Ye Zhao, Zhi Yuan and Fan Chen Kent State University, Ohio, USA

 “Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid” ▪ Taylor and von Kármán 1937

 “Turbulence is an irregular motion which in general makes its appearance in fluids, gaseous or liquid” ▪ Taylor and von Kármán 1937  Model them ?

 Turbulent fluids are “very hard to predict” ▪ Taylor and von Kármán 1937  Very large degree of freedom  Reynolds number (Re) ▪ Kitchen faucet: Re =  Intrinsic fluctuation  Stochastic  Intermittent

 Pure direct numerical simulation  Not practical for high Re number  Limited computational resources  Wind tunnel used in real experiments  Simulation + Synthetic noise  U + u’

 Frequency domain (Fourier)  Stam and Fiume 93, Rasmussen et al. 03  Curl operation on  Perlin noise ▪ Narain et al. 08, Schechter et al. 08  Wavelet noise ▪ Kim et al. 08  Particles in artificial boundary layer  Pfaff et al. 09

 Define energy transport between octaves of noise fields following Kolmogorov 1941 theory (K41): energy cascade  Linear model ▪ Schechter et al. 08  Advection-reaction-diffusion PDE ▪ Narain et al. 08  Locally assembled wavelets ▪ Kim et al. 08  Decay of particles ▪ Pfaff et al. 09

 Relation between u ′ and U following K41  Advect gas by u ′ and U together ▪ Stam and Fiume 93, Rasmussen et al. 03  Artificial seeding ▪ Schechter et al. 08  Local kinetic energy ▪ Kim et al. 08  Viscous hypothesis ▪ Narain et al. 08, Pfaff et al. 09

 Consistent temporal evolution of u ′ with respect to U  Distortion detection ▪ Kim et al. 08  Empirical rotation scalar field ▪ Schechter et al. 08  Special noise particles ▪ Narain et al. 08  Vortex particles ▪ Pfaff et al. 09

 Noise synthesis  Direct Fourier domain generation  Following prescribed energy spectrum  Noise fields as random forces inside a turbulence integration module  Adding forces for animation control

 Divergence free in Fourier domain

 Energy spectrum defines parameter  Gaussian control of spectrum Large variation

 Multiple scale field Kolmogorov Style An arbitrary Choice

 Noise fields as forces so that they are  A small group of force fields is enough  Pre-computed  Randomly selected  Reusable  Introduced turbulence  Continuous energy injection  Model unresolved small-scale effects  Compensate loss in numerical computing

 Enabling a feedback control in the integration  Natural coupling  Control flexibility  Large q: turbulent results close to U  Small q: significant turbulence from U

 Force integration makes it easy  What: different scales and spectra  How: conditions from physical/artificial rules  Where: local, critical, interested regions  When: intermittency

 Determine force magnitude  Velocity condition  Strain rate  Distance to obstacles  Vorticity  Scalar density

 Alternations in time between nearly non- turbulent and chaotic behavior  Extremely hard by direct simulation  We use temporal control in forcing integration  With randomly varied time intervals

 Pros  Turbulence to coarse, existing, ongoing simulation  Natural integration with random forcing  No extra boundary handling  Adaptive, conditional turbulence  Use precomputed, reusable synthetic noise  Generally independent of solvers  Handful control for animators

 Cons.  Not physically exact in spectrum control ▪ Local force integration ▪ Gaussian function in noise scales  Forced integration ▪ Extra computing load ▪ Artificially provided parameters may not always appropriate

 More integration conditions  More noise synthesis schemes  Local random force generation

 U.S. National Science Foundation  Grant IIS  Anonymous reviewers  Theodore Kim and Nils Thuerey  Rama Hoetzlein  Nvidia  Paul Farrel