course organization

workload and requirements

• 5CP, 150h workload
• 13 lectures (90min), 13 exercise sessions (60min)
• $150h - 13\times 2.5h = 117.5h$ self-study
• written exam, 90min
• expected prior knowledge
  • computational fluid dynamics
  • programming (in Python)

resources and communication

• slides and exercises available on GitHub
• suggested reading list (StudIP/data)
• all resources are updated weekly
• lectures and exercises are recorded
• technical support via GitHub issues
• general communication via email, StudIP

exercise time slot options

1. Tuesdays, 4-5pm (original)
2. Tuesdays, 1-2pm (suggestion)

lecture and exercise on April 25 (next week)

1. recorded lecture/exercise
2. alternative appointments
   1. after final lecture (July 25)
   2. Doodle poll

recipe for issue reporting

1. go to the Github repository
2. click on Issues (top left)
3. use the search to check if a similar
   issue has been reported already
4. if not, open a New issue (top left)
5. write the report and click on Submit new issue

an issue with issues

how to report properly

1. precise title
   Incorrect transport equation lecture 2 slides
2. short description
   Comparing the formulas between ... and ..., I found an inconsistency.
3. precise problem description
   The formula reads ... in the script and ... on the slides.

additional (optional) tips

• expected behavior
  I expected a 3x2 tensor but got a 2x3 tensor.
• suggestion for correction
  I believe the correct formula is $\mathbf{XQ}=\mathbf{Q\Lambda}$.
• link related issues using #issue_number
  The problem might be related to issue #6.
• provide details about your setup
  I am running Ubuntu 20.04 using WSL.

see also: 45 Github Issues Dos and Don't

turbulent flows

source: F. X. Trias et al. (link)

Properties of turbulent flows:

source: J. H. Lee et al. (link)

Turbulence is not a well-defined problem awaiting solution but is a state of motion with innumerable different facets which depend on the context in which it occurs. We do not regard the state of laminar flow as a single problem; much less should turbulence be thought of as a single problem. The properties of turbulence are flow dependent, and therein lies the difficulty. Each turbulent flow field which is studied reveals new aspects of turbulence, and we are a very long way from being able to assemble a comprehensive physical description of this many-sided state of motion.

G. K. Batchelor in J. Fröhlich:
LES turbulenter Strömungen (p. 1)

turbulence modeling

Do we really need turbulence modeling?

What do we expect from a turbulence model?

... an ideal model should introduce the minimum amount of complexity while capturing the essence of the relevant physics.

D. C. Wilcox:
Turbulence modeling for CFD (p. 2)

What do we expect from a turbulence model?

• accurate:
• generalizable:
• tested and validated:
• efficient:
• robust:
• transparent:

most common modeling approaches

1. Reynolds-averaged Navier Stokes (RANS):
   attempt to predict mean flow quantities directly
2. Large eddy simulation (LES):
   resolve largest scales, model small scales
3. hybrid RANS-LES
   dedicated zones for LES and RANS

source: Martin Petry (link)

modeling complexity vs. efficiency

fluid flow equations

conservation of mass

• Cartesian coordinates $$ \frac{\partial \rho}{\partial t} + \frac{\partial \rho u}{\partial x} + \frac{\partial \rho v}{\partial y} + \frac{\partial \rho w}{\partial z} = 0$$
• vector notation $$ \frac{\partial \rho}{\partial t} + \nabla\cdot \left(\rho\mathbf{u}\right) = 0$$

$\mathbf{x} = (x, y, z)^T$, $\mathbf{u} = (u, v, w)^T$

• index notation $$\frac{\partial \rho}{\partial t} + \frac{\partial \rho u_i}{\partial x_i} = 0$$
• another alternative $$\frac{\partial \rho}{\partial t} + \mathrm{div}(\rho\mathbf{u}) = 0$$

compressibility condition

$$\frac{\mathrm{D}\rho}{\mathrm{D}t} = \frac{\partial \rho}{\partial t} + \mathbf{u}\cdot \nabla\rho = 0$$

$\begin{aligned} \frac{\partial \rho}{\partial t} + \nabla\cdot \left(\rho\mathbf{u}\right) &= \frac{\partial \rho}{\partial t} + \mathbf{u}\cdot\nabla\rho + \rho\nabla\cdot\mathbf{u}\\ &= \frac{\mathrm{D}\rho}{\mathrm{D}} + \rho\nabla\cdot\mathbf{u} \end{aligned}$

$$\rho\nabla\cdot\mathbf{u} = 0 \rightarrow \nabla\cdot\mathbf{u} = 0$$

conservation of momentum $$\frac{\partial \rho\mathbf{u}}{\partial t} + \nabla\cdot\left(\rho\mathbf{uu}\right) = \mathbf{f}$$

$$\mathbf{f} = \mathbf{f}_s + \mathbf{f}_b$$ external forces = surface forces + body forces

surface forces $$\mathbf{f}_s = \nabla\cdot \mathbf{S}$$

(total) stress tensor $$\mathbf{S} = \underbrace{-p\mathbf{I}}_{\text{pressure}} + \underbrace{\mathbf{\tau}}_{\text{viscous}}$$

$$p=-\mathrm{tr}(\mathbf{S})$$

gravitational body forces $$\mathbf{f}_b = \rho\mathbf{g}$$

$\mathbf{g}$ - gravitational acceleration vector

momentum equation with forces $$\frac{\partial \rho\mathbf{u}}{\partial t} + \nabla\cdot\left(\rho\mathbf{uu}\right) = -\nabla p + \nabla\cdot\mathbf{\tau} + \rho\mathbf{g}$$

Newtonian fluids $$\mathbf{\tau} = \mu \underbrace{\left(\nabla\mathbf{u} + (\nabla\mathbf{u})^T\right)}_{\text{strain rate}} + \lambda (\nabla\cdot \mathbf{u})\mathbf{I}$$

$\mu$ - dynamic viscosity, $\lambda = -2/3\mu$ - bulk viscosity

final (incompressible) form
$$\frac{\partial \rho\mathbf{u}}{\partial t} + \nabla\cdot\left(\rho\mathbf{uu}\right) = -\nabla p + \nabla\cdot \left[\mu \left(\nabla\mathbf{u} + (\nabla\mathbf{u})^T\right)\right] + \rho\mathbf{g}$$

constant density
$$\frac{\partial \mathbf{u}}{\partial t} + \nabla\cdot\left(\mathbf{uu}\right) = -\nabla \tilde{p} + \nabla\cdot \left[\nu \left(\nabla\mathbf{u} + (\nabla\mathbf{u})^T\right)\right] + \mathbf{g}$$
$\tilde{p} = p/\rho$ - kinematic pressure, $\nu = \mu/\rho$ - kinematic viscosity

summary

• turbulence is hard to define
• the resolution of all turbulent scales is typically infeasible or impossible (and not wanted)
• the modeling of turbulence is essential for most applications of interest
• good turbulence models capture the essential physics with a minimum of complexity
• the most common modeling approaches are RANS, LES, hybrid RANS-LES