摘要:

A simple, physical approximation is developed for the effect of viscosity for stable interfacial waves and for the unstable interfacial waves which correspond to Rayleigh‐Taylor instability. The approximate picture is rigorously justified for the interface between a heavy fluid (e.g., water) and a light fluid (e.g., air) with negligible dynamic effect. The approximate picture may also be rigorously justified for the case of two fluids for which the differences in density and viscosity are small. The treatment of the interfacial waves may easily be extended to the case where one of the fluids has a small thickness; that is, the case in which one of the fluids is bounded by a free surface or by a rigid wall. The theory is used to give an explanation of the bioconvective patterns which have been observed with cultures of microorganisms which have negative geotaxis. Since such organisms tend to collect at the surface of a culture and since they are heavier than water, the conditions for Rayleigh‐Taylor instability are met. It is shown that the observed patterns are quite accurately explained by the theory. Similar observations with a viscous liquid loaded with small glass spheres are described. A behavior similar to the bioconvective patterns with microorganisms is found and the results are also explained quantitatively by Rayleigh‐Taylor instability theory for a continuous medium with viscosity.

ISSN:0031-9171    

DOI:10.1063/1.1694570

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

A model for the evolution of an inviscid buoyant thermal into a vortex ring is developed by placing Hill's spherical vortex in a gravitational field. The atmospheric pressure gradient interacts with the density gradient at the edge of the thermal to produce vorticity. This additional effect is treated as an unsteady perturbation to Hill's steady‐state solution and causes the spherical vortex to evolve into a torus. Torus formation times are obtained in terms of the thermal radiusa, the initial rise velocityV0, and the degree of buoyancyg(&Dgr;&rgr;/&rgr;) (&Dgr;&rgr;/&rgr;) arbitrary). AsV0 → 0, the results for a buoyant bubble are recovered. However, whenV02 > ag(&Dgr;&rgr;/&rgr;), the internal circulation of the thermal changes the time scale of torus formation. Denoting the thermal density as&rgr;tand that in the atmosphere as&rgr;∞, the torus formation times are shown to be[3a&rgr;t/g(&rgr;∞ − &rgr;t)]1/2in the buoyant limit andV0/g[(&rgr;∞/&rgr;t)1/2 − 1]in the inertial (largeV0) limit. The analysis is extended to a density‐stratified atmosphere, and it is shown that the expansion of the rising thermal may eliminate the buoyant force before the torus can form. The conditions under which a torus can form are obtained in terms ofa, g, V0 , &Dgr;&rgr;/&rgr;andH, the atmospheric scale height.

ISSN:0031-9171    

DOI:10.1063/1.1694617

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

The hierarchy describing homogeneous and isotropic two‐dimensional turbulence is used to show that one of the effects of(n + 1)‐th order cumulants is to introduce a turbulent relaxation frequency (or eddy‐damping rate) in the equation for the rate of change ofn‐th order cumulants. As a result, it is possible to construct a Markovian eddy‐damped(n + 1)‐th order cumulant‐discard approximation in which the eddy‐damping rate simulates the irreversibility due to the infinite serie of cumulants with order greater thann. In the case of our Markovian eddy‐damped quasi‐normal approximation the governing equations for the evolution of the energy spectrum are compared to those of stochastic models like the generalized random phase approximation or the self‐consistent field approximation.

ISSN:0031-9171    

DOI:10.1063/1.1694580

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

Equations for the fluctuation correlation in an incompressible shear flow are derived on the basis of kinetic theory, utilizing the two‐point distribution function which obeys the BBGKY hierarchy equation truncated with the hypothesis of “ternary” molecular chaos. The step from the molecular to the hydrodynamic description is accomplished by a moment expansion which is a two‐point version of the thirteen‐moment method, and which leads to a series of correlation equations, viz., the two‐point counterparts of the continuity equation, the Navier‐Stokes equation, etc. For almost parallel shearing flows the two‐point equation is separable and reduces to two Orr‐Sommerfeld equations with different physical implications. Solution of an eigenvalue problem for the Blasius boundary layer is obtained in a certain parallelism to the classical stability theory, and is used for predicting the transition Reynolds number of a “quiescent” Blasius flow in which thermodynamic fluctuations alone are the initiating mechanism. Also, the calculated spatial growth rate of fluctuation agrees with the Schubauer‐Klebanoff experiment, which gives an account of unexplained experimental evidence that the fluctuation complex (turbulence bursts plus the Tollmien‐Schlichting wave), as a whole, obeys a certain linear theory.

ISSN:0031-9171    

DOI:10.1063/1.1694592

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

In the closure problem of the theory of turbulent mean flows, the unknown correlations have to be expressed in terms of the other field variables. Estimates for the behavior of the unknowns, both for the steady and nonsteady flows, have been obtained which insure uniqueness of solutions.

ISSN:0031-9171    

DOI:10.1063/1.1694610

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

The stability of an axisymmetric turbulent wake produced behind a stationary sphere in a fluid moving with uniform velocity is investigated. The dispersion relation indicates that the interface is unstable at all wave numbers, if the wake is treated as a viscous medium. From the dispersion relation the surface‐strain energy density spectrum of a known initial disturbance is obtained. The dynamic similarity of the distributions of the phase velocity and the surface‐strain energy density at high Reynolds number is demonstrated. From the energy equation one obtains a critical wavenumber beyond which the energy transport to the wake through the interface is cut off. A critical time is obtained based on the assumption that the momentum deficit in the wake is due to the viscous drag. The critical time and the critical wavenumber are used to explain the existence of an entrainment cycle.

ISSN:0031-9171    

DOI:10.1063/1.1694611

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

Measurements of the spatial development of disturbance pressure waves in a low‐speed axisymmetric turbulent free jet have been carried out. The results show that the wavenumbers of the pressure waves increase monotonically, while the phase velocities decrease as the Strouhal number of the jet increases. The pressure disturbance grows to a maximum at some distance downstream from the nozzle and then decays. The distributions of the amplitude of the pressure waves along the jet are similar if the data are plotted against a normalized distance Stx/D.The most amplified mode is at a Strouhal number of 0.5 for the shear layer and 0.35 for the center line. The wave characteristics follow closely the linear stability theory of an inviscid diverged shear flow.

ISSN:0031-9171    

DOI:10.1063/1.1694612

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

Small amplitude oscillations of a charged fluid globule immersed in a viscous dielectric are studied with special regard for the processes of surface charge relaxation and convection. Perfect dielectrics and conductors behave in identical fashion insofar as oscillation frequencies and stability are concerned whenever viscous effects are small. This similarity is due to interfacial charge convection, in the case of a perfect dielectric, and charge conduction, with a perfect conductor. The damping of stable oscillations depends, however, on the rate of interfacial charge relaxation: With instantaneous relaxation the damping rates are the same as those found with uncharged droplets, while with slower relaxation rates the dissipation is concentrated in an electrohydrodynamic boundary layer outside the drop. If viscous effects predominate then the existence of charge relaxation introduces oscillatory behavior in place of the aperiodic damping characteristic of uncharged drops. Several special circumstances are examined in detail.

ISSN:0031-9171    

DOI:10.1063/1.1694613

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

The electric surface force density caused by imposed electric fields and their concomitant surface charges is the basis for studying monomolecular films on liquid interfaces. The film is modeled as a compressible viscous two‐dimensional fluid characterized by an elasticityE, a surface dilatational viscosity, a surface shear viscosity, surface diffusion, and chemical equilibrium with film molecules diffusing through a liquid substrate. In the first of three configurations considered, the static rise in film surface pressure is experimentally shown to equal the integrated static shear stress. Steady “second‐order” film circulations and film rupture under high electric stress, are discussed. In the second configuration, temporally and spatially periodic electric stresses are used to study the dilatational film dynamics in an “imposed&ohgr;‐k” configuration having angular frequency&ohgr;and wavenumberk. Experiments demonstrate the theoretically predicted dilatational resonance at&ohgr; = (E0k2/√2&rgr;&eegr;)2/3,where&rgr;and&eegr;are, respectively, the mass density and viscosity of the liquid bulk. Effects of surface and bulk diffusion, as well as dilatational viscosity, are shown each to have a characteristic effect on the frequency response. In the third configuration, surface shearing of a film is modeled and sensitivity to surface shearing viscosity predicted.

ISSN:0031-9171    

DOI:10.1063/1.1694614

出版商:AIP    

年代:1974

数据来源: AIP

摘要:

The one‐dimensional unsteady flow problem in the kinetic theory of gases is studied, based directly on the nonlinear Boltzmann equation. Using a singular perturbation technique with a single scaling of time and length, a unified asymptotic theory is developed. It is shown that for any given smooth initial condition, the gas motion can be decomposed into five modes. For the first‐order quantities, these five modes of motion are independent of one another. The three diffusion modes are linear, governed by the diffusion equation for the unsteady heat and shear flows, while the two wave modes are nonlinear, governed by the Burgers equation for the evolution of shock and nonlinear waves in a dissipative medium. The coupling (nonlinear effects) among the five modes starts to appear in higher order. The solution of the distribution function up to the second order is obtained. The difference and relation between the present theory and the theories of Hilbert, of Chapman and Enskog, and of Grad are also discussed.

ISSN:0031-9171    

DOI:10.1063/1.1694615

出版商:AIP    

年代:1974

数据来源: AIP