## Learning objectives

Knowledge and understanding:

During the course the student will acquire the fundamentals of Fluids Mechanics and will develop the ability of understanding and analyzing critically the physical reason of various phenomena involving fluids in motion or at rest in natural or artificial systems. The investigation method is based on mathematical modeling.

Applying knowledge and understanding

The student will gain ability of applying the fundamentals of Hydraulics to basic problems typical of Civil and Environmental Engineering.

Making judgments:

The student will acquire the basic tools and will develop a critical capacity useful to deal in an independent way with simple problems of Hydraulics.

Communication skills:

At the end of the course, the student will be able to present the concepts learned with an adequate fluency and a good speaking ability.

Learning ability:

At the end of the course, the student will have strengthened basic knowledge and competences about the topic of Hydraulics that later will allow him to extend the theoretical and technical knowledge useful to design and verify simple hydraulic structures.

## Prerequisites

It is compulsory to have passed the exam of mathematical Analysis 2.

It is useful to have attended the other courses of Mathematical analysis, Geometry, Physics, and Pure mechanics.

## Course unit content

Fluid properties. The concept of fluid. The fluid as a continuum. Density, compressibility, vapour pressure, viscosity, surface tension. The stress state and the Cauchy theorem.

Fluid statics. Pressure distribution in a fluid at rest. Statics of incompressible and compressible fluids. Manometry. Hydrostatic forces on plane and curves surfaces. Buoyancy and stability. Rigid-body motion.

Fluid kinematics. Lagrangian and Eulerian description of a velocity field. The total derivative. Kinematic analysis of the fluid motion. Flow patterns: pathlines, streamlines and streaklines. The Reynolds transport theorem.

Fundamental of fluid dynamics. Integral and differential analysis. Basic physical laws: the mass conservation equation, the linear momentum equation, the angular momentum equation, the energy equation. Examples: reservoir emptying, computation of dynamic forces.

Frictionless flow. The ideal fluid model. The Euler equation. The Bernoulli theorem and energetic sense. Example: the Pitot-static tube. Flow through orifices; weirs. Power of a flow. Extension of the Bernoulli theorem to a flow. The Venturi meter.

Viscous flow. The viscous fluid model. The Navier-Stokes equations. Analytical solutions of the Navier-Stokes equations: the Couette flow between fixed and moving plates, the Hagen-Poiseuille flow.

Pipe flow. Laminar and turbulent regimes. Equations of motion. Continuous and minor losses. Resistance laws. Energy exchange between fluid and hydraulic machinery: pumps and turbines. Systems of ducts. Verification and project problems.

Open-channel flow. Uniform flow: the Chèzy formula. Specific energy. The critical state. Subcritical and supercritical flows. Steady gradually varied flow equations in prismatic channel. Hydraulic jump. Plotting longitudinal profiles: examples.

Numerical exercises strengthen the competence about the fundamentals of Fluid Mechanics and allow to acquire the ability of quantifying the physical characteristics of a phenomenon.

## Full programme

Chapter 1. Introduction to hydraulics

Lesson No 1: 1.1 General information on fluids, 1.2 The concept of fluid as continuous, 1.3 Dimensions, units of measurement and fundamental and derived quantities

Lesson No 2: 1.4 Tensions in a fluid, 1.4.1 The Cauchy tetrahedron theorem, 1.4.2 Considerations on the Cauchy tetrahedron theorem, 1.4.3 Rotation of the Cauchy tetrahedron, 1.4.4 Corollary of the tetrahedron theorem of Cauchy for isotropic system.

Lesson No 3: 1.5 Compressibility, density and thermal expansion, 1.6 Surface tension, 1.7 Viscosity, 1.8 Other fluid properties.

Chapter 2. Fluid statics

Lesson No 4: 2.1 Undefined equation of the fluid statics, 2.2 Global equation of the fluid statics.

Lesson No 5: 2.3 Stevino's Law and Pressure Considerations, 2.4 On Pressure Distribution.

Lesson No 6: 2.6 Pushes on flat surfaces, 2.7 Pushes on curved surfaces, 2.8 Floating, 2.9 Hydrostatic paradox.

Lesson No 7: 2.10 Measurement of pressure, 2.11 Small specific gravity fluids (outline), 2.12 Mariotte's formula, 2.12 Relative balance (outline)

Chapter 3. Fluid kinematics

Lesson No 8: 3.1 Motion patterns, 3.2 Speed and acceleration.

Lesson No 9: 3.3 The visualization of a field of motion, 3.4 Flow tubes, 3.5 Types of motion, 3.6 Comparison of regimes and types of motion.

Lesson No 10: 3.9 Undefined continuity equation, 3.10 Global continuity equation for fixed control volumes in space, 3.11 Continuity equation applied to currents.

Chapter 4. Fluid dynamics

Lesson No 11: 4.1 Undefined equation of motion, 4.2 Stokes viscosity law (overview), 4.3 Euler and Navier Stokes equation.

Lesson No 12: 4.4 Typical flow conditions, 4.5 Global equation of the dynamic equilibrium, 4.6 Coefficient of the quantity of momentum.

Lesson no 13: 4.7 Bernoulli's theorem, 4.8 Pressure distribution and gradually varied currents, 4.9 Geometric and energetic significance of the Bernoulli theorem.

Lesson no 14: 4.11 Outflow processes, 4.12 Other applications of the Bernoulli theorem, 4.14 Extension to the currents of the Bernoulli theorem and power of a current, 4.15 Some applications of the extensions of the Bernoulli theorem to the various motions and currents (Venturimeter), 4.16 Extension to real fluids.

Lesson no 15: 4.17 Energy exchange between a current and a machine, 4.18 Applications.

Chapter 5. Dimensional analysis and similarity

Lesson No 16: 5.1 The principle of dimensional homogeneity, 5.2 Some classical engineering equations (overview), 5.3 Theorem Π (definition), 5.4 Typical index numbers in hydraulics (overview), 5.5 Dimensional analysis in physical modeling (outline), 5.6 Similarity and self-similarity (outline).Chapter 6. Pressurized currents

Lesson no 17: 6.2 Dragging action of a current, 6.3 Resistance index, 6.4 Analysis of laminar motion in cylindrical ducts with various sections (circular ducts and parallel flat plates only).

Lesson No 18: 6.5 Index of resistance in laminar motion (circular and general flow), 6.6 Overview of turbulence, 6.9 Medium turbulent motion in a circular section duct, 6.11 Turbulent viscosity and mixing length.

Lesson No 19: 6.13 Composite nature of the turbulent boundary layer, 6.14 Average velocity distributions in incompressible fluids on smooth surfaces (viscous substrate, fully turbulent region of the inner region, transition zone (buffer layer) of the inner region, external region), 6.15 Average velocity distributions in incompressible turbulent flows on rough surfaces with zero pressure gradient, The roughness, 6.16 Average velocity distributions in incompressible turbulent flows in smooth circular cylindrical ducts, 6.17 Average velocity distributions in incompressible turbulent flows in rough circular cylindrical ducts ( Absolutely turbulent motorcycles, turbulent motorcycles in transition).

Lesson No 20: 6.19 Laws of resistance in ducts in the presence of turbulent motion. Case of circular ducts, 6.20 Practical formulas, 6.24 Localized leakages.

Lesson No 21: 6.25 Problems of long pipelines, Verification of a laminar pipeline, Serial and parallel pipelines, 6.26 Pipeline with constant diameter with uniform delivery along the path.

Lesson No 22: 6.27 Duct with a lifting system, 6.29 Possible elevation traces of the pipelines, How to highlight the relative piezometric line and the absolute piezometric line

Relative hydrostatic loads line, in which a part of the pipeline is above the relative piezometric line, in which a portion of the pipeline is above the relative piezometric line and at the maximum point a free vent is installed, Vents and drains, a case where a section of the pipeline is above the relative piezometric line and the line of relative hydrostatic loads

A case where a section of the pipeline is above the relative piezometric line and the relative hydrostatic load line and a vent must be installed at the maximum point, in which case a section of the pipeline exceeds a value greater than the patm / the line joining the piezometric values of the upstream and downstream nodes, in which a section of the pipeline exceeds by a value greater than patm / the line joining the piezometric values of the upstream and downstream nodes and the line of hydrostatic loads relative, a case in which a section of the pipeline exceeds by a value greater than patm / the line joining the piezometric measurements of the upstream and downstream nodes and at the maximum of that same section of the pipeline there is a vent that exceeds the line of relative hydrostatic loads, in which case a part of the pipeline exceeds the line of absolute hydrostatic loads, 6.30 Closed networks.

Chapter 8. Free surface currents

Lesson No 23: 8.1 Classification of free-surface motions, 8.2 Classification of motions in the channels, Some early reflections on the varied motion in the channels, 8.3 Laws of resistance for the channels and case of uniform motion.

Lesson No 24: 8.5 Specific energy assessed with respect to the bottom of the canal section, 8.6 Weak and steep slopes. 8.7 Kinematic character of a current: surface wave velocity.

Lesson No 25: 8.8 Gradually changed motion, 8.9 Tracing of current profiles gradually changed, Rules for the qualitative tracking of current profiles gradually changed.

Lesson No 26: 8.10 Hydraulic survey, Concept and calculation of total thrust, Reduction of bottom slope of a riverbed from i> ic to i <ic, 8.11 Permanent motion profile tracking for integration

Numerical. 8.12 Application examples, Passage through a flat bulkhead, Change of roughness.

Lesson No 27: Passage of a current on a bottom threshold, Passage between the stacks of a bridge.

## Bibliography

Recommended books

Mossa, Petrillo, Idraulica, CEI, 2013 (Available at the Library of Engineering and Architecture – 2 copies, of which 1 can be got on loan)

Citrini D., Noseda G. (1987). Idraulica, CEA, Milano. (Available at the Library of Engineering and Architecture – 4 copies, of which 3 can be got on loan)

Additional books

Marchi E., Rubatta A. (1981). Meccanica dei fluidi, UTET, Torino. (Available at the Library of Engineering and Architecture – 2 copies, of which 1 can be got on loan)

White F.M. (1999). Fluid mechanics, McGraw-Hill, Singapore. (Available at the Library of Engineering and Architecture – 4 copies, of which 2 can be got on loan)

Ghetti A. (1980), Idraulica, Libreria internazionale Cortina, Padova. (Available at the Library of Engineering and Architecture – 3 copies, of which 2 can be got on loan)

Books of exercises

Longo, S., Tanda, M. G., Chiapponi, L., 2021, Problems in Hydraulics and Fluid Mechanics. Springer, https://doi.org/10.1007/978-3-030-51387-0 Print ISBN 978-3-030-51386-3, Online ISBN 978-3-030-51387-0, XVII+391 pp. (Available for loan at the Library of Engineering and Architecture)

Alfonsi G., Orsi E. (1984). Problemi di Idraulica e Meccanica dei fluidi, CEA, Milano. (Available at the Library of Engineering and Architecture – 2 copies, of which 1 can be got on loan)

Longo S., Tanda M.G. (2009). Esercizi di Idraulica e di Meccanica dei fluidi, Springer, Milano. (Available at the Library of Engineering and Architecture – 2 copies, of which 1 can be got on loan)

Lecture slides and additional educational material (downloadable from the webpage of the course on the University web site "Web LEArning in Ateneo")

## Teaching methods

The course is structured in frontal lessons on the blackboard (with projection of slides, if need) in order to explain theoretical aspects and complementary subjects. During the practice exercises, numerical problems are solved. Usually, visits to the hydraulics laboratory are planned so that the student can see the application value of the subject first hand.

## Assessment methods and criteria

The exam consists of an written part followed by an oral part. To gain access to the oral exam, it is necessary to pass the written test. The two parts of the examination must be passed successfully in the same exam session.

Evaluation criteria:

Written examination 50% including:

- Resolution of no.3 exercises (knowledge/proficiency)

Oral examination 50% including:

- Theory questions (knowledge)

- Applications of theory and exercises (proficiency/making judgments)

- Speaking ability (communication skills)