Structure and dynamics of turbulent pipe flow

Structure and dynamics of turbulent pipe flow

Andrew DugglebyMechanical EngineeringTexas A&M University

Isaac Newton Institute2008

Describe…

Model…

Predict…

Collaborators:Ken Ball, Mark PaulMechanical Engineering Virginia Tech

Markus SchwaenenMechanical EngineeringTexas A&M University

Paul FischerArgonne National Lab.

Proper Orthogonal Decomposition Translational invariance vs. method of snapshots What POD is, and what it is not (3 misnomers)

Turbulent pipe flow (spectral element DNS) Describe, Model, Predict Applications to drag reduction

Structure and dynamics of turbulent pipe flow

80 ,10 ,150Re L

tUDL b

Karhunen-Loève (KL) Decomposition is a powerful tool that generates an optimal basis set for dynamical data

Proper Orthogonal Decomposition (POD)

Empirical (or dynamical) Eigenfunctions

Optimally fast convergence Maximizes “energy”

Originates as a variational problem

In order to reduce the size of the problem, the translational invariance of the system is taken into account.

By translational invariance

The POD mode is then

And the Fredholm integral reduces to

POD modes are labeled by the triplet (m,n,q) with

(1,3,1)

Misnomer 1:This is a mode

Method of Snapshots

Define c(t) and rewrite

Take inner product with velocity at a different time

Solve Fredholm equation for coefficient c(t)

)8(every t snapshots 2100

Translational invariance vs. snapshots for turbulent pipe flow

Dimension vs. time for turbulent pipe flow

L)tU 4000~(

000,850at

9.0

9.0

m

00

T

DDKL

kk

D

kk

KL

Translational Invariance and Method of Snapshots agree at infinite time – shown using Rayleigh-Bénard convection

Misnomer 2:The basis set is only optimal for “recorded events”

Misnomer 3:The basis set is only optimal for energy dynamics

"Goal-oriented, model-constrained optimization for reduction of large-scale systems“ T. Bui-Thanh, K. Willcox, O. Ghattas, B. van Bloemen WaandersJ. of Comp. Phys. (2006)

POD reduces the order of the system to a much more manageable level(from 107 to 104) whereby one can examine the system

Insight gained through examining: The energy ordering of the modes The structure of the modes The dynamics of the modes (1,5,1) mode – travelling wave

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Example modes (wall modes)(1,3,1)

(1,5,1)

(2,4,1)

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

…more modes (lift modes)(2,2,1)

(3,2,1)

(3,3,1)

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

…and yet even more modes (roll mode)

(0,6,1)

(0,5,1)

(0,3,1)

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Karhunen-Loève decomposition is a very powerful tool in helping to understand large scale energy dynamics.

= Streamwise Roll

Travelling wave interpretation of turbulence

+ Travelling wave packet

Energy content of the modes

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

38.34% energy in structures >8R (m<2)

Model: to understand drag reduction, two sets of DNS calculations were analyzed, one with spanwise wall oscillation and one without.

Wall mode (vorticity starts and stays near the wall) is pushed away from the wall in the presence of oscillation

Non-oscillated Oscillated

(1,2,1)

(1,5,1)

8.6shifty

9.9shifty

Duggleby et al., 2007, Phys. Fluids, Vol. 19, 125107

Lift mode (vorticity starts near the wall and lifts away from the wall ) is also pushed away from the wall in the presence of oscillation

Non-oscillated Oscillated

(2,2,1)

(3,2,1)

6.10shifty

2.11shifty

Duggleby et al., 2007, Phys. Fluids, Vol. 19, 125107

Roll mode (no spanwise dependence, “streamwise vortices”) is also pushed away from the wall due to spanwise wall oscillation

Non-oscillated Oscillated

(0,6,1)

(0,2,1)

1.10shifty

8.7shifty

Duggleby et al., 2007, Phys. Fluids, Vol. 19, 125107

Model: Drag reduction mechanism

Duggleby et al., 2007, Phys. Fluids, Vol. 19, 125107

Prediction: Drag reduction by sectional rotation

Conclusions

Describe: POD is a great way to visualize the large scale energy dynamics Method of Snapshots and Translational invariance agree at

late time L=100D pipe simulations underway

Model: drag reduction model

Predict: Drag reduction Experimental testing under way

Acknowledgements

System X Teragrid Paul Fischer Markus Schwaenen Travis Thurber Ken Ball Mark Paul

Appendix

Top 15 POD modes for various flows

Convergence for Re_tau=180

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Statistics

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Stress

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Mean velocity profile (Reτ=150) shows a 26.9% increase in the mean flow rate due to spanwise wall oscillation.

26.9% increase in mean flowrate

Duggleby et al., Phys. Fluids (in review), 2007

The peaks of root-mean-square velocity fluctutations and Reynolds stress profiles shift due to the oscillation.

38,627.031,68.0

61,78.055,81.0

29,03.140,99.0

22,48.216,68.2

,

,

,

yyuu

yyu

yyu

yyu

zr

rmsr

rms

rmsz

Duggleby et al., Phys. Fluids (in review), 2007

Turbulent Pipe Flow examinations

Pipe Flow: L/D=10 Reτ=150 Total simulation time:

t+=16800 80 flow through times

Rayleigh-Benard R=6000 σ=1 Γ=10

H.M. Tufo and P.F. Fischer, in Proc. Of the ACM/IEEE SC99 Conf. on High Performance Networking and Computing (IEEE Computer Soc., 1999), Gordon Bell winning paper

TTu

zTuPuu

u

t

t

2

21 ˆR

0

Red: hot rising fluid, Blue: cold falling fluid

Translational invariance vs. Snapshots

First mode from translational invariance (18,1)

First mode from method of snapshots

Propagating Subclasses: Asymmetric mode

(1,1,1)

(2,1,1)

(3,1,1)

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Propagating Subclasses: Ring mode

32

(1,0,1)

(1,0,2)

(2,0,1)

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Effect of quantum number

(2,6,3) (2,6,5)

(6,2,3) (6,2,5)

Duggleby et al., 2007, Journal of Turbulence, Vol. 8 no. 43

Model & Prediction: relaminarization (?)

Or L=10D is too short!

Spectral Elements combines geometrical flexibility, efficient parallelization, and exponential convergence

Spectral Element Legendre Lagrangian interpolants 3rd order in time Jacobi w/ Schwarz

multigrid and GMRES Scalable

1.26 TFLOPS on 2048 proc. (BG/L) 108 GFLOPS on 128 proc. (SysX)

Avoids the singularity at the origin inpolar-cylindrical coordinates

tU/L=80

H.M. Tufo and P.F. Fischer, in Proc. Of the ACM/IEEE SC99 Conf. on High Performance Networking and Computing (IEEE Computer Soc., 1999), Gordon Bell winning paper

Channel modes