifies the ``rippling'' and gravitational modes, and may permit overstable modes to appear. The existence of overstable modes depends also on increasingly large zero-order resistivity gradients as S → ∞. The three unstable modes merely require increasingly large gradients of the first-order fluid velocity; but even so, the hydromagnetic approximation breaks down as S → ∞. Allowance for isotropic viscosity increases the effective mass density of the fluid, and the growth rates of the ``tearing'' and ``rippling'' modes then scale as τR−2/3τH−1/3. In plasmas, allowance for thermal conductivity suppresses the ``rippling'' mode at moderately high values of S. The ``tearing'' mode can be stabilized by conducting walls. The transition from the low-g ``resistive'' gravitational mode to the familiar high-g infinite conductivity mode is examined. The extension of the stability analysis to cylindrical geometry is discussed. The relevance of the theory to the results of various plasma experiments is pointed out. A nonhydromagnetic treatment will be needed to achieve rigorous correspondence to the experimental conditions.

The effects of couple stresses in fluids are considered. Linearized constitutive equations are proposed for force and couple stresses. A series of boundary-value problems are solved to indicate the effects of couple stresses as well as for experiments measuring the various material constants. It is found that a size effect comes in which is not present in the nonpolar case (couple stresses absent).

It is well known that the linear theory of collisionless damping breaks down after a time τ ≡ (m/eεκ)½, where κ is the wavenumber and ε is the amplitude of the electric field. Jacobi elliptic functions are now used to provide an exact solution of the Vlasov equation for the resonant electrons, and the damping coefficient is generalized to be valid for times greater than t = τ. This generalized damping coefficient reduces to Landau's result when t/τ ≪ 1; it has an oscillatory behavior when t/τ is of order unity, and it phase mixes to zero as t/τ approaches infinity. The above results are all shown to have simple physical interpretations.

Rosenbluth’s nonlinear, approximate tokamak equations of motion are generalized to three dimensions. The equations describe magnetohydrodynamics in the low β, incompressible, large aspect ratio limit. Conservation laws are derived and a well-known form of the energy principle is recovered from the linearized equations. The equations are solved numerically to study kink modes in tokamaks with rectangular cross section. Fixed-boundary kink modes, for which the plasma completely fills the conducting chamber, are considered. These modes, which are marginally stable to lowest order in circular tokamaks, become unstable with large growth rates, comparable to the growth rates of free boundary kink modes. The unstable modes are found using linearized, two-dimensional equations. The linear results are used as initial values in the nonlinear, three-dimensional computations. The nonlinear results show that the magnetic field is perturbed only slightly, while a large amount of plasma convection takes place carrying plasma from the center of the chamber to the walls.

The nature and significance of large-scale coherent structures in turblent shear flows are addressed. A definition for the coherent structure is proposed and its implications discussed. The characteristic coherent structure properties are identified and the analytical and experimental constraints in the eduction of coherent structures are examined. Following a few comments on coherent motions in wall layers, the accumulated knowledge from a number of recent and ongoing coherent structure investigations in excited and unexcited free shear flows in the author’s laboratory is reviewed. Also briefly addressed are effects of initial conditions, the role of coherent structures in jet noise production and broadband noise amplification, the feedback effect of coherent structures, the use of the Taylor hypothesis in coherent structure description, negative production, turbulence suppression via excitation, validity of the Reynolds number similarity hypothesis, etc. From the detailed quantitative results, a picture of the state of the art in coherent structure studies emerges. While coherent structures are highly interesting characteristic features of (perhaps all) turbulent shear flows, it is argued that their dynamical significance has been overemphasized. These are predominant only in their early stages of formation following instability, or in resonant situations and excited flows, or in regions adjacent to a wall of a turbulent boundary layer. The coherent Reynolds stress, vorticity, and production are comparable to (and not an order of magnitude larger than) the time-average Reynolds stress, vorticity, and production, respectively, in fully developed states of turbulent shear flows, where incoherent turbulence is also important and cannot be ignored. The concept and importance of coherent structures are here to stay; understanding and modeling of turbulent shear flows will be incomplete without them; but they are not all that matter in turbulent shear flows.

In the atomization regime of a round liquid jet, a diverging spray is observed immediately at the nozzle exit. The mechanism that controls atomization has not yet been determined even though several have been proposed. Experiments are reported with constant liquid pressures from 500 psia (33 atm) to 2500 psia (166 atm) with five different mixtures of water and glycerol into nitrogen, helium, and xenon with gas pressures up to 600 psia (40 atm) at room temperature. Fourteen nozzles were used with length-to-diameter ratios ranging from 85 to 0.5 with sharp and rounded inlets, each with an exit diameter of about 340 μm. An evaluation of proposed jet atomization theories shows that aerodynamic effects, liquid turbulence, jet velocity profile rearrangement effects, and liquid supply pressure oscillations each cannot alone explain the experimental results. However, a mechanism that combines liquid–gas aerodynamic interaction with nozzle geometry effects would be compatible with our measurements but the specific process by which the nozzle geometry influences atomization remains to be identified.

Calculations of rotational and vibrational relaxation times for gases composed of homonuclear diatomic molecules have been carried out. The model used for the molecular interaction potential consists of an attractive component, which acts between geometrical centers of the molecules, and a repulsive component which is assumed to originate from two centers of force in each of the molecules. For large intermolecular separations, the attractive forces prevail while at close distances the repulsive forces control. Using this model, the number of collisions to establish rotational equilibrium ZR and also the number for vibraton ZV are calculated. Both ZR and ZV contain some of the same molecular parameters and are therefore dependent on each other. From the analysis it turns out that ZR is a gradually increasing function of increasing temperature and ZV is a rapidly decreasing function of increasing temperature. Comparison with experiment for the gases chlorine, nitrogen, and oxygen indicates that the calculations are for the most part reliable.

In toroidal systems with geodesic curvature an electrostatic acoustic mode occurs with plasma motion in the magnetic surfaces, perpendicular to the field. In typical stellarators this mode should dominate ordinary sound waves associated with motion along the field.

A statistical theory for compressible turbulent shear flows subject to buoyancy effects is developed. Important correlation functions in compressible shear flows are calculated with the aid of a multiscale direct-interaction approximation. They are expressed in the gradient-diffusion form similar to the eddy-viscosity representation for the Reynolds stress in incompressible flows. The results obtained are applicable to subgrid modeling, and a Smagorinsky-type model in compressible flows is constructed.