Mathematical Theory of turbulent flows

Question

I am a mathematician and I'm going to start working in modeling turbulent flows. Mainly with computational simulation, but I want to know the mathematical theory used in turbulence theory. I know that probability is an important field, but I would like to know if differential equations theory, analysis theory, etc., are also useful to study and model turbulent flow.

Finally, can you recommend any good reference about this field?

Answer 1

Any fluid equation is fundamentally going to be studied to some degree via analytic PDE. I should preface this with I don't look much at the statistical, probabilistic, or numerical side of things, but rather the analytic side of things. However it seems to me that the people who do numerics are well versed in the analytic theory. In fact, numerics and analytic theory tend to work together quite often. For example, there is a blowup criterion by Beale, Kato, and Majda which was proved analytically but is used in numerics to check if solutions are blowing up.

Standard introductory texts to mathematical fluid mechanics (prerequisite to studying turbulence in my opinion) are

• Chorin Marsden, A Mathematical Introduction to Fluid Mechanics
• Batchelor, An Introduction to Fluid Dynamics

You will also need a solid foundation in functional analytic PDE theory

• Rudin, Real and Complex Analyis
• Rudin, Functional Analysis
• Evans, Partial Differential Equations
• Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations
• Tsai, Lectures on the Navier Stokes Equations

More advanced texts which are going to be (probably) necessary

• Majda and Bertozzi, Vorticity and Incompressible Flow
• Ladyzhenskaya, Mathematical Theory of Viscous Incompressible Flow

Some standard turbulence specific books are

• Batchelor, The Theory of Homogeneous Turbulence
• Frisch, Turbulence

For Numerics

• Temam, Navier Stokes Equations, Theory and Numerical Analysis

I would say for mathematicians studying fluid flows/turbulence, the analytic stuff is very important, and understanding Sobolev spaces, norms, approximating solutions to prove existence, ect.