on the forcing term in the dns of a turbulent channel flow · on the forcing term in the dns of a...
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On the forcing term in the DNS of a turbulentchannel flow
Maurizio Quadrio1, Bettina Frohnapfel2, Yosuke Hasegawa3
1Politecnico di Milano2Karlsruhe Institute of Technology
3University of Tokyo
Rome, Sept 20, 2014My best wishes to P.O.!
The need for a forcing term in DNS
• NS equations alone cannot push fluid through the duct• Forcing term must be added to mimick pump / gravity / etc
Forcing term is "arbitrary"
• Popular choices are constant flow rate (CFR) and constantpressure gradient (CPG)
• Often equivalent on physical grounds• Known difference on practical grounds• Different realizations, statistics are the same
Important when comparing two different flowsExample: turbulent drag reduction by spanwise oscillating walls
"Turbulence intensity is destroyed"
Important when comparing two different flowsExample: turbulent drag reduction by spanwise oscillating walls
"Turbulence intensity is destroyed"
Important when comparing two different flowsExample: turbulent drag reduction by spanwise oscillating walls
"Turbulence intensity is destroyed"
Important when comparing two different flowsExample: turbulent drag reduction by spanwise oscillating walls
"Turbulence intensity is destroyed"
CFR or CPG?Pre-determines the global energy budget for drag reduction
• Potential source of confusion• Concerns both DNS and experiments• CFR: pumping power is reduced with drag reduction• CPG: pumping power is increased with drag reduction
A further option: CPIThe Money-vs-Time plane (JFM 2012, 2014)
Question
Does the choice of the forcing termaffect the statisticsof the same flow?
Finding the answer
• Large spatio-temporal DNS channel databases for CFR,CPG, CPI
• DNS code: mixed-discretization solver• Channel flow at Reτ ≈ 200• Lx ×Ly ×Lz = 4πh×2h×2πh• ∆x+ = 9.6 ∆z+ = 4.8 ∆y+ = 0.8−4.9• Sample size: T + = 100,000 at ∆t+ = 1
No obvious changes (obviously!)
forcing term flow driven with measuredCFR Reb = 3173 Reτ = 199.01CPG Reτ = 200 Reτ = 199.89CPI ReΠ = 6500 Reτ = 199.49
y+
U
0 50 100 1500
5
10
15
20
CFR
CPGCPI
y+
u’ rm
s
0 50 100 150 2000
0.5
1
1.5
2
2.5
3
CFR
CPGCPI
Focus on wall frictionComparison with Lenaers et al, PoF 2012
σ
P(τ
w’)
4 2 0 2 410
6
105
104
103
102
101
100
101
102
CFR
CPG
CPI
An in-depth look
σ
P(τ
w’)
15 10 5 0 5 10 1510
9
108
107
106
105
104
103
102
101
100
CFR
CPG
CPI
Space-time autocorrelation of wall frictionRed: CFR; black: CPG; green: CPI
∆t+
∆x/h
0 50 100 150 200 250 3000
2
4
6
8
10
12
Differences appear in Lagrangian frame only!One-dimensional space or time correlations are mostly unaffected
∆x/h
R
0 2 4 60
0.2
0.4
0.6
0.8
1
CFR
CPG
CPI
∆t+
R
0 50 100 150 200 250 3000
0.2
0.4
0.6
0.8
1
CFR
CPG
CPI
Statistical significance?
∆t+
∆x/h
0 50 100 150 200 250 3000
2
4
6
8
10
12
Link to vortical structures?
Integral timescale of "lagrangian" correlation: lifetime ofnear-wall structures
∆t+
R
0 50 100 150 200 2500
0.2
0.4
0.6
0.8
1
CFR
CPG
CPI
Conclusions
• Choice of forcing term does leave a statistical footprint• Most evident (so far) in lagrangian frame• Relevance?
A 18-years-old pair of skiesGratefully remembering my first workshop in Aussois (1997), organized by P.O.