Analysis of Lagrangian stretching in turbulent channel...
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Analysis of Lagrangian stretching in turbulent channel flow using a task-parallel particle tracking method in the Johns Hopkins Turbulence Databases Perry L. Johnson 1 , Stephen S. Hamilton 2 , Randal Burns 2 and Charles Meneveau 1 1 Department of Mechanical Engineering, Johns Hopkins University; 2 Department of Computer Science, Johns Hopkins University Johns Hopkins Turbulence Databases (JHTDB) I http://turbulence.pha.jhu.edu/ I access via web services I Fortran, C, Matlab, HDF5 cutout I built-in functions I e.g. getVelocity, getPressureHessian I interpolation & finite-differencing I Currently hosts four datasets: I Isotropic: 1024 3 × 5024 I Magnetohydrodynamics: 1024 4 I Channel: 2048 × 512 × 1536 × 4000 I Mixing: 1024 3 × 1012 Homogeneous Isotropic Turbulence vs. Turbulent Channel Flow incompressible Navier-Stokes: ∂ u i ∂ t + u j ∂ u i ∂ x j = - 1 ρ ∂ p ∂ x i + ν ∂ 2 u i ∂ x j ∂ x j + f i , ∂ u j ∂ x j = 0 I isotropic forcing in a periodic box I for studying small-scale turbulent motions I flow between infinite flat plates (u = 0 B.C.) I for studying turbulence near boundaries I Local isotropy hypothesis: zoom-in to any flow, statistically like isotropic turbulence Stretching of Fluid Elements and Vorticity x i ( t; X ,t 0 ) σ 1 σ 2 σ 3 X i ,t 0 D ij ( t; X ,t 0 )= ∂ x i ∂ X j dx i dt =u i ( x ,t ) dD ij dt = A ik D kj x i ( t; X ,t 0 ) X i ,t 0 dx i dt =u i ( x ,t ) ω i ( t 0 ) ω i ( t ) d ω i dt = A ij ω j +ν∇ 2 ω i I fluid elements and vorticity both stretched and rotated by same mechanism I BUT, different alignments with strain-rate → different stretching rates d σ i dt = b S (ii ) σ i where b S (ii ) is strain-rate along i th semi-axis σ 1 σ 2 σ 3 ^ S 11 ^ S 22 ^ S 33 ^ S ω ω d ω dt = b S ω ω where b S ω is strain-rate along vorticity axis Fluid Particle Tracking I Fluid particles follow the local flow velcity: ˙ x = u(x(t ), t ) I 2 nd -order method prevents particles from crossing physical boundaries Mediator Approach Task-Parallel Approach I Kanov & Burns (SC ‘15) showed that task-parallel approach is faster I here: implement for channel Stretching Rates: Mean Strength of strain-rate S ij ∼ τ -1 η varies with distance from boundary. τ η using bulk dissipation 10 0 10 1 10 2 10 3 y + 10 -3 10 -2 10 -1 10 0 | S (ii) |y + |τ η,bulk S 11 S 22 S 33 S ω I Peak stretching close to boundary: 10 < y + < 50 τ η using local dissipation 10 0 10 1 10 2 10 3 y + 10 -3 10 -2 10 -1 10 0 | S (ii) |y + |τ η S 11 S 22 S 33 S ω I Local isotropy in core: y + > 100 I Near boundary: less efficient I Investigate alignment between strain-rate and fluid element/vorticity. Stretching Rates: Full Probability Distribution All locations y + > 100, normalized by local dissipation. Fluid Elements -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 S (ii) τ η 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 Vorticity -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 S ω τ η 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 10 1 I More detailed support for local isotropy hypothesis in the core of the channel. Alignment with Strain-Rate: Mean 10 0 10 1 10 2 10 3 y + 0.0 0.2 0.4 0.6 0.8 1.0 cos 2 (θ 1j )|y + θ 11 θ 12 θ 13 10 0 10 1 10 2 10 3 y + 0.0 0.2 0.4 0.6 0.8 1.0 cos 2 (θ 3j )|y + θ 31 θ 32 θ 33 10 0 10 1 10 2 10 3 y + 0.0 0.2 0.4 0.6 0.8 1.0 cos 2 (θ 2j )|y + θ 21 θ 22 θ 23 10 0 10 1 10 2 10 3 y + 0.0 0.2 0.4 0.6 0.8 1.0 cos 2 (θ ωj )|y + θ ω1 θ ω2 θ ω3 Alignment with Strain-Rate: Full Probability Distribution 0.0 0.2 0.4 0.6 0.8 1.0 cos(θ 1j ) 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 cos(θ 3j ) 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 cos(θ 2j ) 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 cos(θ ωj ) 0 1 2 3 4 5 Conclusions I Local isotropy in core (y + > 100), less favorable alignments near the wall. I Overall, channel has 50% lower mean stretching rates per unit dissipation. Department of Mechanical Engineering - Johns Hopkins University - Baltimore, MD – [email protected] – http://turbulence.pha.jhu.edu/
Transcript of Analysis of Lagrangian stretching in turbulent channel...
Analysis of Lagrangian stretching in turbulent channel flow using a task-parallelparticle tracking method in the Johns Hopkins Turbulence Databases
Perry L. Johnson1, Stephen S. Hamilton2, Randal Burns2 and Charles Meneveau1
1Department of Mechanical Engineering, Johns Hopkins University; 2Department of Computer Science, Johns Hopkins University
Johns Hopkins Turbulence Databases (JHTDB)
I http://turbulence.pha.jhu.edu/I access via web servicesI Fortran, C, Matlab, HDF5 cutoutI built-in functions
I e.g. getVelocity, getPressureHessianI interpolation & finite-differencing
I Currently hosts four datasets:I Isotropic: 10243 × 5024I Magnetohydrodynamics: 10244
I Channel: 2048× 512× 1536× 4000I Mixing: 10243 × 1012
4 Y. Li et al.
Figure 1. The architecture of the turbulence database cluster.
a high degree of parallelism. The details of the turbulence data set and several components of thedatabase cluster are described in the following several subsections.
2.1. Turbulence data set
The data are from a DNS of forced isotropic turbulence on a 10243 periodic grid in a [0, 2π ]3
domain, using a pseudo-spectral parallel code. Time integration of the viscous term is doneanalytically using the exact integrating factor. The other terms are integrated using a second-orderAdams–Bashforth scheme and the nonlinear term is written in vorticity form [26]. The simulationis de-aliased using phase shift and 2
√2/3 spherical truncation [27], so that the effective maximum
wave number is about kmax = 1024√
2/3 ≈ 482. Energy is injected by keeping constant the totalenergy in modes such that their wave number magnitude is less or equal to 2. The simulationtime step is �t = 0.0002. After the simulation has reached a statistical stationary state, 1024frames of data, which include the three components of the velocity vector and the pressure, aregenerated in physical space and ingested into the database. The data are stored at every 10 DNStime steps, i.e. the samples are stored at time step δt = 0.002. The duration of the stored datais 1024 × 0.002 = 2.048, i.e. about one large-eddy turnover time since T = L/u� ≈ 2.02 (Lis the integral scale). The parameters of the simulation are given in Table 1. The total kineticenergy, dissipation rate and energy spectra are computed by averaging in time between t = 0 andt = 2.048.
Figure 2 shows the radial energy spectrum computed from the simulation and averagedbetween t = 0 and t = 2.048. Figures 3 and 4 show time series of the total turbulent kineticenergy and Taylor-scale-based Reynolds numbers starting near the simulation initial condition.The values corresponding to the data in the database are for t > 0 and are shown in solid lineportions.
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Homogeneous Isotropic Turbulence vs. Turbulent Channel Flow
incompressible Navier-Stokes:∂ui∂t
+ uj∂ui∂xj
= −1ρ
∂p∂xi
+ ν∂2ui∂xj∂xj
+ fi ,∂uj∂xj
= 0
I isotropic forcing in a periodic boxI for studying small-scale turbulent motions
I flow between infinite flat plates (u = 0 B.C.)I for studying turbulence near boundaries
I Local isotropy hypothesis: zoom-in to any flow, statistically like isotropic turbulence
Stretching of Fluid Elements and Vorticity
xi(t ; X , t 0)
σ1
σ2
σ3X i ,t 0
Dij(t ;X ,t 0)=∂ x i
∂ X j
dxi
dt=ui(x , t)
dDij
dt=A ik D kj xi(t ; X , t 0)X i ,t 0
dxi
dt=ui(x , t)
ωi(t 0)ωi(t)
dω i
dt=A ijω j+ν∇
2ω i
I fluid elements and vorticity both stretched and rotated by same mechanismI BUT, different alignments with strain-rate→ different stretching rates
dσidt
= S(ii)σi
where S(ii) is strain-ratealong i th semi-axis
σ1
σ2
σ3
S11
S22
S33
Sω
ω
dωdt
= Sωω
where Sω is strain-ratealong vorticity axis
Fluid Particle Tracking
I Fluid particles follow the local flow velcity: x = u(x(t), t)I 2nd-order method prevents particles from crossing physical boundaries
Mediator Approach Task-Parallel Approach
I Kanov & Burns (SC ‘15) showed that task-parallel approach is fasterI here: implement for channel
Stretching Rates: Mean
Strength of strain-rate Sij ∼ τ−1η varies with distance from boundary.
τη using bulk dissipation
100 101 102 103
y+
10−3
10−2
10−1
100
|〈S(ii)|y
+〉|τ
η,bulk
S11
S22
S33
Sω
I Peak stretching close to boundary:10 < y+ < 50
τη using local dissipation
100 101 102 103
y+
10−3
10−2
10−1
100
|〈S(ii)|y
+〉|τ
η
S11
S22
S33
Sω
I Local isotropy in core: y+ > 100I Near boundary: less efficient
I Investigate alignment between strain-rate and fluid element/vorticity.
Stretching Rates: Full Probability Distribution
All locations y+ > 100, normalized by local dissipation.
Fluid Elements
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
S(ii)τη
10−5
10−4
10−3
10−2
10−1
100101
Vorticity
−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5
Sωτη
10−5
10−4
10−3
10−2
10−1
100101
I More detailed support for local isotropy hypothesis in the core of the channel.
Alignment with Strain-Rate: Mean
100 101 102 103
y+
0.0
0.2
0.4
0.6
0.8
1.0
〈cos
2(θ
1j)|y
+〉
θ11
θ12
θ13
100 101 102 103
y+
0.0
0.2
0.4
0.6
0.8
1.0
〈cos
2(θ
3j)|y
+〉
θ31
θ32
θ33
100 101 102 103
y+
0.0
0.2
0.4
0.6
0.8
1.0
〈cos
2(θ
2j)|y
+〉
θ21 θ22
θ23
100 101 102 103
y+
0.0
0.2
0.4
0.6
0.8
1.0
〈cos
2(θ
ωj)|y
+〉
θω1
θω2
θω3
Alignment with Strain-Rate: Full Probability Distribution
0.0 0.2 0.4 0.6 0.8 1.0cos(θ1j)
0
1
2
3
4
5
0.0 0.2 0.4 0.6 0.8 1.0cos(θ3j)
0
1
2
3
4
5
0.0 0.2 0.4 0.6 0.8 1.0cos(θ2j)
0
1
2
3
4
5
0.0 0.2 0.4 0.6 0.8 1.0cos(θωj)
0
1
2
3
4
5
Conclusions
I Local isotropy in core (y+ > 100), less favorable alignments near the wall.I Overall, channel has 50% lower mean stretching rates per unit dissipation.
Department of Mechanical Engineering - Johns Hopkins University - Baltimore, MD – [email protected] – http://turbulence.pha.jhu.edu/
