Teaching

16.003 Unified Engineering: Fluid Dynamics (Spring)

Presents fundamental principles and methods of fluid dynamics for aerospace engineering, and engineering analysis and design concepts applied to aerospace systems. Topics include aircraft and aerodynamic performance, conservation laws for fluid flows, quasi-one-dimensional compressible flows, shock and expansion waves, streamline curvature, potential flow modeling, an introduction to three-dimensional wings and induced drag. Experiential lab and aerospace system projects provide additional aerospace context.

16.18 Fundamentals of Turbulence (Fall)

Aims at introducing the fundamentals of turbulent flows along with the mathematical tools for turbulence research. Topics range from the classic viewpoint of turbulence to the theories developed in the last decade. The content combines theory, data science and numerical simulations, and is designed for a wide audience of students in the areas of aerospace, mechanical engineering, geophysics, and astrophysics. The main goal of the course is to give the students a solid background in 1) the fundamental concepts of turbulence, 2) the tools utilized in turbulence research, and 3) the numerical methods with different degree of fidelity to compute turbulent flows. The course also breaks new ground in discussing the latest advances in fluid mechanics using machine learning, operator theory, modal decomposition, etc.

Syllabus:

Introduction:

- The origin of turbulence: what is turbulence?

- Reynolds experiment

- Properties of turbulent flows

- The coherent structure of turbulence

The basic equations of fluid motion:

- Recap of fluid mechanics

- The continuum hypothesis

- The equations of motion for fluids

- Symmetries in the equations

- Experiments of Hagens and Darcy

- The continuum hypothesis revisited

- The Millennium problem

- Navier--Stokes Eqs. for small viscosity vs. Euler Eqs.

- Non-linearity in Navier--Stokes Eqs.

- Fractals and velocity gradients

- The role of the pressure

- The intractability of turbulence: The three N's:
   - Non-integrability
   - Non-linearity
   - Non-locallity

- Characteristic scales of turbulence

- Vortex dynamics:
    - Vorticity equation
    - Interpretation: diffusion + advection
    - 2D 'turbulence' vs. 3D turbulence

Statistical description of turbulence:

- History timeline

- What does 'random' mean in turbulence?

-Probability theory:
    - Probability mass/density functions
    - Statistical moments.  
    - Joint probability distributions. 

- Stationarity, homogeneity, isotropy

- The RANS equations

- The fluctuations equations

- Interactions mean-fluctuations

- Structure functions

- Correlations

- Spectra
    - Fourier transform/series. 
    - Parseval's theorem. 
    - Kinetic energy and dissipation spectra.
    - Premultiplied spectra. 
    - Scale-invariance and power laws.

Isotropic turbulence:

- Canonical turbulent flows.

- Fourier series representation of Navier--Stokes eqs.

- The energy cascade in Fourier space

- The energy cascade in Fourier space

- Kolmogorov hypothesis

- Spectral ranges: energy-containing, inertial, dissipation

- Intermittency & refined Kolmogorov hypothesis

- Comments about 2-D spectra

Free-shear turbulence:

- Typical free-shear flows

- Homogeneous shear turbulence

- Conditions for self-similarity

- Self-similarity in free-shear flows

Wall-bounded turbulence:

- Typical wall-bounded turbulent flows

- Turbulent channel flow

- Characteristic scales of wall turbulence

- Normalizations and layer structure

- The energy and momentum cascades

- Dimensional analysis and layer structure

- The energy and momentum cascades

- Turbulent kinetic energy budget

- Prandtl's mixing-length hypothesis

- Townsend's wall-attached eddy model

- Coherent structures in wall turbulence

- Structural models of wall turbulence: 

    - Hairpin packet model.
    - Townsend attached-eddy model.
    - Current data-driven models.

- Turbulent boundary layers:
    - Flow set-up.
    - Key features. 
    - Boundary layer thickness.     
    - Mean momentum equation.   
    - von Karman integral momentum equation. 
    - Favorable and adverse pressure gradients.
    - Turbulent/non-turbulent interface.    
    - Inner layer of TBL vs. turbulent channel flows. 

    - Kinetic energy budget TBL vs. channel flows 

- Effect of wall roughness in turbulence:

    - All walls are "rough"

    - Equivalent sand-grain roughness

    - Cf in laminar vs. smooth/rough wall turbulence

    - Wall roughness regimes: 

       -Hydraulically smooth regime
      - Fully rough regime
      - Transitional regime

    - The law of the wall with roughness

Laminar-to-turbulent transition:

- Transition stages in boundary layer transition
    - Modal instability
    - Non-modal instability (transient growth)
    - Bypass transition

- Non-linear disturbance equation

- Stability analysis
    - Definition of stability 
    - Critical Reynolds number
    - The Reynolds-Orr equation

- Linear stability analysis

- Inviscid linear analysis for periodic disturbance
    - The Rayleigh equation
    - Rayleigh inflexion point criterion
    - Fjortoft's criterion

- Viscous linear analysis for periodic disturbance
    - Orr-Sommerfeld and Squire equations
    - Squire's theorem
    - Tollmien-Schlichting waves

- Viscous linear analysis for periodic disturbance
    - Neutral curves

- Viscous linear analysis for initial value problem:
    - The disturbance equation
    - Transient growth and non-normality
    - Optimal growth and Singular value decomposition

Simulation and modeling of turbulent flows:

   - Reasons for computing turbulence

   - Computational cost of simulating turbulence

   - Cost of isotropic turbulence

   - The role of supercomputers

   - Degrees of fidelity in simulation of wall turbulence

        - Direct numerical simulation
        - Large-eddy simulation (wall-resolved/modeled)
        - Reynolds-Averaged NS eqs.

Modal analysis of turbulent flows:

   - Motivation: flow decomposition in coherent structures

   - Proper orthogonal decomposition (POD)

   - Applications to dimensionality reduction

   - Dynamic mode decomposition (DMD)

   - Resolvent analysis

   - Applications resolvent analysis to flow structure analysis and reduced-order modeling 

Machine learning for fluid mechanics:

- Historical overview

- Fundamentals of machine learning:
    - The perceptron and neural networks
    - Universal approximation theorem 
    - Gradient descent and back propagation

- Machine learning methods:
    - Supervised learning
    - Unsupervised learning
    - Reinforcement learning

- Examples: applications to fluid mechanics

- Neural network architecture and applications to turbulence:
    - Fully connected feedforward NN
    - Recurrent NN
    - Convolutional NN
    - Generative adversarial NN
    - Autoencoders NN

- Limitations:
    - Overfitting & robusteness
    - Training
    - Interpretability

Compressible turbulence:

- Examples

- The incompressible Navier-Stokes equations:

    - Incompressible vs. compressible eqs.

    - Compressible RANS

    - Favre average

   - The kinetic energy budget

- Wall-bounded compressible turbulence:

   - The Reynolds analogy

   -  Morkovin's hypothesis

- Shockwaves in turbulent flows

Basics of Magnetohydrodynamics:

- What is MHD?

- Communities studying MHD

 - The governing equations of MHD:
    - Navier-Stokes + Maxwell equations
    - Non-dimensional groups
    - Advection, diffusion, and intensification of the magnetic field
    - The energy cascade in MHD

- Applications: astrophysics, geophysics, plasma physics, metallurgy. 

Assignments

• 3 Psets, 50% of the grade.

• 1 Project, 50% of the grade.