Experiments were performed in an Couette ring shear cell with particles from the food industry of nearly spherical geometry and nominal diameters between 2.0 and 6.7 mm and with quartz grains with 2.1 to 3.3 mm diameter to examine whether rapid shear flow would give rise to an appreciable change in the granular properties. In particular, it was suspected that surface roughness would change during shearing and lead to changes in the values of the internal and wall friction angles. Measuring these angles corroborated the supposition that abrasion of the particles reduced their frictional resistance, both via the internal and wall friction angles. This demonstrates that roughness changes may occur in granular flows and should in those situations not be ignored in the theoretical descriptions.

Fluid viscosity is a main feature of fluids; an inviscid fluid does not exist even though a large number of theories has been advanced for flows of such fluids. The velocity of fluid flow may considerably be reduced due to the presence of fluid viscosity both for laminar and turbulent flows. The letters exchanged between de Saint-Venant (dSV) and Boussinesq mainly refer to laminar flow. The most peculiar statement of dSV is that, if the flow in a typical lowland river would be laminar, its velocity would be larger than the speed of sound. It is evident that this statement is wrong because laminar flow has stringent limitations typically expressed by the Reynolds number. The hydraulic resistance is another peculiar feature in hydrodynamics, given that several theories have been advanced which do not at all reflect the everyday experience. One of these is the d'Alembert paradox, stating that the resistance is equal to zero for a steady flow of an inviscid and incompressible fluid. Typically, a body suspended in a large pipe is considered. Applying the momentum equation in the axial direction sufficiently up- and downstream of the body, the resistance would indeed become zero, as occurs for potential flows. Again, this is far away from everybody's experience, such as for a swimmer or a walker under heavy wind. This paper mainly explores the letters exchanged between the two late 19th century scientists dSV and Boussinesq. Their considerations indirectly advanced the turbulence theory later developed by Boussinesq, based on the Boussinesq turbulent exchange coefficient. It also rectifies the role of dSV in improving the fundamental equations currently referred to as the Navier-Stokes equations. Both aspects of turbulence and hydraulic resistance are by now still under considerable research activity.

Even though applications of direct numerical simulations are on the rise, today the most usual method to solve turbulence problems is still to apply a closure scheme of a defined order. It is not the case that a rising order of a turbulence model is always related to a quality improvement. Even more, a conceptual advantage of applying a lowest order turbulence model is that it represents the analogous method to the procedure of introducing a constitutive equation which has brought success to many other areas of physics. First order turbulence models were developed in the 1920s and today seem to be outdated by newer and more sophisticated mathematical-physical closure schemes. However, with the new knowledge of fractal geometry and fractional dynamics, it is worthwhile to step back and reinvestigate these lowest order models. As a result of this and simultaneously introducing generalizations by multiscale analysis, the first order, nonlinear, nonlocal, and fractional Difference-Quotient Turbulence Model (DQTM) was developed. In this partial review article of work performed by the authors, by theoretical considerations and its applications to turbulent flow problems, evidence is given that the DQTM is the missing (apparent) constitutive equation of turbulent shear flows.

The original version of the book was inadvertently published with an error in Table 11.2. Owing to an oversight Table 7.2 was mistakenly duplicated as Table 11.2. The chapter has now been corrected and approved by the author. The correct presentation is included now.

The Shallow-Water Equations (SWEs), also referred to as the de Saint-Venant equations, constitute the current governing mathematical tool for free-surface water flows. These include, e.g., flood flows in rivers and in urban zones, flows across hydraulic structures as dams or wastewater facilities, flows in the environmental fields, glaciology, or meteorology. Despite this attractiveness, the system of two partial differential equations has an exact mathematical solution only for a limited number of problems of practical relevance. This historical work on the SWEs is based on a correspondence between two 19th-century scientists, de Saint-Venant and Boussinesq. Their well-known papers are thus commented from the point of development of their theory; the input of both scientists is evidenced by their writings, and comments of both to each other that led to what is commonly known as the SWEs. Given the age difference of the two of 45 years, the experienced engineer de Saint-Venant, and the mathematician Boussinesq, two eminent researchers, met to discuss not only problems in hydraulics, but in physics generally. In addition, their correspondence embraced also questions in ethics, religion, history of sciences, and personal news. The results of the SWEs cease to hold if streamline curvature effects dominate; this includes breaking waves, solitary and cnoidal waves, or non-linear waves in general. In most other cases, however, the SWEs perfectly apply to typical flows in engineering practice; they are considered the fundamental system of equations describing open channel flows. This work thus provides a background to its birth, including lots of comments as to its improvement, physical meanings, methods of solution, and a discussion of the results. This paper also deals with the steady flow equations, gives a short account on the main persons mentioned in the Correspondence, and provides a summary of further developments of the SWEs until 1920. (C) 2019 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.

In zero-equation turbulence modeling the Reynolds shear stress is usually assumed to be an order of magnitude larger than the viscous shear stress. Therefore, this kind of modeling is only accurate for flows at Reynolds numbers Re well above criticality, Rec. New nonlocal and fractional turbulence models, as e.g. the Difference-Quotient Turbulence Model (DQTM), are based on a self-similar cascade of turbulent eddies of diameters which are related to Lévy flights of corresponding jump sizes. The smaller the Reynolds number is, the smaller will be the number of eddy classes and their total momentum transfer by eddy motion. By tending from above to a Reynolds number close to its critical value, only a single class of eddies with the largest eddy diameter remains. In this special case the diameter of these remaining eddies is identical to the Kolmogorov dissipation length that now equals the characteristic overall length scale of the flow domain. Below criticality even these largest eddies diminish and the flow lacks eddies of all sizes and, therefore, is laminar. Based on these ideas and Kolmogorov’s microscales the DQTM is generalized. This new extension leads to a vanishing Reynolds shear stress at criticality. The derived nonlocal multiplicative correction term in the DQTM is our main result and can be easily split off and combined with other turbulence models.

This paper provides short excerpts of the beginnings of glacier ice research. It is restricted to the period from the beginnings in the 17th century up to the formulation of a flow law in the fifties of the forgoing century. Its role in climate research is not touched; it will be discussed in another publication.

Goodman Goodman and Cowin Cowin (1972) (Goodman and Cowin (1971). J. Fluid Mech. 45, 321–339. [6]) proposed a continuum theory of a dry cohesionless granular material in which the solid volume fraction $$\nu $$ is treated as an independent kinematic field for which an additional balance law of equilibrated forces is postulated. They motivated this additional balance law as an equation describing the kinematics of the microstructure and employed a variational formulation for its derivation. By adopting the müller Müller –Liu Liu approachMüller–Liu approach to the exploitation of the entropy inequality we show that in a constitutive model containing $$\nu , \dot{\nu }$$ and $$\mathrm {grad}\,\nu $$ as independent variables, results agree with the classical Coleman Coleman –Noll Noll approachColeman–Nollapproach only, provided the Helmholtz Helmholtz free energyHelmholtz free energy does not depend on $$\dot{\nu }$$ , for which the Goodman–Cowin equations are reproduced. This reduced theory is then applied to analyses of steady fully developed horizontal shearing flows and gravity flows of granular materials down an inclined plane and between parallel plates. It is demonstrated that the equations and numerical results presented by Passman Passman et al. (1980) (Passman, Nunziato, Bailey and Thomas (1980). J. Eng. Mech. Div. ASCE 106, 773–783. [15]) are false, and they are corrected. The results show that the dynamical behavior of these materials is quite different from that of a viscous fluid. In some cases, the dilatant shearing layers exist only in the narrow zones near the boundaries. They motivated this additional balance law as an equation describing the kinematics of the microstructure and employed a variational formulation for its derivation. In an appendix, we present a variational formulation, treating the translational velocity and solid volume fraction as generalized coordinates of a Lagrange Lagrange an formulation.

The extended thermodynamics of Tsallis is reviewed in detail and applied to turbulence. It is based on a generalization of the exponential and logarithmic functions with a parameter q. By applying this nonequilibrium thermodynamics, the Boltzmann-Gibbs thermodynamic approach of Kraichnan to 2-d turbulence is generalized. This physical modeling implies fractional calculus methods, obeying anomalous diffusion, described by Lévy statistics with q < 5/3 (sub diffusion), q = 5/3 (normal or Brownian diffusion) and q > 5/3 (super diffusion). The generalized energy spectrum of Kraichnan, occurring at small wave numbers k, now reveals the more general and precise result k−q. This corresponds well for q = 5/3 with the Kolmogorov-Oboukov energy spectrum and for q > 5/3 to turbulence with intermittency. The enstrophy spectrum, occurring at large wave numbers k, leads to a k−3q power law, suggesting that large wave-number eddies are in thermodynamic equilibrium, which is characterized by q = 1, finally resulting in Kraichnan’s correct k−3 enstrophy spectrum. The theory reveals in a natural manner a generalized temperature of turbulence, which in the non-equilibrium energy transfer domain decreases with wave number and shows an energy equipartition law with a constant generalized temperature in the equilibrium enstrophy transfer domain. The article contains numerous new results; some are stated in form of eight new (proven) propositions.