Application of the integral diffusion equations to the investigation of turbulent transport

In recent years there have appeared several experimental studies [1–5] which have shown that there are cases of turbulent flow with an asymmetric distribution of the flow velocity and in which at the point where the velocity derivative is zero the turbulent shear stress is not zero. This raises the question of the connection of the Reynolds stress tensor with the characteristics of the average flow. The relationships used in the usual mixing length theory connect the shear stress with the local value of the flow velocity derivative and are not consistent with the experimental results mentioned above. These relationships are based on the assumption that the mixing length is small in comparison with the characteristic length of the flow. Experiment shows that this assumption is not justified [6].Thus, turbulent diffusion refers to the case of diffusion with a large mean free path. In addition to the concept of gradient diffusion, there is also the concept of bulk convection or integral diffusion [10], which means a transfer mechanism in which the shear stress is not expressed in terms of the velocity gradient. The generalization of mixing length theory proposed in [11–14] is based on the very simple kinetic equation which was used for the examination of turbulent transfer problems in [8] and which is encountered in the treatment of transport problems in gases, neutron diffusion, and radiative energy transfer.The proposed generalization of mixing length theory employs an analogy with the indicated processes and permits the derivation of formulas which are valid for large mean free paths. In the case of small mean free paths the obtained relationships lead to the relationships for diffusion in a continuous medium and, in particular, to the relationships of the Prandtl mixing length theory. The integral diffusion model is a phenomenological semiempirical theory in which empirical constants and several hypotheses common in mixing length theory are used. A very general analysis of the expression for the shear stress leads to the conclusion that if the flow is asymmetric over a distance comparable with the mixing length the points at which the velocity derivative and the turbulent shear stress are zero do not coincide [12]. Hence, it is to be hoped that the integral diffusion model will allow treatment of the above questions, which cause difficulty in the case of ordinary mixing length theory. Incompressible turbulent flow is considered.     

保持顶点多样性的复合形法及其在边坡稳定分析中的应用

针对基本复合形法全局搜索能力不强的缺陷,改进了基本复合形法的寻优方法。在关于各个顶点的寻优直线上分别找出比各个顶点优异的点并替换掉各顶点构成多个新复形,并以各个复合形中心点到各个顶点的海明距离之和为中心距指标,找出最大中心距的复形为下一次迭代之新复形。依次迭代,直到没有新的复形产生。通过对两个复杂边坡最小安全系数的搜索,发现本文新复合形法的全局搜索能力有较大提高。     

Some Remarks on the Rule of the Middle Third

The so-called rule of the middle third states that an inclined force applied at the top of a vertical pillar with rectangular cross-section must intersect the bottom within the middle third of the height of the rectangle in order that the normal stress on the base is one-signed. This rule has been extended by Michell (1900) to the case of a plane elastic wedge loaded at its vertex. We here study plane elastic pillars with other profiles, like a trapezoid, the plane region bounded by two branches of an equilateral hyperbola, a blunt pillar. The result is that the rule is only partially valid.     

On the linear instability of the Hall-Stewartson vortex

It is demonstrated that the Hall-Stewartson leading-edge vortex is linearly unstable to viscous perturbations of the center-mode type. Center modes are found to occur in two reigons of Reynolds-number-wave-number space, in limits in which the axial wave number is large. The appropriate center-mode equations in these neighborhoods are established, and it emerges that the two sets are identical. The single system of equations, which depends on the azimuthal wave number m and a distance parameter only, is solved numerically for various values of m and . Highly unstable modes are found for large positive , and the results are shown to be in good agreement with proposed asymptotic expansions when >1. To lowest order, unstable modes have phase surfaces that rotate with the fluid: in addition constant phase surfaces propagate upstream but the group velocity is directed downstream. The growth rate of the instability increases faster than Reynolds number to the quarter power. This, together with the finding that the length scale of the unstable modes found goes to zero as the Reynolds number tends to infinity, makes this instability an unusual one.This work was supported by the Air Force Office of Scientific Research under contract AFOSR-89-0346 monitored by Dr. L. Sakell, and by the U.S. Army Research Office at the Mathematical Sciences Institute of Cornell University.     

On the Possibility of Delaying Laminar-Turbulent Transition at High Reynolds Numbers by the Optimal Choice of Body Forces

The possibility of delaying laminar-turbulent transition on a flat plate in a longitudinal viscous incompressible flow by the optimal choice of the body force distribution in the boundary layer is discussed. It is shown that even for very high Reynolds numbers, Re 10¹⁰, a body force distribution can be found such that the corresponding boundary layer flow is absolutely stable, while the total drag of the body is less than that in the absence of body force action on the flow.     

Some conditions for the existence of a “Pinch” structure of the current sheet in plasma

In pulsed plasma accelerators of various configurations it is frequently possible to observe an instability of the current sheet characterized by separation into individual narrow channels (local pinches). In this paper the flow of plasma over these channels is qualitatively investigated. It is shown that a large part of the discharge current can flow through a series of narrow channels only when they completely decelerate the plasma flow. This places upper and lower bounds with respect to pressure on the region of existence of the pinch structure, if it is assumed that deceleration by the magnetic field predominates over viscous deceleration. The upper bound is obtained from the condition of deceleration of the plasma by the magnetic field near the pinches, the lower bound from the condition that the ratio of specific heats of the gas must be small. As the initial gas pressure decreases, so does the plasma deceleration time, but the characteristic times of the excitation and ionization processes increase; therefore the condition 1 ceases to be satisfied. The results are compared with experiment. The induced currents in the plasma near the pinches are also a reason for the stability of the network of current filaments.According to [1] a pulsed high-current discharge in a low-pressure gas develops as follows. After breakdown and the attainment of a certain uniform value of the gas conductivity the current continues to increase only in a surface layer (skin effect), which, if the particle density is sufficient, acts as an impermeable piston on the gas in front of it (snowplow model). Clearly, to judge from the collapse time of the current cylinder [2], this gives a good picture of the situation in the Z pinch. However, in coaxial and rail guns the gas is by no means always completely raked up by the current sheet (see, for example, [3]). This is also indicated by the observed broad spectrum of plasmoid velocities.It has been noted in high-speed photographs of pulsed discharges of very different configurations (Z and pinches, coaxial and rail plasma guns) that the current sheet is often divided into a series of channels [4–12], through which the main current flows. This follows from their high acceleration as compared with the rest of the plasma [4, 5], from the arrangement of the cathode spots along the electrodes, and from the large value of the plasma density in the channels [11]. This effect is well reproduced from discharge to discharge.Experiments with the Z pinch [7] have shown that the division of the current sheet into pinches has almost no effect on its velocity, which is in good agreement with calculations based on the snowplow model, although the diameter of the channels (determined, it is true, from the luminescence distribution and not the current density) is much less than the distance between them. On the other hand, in rail guns the pinches only slightly entrain the gas filling the tube [11]. The surface layer pinch instability is observed in the pressure range from several mm to several Hg [9]. Both pressure limits depend heavily on the nature of the gas in which the discharge takes place, the upper limit being approximately inversely proportional to the molecular weight of the gas [9], In a rail gun at low initial gas pressures (p < <20 Hg) pinches appear at some optimal rate of gas release from the electrodes and the walls [12].A uniform transverse magnetic field has only a slight influence on the pinches, since it is smaller in magnitude than the magnetic field of the pinches themselves [11]. A transverse magnetic field nonuniform along the length of the pinches causes them to decay, if the magnitude of the external field is comparable with the pinch field. Pinch decay was observed by the author in a rail gun when one of the electrodes was cut out so that on a certain section of the rail its transverse dimension was reduced to the diameter of the pinch, which was much less than the distance between electrodes (this setup was described in [11]).The instability of a plane current sheet in a plasma has been demonstrated on several occasions [13, 14]. Under experimental conditions the pinches move through the plasma. Therefore we will consider the case when a considerable part of the discharge current flows through a series of narrow channels (assuming that for this case the conductivity of the pinches is infinite).In conclusion, the author thanks A. K. Musin for his helpful suggestions.     

On the effect of a high-frequency magnetic field on the instability of a plasma

The action of a high-frequency magnetic field on low-frequency instabilities of a plasma is considered. The harmonics of the high-frequency field that appear under these conditions are taken into account. It is shown that their effect on the reduction of the growth rate of the instability is weak. However, in analyzing the oscillation spectrum consideration of these harmonics is necessary, since they have the same growth rate as does the suppressed instability. It is well known that in a plasma situated in a strong magnetic field H_0Z unstable oscillations develop, the most dangerous being electrostatic oscillations that propagate almost perpendicular to the magnetic field (k k_¦). These oscillations have the form of troughs extended almost along H_0Z. If it were possible by some method to create conditions under which particles of one species (either electrons or ions) would have time to traverse the distance between the humps of the troughs in a time considerably shorter than the period 2/ of the unstable oscillations, then the potential of the instability would be smoothed out (i.e., the instability would be suppressed). This may be achieved by exciting a high-frequency magnetic field H₁(t) = H₁ cos t( ) in the plasma, which is oriented perpendicular to the constant field H_0z. Then the crossing of the humps by the particles is achieved as a result of motion with thermal velocities along the resultant curved magnetic field.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 10–14, March–April, 1971.The author thank A. V. Gordeev for fruitful discussions.     

Analytic Tools to Bound the Criticality at the Outer Boundary of the Period Annulus

In this paper we consider planar potential differential systems and we study the bifurcation of critical periodic orbits from the outer boundary of the period annulus of a center. In the literature the usual approach to tackle this problem is to obtain a uniform asymptotic expansion of the period function near the outer boundary. The novelty in the present paper is that we directly embed the derivative of the period function into a collection of functions that form a Chebyshev system near the outer boundary. We obtain in this way explicit sufficient conditions in order that at most \(n\geqslant 0\) critical periodic orbits bifurcate from the outer boundary. These theoretical results are then applied to study the bifurcation diagram of the period function of the family \(\ddot{x}=x^p-x^q,\) \(p,q\in {\mathbb {R}}\) with \(p>q\).     

Large-eddy simulation of the temporal mixing layer using the Clark model

The Clark model for the turbulent stress tensor in large-eddy simulation is investigated from a theoretical and computational point of view. In order to be applicable to compressible turbulent flows, the Clark model has been reformulated. Actual large-eddy simulation of a weakly compressible, turbulent, temporal mixing layer shows that the eddy-viscosity part of the original Clark model gives rise to an excessive dissipation of energy in the transitional regime. On the other hand, the model gives rise to instabilities if the eddy-viscosity part is omitted and only the gradient part is retained. A linear stability analysis of the Burgers equation supplemented with the Clark model is performed in order to clarify the nature of the instability. It is shown that the growth-rate of the instability is infinite in the inviscid limit and that sufficient (eddy-)viscosity can stabilize the model. A model which avoids both the excessive dissipation of the original Clark model as well as the instability of the gradient part, is obtained when the dynamic procedure is applied to the Clark model. Large-eddy simulation using this new dynamic Clark model is found to yield satisfactory results when compared with a filtered direct numerical simulation. Compared with the standard dynamic eddy-viscosity model, the dynamic Clark model yields more accurate predictions, whereas compared with the dynamic mixed model the new model provides equal accuracy at a lower computational effort.     

Transformation and some solutions of the equations of motion of an ideal gas

The equations of one-dimensional and plane steady adiabatic motion of an ideal gas are transformed to a new form in which the role of the independent variables are played by the stream function and the function introduced by Martin [1, 2], It is shown that the function retains a constant value on a strong shock wave (and on a strong shock for plane flows). For one-dimensional isentropic motions the resulting transformation permits new exact solutions to be obtained from the exact solutions of the equations of motion. It is shown also that the one-dimensional motions of an ideal gas with the equation of state p=f(t) and the one-dimensional adiabatic motions of a gas for which p=f() are equivalent (t is time, is the stream function). It is shown that if k=s=–1, m and n are arbitrary (m+n0) and =1, the general solution of the system of equations which is fundamental in the theory of one-dimensional adiabatic self-similar motions [3] is found in parametric form with the aid of quadratures. Plane adiabatic motions of an ideal gas having the property that the pressure depends only on a single geometric coordinate are studied.     

On the principle of exchange of stabilities for the viscous flow between two co-axial rotating cylinders

Summary In the first part of this paper we have derived an adjoint system of equations for the set of equations characterising the solution of the stability of viscous flow between two rotating cylinders, when the marginal stability is assumed not to be stationary. Then the adjoint system of differential equations has been solved to arrive at a simpler secular equation than the one obtained by Chandrasekhar. By a different approach than that of Chandrasekhar's, an attempt is made to show that for , which is defined as the ratio of the velocities ₁ and ₂ with which the inner and outer cylinders are rotated, greater than zero, there is no possibility of the instability setting in as overstability.     

Effect of the interlayer connectivity of nonuniform reservoirs on the volume of the residual oil pillars

The limiting-equilibrium pillars of residual oil with an initial pressure gradient are calculated for reservoirs of the layer cake type whose nonuniformity conforms to a random (Pearson) distribution. It is assumed that hydrodynamically in the direction of stratification the layers are completeiy disconnected. A parametric investigation of the volume of the pillars is carried out for two problems (fivepoint area flooding scheme and a single well in a circular reservoir) and the results are compared with those for reservoirs with ideal interlayer connectivity.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5. pp. 184–186, September–October, 1989.     

Kinematics of the flow of dry powders and bulk solids

Summary Although qualitative pictures of the flow pattern in moving powders and bulk solids have been published, there is no quantitative evidence on the velocity profiles. This was needed in developing an energy theorem determining flow rates (Brown, Nature, 1961, 191, 458–461) and the experiments described here were undertaken to test various suppositions made at that time. It is shown that the assumptions made were correct.The gravity flow of dry powders and bulk solids through apertures cannot begin until the powder has dilated. The friction caused by the motion then determines the flow pattern.In this paper only those materials that flow freely under gravity are considered. In such materials cohesive forces between the grains are inappreciable. The results are expected to apply to all bulk solids that do not contain large quantities of finely-divided powders in the range of sizes below 20 to 30 microns, and to closely graded powders with particles larger than about 100 microns. In all cases it is assumed that the moisture content is in equilibrium with the ambient air.Measurements of the voidage in a single layer of ballbearings flowing down an inclined plate show that dilatant waves pass upward through the bed. Below the aperture the emergent stream forms a vena contracta. At the aperture there is a statistically empty space adjacent to the edges, through which few balls pass. The existence of a free-fall arch at the aperture is demonstrated, below which the grains fall freely under gravity.By fixing a flowing system in wax it is shown that the flow just above the free-fall arch is radial, converging to an apex below the aperture, and that the flowing region is bounded by the surfaces of sliding that separate flowing from non-flowing material.By observing the flow of powder through a transparent end face, the angle to the vertical ( degrees) of the surface of sliding at the aperture has been measured for (a) flow through a central slot (_c ^o), (b) flow through an edge slot adjacent to a vertical wall (_E ^o). A different method was used to determine the inclination of the surface of sliding through a central circular aperture (₃ ^o). It was found that ₃ ^o < _c ^o _E ^o.The surfaces of sliding are steeper than the drained angle of repose of a conical heap.Paper presented at a meeting of the British Society of Rheology, University of Nottingham, April 6–8, 1965.     

Effect of the Nature of the Interaction between a Gas and a Surface on the Relations for the Knudsen Layer in the Case of Strong Evaporation

The effect of the temperature accommodation coefficient _T on the relations at the Knudsen layer edge is investigated for strong evaporation using the moment method. An explicit expression for the dimensionless density as a function of the temperature and the Mach number M is obtained for 0 < _T < 1. For _T = 0 the entire solution is obtained in explicit form. It is shown that for = 0 and a condensation coefficient << 1 the temperature outside the Knudsen layer changes sharply as M varies from 0 to a certain value much less than unity after which the temperature ceases to depend on . For the model of specular reflection of the molecules from the surface the density and the temperature outside the Knudsen layer are found in explicit form as functions of the Mach number.     

Determination of the Closure of the Set of Elasticity Functionals

We determine the closure for Mosco-convergence in L ² (,³) of the set of elasticity functionals. We prove that this closure coincides with the set of all non-negative lower-semicontinuous quadratic functionals which are objective, i.e., which vanish for rigid motions. The result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed Poisson coefficient. This shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients.     

Stability of the tumbled state in shear flow of a nematic liquid crystal

In shear flow of a nematic liquid crystal with ₃ 0 flow alignment cannot occur. The stability of the stationary in-plane solution, the tumbled state, is investigated using abstract techniques. Employing the existence of an elastic energy a sufficient criterion for stability is formulated. This criterion depends on the in-plane solution which is obtainable as a quadrature that is non-elementary except in special cases. It is shown that the tumbled state is stable and asymptotically stable for some physical configurations. The criterion presented is not a necessary condition for stability and thus only gives a lower bound.