EU Fluid Mechanics - 4PUFMFLU

Goal(s)

Mathematics section (8h)
The objective is to develop the student’s ability to apply the fundamental tools of vector analysis and ordinary differential equations in typical mathematical situations, in particular:
– correctly manipulating the operators (gradient, divergence, curl, Laplacian),
– appropriately using the major integral theorems (Green’s Theorem, Stokes’ Theorem, Gauss’ Divergence Theorem),
– solving first?order ordinary differential equations,
– performing and using a Taylor expansion.

Fluid Mechanics (28 h)
This course introduces the fundamentals of fluid mechanics for engineers. The following topics are covered: fluid properties and mathematical description, hydrostatic, kinematics, integral relations for a fluid flow (conservation laws), differential relations for a fluid flow, dynamic similarity, irrotational flow, laminar flow, boundary layers and a brief introduction to turbulence. This course in taught in English.

Content(s)

Vector analysis (gradient, divergence, curl, as well as the derived operator Laplacian, together with the main integral theorems: Green’s Theorem, Stokes’ Theorem, and the Divergence (Gauss) Theorem) + first?order ordinary differential equations, Taylor expansion.

Fluid Mechanics:
Chapter 1 – Introduction (Review):
Solid, liquid and gases. Concept of fluid. Newtonian fluid. Continuum hypothesis. Fluid properties. Types of fluid forces. Mathematical description of a fluid. Scalars, Vector and Tensors. Symmetric and antisymmetric tensors. Basic operations. Gradient, Divergence and Curl. Gauss’ and Stokes’ Theorems. Force on a surface and the stress tensor.

Chapter 2 – Hydrostatic (Review):
Statics Fluid: examples. Pressure and pressure gradient. Equilibrium of a fluid element. Hydrostatic pressure distribution. Pressure distribution in a rigid body motion. Surface tension. Buoyancy and stability.

Chapter 3 - Fluid kinematics:
Eulerian and Lagragian descriptions of a fluid. Flow lines. Material derivate: flow acceleration, local and convective accelerations. Euler equation. Motion equations for inviscid flow: Bernoulli equation. Strain and vorticity tensors.

Chapter 4 - Integral relations for a fluid flow (conservation laws):
Basic physical laws of fluid mechanics. Control volume: fixed, moving and deformable. Rule of Leibniz. Reynolds transport theorem. Conservation of mass. Linear momentum equation. Angular momentum equation. Energy equation.

Chapter 5: Differential relations for fluid flow:
Differential equation of mass conservation. Incompressible flow. Differential equation of linear momentum. Navier-stokes equations: compressible and incompressible cases. Differential equation of angular momentum. Differential equation of energy. Boundary conditions for the basic equations.

Chapter 6 - Dimensional analysis and similarity:
Geometric, kinematic and dynamic similarities. Non-dimensionalization of the basic equations. Principle of dimensional homogeneity. The Pi Theorem. Important dimensionless parameters.

Chapter 7 - Ideal flow:
Introduction. Relevance of irrotational constant density flow theory. Stream function. Geometric interpretation. Velocity potential. Orthogonality of streamlines and potential lines. Frictionless irrotational flows: Bernoulli equation. Generation of rotationality. Construction of elementary flows in two dimensions. Complex potential.

Chapter 8 - Laminar flow:
Introduction. Development length. Exact solutions for steady incompressible viscous laminar flow in very simple geometries: Couette flow, Hagen-Poiseville flow in a pipe. Creeping flow and theory of lubrication. Laminar boundary layer. Boundary layer on a flat plate: Blasius solution. Transition, pressure gradient and boundary layer separation.

Chapter 9 – Introduction to Turbulence and Computational Fluid Dynamics (CFD) (Optional)
Flow governing equations. Boussinesq Approximation. Turbulence versus laminar flow. Turbulence characteristics. Turbulence RANS equations. Reynolds stress and wall functions. Overview of Computational Fluids Dynamics (CFD). Use of CFD in nuclear reactors. General guidelines.

Mathematics section (8h)
The prerequisites for this part are a solid command of differential and integral calculus, knowledge of standard elementary functions, basic linear algebra, as well as fundamental notions concerning vector fields.

Fluid Mechanics
• Continuum mechanics
• Linear algebra
• Vector analysis
• Advanced calculus, including: Gauss’s and Stokes’s theorems, Partial differential equations (PDEs), Multivariable integration and differentiation and Complex variables.

Test

Mathematics section: written exam (1h).

Fluid Mechanics: 2-hour written examination. Students are permitted one handwritten double-sided A4 sheet and a calculator. The same conditions apply for the second examination session.

Additional Information

This course brings 2.5 ECTS to students in Engineer APPRENTISSAGE MEP 2Y

Bibliography

[1] “Fluid mechanics”, P. Kundu, I. Cohen and D. Dowling, 5th ed. (2012), Elsevier, Waltham, USA.
[2] “Fluid mechanics”, F. White, 5th ed. (2005), McGraw-Hill education, New York, USA.
[3] “Computational Fluid Dynamics: the basics with applications”, J. D. Anderson, 1995, McGraw-Hill education, New york, USA.
[4] “Essential Computational Fluid Dynamics”, Zikanov, Wiley & Son (2010).