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

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 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

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 Structures in Turbulent Boundary Layers At Large Reynolds Numbers

Processing the data from a large variety of zero-pressure-gradient boundary layer flows shows that the Reynolds-number-dependent scaling law, which the present authors obtained earlier for pipes, gives an accurate description of the velocity distribution in a self-similar intermediate region of distances from the wall adjacent to the viscous sublayer. The appropriate length scale that enters the definition of the boundary layer Reynolds number is found for all the flows under investigation. Another intermediate self-similar region between the free stream and the first intermediate region is found under conditions of weak free stream turbulence. The effects of turbulence in the free stream and of wall roughness are assessed, and conclusions are drawn.Comment: 77 pages, (includes 61 charts and graphs