Simulation and modal analysis of transonic shock buffets on a NACA-0012 airfoil

Andre Weiner, Richard Semaan
TU Braunschweig, ISM, Flow Modeling and Control Group

AIAA SciTech Jan. 7, 2022

Copyright © by Andre Weiner and Richard Semaan, TU Braunschweig. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Outline

  • A brief literature survey
  • Simulation approach
  • Dynamic mode decomposition
  • Results
  • Summary and outlook

A brief literature survey

Contradictions in literature about:

  • onset angle of attack
  • shock position
  • buffet frequency
  • dependency on $Ma_\infty$ and $\alpha$
  • ...

Main issues:

  • high sensitivity to model constants
  • incomplete/missing description
  • no access to implementation details

Goal: workflow that is fully

  • open-source
  • reproducible
  • transparent

github.com/AndreWeiner/naca0012_shock_buffet

Simulation approach

Simulation approach in a nutshell:

  • OpenFOAM-v2012
  • rhoCentralFoam
  • IDDES turbulence modeling
  • Spalart-Allmaras closure
  • 2D and 3D simulations
sigma

Various views of the computational mesh.

Dyanmic mode decomposition

Definition of data matrices:

$$ \mathbf{X} = \left[ \begin{array}{cccc} | & | & & | \\ \mathbf{x}_1 & \mathbf{x}_2 & ... & \mathbf{x}_{N-1} \\ | & | & & | \\ \end{array}\right],\quad \mathbf{X}^\prime = \left[ \begin{array}{cccc} | & | & & | \\ \mathbf{x}_2 & \mathbf{x}_3 & ... & \mathbf{x}_{N} \\ | & | & & | \\ \end{array}\right] $$

$\mathbf{x}_n$ - state vector snapshot at timestep $n$

DMD in a nutshell:

  1. $\mathbf{X}^\prime = \mathbf{A} \mathbf{X}$
  2. $\mathbf{A} = \mathbf{\Phi\Lambda\Phi}^{-1}$
  3. $\mathbf{\Phi}$ - DMD modes
  4. $\mathbf{\Lambda}$ - yields freq., stability

Key difference to previous studies:

  1. volume-weighted state vector
  2. thermodynamic variables in state vector$^*$
    $$ \mathbf{x}_n = \left[ \mathbf{u}_n, \mathbf{v}_n,\mathbf{w}_n, 2\mathbf{a}_n/\left(\gamma -1\right) \right]^T $$
  3. $^*$ Rowley et al. 2004; $\gamma$ - adiabatic index, $a$ - local speed of sound, $u/v/w$ - velocity components

Practical DMD details:

  • about 250 snapshots per cycle
  • >3 cycles
  • $-1 \le x/c \le 3$; $-1 \le y/c \le 1$
  • only slice at $\tilde{z}=0$
  • flowTorch for data processing and DMD

Results

Setup motivated by exp. investigations of McDevitt and Okuno 1985:

  • NACA-0012 airfoil
  • $Re_\infty = 10^7$
  • $Ma_\infty = 0.75$
  • pre-onset: $\alpha = 2^\circ$
  • post-onset: $\alpha = 4^\circ$
sigma

Pressure coefficient $c_p$ at pre-onset conditions, $\alpha = 2^\circ$.

sigma

Pressure coefficient $c_p$ at post-onset conditions, $\alpha = 4^\circ$.

Local Mach number $Ma$, 2D, $\alpha=4^\circ$.

Slice of local Mach number $Ma$, 3D, $\alpha=4^\circ$.

sigma

DMD spectrum for 2D and 3D datasets/simulations; $\bar{f}=2\pi c f/U_\infty$.

sigma

Real parts of DMD buffet mode and first harmonic; $u$ and $v$ are the velocity components.

DMD mode at approx. $20f_{buffet}$, $u$-component.

Summary and outlook

Summary

  • 3D simulations yield more realistic results
  • volume-weighted state vector
  • thermodynamic variables in state vector
  • workflow and data publicly available
  • further 3D investigations

https://github.com/AndreWeiner/naca0012_shock_buffet

THE END

Thank you for you attention!

for2895

{a.weiner|r.semaan}@tu-braunschweig.de