Lecture notes on Fluid Mechanics (in Italian)

Theory

1. Introduzione v2.9
2. Statica v1.2
3. Cinematica v1.3
4. Dinamica v1.2
5. Fluidi ideali v1.3
6. Analisi dimensionale v1.0
7. Fluidi viscosi: moto in condotte v1.5
8. Fluidi viscosi: equazioni generali v2.3
9. Turbolenza v1.3
10. Flussi esterni v1.0

Exercise

1. Statica v3.2
2. Dinamica v3.2

 Dry Granular Flows

Granular flows are collections of many solid moving particles, whose mass is so large (at least 1020 molecular masses) that Brownian motion is irrelevant. Also, the particles dissipate energy whenever they interact. Granular flows can be considered dry when the interstitial fluid, even if present, exerts negligible forces on the particles.

Examples of dry granular flows range from Civil (landslides, snow avalanches) to Industrial (transport of seeds, minerals and pills) Engineering to Astrophysics (planetary rings).

The topic is still relatively young from the Scientific point of view, so that many fundamental questions are still open. In particular, here at the Politecnico di Milano, we have focused on relaxing some of the demanding assumptions of kinetic theories of granular gases to deal with more realistic particles.

Role of velocity correlation

At volume fractions larger than 0.49, the molecular chaos assumption fails. Velocity correlation induced by the particle inelasticity leads to overestimate the collisional dissipation rate. A phenomenological correction to the dissipation rate of classical kinetic theory of granular gases allows to quantitatively reproduce the results of numerical simulations and physical experiments in a number of flow configurations.

•  Berzi, D. & Jenkins, J.T. 2018. Fluidity, anisotropy, and velocity correlations in frictionaless, collisional grain flows. Phys. Rev. Fluids, 3 (9), 094303. Copyright (2018) APS. The article may be found at http://dx.doi.org/10.1103/PhysRevFluids.3.094303.
•  Berzi, D. 2014. Extended kinetic theory applied to dense, granular, simple shear flows. Acta Mech., 225(8), 2191-2198. The final publication is available at Springer via http://dx.doi.org/10.1007/s00707-014-1125-1.
•  Jenkins, J.T. & Berzi, D. 2012. Kinetic Theory applied to Inclined Flows. Granul. Matter, 14(2), 79 84. The final publication is available at Springer via http://dx.doi.org/10.1007/s10035-011-0308-x.
•  Berzi, D. & Jenkins, J.T. 2011. Surface Flows of Inelastic Spheres. Phys. Fluids, 23(1), 013303. Copyright (2011) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article may be found at http://dx.doi.org/10.1063/1.3532838.
•  Jenkins, J.T. & Berzi, D. 2010. Dense Inclined Flows of Inelastic Spheres: Tests of an Extension of Kinetic Theory. Granul. Matter, 12 (2), 151 158. The final publication is available at Springer via http://dx.doi.org/10.1007/s10035-010-0169-8.

Role of friction

Surface friction determines the volume fraction at which both pressure and shear stress are singular in the random collisional, shearing flow of spheres. This is linked with the development of yield stress in granular materials. 

•  Berzi, D. & Vescovi, D. 2015. Different singularities in the functions of extended kinetic theory at the origin of the yield stress in granular flows. Phys. Fluids, 27(1), 013302. Copyright (2015) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article may be found at http://dx.doi.org/10.1063/1.4905461.

Role of stiffness

For soft particles, the frequency of collisions decreases and rate-independent components to the stresses are possible. Our predictions for simple shearing are in agreement with numerical simulations for stiffnesses ranging from that of 1 mm glass spheres to that appropriated for soccer balls or even softer particles.

•  Vescovi, D., Berzi, D. & di Prisco, C.G. 2018. Fluid-solid transition in unsteady, homogeneous, granular shear flows. Granul. Matter, 20(2), 27. The final publication is available at Springer via http://dx.doi.org/10.1007/s10035-018-0797-y.
•  Berzi, D. & Jenkins, J.T. 2015. Steady shearing flows of deformable, inelastic spheres. Soft Matter, 11(24), 4799-4808. Copyright (2015) Royal Society of Chemistry. The article may be found at http://dx.doi.org/10.1039/C5SM00337G.
•  Berzi, D. & Jenkins, J.T. 2015. Correction: Steady shearing flows of deformable, inelastic spheres. Soft Matter, 11(29), 5970-5970. Copyright (2015) Royal Society of Chemistry. The article may be found at http://dx.doi.org/10.1039/C5SM90114F.
•  Vescovi, D., Di Prisco, C.G. & Berzi, D. 2013. From solid to granular gases: the steady state for granular materials. Int. J. Numer. Anal. Met., 37, 2937 2951. This is the peer reviewed version of the article, which has been published in final form at http://dx.doi.org/10.1002/nag.2169. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.
•  Berzi, D., Di Prisco, C.G. & Vescovi, D. 2011. Constitutive relations for steady, dense granular flows. Phys. Rev. E, 84 (3), 031301. Copyright (2011) APS. The article may be found at http://dx.doi.org/10.1103/PhysRevE.84.031301.

Role of boundaries

Different types of boundaries (rigid, bumpy and erodible) have a profound influence on the characteristics of the flow.

•  Berzi, D., Jenkins, J.T. & Richard, P. 2020. Extended kinetic theory for granular flow over and within an inclined erodible bed. J. Fluid Mech., 885, A27. Copyright (2020) Cambridge University Press. The article may be found at http://dx.doi.org/10.1017/jfm.2019.1017.
•  Berzi, D., Jenkins, J.T. & Richard, P. 2019. Erodible, granular bed are fragile. Soft Matter, 15, 7173. Copyright (2019) Royal Society of Chemistry. The article may be found at http://dx.doi.org/10.1039/C9SM01372E.
•  Berzi, D. & Vescovi, D. 2017. Shearing flows of frictionless spheres over bumpy planes: slip velocity. Comp. Part. Mech., 4(4), 373-377. The final publication is available at Springer via http://dx.doi.org/10.1007/s40571-016-0115-6.
•  Gollin, D., Berzi, D. & Bowman, E.T. 2017. Extended kinetic theory applied to inclined granular flows: role of boundaries. Granul. Matter, 19(3), 56. The final publication is available at Springer via http://dx.doi.org/10.1007/s10035-017-0738-1.
•  Berzi, D. & Jenkins, J.T. 2016. Erosion and deposition in depth-averaged models of dense, dry, inclined, granular flows. Phys. Rev. E, 94 (5), 052904. Copyright (2016) APS. The article may be found at http://dx.doi.org/10.1103/PhysRevE.94.052904.
•  Berzi, D. & Jenkins, J.T. 2015. Inertial shear bands in granular materials. Phys. Fluids, 27(3), 033303. Copyright (2015) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article may be found at http://dx.doi.org/10.1063/1.4914920.
•  Vescovi D., Berzi D., Richard P. and Brodu N. 2014. Plane shear flows of frictionless spheres: Kinetic theory and 3D soft-sphere discrete element method simulations. Phys. Fluids, 26(5), 053305. Copyright (2014) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article may be found at http://dx.doi.org/10.1063/1.4879267.

Role of shape

Non-spherical particles may present peculiar features such as alignment ordering typical of liquid crystals.

•  Berzi, D., Thai-Quang, N., Guo, Y. & Curtis, J. 2017. Collisional dissipation rate in shearing flows of granular liquid crystals. Phys. Rev. E, 95, 050901(R). Copyright (2017) APS. The article may be found at http://dx.doi.org/10.1103/PhysRevE.95.050901.
•  Berzi, D., Thai-Quang, N., Guo, Y. & Curtis, J. 2016. Stresses and orientational order in shearing flows of granular liquid crystals. Phys. Rev. E, 93, 040901(R). Copyright (2016) APS. The article may be found at http://dx.doi.org/10.1103/PhysRevE.93.040901.

 Granular-Fluid Flows

In granular-fluid flows, the role of the intertitial fluid cannot be neglected. Examples of granular-fluid flows range from Civil (debris flows, wind-blown sand and sediment transport in water) to Industrial (slurries, aerosols) to Bio-Engineering (transport of red blood cells).

Here at the Politecnico di Milano, we have focused on the transport of particles in a shearing fluid, driven by either gravity, pressure-gradient or a moving boundary. 

Bedload: Saltation

Saltation (motion of particles through successive jumps) is the main mode of transport of sand particles in air (Aeolian saltation) and might have a role in the case of aquatic sediment transport. Saltation on other planets (Exo-saltation) can also be described.

•  Berzi, D., Jenkins, J.T. & Valance, A. 2017. The threshold for continuing saltation on Earth and other solar system bodies. J. Geophys. Res.: Earth Surf., 122(7), 1374-1388. Copyright (2017) American Geophysical Union. The edited version of the article may be found at http://dx.doi.org/10.1002/2016JF003982.
•  Berzi, D., Jenkins, J.T. & Valance, A. 2016. Periodic saltation over hydrodynamically rough beds: aeolian to aquatic. J. Fluid Mech., 786, 190-209. Copyright (2016) Cambridge University Press. The article may be found at http://dx.doi.org/10.1017/jfm.2015.601.

Collisional suspensions (and debris flows)

Collisional suspensions are granular-fluid flows in which the weight of the particles is balanced by repulsive forces due to interparticle-collisions. A particular case of collisional suspension is debris flow, a gravity-driven flow in which the height of the particles and the height of the fluid with respect to a boundary substantially coincides.

•  Berzi, D. & Fraccarollo, L. 2015. Turbulence Locality and Granularlike Fluid Shear Viscosity in Collisional Suspensions. Phys. Rev. Lett., 115, 194501. Copyright (2015) American Physical Society. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Physical Society. The article may be found at  http://dx.doi.org/10.1103/PhysRevLett.115.194501.
•  Berzi, D. & Fraccarollo, L. 2013. Inclined, collisional sediment transport. Phys. Fluids, 25(10), 106601. Copyright (2013) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article may be found at http://dx.doi.org/10.1063/1.4823857.
•  Berzi, D. 2013. Simple shear flow of collisional granular-fluid mixtures. J. Hydraul. Eng.-ASCE, 139(5), 547 549. Copyright (2013) American Society of Civil Engineers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Society of Civil Engineers. The article may be found at http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000701.
•  Berzi, D. & Larcan, E. 2013. Flow resistance of inertial debris flows. J. Hydraul. Eng.-ASCE, 139(2), 187 194. Copyright (2013) American Society of Civil Engineers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Society of Civil Engineers. The article may be found at http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000664.
•  Berzi, D., Bossi, F.C. & Larcan, E. 2012. Collapse of granular-liquid mixtures over rigid, inclined beds. Phys. Rev. E, 85 (5), 051308. Copyright (2012) APS. The article may be found at http://dx.doi.org/10.1103/PhysRevE.85.051308.
•  Berzi, D. 2011. Analytical solution of collisional sheet flows. J. Hydraul. Eng.-ASCE, 137(10), 1200 1207. Copyright (2011) American Society of Civil Engineers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Society of Civil Engineers. The article may be found at http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000420.
•  Berzi, D., Jenkins, J.T. & Larcher, M. 2010. Debris Flows: Recent Advances in Experiments and Modeling. Adv. Geophys., 52 (C), 103 138. The final publication is available at Elsevier via http://dx.doi.org/10.1016/S0065-2687(10)52002-8.
•  Berzi, D. & Jenkins, J.T. 2009. Steady inclined flows of granular-fluid mixtures. J. Fluid Mech., 641, 359 387. Copyright (2009) Cambridge University Press. The article may be found at http://dx.doi.org/10.1017/S0022112009991510.
•  Berzi, D. & Jenkins, J.T. 2008. Approximate analytical solutions in a model for highly concentrated granular-fluid flows. Phys. Rev. E, 78 (1), 011304. Copyright (2008) APS. The article may be found at http://dx.doi.org/10.1103/PhysRevE.78.011304.
•  Berzi, D. & Jenkins, J.T. 2008. A theoretical analysis of free-surface flows of saturated granular-liquid mixtures. J. Fluid Mech., 608, 393 410. Copyright (2008) Cambridge University Press. The article may be found at http://dx.doi.org/10.1017/S0022112008002401.

Turbulent-Collisional suspensions

In turbulent-collisional suspensions the weight of the particles is partially balanced by repulsive forces due to interparticle-collisions and turbulent lift.

•  Berzi, D. & Fraccarollo, L. 2016. Intense sediment transport: Collisional to turbulent suspension. Phys. Fluids, 28, 023302. Copyright (2016) American Institute of Physics. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Institute of Physics. The article may be found at http://dx.doi.org/10.1063/1.4941770.
•  Berzi, D. 2013. Transport formula for collisional sheet flows with turbulent suspension. J. Hydraul. Eng.-ASCE, 139(4), 359 363. Copyright (2013) American Society of Civil Engineers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the American Society of Civil Engineers. The article may be found at http://dx.doi.org/10.1061/(ASCE)HY.1943-7900.0000686.

Freely available Matlab codes

•  Codes to solve for steady, inclined, dry granular flows over rigid, bumpy beds with and without frictional sidewalls.
•  Codes to solve for steady, inclined, dry granular flows over and within an erodible granular bed.