A Model of a Turbulent Boundary Layer With a Non-Zero Pressure Gradient

According to a model of the turbulent boundary layer proposed by the authors, 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 $\phi = u/u_*=A\eta^{\alpha}$, $A=\frac{1}{\sqrt{3}} \ln Re_{\Lambda}+ \frac 52$, $\alpha=\frac{3}{2\ln Re_{\Lambda}}$, $\eta = u_* y/\nu$. (Here $u_*$ is the dynamic or friction velocity, y is the distance from the wall, $\nu$ the kinematic viscosity of the fluid, and the Reynolds number $Re_{\Lambda}$ is well defined by the data) In the region adjacent to the external flow the scaling law is different: $\phi= B\eta^{\beta}$. The power $\beta$ 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 non-zero-pressure-gradient boundary layers, the power $\beta$ is larger than 0.2 in the case of adverse pressure gradient and less than 0.2 for favourable pressure gradient. Similarity analysis suggests that both the coefficient B and the power $\beta$ depend on $Re_{\Lambda}$ and on a new dimensionless parameter P proportional to the pressure gradient. Recent experimental data of Perry, Maru\v{s}i\'c and Jones (1)-(4) were analyzed and the results are in agreement with the model we propose.Comment: 10 pages, 9 figure

The 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, 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".Comment: 11 pages, includes 2 tables and 3 figure

A Note on the Intermediate Region in Turbulent Boundary Layers

We demonstrate that the processing of the experimental data for the average velocity profiles obtained by J. M. \"Osterlund (www.mesh.kth.se/$\sim$jens/zpg/) presented in [1] was incorrect. Properly processed these data lead to the opposite conclusion: they confirm the Reynolds-number-dependent scaling law and disprove the conclusion that the flow in the intermediate (`overlap') region is Reynolds-number-independent.Comment: 8 pages, includes 1 table and 3 figures, broken web link in abstract remove

Self-Similar Intermediate Asymptotics for a Degenerate Parabolic Filtration-Absorption Equation

The equation $$\partial_tu=u\partial^2_{xx}u-(c-1)(\partial_xu)^2$$ 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.Comment: 19 pages, includes 7 figure

Approach to equilibrium of diffusion in a logarithmic potential

The late-time distribution function P(x,t) of a particle diffusing in a one-dimensional logarithmic potential is calculated for arbitrary initial conditions. We find a scaling solution with three surprising features: (i) the solution is given by two distinct scaling forms, corresponding to a diffusive (x ~ t^(1/2)) and a subdiffusive (x ~ t^{\gamma} with a given {\gamma} < 1/2) length scale, respectively, (ii) the overall scaling function is selected by the initial condition, and (iii) depending on the tail of the initial condition, the scaling exponent which characterizes the scaling function is found to exhibit a transition from a continuously varying to a fixed value.Comment: 4 pages, 3 figures; Published versio

Effective capillary interaction of spherical particles at fluid interfaces

We present a detailed analysis of the effective force between two smooth spherical colloids floating at a fluid interface due to deformations of the interface. The results hold in general and are applicable independently of the source of the deformation provided the capillary deformations are small so that a superposition approximation for the deformations is valid. We conclude that an effective long--ranged attraction is possible if the net force on the system does not vanish. Otherwise, the interaction is short--ranged and cannot be computed reliably based on the superposition approximation. As an application, we consider the case of like--charged, smooth nanoparticles and electrostatically induced capillary deformation. The resulting long--ranged capillary attraction can be easily tuned by a relatively small external electrostatic field, but it cannot explain recent experimental observations of attraction if these experimental systems were indeed isolated.Comment: 23 page

Asymptotically self-similar propagation of the spherical ionization waves

It is shown that a new type of the self-similar spherical ionization waves may exist in gases. All spatial scales and the propagation velocity of such waves increase exponentially in time. Conditions for existence of these waves are established, their structure is described and approximate analytical relationships between the principal parameters are obtained. It is notable that spherical ionization waves can serve as the simplest, but structurally complete and physically transparent model of streamer in homogeneous electric field.Comment: Corrected typos, the more precise formulas are obtaine

Finite-distance singularities in the tearing of thin sheets

We investigate the interaction between two cracks propagating in a thin sheet. Two different experimental geometries allow us to tear sheets by imposing an out-of-plane shear loading. We find that two tears converge along self-similar paths and annihilate each other. These finite-distance singularities display geometry-dependent similarity exponents, which we retrieve using scaling arguments based on a balance between the stretching and the bending of the sheet close to the tips of the cracks.Comment: 4 pages, 4 figure