A note concerning the Lighthill "sandwich model" of tropical cyclones.

The basic element of Lighthill's "sandwich model" of tropical cyclones is the existence of "ocean spray," a layer intermediate between air and sea made up of a cloud of droplets that can be viewed as a "third fluid." We propose a mathematical model of the flow in the ocean spray based on a semiempirical turbulence theory and demonstrate that the availability of the ocean spray over the waves in the ocean can explain the tremendous acceleration of the wind as a consequence of the reduction of the turbulence intensity by droplets. This explanation complements the thermodynamic arguments proposed by Lighthill.

The turbulent wall jet: a triple-layered structure and incomplete similarity.

We demonstrate using the high-quality experimental data that turbulent wall jet flows consist of two self-similar layers: a top layer and a wall layer, separated by a mixing layer where the velocity is close to maximum. The top and wall layers are significantly different from each other, and both exhibit incomplete similarity, i.e., a strong influence of the width of the slot that had previously been neglected.

Transfer of a passive additive in a turbulent boundary layer at very large Reynolds numbers.

We formulate the mass transfer problem for a passive additive in a turbulent boundary layer based on the recently proposed model of the turbulent boundary layer at very large Reynolds numbers. The solutions of three basic problems are obtained. These solutions are self-similar asymptotics describing the mass exchange at its initial stages. The solutions obtained can be used for the construction (in particular, the numerical construction) of the solution to the more general problems of passive admixture transfer in the developed turbulent wall-bounded shear flows.

A model of a turbulent boundary layer with a nonzero pressure gradient.

According to a model of the turbulent boundary layer that we propose, in the absence of external turbulence the intermediate region between the viscous sublayer and the external flow consists of two sharply separated self-similar structures. The velocity distribution in these structures is described by two different scaling laws. The mean velocity u in the region adjacent to the viscous sublayer is described by the previously obtained Reynolds-number-dependent scaling law Φ = u / u(*) = Aη(α), A = 1/√3 In ReΛ + 5/2, α = 3/2 in ReΛ Î· = u(*)y/v. (Here u(*) is the dynamic or friction velocity, y is the distance from the wall, ν the kinetic viscosity of the fluid, and the Reynolds number ReΛ is well defined by the data.) In the region adjacent to the external flow, the scaling law is different: Φ = Bη(ß). The power ß for zero-pressure-gradient boundary layers was found by processing various experimental data and is close (with some scatter) to 0.2. We show here that for nonzero-pressure-gradient boundary layers, the power ß is larger than 0.2 in the case of an adverse pressure gradient and less than 0.2 for a favorable pressure gradient. Similarity analysis suggests that both the coefficient Β and the power ß depend on ReΛ and on a new dimensionless parameter P proportional to the pressure gradient. Recent experimental data of Perry, Marusic, and Jones were analyzed, and the results are in agreement with the model we propose.

Self-similar intermediate asymptotics for nonlinear degenerate parabolic free-boundary problems that occur in image processing.

In the boundary layers around the edges of images, basic nonlinear parabolic equations for image intensity used in image processing assume a special degenerate asymptotic form. An asymptotic self-similar solution to this degenerate equation is obtained in an explicit form. The solution reveals a substantially nonlinear effect-the formation of sharp steps at the edges of the images, leading to edge enhancement. Positions of the steps and the time shift parameter cannot be determined by direct construction of a self-similar solution; they depend on the initial condition of the pre-self-similar solution. The free-boundary problem is formulated describing the image intensity evolution in the boundary layer.

Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation.

The equation partial differential(t)u = u partial differential(xx)(2)u -(c-1)( partial differential(x)u)(2) is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water-absorbing fissurized porous rock; therefore, we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.

Characteristic length scale of the intermediate structure in zero-pressure-gradient boundary layer flow.

In a turbulent boundary layer over a smooth flat plate with zero pressure gradient, the intermediate structure between the viscous sublayer and the free stream consists of two layers: one adjacent to the viscous sublayer and one adjacent to the free stream. When the level of turbulence in the free stream is low, the boundary between the two layers is sharp, and both have a self-similar structure described by Reynolds-number-dependent scaling (power) laws. This structure introduces two length scales: one-the wall-region thickness-determined by the sharp boundary between the two intermediate layers and the second determined by the condition that the velocity distribution in the first intermediate layer be the one common to all wall-bounded flows and in particular coincide with the scaling law previously determined for pipe flows. Using recent experimental data, we determine both these length scales and show that they are close. Our results disagree with the classical model of the "wake region."

Very intense pulse in the groundwater flow in fissurized-porous stratum.

An asymptotic solution is obtained corresponding to a very intense pulse: a sudden strong increase and fast subsequent decrease of the water level at the boundary of semi-infinite fissurized-porous stratum. This flow is of practical interest: it gives a model of a groundwater flow after a high water period or after a failure of a dam around a collector of liquid waste. It is demonstrated that the fissures have a dramatic influence on the groundwater flow, increasing the penetration depth and speed of fluid penetration into the stratum. A characteristic property of the flow in fissurized-porous stratum is the rapid breakthrough of the fluid at the first stage deeply into the stratum via a system of cracks, feeding of porous blocks by the fluid in cracks, and at a later stage feeding of advancing fluid flow in fissures by the fluid, accumulated in porous blocks.

The thermal explosion revisited.

The classical problem of the thermal explosion in a long cylindrical vessel is modified so that only a fraction alpha of its wall is ideally thermally conducting while the remaining fraction 1-alpha is thermally isolated. Partial isolation of the wall naturally reduces the critical radius of the vessel. Most interesting is the case when the structure of the boundary is a periodic one, so that the alternating conductive alpha and isolated 1-alpha parts of the boundary occupy together the segments 2pi/N (N is the number of segments) of the boundary. A numerical investigation is performed. It is shown that at small alpha and large N, the critical radius obeys a scaling law with the coefficients depending on N. For large N, the result is obtained that in the central core of the vessel the temperature distribution is axisymmetric. In the boundary layer near the wall having the thickness approximately 2pir0/N (r0 is the radius of the vessel), the temperature distribution varies sharply in the peripheral direction. The temperature distribution in the axisymmetric core at the critical value of the vessel radius is subcritical.

Structure of the zero-pressure-gradient turbulent boundary layer.

A processing of recent experimental data by Nagib and Hites [Nagib, H. & Hites, M. (1995) AIAA paper 95-0786, Reno, NV) shows that the flow in a zero-pressure-gradient turbulent boundary layer, outside the viscous sublayer, consists of two self-similar regions, each described by a scaling law. The results concerning the Reynolds-number dependence of the coefficients of the wall-region scaling law are consistent with our previous results concerning pipe flow, if the proper definition of the boundary layer Reynolds number (or boundary layer thickness) is used.

Scaling laws for fully developed turbulent flow in pipes: discussion of experimental data.

We compare mean velocity profiles measured in turbulent pipe flows (and also in boundary layer flows) with the predictions of a recently proposed scaling law; in particular, we examine the results of the Princeton "super-pipe" experiment and assess their range of validity.

The problem of the spreading of a liquid film along a solid surface: a new mathematical formulation.

A new mathematical model is proposed for the spreading of a liquid film on a solid surface. The model is based on the standard lubrication approximation for gently sloping films (with the no-slip condition for the fluid at the solid surface) in the major part of the film where it is not too thin. In the remaining and relatively small regions near the contact lines it is assumed that the so-called autonomy principle holds-i.e., given the material components, the external conditions, and the velocity of the contact lines along the surface, the behavior of the fluid is identical for all films. The resulting mathematical model is formulated as a free boundary problem for the classical fourth-order equation for the film thickness. A class of self-similar solutions to this free boundary problem is considered.

The influence of the flow of the reacting gas on the conditions for a thermal explosion.

The classical problem of thermal explosion is modified so that the chemically active gas is not at rest but is flowing in a long cylindrical pipe. Up to a certain section the heat-conducting walls of the pipe are held at low temperature so that the reaction rate is small and there is no heat release; at that section the ambient temperature is increased and an exothermic reaction begins. The question is whether a slow reaction regime will be established or a thermal explosion will occur. The mathematical formulation of the problem is presented. It is shown that when the pipe radius is larger than a critical value, the solution of the new problem exists only up to a certain distance along the axis. The critical radius is determined by conditions in a problem with a uniform axial temperature. The loss of existence is interpreted as a thermal explosion; the critical distance is the safe reactor's length. Both laminar and developed turbulent flow regimes are considered. In a computational experiment the loss of the existence appears as a divergence of a numerical procedure; numerical calculations reveal asymptotic scaling laws with simple powers for the critical distance.

Small viscosity asymptotics for the inertial range of local structure and for the wall region of wall-bounded turbulent shear flow.

The small viscosity asymptotics of the inertial range of local structure and of the wall region in wallbounded turbulent shear flow are compared. The comparison leads to a sharpening of the dichotomy between Reynolds number dependent scaling (power-type) laws and the universal Reynolds number independent logarithmic law in wall turbulence. It further leads to a quantitative prediction of an essential difference between them, which is confirmed by the results of a recent experimental investigation. These results lend support to recent work on the zero viscosity limit of the inertial range in turbulence.

Similarity principles for the biology of pelagic animals.

A similarity principle is formulated according to which the statistical pattern of the pelagic population is identical in all scales sufficiently large in comparison with the molecular one. From this principle, a power law is obtained analytically for the pelagic animal biomass distribution over the animal sizes. A hypothesis is presented according to which, under fixed external conditions, the oxygen exchange intensity of an animal is governed only by its mass and density and by the specific absorbing capacity of the animal's respiratory organ. From this hypothesis a power law is obtained by the method of dimensional analysis for the exchange intensity mass dependence. The known empirical values of the exponent of this power law are interpreted as an indication that the oxygen-absorbing organs of the animals can be represented as so-called fractal surfaces. In conclusion the biological principle of the decrease in specific exchange intensity with increase in animal mass is discussed.

Model of clustering of earthquakes.

A two-dimensional model of the expansion of a crack in an elastic medium is considered in which friction depends on the slip rate and the modulus of cohesion depends on the speed of expansion of the crack. Elastic waves are neglected (quasi-static model). Under some conditions, the expansion of the crack is realized by the alternation of slow and fast episodes ("shocks") of slip. This offers a possible qualitative explanation of several forms of earthquake clustering, including clustering that is premonitory to strong earthquakes.