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
Outline Motivation for Current Problem 1 Background 2 Effects of Forcing Scheme in DNS on Inertial Particles 3 Parallel Performance of DNS code 4 DNS of Inertial Particles May 16, 2017 2 / 27
Motivation for Current Problem DNS of Inertial Particles May 16, 2017 3 / 27
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
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
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 ( U r ) Through these two statistics, one can study the role of turbulent fluctuations in driving particle collision frequency: � 0 Collision frequency N c = 4 πσ 2 g ( σ ) U r P ( U r | σ ) dU r −∞ DNS of Inertial Particles May 16, 2017 6 / 27
Particle Preferential Concentration I Particle response to turbulence is controled by its inertia, as quantified by the Stokes number St = τ v /τ flow ◮ τ v is particle viscous relaxation time and τ flow is a flow time scale When particle Stokes number St η = τ v τ η � 1 ◮ Denser-than-fluid particles accumulate in regions of excess strain-rate over rotation-rate, i.e. where S 2 − Ω 2 > 0 DNS of Inertial Particles May 16, 2017 7 / 27
Particle Preferential Concentration II DNS of isotropic turbulence by Reade and Collins 1 demonstrates the effects of St η on clustering h ( r ) > 0 is indicative of particle preferential concentration DNS of isotropic turbulence Residual RDF ( g ( r ) − 1 )vs � r 1 Reade and Collins, Phys. Fluids, Vol. 12, 2000. DNS of Inertial Particles May 16, 2017 8 / 27
Pair Relative Velocities DNS of Sundaram and Collins 2 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 2 Sundaram and Collins, JFM, Vol. 335, 1997. DNS of Inertial Particles May 16, 2017 9 / 27
Background DNS of Inertial Particles May 16, 2017 10 / 27
Stochastic Theory ( St r = τ v /τ r ≫ 1) In a recent study 3 , 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 St r ≫ 1 ◮ St r = τ v /τ r , τ v is particle response time and τ r is time scales of eddies whose size scales with pair separation r For St r ≫ 1 particles, the pair PDF Ω( r , U ) is governed by: ∂ Ω ∂ t + ∇ r • ( U Ω) − 1 ∇ U • ( U Ω) − ∇ U • ( D UU • ∇ U Ω) = 0 τ v 3 Rani, Dhariwal, and Koch, JFM, Vol. 756, 2014 DNS of Inertial Particles May 16, 2017 11 / 27
Stochastic Theory ( St r = τ v /τ r ≫ 1) For St r ≫ 1 particles, it was shown that diffusivity � 0 1 = � ∆ u ( r , x , 0) ∆ u ( r , x , t ) � dt D UU τ 2 v −∞ ◮ 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 D UU 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
Why Blue Waters If not for Blue Waters, we would probably have not been able to compute D UU 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 10 6 particles or ∼ 5 × 10 11 pairs DNS of Inertial Particles May 16, 2017 13 / 27
Why Blue Waters Over ∼ 20 large-eddy turnover times T E , 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 D UU DNS of Inertial Particles May 16, 2017 14 / 27
Effects of Forcing Scheme in DNS on Inertial Particles DNS of Inertial Particles May 16, 2017 15 / 27
Governing Equations Fluid phase governing equations ∇ · u = 0 � � ∂ u p /ρ f + u 2 / 2 + ν ∇ 2 u + f f ∂ t + ω × u = −∇ f f is external forcing to maintain a statistically stationary turbulence Particle phase governing equations d x p dt = v p d v p dt = u ( x p , t ) − v p τ v u ( x p , t ) obtained using 8 t h order Lagrange interpolation DNS of Inertial Particles May 16, 2017 16 / 27
Forcing Schemes Recall, large scale external forcing is added to N-S equation to maintain statistically stationary turbulence Deterministic forcing 4 : Turbulent kinetic energy dissipated during a time step is added back to the velocity field Stochastic forcing 5 : Random forcing acceleration based on Ornstein-Uhlenbeck process is added to the velocity components ◮ Two important parameters: acceleration variance, σ 2 f and forcing time-scale, T f Both forcing schemes add energy to a low-wavenumber band 4 Witkowska et al. , J Comput Acoust 1997;5:317–36 5 Eswaran & Pope, Comput Fluids 1988;16:257–78 DNS of Inertial Particles May 16, 2017 17 / 27
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 � ∆ E diss (∆ t ) � u ( κ , t + ∆ t ) = � � κ max u ( κ , t + ∆ t ) 1 + κ min E ( κ, t + ∆ t ) d κ √ κ = | κ | such that κ ∈ (0 , 2], [ κ min , κ max ] is the entire wavenumber range of the DNS DNS of Inertial Particles May 16, 2017 18 / 27
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 ) � � � 2 σ 2 ∆ T � 1 / 2 1 − ∆ t b ( κ , t + ∆ t ) = � � b ( κ , t ) + θ T f T f b ( κ , t ) is an UO process having σ 2 as the variance and T f time-scale � Forcing time-scale T f is a key parameter, whose effects are studied Forcing time scale T f = 4 T E , 2 T E , T E , T E / 2, and T E / 4 considered DNS of Inertial Particles May 16, 2017 19 / 27
Parallel Performance of DNS code DNS of Inertial Particles May 16, 2017 20 / 27
1D Domain Decomposition Figure: (a) XZ slabs; (b) YZ slabs Domain decomposition along one direction N 3 simulations can be run on up to N processors Limited to small Re λ DNS of Inertial Particles May 16, 2017 21 / 27
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