Why Should We Be Interested in Hydrodynamics?

Transcription

1 Why Should We Be Interested in Hydrodynamics? Li-Shi Luo Department of Mathematics and Statistics Center for Computational Sciences Old Dominion University, Norfolk, Virginia 23529, USA URL: Math Club Department of Mathematics & Statistics Old Dominion University Thursday, February 0, 20 Luo (Math Dept, ODU) Hydrodynamics / 33

2 Outline Prelude: Some examples; Some Mathematics Modeling Flows Our Research Activities Conclusions Luo (Math Dept, ODU) Hydrodynamics 2 / 33

3 Medical Applications Luo (Math Dept, ODU) Hydrodynamics 3 / 33

4 Medical Applications Luo (Math Dept, ODU) Hydrodynamics / 33

5 How Owl Catch Mouse? Luo (Math Dept, ODU) Hydrodynamics 5 / 33

6 Owl Quiet UAV Luo (Math Dept, ODU) Hydrodynamics 6 / 33

7 Owl Quiet UAV, Car Luo (Math Dept, ODU) Hydrodynamics 7 / 33

8 Owl Quiet UAV, Car, and Airplane Luo (Math Dept, ODU) Hydrodynamics 8 / 33

9 How Dolphin Swim and Swim Fast? Luo (Math Dept, ODU) Hydrodynamics 9 / 33

10 Dolphin Speedo Swimsuit Luo (Math Dept, ODU) Hydrodynamics 0 / 33

11 Dolphin Speedo Swimsuit and Submarine Luo (Math Dept, ODU) Hydrodynamics / 33

12 Weather Prediction Luo (Math Dept, ODU) Hydrodynamics 2 / 33

13 Weather Prediction Luo (Math Dept, ODU) Hydrodynamics 3 / 33

14 Big Bang, Cosmology Luo (Math Dept, ODU) Hydrodynamics / 33

15 Big Bang, Cosmology Luo (Math Dept, ODU) Hydrodynamics 5 / 33

16 Navier-Stokes Equations: Continuum Theory Conservation laws of mass, momentum, and energy: t ρ + ρu = 0 ρ u + ρu u = P ρ t e + ρu e = P : u q ( P αβ := pδ αβ µ α u β + β u α 2 ) 3 δ αβ u ζδ αβ u (a) (b) (c) q = κ T, e = e(t ), p = p(ρ, T ) Dimensionless Navier-Stokes equations (similarity law): ρ u + ρu u = γma 2 p + Re S ρ t e + ρu e = γma 2 p u + α 2 T + Re S : u α = (γ )PrMaRe (2a) (2b) Luo (Math Dept, ODU) Hydrodynamics 6 / 33

17 Hierarchy of Scales and PDEs Microscopic Scale Mesoscopic Scale Macroscopic Scale q k = H, ṗ k = H p k q k H = (D+K)N k= p 2 k + V i ψ = Hψ H = 2 N j= 2m 2 j +V h (J s) c (m/s) a 5 0 (m) t a (s) m 0 27 (kg) N =, 2,..., N 0 t f + ξ f = Q(f, f) ε f = f(x, ξ, t) ε = Kn = l L, Ma = U c s k B (J/ K) l (Å) 0 00 (nm) τ 0 0 (s) c s 300 (m/s) N ρd t u = p + Re σ D t = t + u σ = ρν 2 [( u)+( u) ] + 2ρζ D I( u) Re δ = UL ν Ma Kn ν (m 2 /s) Kn 0 Ma < 0 3 L 0 5 (m) = 0(µm) T 0 (s) N N A Luo (Math Dept, ODU) Hydrodynamics 7 / 33

18 In the Course of Hydrodynamic Events... Ma (Mach) and Kn (Knudsen) characterize nonequilibrium Ma Ma Ma Van Karmen relation based on Navier-Stokes equation (Kn=O(ε)): Ma=Re Kn Re Re Re Stokes Flows Incompressible Navier-Stokes Flows Ma=O(ε 2 ), Re=O(ε) Ma=O(ε), Re=O() Ma=O(ε α ), Re=O(ε α ) Sub/Transonic Flows Ma=O(), Re=O(ε ) Super/Hypersonic Flows Ma=O(ε ), Re=O(ε 2 ) With the framework of kinetic theory (Boltzmann equation) Hydrodynamics Slip Flow Transitional Free Molecular Kn< <Kn<0 0 <Kn<0 0<Kn Luo (Math Dept, ODU) Hydrodynamics 8 / 33

19 Evolution of LGA: Collision + Advection An intuitive system of fictitious particles on a Lattice Evolution from t (solid arrows) to t + (hollow arrows): N α (x i + e α, t + ) = N α (x i, t) + C α ({N β }) FHP-LGA Collision Rules Input State Output State Luo (Math Dept, ODU) Hydrodynamics 9 / 33

20 Binary Collision Table for 6-Velocity FHP Model FHP-LGA Collision Rules Binary Representation of FHP LGA Collision Rules Input State Output State Input State Output State Luo (Math Dept, ODU) Hydrodynamics 20 / 33

21 Why should LGCA work? We have noticed in nature that the behavior of a fluid depends very little on the nature of the individual particles in that fluid. For example, the flow of sand is very similar to the flow of water or the flow of a pile of ball bearings. We have therefore taken advantage of this fact to invent a type of imaginary particle that is especially simple for us to simulate. This particle is a perfect ball bearing that can move at a single speed in one of six directions. The flow of these particles on a large enough scale is very similar to the flow of natural fluids. Richard P. Feynman (98 988) Luo (Math Dept, ODU) Hydrodynamics 2 / 33

22 ODU Research: Complex Bio-Fluids, Lab-on-a-Chip Prof. S. Qian et al. (Dept. of Mech. & Aerospace Eng.): Experiments (MEMS) to separate particles by size or charge; Prof. Y. Peng et al. (Dept. of Math. & Stat.): Simulations of a particle moving in sediment or in a channel. Luo (Math Dept, ODU) Hydrodynamics 22 / 33

23 ODU Research: Gas Flow in MEMS Luo (Math Dept, ODU) Hydrodynamics 23 / 33

24 Gas Flow in MEMS: What s the problem? The scales in this problem: Device size: 0 6 m = µm The mean-free-path of air: m = 70nm The effective air molecular diameter: σ m = 0.nm Typical hydrodynamic time scale: 0 3 s 0 9 s To consider molecule-surface interaction, one must use Molecular Dynamics (MD) simulations. Typical time step size for MD: τ σ m/ɛ 0 2 s = ps Luo (Math Dept, ODU) Hydrodynamics 2 / 33

25 Gas Flow in MEMS: A Possible Solution Not to directly couple MD and CFD! Use MD to compute the mean-field potential near the wall; Use kinetic schemes to include the mean-field potential obtained from the MD; The goal is to drastically reduce computational time. Luo (Math Dept, ODU) Hydrodynamics 25 / 33

26 ODU Research: Hypersonic Rarefied Gas Flow Luo (Math Dept, ODU) Hydrodynamics 26 / 33

27 Hypersonic Rarefied Gas Flow The hypersonic rarefied gas flows are strongly nonequilibrium: Large Ma: strong and complex shock-shock and shock-boundary layer interactions; Large Kn: Navier-Stokes equations no longer valid; Established method for nonequilibrium flows: Direct Monte Carlo simulations (DSMC), a stochastic method; Deterministic solution methods of the Boltzmann equation. Our strategy: kinetic methods Extending the Navier-Stokes equations based on the Boltzmann equation. Luo (Math Dept, ODU) Hydrodynamics 27 / 33

28 Hypersonic Rarefied Gas Flow: Shock at Ma = 8.0 The gas-kinetic scheme based on the Boltzmann equation can accurately and efficiently predict shock thickness (about a few mean-free paths), the stress and heat flux across a shock. Luo (Math Dept, ODU) Hydrodynamics 28 / 33

29 A New Direction: Energy Storage Luo (Math Dept, ODU) Hydrodynamics 29 / 33

30 A New Direction: Energy Storage Nanoporous energy absorption system (NEAS), e.g., silica nanoporous material: Porosity 30% 90%, high specific pore area >, 000 (m 2 /g) Extremely high specific energy absorption: > 0.0 (J/g), compared to Li-Battery 2, 500 (J/g), TNT, 60 (J/g), gasoline 6, 00 (J/g). Luo (Math Dept, ODU) Hydrodynamics 30 / 33

31 Conclusions Why hydrodynamics? Interesting involving modeling (PDE) of multi-physics and multi-scale problems; Useful wide variety of applications; Challenging many difficult problems remained solved. Luo (Math Dept, ODU) Hydrodynamics 3 / 33

32 Other Applications Ladd, Univ. Florida: Cluster of 82 particles Flow through Porous Media (Krafczyk et al., TUB): Air through fluids in a packed-sphere bio-filter Multi-component flow past porous media: imbibe, drain, re-imbibe Free-Surface Flow (Krafczyk et al., TUB): A single drop impact on a liquid surface Flow coming down from a dam Flow past through/over a bridge Flow interacts with a column Droplet collision (Frohn et al., Univ. Stuttgart): Merge, Separate, and one extra; LBE vs. Experiment. More (Thüry et al., Univ. Erlangen): Bubbles rising in water tank Metal foams Luo (Math Dept, ODU) Hydrodynamics 32 / 33

33 Questions? Luo (Math Dept, ODU) Hydrodynamics 33 / 33