Valeriy M. Prostokishin

On the problem of turbulent flows in pipes at very large Reynolds numbers (reply to comment by I I Vigdorovich [Phys. Usp. 58 (2015); Usp. Fiz. Nauk 185 213 (2015)] on "Turbulent flows at very large Reynolds numbers: new lessons learned" [Phys. Usp. 57 250 (2014); Usp. Fiz. Nauk184 265 (2014)]), Barenblatt G.I., Chorin A.J., Prostokishin V.M.

In [Comm. Appl. Math. Comput. Sci., 4 (2009), pp. 153--175], Barenblatt presents a model for partial laminarization and acceleration of shear flows by the presence of suspended particles of different sizes, and provides a formal asymptotic analysis of the resulting velocity equation. In the present paper we revisit the model. In particular we allow for a continuum of particle sizes, rewrite the velocity equation in a form which involves the Laplace transform of a given function or measure, and provide several rigorous asymptotic expansions for the velocity. The model contributes to a better insight to the extreme velocities in hurricanes, fire storms, and dust storms, and the analysis confirms Barenblatt's conclusion that often the smallest suspended particles are responsible for the extreme flow acceleration at large altitudes.

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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.

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.

We demonstrate that the processing of the experimental data for the average velocity profiles obtained by J. M. Österlund presented in Phys. Fluids 12, 1–4 (2000) 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.

The equation ∂ _t u = u∂ _xx ² u − ( c − 1)(∂ _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.

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 adjacent to the viscous sublayer next to the wall. 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.