Pushing the Limits of Continuum Fluid Mechanics and Going beyond the Navier-Stokes-Fourier

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1 Pushing the Limits of Continuum Fluid Mechanics and Going beyond the Navier-Stokes-Fourier March 30, 2011 (4:00-5:00PM) R. S. Myong Dept. of Mechanical and Aerospace Engineering Gyeongsang National University South Korea Presented at Mechanical Engineering Colloquium in McGill Montreal, Canada

2 Many thanks to Prof. W. G. Habashi for making this opportunity possible and his generous support and Prof. Damiano Pasini, Antonella Fratino for all the arrangements.

3 Introduction to Gyeongsang National University Seoul Jinju Busan 9 Provinces in South Korea One of the major national universities Located in Gyeongnam province (manufacturing industrial belt)

4 Overview of GNU 3 campus, 11 colleges, 67 divisions and departments 23,000 students 730 full-time faculty members Founded in 1948 Aerospace program (9 faculty; 180 undergraduates; 120 graduates)

5 Research areas of Aerospace Comp. Modeling Lab Impact on the field No. of papers JCP,JFM, JFM Rarefied micro/nanoscale gases Aircraft survivability Amount of fund $$$$$ PoF CFD MHD space plasmas Applied aerodynamics Aircraft icing Basic Idea intensive Application Labor intensive

6 Research goal: Develop a unified computational model for rarefied and micro- & nano- scale gases upon which others can build efficient CFD codes <Open knowledge>

7 Part I. Fundamentals

of unified framework Shear-dominated Low M, high Kn Micro and nanoscale cylinder Gas (liquid) flow + MN solid devices Molecular interaction

8 Introduction to rarefied and micro/nanoscale gases Compression- dominated High M, low Kn Intermediate Experimental Vehicle Gas flows around hypersonic vehicles and plume flows Continuous shift of continuum, transition, and free-molecular regimes Coexistence of various regimes Need of unified framework Shear-dominated Low M, high Kn Micro and nanoscale cylinder Gas (liquid) flow + MN solid devices Molecular interaction between gas (liquid) particles and solid atoms Gas (liquid) flows in thermal (trans., rot.) nonequilbrium regimes Electrokinetics, surface tension etc.

9 Approaches for modeling rarefied and micro/nanoscale gases Molecular approach DSMC (Direct Simulation Monte Carlo) (Bird) Linearized Boltzmann equation (Cercignani, Sone) Lattice-Boltzmann method Continuum approach Chapman-Enskog: Burnett (1935) etc. Moment method: Grad (1949), Eu (1992), regularized-13 (2005) Constitutive equations: the only ingredient in the conservation laws in which the microscopic nature of gas molecules is taken into account. Hybrid approach DSMC-continuum coupling (Nie 2004, Schwartzentruber 2007) Seamless multi-scale method: viscous stress calculated by MD (Weinan 2009)

Modeling micro and nanomechanics of fluids and rarefied gases Bottom-up: only a molecular-

Top-down: the classical linear (fluid mechanics) theories can account for virtually everything about

1/ ρ u D ρ p Dt u + I+ Π = 0 E t ( p + ) + I Π u Q Π (2) = η u, Q = k T Navier Linear uncoupled

10 Modeling micro and nanomechanics of fluids and rarefied gases Bottom-up: only a molecular- statistical theory of the structure of fluids can provide understanding of their true behavior. Top-down: the classical linear (fluid mechanics) theories can account for virtually everything about materials (fluids). 1/ ρ u D ρ p Dt u + I+ Π = 0 E t ( p + ) + I Π u Q Π (2) = η u, Q = k T Navier Linear uncoupled constitutive relations Fourier A critical observation: an efficient way of including the molecular nature of gases is to develop nonlinear coupled constitutive relations but to retain the conservation laws.

11 Linear uncoupled Navier-Fourier equations Navier (1822) Fourier (1822) Du ρ Dt DE + p+ Π = 0, ρ t + pu+ + = Π u Q 0 Dt (2), Π = η u Q = k T Π xx Πxy ux ( ux + vy )/3 ( uy + vx )/2 2η Πyx Π yy ( vx + uy )/2 vy ( ux + vy )/3 u x case (compression and expansion) case (velocity shear only) 2 ux /3 0 2η 0 uy /2 2η 0 ux /3 Uncoupled uy /2 0 2 Not like ( u ) Newtonian or Always vanishing normal stress Π or x linear u y xx Π yy Qx Tx k. Not like ( Tx ) Qy Ty 2

12 Physics of rarefied and micro/nanoscale gases Kn=M/Re Kn = 0.1 Nan o/micro Rare efied Increase of thermal non-equilibrium Solid-gas interaction Kn = 0.03 Coupling Non-isothermal Nonlinear Rarefied Hypersonic Re-entry trajectory M= 30 High temperature effect boundary 2 M / Re=O(1) 2-1 M/Re=O(10 ) M [ ] v f (, r v) = C f, f ρu u+ pi+ Π = 0 2 Two terms: Kn Three terms: M, Kn Main parameter (not Kn alone!) Π / p ~ Kn M

13 Modelling of nonequilibrium gas system Molecular (Probabilistic) Phase Space Boltzmann ( ) f ( t, r; v) t + v + a f ( t, r; v) C f v = [ f, ] ρ = mf ( t, r; v) Thermodynamics Statistical average ρu = mvf ( t, r; v) (Reduction of β = k B 1T L = L dv Information) xdvydvz Continuum (, u, T, Π, Q, L) ( t, r) Du Dt Conservation Laws ρ ρ + ( pi + Π) = ρa Thermodynamic (Hydrodynamic) Space Moment Equation (Constitutive Relation) Not far from LTE Navier-Stokes-Fourier 2

14 A modified moment method [Eu, 1992] ρ mf( t, rv ; ), ρu mvf( t, rv ; ), + v + a v f( t, rv ; ) = C f, f tt ρ + = t [ ] Differentiating the statistical definition ρ mf ( t, rv ; ) with time and then combining with the Boltzmann equation f ρ = mf (, t rv ; ) = m = mc [ f, f2 ] mv f t t t ρ + mv f = mc[ f, f2 ] = 0 t ρ ρ + m fv mf v = + m fv = t t ρ mfv t + v = 0 ( ρ u ) 0 ( ) ( ) 0 2

15 The modified moment method (continued) Moment equations in vector form D ρ Dt Unsteady 1/ ρ u 0 0 p ρ u + I + Π = a E u Π u + au t p Q ρ Convection ( Π) Π Π ψ + = ( Q) ( P) t u Q Q ψ + ψ : u Higher-order ( + v + a ) (, r; v) = [, ] t v f t C f f (2) [ Π u ] [ ] (2) ( Π) p u Λ Du + + ( Q) C C p T + + p p T Π Q u Π a Π Λ Dt Kinematic Force Thermo. driving i Dissipation i 2 [ ] (2) 2 Π m cc f, Q mc cf /2, [ ] ( Π ) (2) ( P) ( Q) 2 [ cc],, ψ m cc cf ψ mcccf ψ mc ccf Λ C[ ], Λ C[ ] ( Π) (2) ( Q) 2 m f, f2 mc c /2 f, f2 /2

16 A physically motivated closure By noting the relative importance of various terms, for example, Higher-order [ ], p[ ] ( Π) ψ < Π u (2) u (2) Kinematic Thermo. driving one may neglect the higher-order term as an approximation. ψ ( Π) 0, ψ ψ : u 0 ( Q) ( P) ψ + ψ Then we have a nonlinear coupled algebraic constitutive relation (NCCR) 0 (2) [ Π u] (2) ( Π) 0 2 p[ u] Λ 2 = Du + + ( Q) C C T + + a Π p p T Λ Π Q u p Dt Π Kinematic Force Thermo. driving Dissipation

17 A computational framework based on NCCR Compression Shear (expansion) Edge based finite volume formulation in general coordinates (JCP 2004) + = 0 UdV F nds t V S Δt U U R F n+ 1 n N 1, =, n i j i j k kδlk Ai, j k = 1 (,,, ), (,,, ) Π f Π Q p T Q f Π Q p T = Π NSF NSF = Q NSF NSF

18 NCCR property in compression- expansion and velocity shear case 2ηη u k T (2) Thermodynamic driving force p p p CT p 2Pr Algebraic NCCR p CT p 2Pr Π Q Shear stress heat flux 6 Π xx p 1.5 Π xx, xy p 4 Navier Stokes 1 Shear stress (Navier Stokes) 2 NCCR (Monatomic) 0.5 Shear stress (monatomic NCCR) 0 NCCR (Diatomic) 0 Normal stress (Navier Stokes) 2 4 Karlin EH (Monatomic) Nonlinear (non-newtonian) Coupled Normal stress (monatomic NCCR) ηdu / dx / p expansion compression velocity shear ηdu / dy / p

19 Non-Fourier law by the coupling of force and shear stress Heat fl lux Q y Non-Fourier behavior! ) ) ) Q = Q + aπ ; a is force 3 ( ) ) ) ( 3+ 2Π xy ) y y xy xy 0 Πˆ ˆ xy Πxy / p ) Qxy, Qxy, / p CpT /(2Pr) ( ) Fourier law ) ) Q = Q y y Vel. gradient 0 Π xy Q y0 0 1 Temp. gradient

20 Part II. Validation

Validation I: Celebrated shock structure Major

long time is found that the solution breaks

stronger shocks (specifically, at Mach number

21 Validation I: Celebrated shock structure Major stumbling blocks for theoreticians ti i for a long time is found that the solution breaks down completely, and no solution exists for stronger shocks (specifically, at Mach number M=1.65) (H. Grad 1952) It was solved by B. C. Eu in 1997 (Phys. Rev. E) How his theory works is explained here. Graveyard of theoreticians ti i

22 Validation I (continued) Unsteady Convection Higher-order Π Π + ( u ) Π = ψ 2[ Π u ] 2p[ u ] + Λ t ( Π ) (2) (2) ( Π ) Kinematic Thermo. driving Dissipation Comment: In steady-state t t case, one has to deal with 5 terms, which is really difficult juggling. Even in simple 3 ball juggling, there can be as many as 17 methods. Key finding: Gases are compressed rapidly across the shock wave and so accurate treatment of dissipation term will be critical. What B. C. Eu did: Assume the distribution function f in exponential form (not in usual polynomial form) and then apply the cumulant expansion method: the celebrated Eu s hyperbolic sine factor.

23 Validation I (continued) Essential terms are three in the right-hand side Λ ( Π ) Π = η / p Cf. q(kappa)=1; BGK or relaxation approx. [ ] [ ] (2) (2) ( Π ) = 0 2 Π u 2p u + Λ q( κ ), Kinematic Thermo. driving Dissipation ( mk T ) sinhκ B Π : Π Q Q q( κ) whereκ = + κ 2pd 2η kt q(x) q(0)=1 equil f f C [ f, f 2 ] x τ Comment: No 1st-order term in the series expansion of q(kappa) explains why the NSF approximation is so successful. Cf. Drag polar in aerodynamics 1/ 4 1/ 2

24 Before Validation I (continued) With sinhκ q( κ) κ singularity removed! Π p Π xx p xx 10 After 8 Navier Stokes 6 4 I A Normal Stress II III 4 I B Singularity Thermodynamic Force ηdu / dx / p ηdu / dx / p expansion compression expansion compression

for a diatomic gas versus the Mach number (o: experimental

25 Validation I (continued) kness se Shock Density Thick Invers Mach Number Shock density thickness for a diatomic gas versus the Mach number (o: experimental data) Normalized temperature contours (NSF vs NCCR) in multi-species flowfield around a reentry vehicle (M=23.47, Kn=0.2238; courtesy of J. H. Ahn)

26 Validation II: 1-d force-driven compressible Poiseuille gas flow (velocity shear dominated; PoF 2011) y T wall h T wall T wall ( pt, ) reservoir ( pt, ) reservoir x T wall uniform driving force a ah π η RT ε hwall RT 2 p h wall wall wall : Richardson no., Kn : Knudsen no. reservoir

27 Different role of q(kappa) term in velocity shear With q( κ) = 1 Π xy p Π xx p With Π p xy sinhκκ q( κ) κ 2 = 3 Π 1+ 2 p yy Local kinematic stress constraint Π p yy ηdy / dy / p BGK or relaxation is fairly good approximation, in stark contrast with the shock structure case!

28 Validation II: Fully analytical solution (Kn=0.1, ε = 0.6 ) h w 3/5 * Factor sec S p( S * p(0 ) ) 1 tan 2 * = + S minimum * ( ) p S p(0 ) = 1 concave NSF: S *4 * Factor F( S ) Pressure profile across channel (NSF: blue, NCCR: cyan) y / h y / h TS ( ) Pr ε ( ) 2 w * 2 4 * T(0) 816 T (0) Kn S 3 w * 5 * 2 * 3/5 * π Tw h FS * * * = sec S 1, S Twε hs * ( ) FS cos 17/5 * 7cos 7/5 * S S 17 7 Temperature profile across channel (NSF: blue, NCCR: cyan) w

29 Validation II: comparison with other solutions R 13 DSMC 1.06 NCCR 1 p(y/h)/p(0) DSMC NSF R 13 T(y/h)/T(0) NCCR NSF y/h 0.02 Pressure profile across channel Normal heat flux profile across channel y/h Temperature profile across channel Capturing all the qualitative features predicted by the DSMC Cf. DSMC (Bird), Regularized-13 (Struchtrup)

30 An exotic prediction: heat conduction from cold to hot Non-Fourier behavior ) 3 ) ) ) Qy = ) Q ; is force 2 y + aπ NSF xy a NSF 3+ 2Π in NCCR ( ) ( ) xy NSF Temperature contours (Courtesy of E. Roohi) Heat transfer from Cold (center) to hot H function contours Nh() v Nh() v Cf. H () t = Δv ln in DSMC N N bins N ( v) : the number of particles in a histogram bin of width Δv h

31 Another example of exotic prediction: lid-driven cavity 2-D gas flows (no force; DSMC; Kn=0.5) Temperature contours (Courtesy of E. Roohi) Lid-driven cavity gases Non-Fourier behavior in NCCR H function contours Heat transfer from Cold (center) to hot in 2-D

32 Part III. New Potential ti Applications of the Present Framework

Loss of convergence of viscoelastic flow solutions in complex fluids (polymer

state-of-stress depends on the flow history) Visco: friction, irreversibility, loss

33 Loss of convergence of viscoelastic flow solutions in complex fluids (polymer solutions; rheology) Viscoelastic fluids are complex fluids that have memory (the state-of-stress depends on the flow history) Visco: friction, irreversibility, loss of memory Elastic: recoil, internal energy storage The high-weissenberg number problem: A 30 year old mystery. All methods, without exception, were found to break down at a frustratingly low value of the Weissenberg number around We=1. (R. Fattal, R. Kupferman 2005)

34 Analogy of viscoelastic flow and rarefied gases We τ / T ~ ( l / a ) /( L / u ) ~ Kn M vs N δ Π / p ~ Kn M Upper-convected Maxwell equation derived from the stochastic model of dumbbells Nonlinear coupled constitutive equation derived from the Boltzmann equation Key observation: almost complete parallel with the gas dynamic problem -> on-going g topic

35 Occurrence ce of the singularity in the stationary a solution o of continuum models of carrier transport in semiconductors The mathematic singularity problem in the ballistic regime: Concerning the second effect, the spike is enhanced by a smaller mesh size, which is an indication of the occurrence of some sort of singularity in the stationary solution. We remark that such phenomena is not a numerical artifact. Indeed, a similar behavior has been observed with different discretization based on kinetic schemes. (A. M. Anile et al. 2000)

36 Concluding remarks Pushing the limits of continuum theory: -algebraic nonlinear coupled constitutive relation (NCCR) - persistent attack on the key problems the shock structure problem - delicate balancing in juggling terms Going beyond Navier-Stokes-Fourier: - not easy due to no appearance of 1 st order term - nothing taken for granted such as the possibility of heat transfer from cold to hot Applicable to other unsolved issues: defeating the high h Weissenberg number problem in computational ti rheology and the ballistic transport of carriers in semiconductors.

37 Acknowledgements National Research Foundation (Korea), EPSRC (UK), NASA (US) B. C. Eu (McGill Chemistry), E. Roohi (Ferdowsi Univ. of Mashhad), J. Reese (Strathclyde Univ.)

Revisit to Grad s Closure and Development of Physically Motivated Closure for Phenomenological High-Order Moment Model

Revisit to Grad s Closure and Development of Physically Motivated Closure for Phenomenological High-Order Moment Model R. S. Myong a and S. P. Nagdewe a a Dept. of Mechanical and Aerospace Engineering