Structure and dynamics of turbulent pipe flow Andrew Duggleby Mechanical Engineeri Texas A&M University Isaac Newton Institu 2008 Describe… Model… Predict… Collaborators: Ken Ball, Mark Paul Mechanical Engineering Virginia Tech Markus Schwaenen Mechanical Engineering Texas A&M University Paul Fischer Argonne National Lab.
Describe… Model…. Andrew Duggleby Mechanical Engineering Texas A&M University Isaac Newton Institute 2008. Structure and dynamics of turbulent pipe flow. Collaborators: Ken Ball, Mark Paul Mechanical Engineering Virginia Tech Markus Schwaenen Mechanical Engineering - PowerPoint PPT Presentation
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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