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Theoretical and numerical aspects in fluid dynamics and turbulent flow

General details of the subject

Face-to-face degree course

Description and contextualization of the subject

This course is devoted to the modeling of equations of fluid dynamics in the presence of turbulence, vortices and stochastic flows. The knowledge about these dynamics is essential for the design of durable mechanisms that optimize the capture of energy from either wind or sea forces. Some nonlinear PDE which are susceptible of being studied analytically and numerically are relevant within this context. Moreover, the stochastic Burgers and Navier-Stokes equations are useful in the presence of nondeterministic forces.

The objective of the course is to provide students with a mathematical description of the emergence and propagation of some types of singularities in fluid dynamics. We show the equations modeling these phenomena and we analyze some particular solutions of special interest. Due to the limitations of analytical methods for solving nonlinear partial differential equations and stochastic differential equations that appear in this framework, some sophisticated numerical schemes are proposed. Students will focus on the programming of numerical methods to make an efficient use of them.

Teaching staff

NameInstitutionCategoryDoctorTeaching profileAreaE-mail
DE LA HOZ MENDEZ, FRANCISCOUniversity of the Basque CountryProfesorado AgregadoDoctorBilingualApplied
GORRIA CORRES, CARLOSUniversity of the Basque CountryProfesorado AgregadoDoctorBilingualApplied
VEGA GONZALEZ, LUISUniversity of the Basque CountryProfesorado Catedratico De UniversidadDoctorNot bilingualMathematical


Learning about the partial differential equations of fluid dynamics, the physical laws that lead into these equations and the assumptions taken in the formulation25.0 %
Knowing the mathematical concepts of vortex filaments, sheets and patches and visualizing the evolution of these types of solutions25.0 %
Understanding the concept of stochastic forces in fluids and introducing its effect into the equations25.0 %
Being able to program nontrivial numerical schemes to solve partial differential equations with singularities and stochastic differential equations25.0 %

Study types

TypeFace-to-face hoursNon face-to-face hoursTotal hours
Applied classroom-based groups404
Applied computer-based groups81220

Training activities

NameHoursPercentage of classroom teaching
Application Workshops12.00 %
Classroom/Seminar/Workshop4.0100 %
Exercises33.00 %
Expositive classes18.0100 %
Working with it equipment8.0100 %

Assessment systems

NameMinimum weightingMaximum weighting
Drawing up reports and presentations40.0 % 60.0 %
Practical tasks40.0 % 60.0 %


Lesson 1 Basic notions of turbulent flows

From the Euler equations and by taking some assumptions, the formulation can be transformed into some nonlinear PDE suitable for being studied analytically and numerically.

Lesson 2 Vortex filaments, sheets and patches

The analysis of the self-similar solutions of the PDE involved in fluid dynamics and the numerical simulations help to understand the origin and evolution of singularities.

Lesson 3 Burgers and Stokes equations under stochastic forces

In the presence of nondeterministic forces the model turns to the stochastic Burgers and Navier-Stokes equations. Design of implicit numerical methods for approximate solutions.


Basic bibliography

Uriel Frisch, Turbulence, the Legacy of A. N. Kolmogorov, Cambridge University Press, 1995.

Andrew J. Majda, Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge University Press, 2002.

Philip G. Saffman, Vortex dynamics, Cambridge University Press, 1992.

Alexandre J. Chorin, Vorticity and Turbulence, Springer, 1994.

Hiroshi Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990.

Peter E. Kloeden, Eckhard Platen, Numerical Solution of Stochastic Differential Equations, Springer, 1992.

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