Effects of Forcing Scheme on the Relative Motion of Inertial - - PowerPoint PPT Presentation

▶
effects of forcing scheme on the relative motion of
SMART_READER_LITE
LIVE PREVIEW

Effects of Forcing Scheme on the Relative Motion of Inertial - - PowerPoint PPT Presentation

Jul 03, 2023 285 likes •559 views

Effects of Forcing Scheme on the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence Sarma L. Rani and Rohit Dhariwal PI: Sarma L. Rani Department of Mechanical and Aerospace Engineering University of Alabama in Huntsville

slide-1
SLIDE 1

Effects of Forcing Scheme on the Relative Motion of Inertial Particles in DNS of Isotropic Turbulence

Sarma L. Rani and Rohit Dhariwal PI: Sarma L. Rani

Department of Mechanical and Aerospace Engineering University of Alabama in Huntsville

NCSA Blue Waters Symposium June 4-7, 2018 Sunriver, OR

DNS of Inertial Particles May 16, 2017 1 / 27

slide-2
SLIDE 2

Outline

1

Motivation for Current Problem

2

Background

3

Effects of Forcing Scheme in DNS on Inertial Particles

4

Parallel Performance of DNS code

DNS of Inertial Particles May 16, 2017 2 / 27

slide-3
SLIDE 3

Motivation for Current Problem

DNS of Inertial Particles May 16, 2017 3 / 27

slide-4
SLIDE 4

Particle-Laden Turbulent Flows I

Particle-laden turbulent flows are important both in natural and engineering applications such as:

◮ Warm-Cloud Precipitation: We are interested in understanding if

turbulence augments water-droplet growth rates by increasing droplet collision rates, which may hasten rainfall initiation

◮ Planetesimal Formation: Astrophysicists are interested in knowing if

turbulence-driven dispersion and collisional coalescence of dust particles impact planetesimal formation

Warm-Cloud Precipitation Planetesimal Formation DNS of Inertial Particles May 16, 2017 4 / 27

slide-5
SLIDE 5

Particle-Laden Turbulent Flows II

◮ Volcanic Eruption: Quantifying the dispersion of volcanic particles in

the atmosphere is of interest

◮ Spray Dynamics in Engines: Effects of turbulence on atomization,

dispersion, and evaporation of fuel droplets is the relevant physics

Volcanic Eruption Spray Dynamics in Engines

In these applications, we are interested in quantifying the effects of turbulence on particle-pair relative motion.

DNS of Inertial Particles May 16, 2017 5 / 27

slide-6
SLIDE 6

Particle-Pair Relative Motion

Pair relative motion refers to the temporal and spatial dynamics of separations r and relative velocities U of disperse particle pairs Turbulence is known to spatially homogenize passive scalars However, it induces strong inhomogeneities in inertial particle relative motion, which are of two kinds:

◮ Spatial Inhomogeneities: Particle clustering, quantified by Radial

Distribution Function (RDF) g(r)

◮ Relative Velocity Inhomogeneities: Non-Gaussian relative velocity

distribution, described by pair relative velocity PDF P(Ur)

Through these two statistics, one can study the role of turbulent fluctuations in driving particle collision frequency: Collision frequency Nc = 4πσ2g(σ)

−∞

Ur P(Ur|σ) dUr

DNS of Inertial Particles May 16, 2017 6 / 27

slide-7
SLIDE 7

Particle Preferential Concentration I

Particle response to turbulence is controled by its inertia, as quantified by the Stokes number St = τv/τflow

◮ τv is particle viscous relaxation time and τflow is a flow time scale

When particle Stokes number Stη = τv

τη 1

◮ Denser-than-fluid particles accumulate in regions of excess strain-rate

  • ver rotation-rate, i.e. where S2 − Ω2 > 0

DNS of Inertial Particles May 16, 2017 7 / 27

slide-8
SLIDE 8

Particle Preferential Concentration II

DNS of isotropic turbulence by Reade and Collins1 demonstrates the effects of Stη on clustering h(r) > 0 is indicative of particle preferential concentration

DNS of isotropic turbulence Residual RDF (g(r) − 1)vs r

1Reade and Collins, Phys. Fluids, Vol. 12, 2000. DNS of Inertial Particles May 16, 2017 8 / 27

slide-9
SLIDE 9

Pair Relative Velocities

DNS of Sundaram and Collins2 illustrates the nature of relative velocity PDF at various separations:

◮ Gaussian relative velocity PDF at integral-scale pair separations ◮ Non-Gaussian relative velocity PDF with a peak and a long tail at

smaller separations; σ = sum of particle radii (at contact)

Therefore, a closure theory should capture both preferential concentration and Gaussian to Non-Gaussian PDF transition

2Sundaram and Collins, JFM, Vol. 335, 1997. DNS of Inertial Particles May 16, 2017 9 / 27

slide-10
SLIDE 10

Background

DNS of Inertial Particles May 16, 2017 10 / 27

slide-11
SLIDE 11

Stochastic Theory (Str = τv/τr ≫ 1)

In a recent study3, we derived a closure for diffusion current in the PDF kinetic equation for the relative motion of high-Stokes-number particle pairs in isotropic turbulence

◮ Probability density function (PDF) of interest is Ω(r, U) ◮ r and U are pair separation and relative velocity ◮ Stokes number regime of interest is Str ≫ 1 ◮ Str = τv/τr, τv is particle response time and τr is time scales of eddies

whose size scales with pair separation r

For Str ≫ 1 particles, the pair PDF Ω(r, U) is governed by: ∂Ω ∂t + ∇r • (UΩ) − 1 τv ∇U • (UΩ) − ∇U • (DUU • ∇UΩ) = 0

3Rani, Dhariwal, and Koch, JFM, Vol. 756, 2014 DNS of Inertial Particles May 16, 2017 11 / 27

slide-12
SLIDE 12

Stochastic Theory (Str = τv/τr ≫ 1)

For Str ≫ 1 particles, it was shown that diffusivity DUU = 1 τ 2

v −∞

∆u(r, x, 0) ∆u(r, x, t) dt

◮ In St ≫ 1 regime, pair separation r and center of mass position x

remain essentially fixed during fluid time scales

◮ Therefore, ∆u(r, x, 0) ∆u(r, x, t) is a Eulerian two-time correlation

DUU derived from theory can be validated by computing ∆u(r, x, 0) ∆u(r, x, t) from DNS

DNS of Inertial Particles May 16, 2017 12 / 27

slide-13
SLIDE 13

Why Blue Waters

If not for Blue Waters, we would probably have not been able to compute DUU from DNS ∆u(r, x, 0) ∆u(r, x, t) was evaluated using DNS of forced isotropic turbulence with disperse, fixed particles Correlations computed using 20,000 to 40,000 processor cores Evaluating this correlation is computationally very expensive. Why? Flow is seeded with 106 particles or ∼ 5 × 1011 pairs

DNS of Inertial Particles May 16, 2017 13 / 27

slide-14
SLIDE 14

Why Blue Waters

Over ∼ 20 large-eddy turnover times TE, we write out the fluid velocities at the locations of fixed particles Fluid velocities are written at intervals of 2 − 5 Kolmogorov time scales Consider two snapshots of flow separated by a time interval τ in a DNS run The longitudinal and transverse components of ∆u(r, x, t)∆u(r, x, t + τ) for a particle pair are stored in the appropriate r bin, and then averaged over all pairs within a bin. Next, we average the two components over pairs of flow snapshots with the same time separation τ For each value of τ, we averaged over 200 such pairs of flow snapshots The correlations at various separations are then integrated in time to yield DUU

DNS of Inertial Particles May 16, 2017 14 / 27

slide-15
SLIDE 15

Effects of Forcing Scheme in DNS on Inertial Particles

DNS of Inertial Particles May 16, 2017 15 / 27

slide-16
SLIDE 16

Governing Equations

Fluid phase governing equations ∇· u = ∂u ∂t + ω × u = −∇

  • p/ρf + u2/2
  • + ν∇2u + ff

ff is external forcing to maintain a statistically stationary turbulence Particle phase governing equations dxp dt = vp dvp dt = u(xp, t) − vp τv u(xp, t) obtained using 8th order Lagrange interpolation

DNS of Inertial Particles May 16, 2017 16 / 27

slide-17
SLIDE 17

Forcing Schemes

Recall, large scale external forcing is added to N-S equation to maintain statistically stationary turbulence Deterministic forcing4: Turbulent kinetic energy dissipated during a time step is added back to the velocity field Stochastic forcing5: Random forcing acceleration based on Ornstein-Uhlenbeck process is added to the velocity components

◮ Two important parameters: acceleration variance, σ2

f and forcing

time-scale, Tf

Both forcing schemes add energy to a low-wavenumber band

4Witkowska et al., J Comput Acoust 1997;5:317–36 5Eswaran & Pope, Comput Fluids 1988;16:257–78 DNS of Inertial Particles May 16, 2017 17 / 27

slide-18
SLIDE 18

Deterministic Forcing (DF) Scheme

Turbulence is initialized with a certain amount of turbulent kinetic energy (TKE) In our DF, we maintain TKE constant as turbulence evolves temporally Energy dissipated during ∆t is resupplied to the spectral velocity components in the range κ ∈ (0, √ 2] This is done by scaling velocity components in the forcing wavenumber band

  • u(κ, t + ∆t) =

u(κ, t + ∆t)

  • 1 +

∆Ediss(∆t) κmax

κmin E(κ, t + ∆t)dκ

κ = |κ| such that κ ∈ (0, √ 2], [κmin, κmax] is the entire wavenumber range of the DNS

DNS of Inertial Particles May 16, 2017 18 / 27

slide-19
SLIDE 19

Stochastic Forcing (SF) Scheme

Here TKE is not kept constant in the stochastic scheme Instead, a random acceleration term f is added to N-S equations

  • f computed from six independent Uhlenbeck-Ornstein processes
  • f =

b(κ, t) − κκ · b(κ, t)/(κ · κ)

  • b(κ, t + ∆t) =

b(κ, t)

  • 1 − ∆t

Tf

  • + θ

2σ2∆T Tf 1/2

  • b(κ, t) is an UO process having σ2 as the variance and Tf time-scale

Forcing time-scale Tf is a key parameter, whose effects are studied Forcing time scale Tf = 4TE, 2TE, TE, TE/2, and TE/4 considered

DNS of Inertial Particles May 16, 2017 19 / 27

slide-20
SLIDE 20

Parallel Performance of DNS code

DNS of Inertial Particles May 16, 2017 20 / 27

slide-21
SLIDE 21

1D Domain Decomposition

Figure: (a) XZ slabs; (b) YZ slabs

Domain decomposition along one direction N3 simulations can be run on up to N processors Limited to small Reλ

DNS of Inertial Particles May 16, 2017 21 / 27

slide-22
SLIDE 22

2D Domain Decomposition

Figure: (a) X ; (b) Y; (c) Z pencils

Domain decomposition along two directions N3 simulations can be run on up to N2 processors Allows higher flow Reλ

DNS of Inertial Particles May 16, 2017 22 / 27

slide-23
SLIDE 23

Strong Scaling

Strong scaling for 2D parallel code

Processors Speedup vs 1024 Processors

5000 10000 15000 20000 5 10 15 20 Ideal Speedup Measured Speedup

DNS of Inertial Particles May 16, 2017 23 / 27

slide-24
SLIDE 24

DNS parameters

Two grid resolutions considered: 1283 and 5123 Reλ achieved: 80 and 210 Forced wavenumbers range for deterministic and stochastic schemes, |κ| ∈ (0, √ 2] Five Tf considered: TE/4, TE/2, TE, 2TE and 4TE

◮ Te is the eddy turnover time obtained using deterministic forcing

Thus a total of 6 × 2 = 12 DNS runs were performed for both Reλ’s

DNS of Inertial Particles May 16, 2017 24 / 27

slide-25
SLIDE 25

Reλ = 80: (a) r/L = 0.56, (b) r/L = 1.12, (c) r/L = 2.24, (d) r/L = 3.36

2 4 6 8 10 12

  • 0.2

0.2 0.4 0.6 0.8 1 DF SF1 SF2 SF3 SF4 SF5

τ ∗ R⊥(r, τ)/R||(r, 0)

2 4 6 8 10 12

  • 0.2

0.2 0.4 0.6 0.8 1

τ ∗ R⊥(r, τ)/R||(r, 0)

2 4 6 8 10 12

  • 0.2

0.2 0.4 0.6 0.8 1

τ ∗ R⊥(r, τ)/R||(r, 0)

2 4 6 8 10 12

  • 0.4
  • 0.2

0.2 0.4 0.6 0.8 1

τ ∗ R⊥(r, τ)/R||(r, 0)

DNS of Inertial Particles May 16, 2017 25 / 27

slide-26
SLIDE 26

Reλ = 210: (a) r/L = 0.56, (b) r/L = 1.12, (c) r/L = 2.24, (d) r/L = 3.36

2 4 6 8

  • 0.2

0.2 0.4 0.6 0.8 1 DF SF1 SF2 SF3 SF4 SF5

τ ∗ R⊥(r, τ)/R||(r, 0)

2 4 6 8

  • 0.2

0.2 0.4 0.6 0.8 1

τ ∗ R⊥(r, τ)/R||(r, 0)

2 4 6 8

  • 0.2

0.2 0.4 0.6 0.8 1

τ ∗ R⊥(r, τ)/R||(r, 0)

2 4 6 8

  • 0.2

0.2 0.4 0.6 0.8 1

τ ∗ R⊥(r, τ)/R||(r, 0)

DNS of Inertial Particles May 16, 2017 26 / 27

slide-27
SLIDE 27

Acknowledgements

We are grateful to NSF for funding this research (Grant # 1436100), as well as providing access to Blue Waters through the PRAC grant We also acknowledge our collaborator Prof. Don Koch of Cornell University

DNS of Inertial Particles May 16, 2017 27 / 27