Scaling Parameters in Rarefied Flow and the Breakdown of the Navier-Stokes Equations Mechanical Engineering Research Report No: 2004/09

Transcription

1 Scaling Parameters in Rarefied Flow and the Breakdown of the Navier-Stokes Equations Mechanical Engineering Research Report No: 2004/09 Michael Macrossan, Centre for Hypersonics, University of Queensland Notes for a seminar delivered at University of Southern California (Jan 2004) Aeronautical Society of India at Bangalore (June 2004) July 28, 2004 Typeset by FoilTEX

2 Abstract In high altitude flight the average spacing between the molecules of the flow gas is not negligible compared to a typical dimension of the flow field. In this case, the gas does not behave like a continuum and its discrete particle nature must be considered. The continuum assumption breaks down and the Navier-Stokes equations can, in theory, no longer be shown to be valid, particularly for high speed flight. The fundamental equation describing the flow at the particle level is the Boltzmann equation, from which the Euler equations, the Navier-Stokes equations and more accurate Burnett equations may be derived under various assumptions. Various scaling parameters have been suggested to identify the regimes in which these different equations are valid, ranging from the Knudsen number, Tsein s (1946) parameter Cheng s rarefaction parameter (1961), Bird s breakdown parameter (1971), and a form of the viscous interaction parameter derived from shock-boundary layer theory. All these depend primarily on the non-dimensional group, which must be small for the Navier-Stokes equations to remain valid.

3 We show how all these parameters may be derived from the Boltzmann equation and interpreted in terms of the ratio of mean time between molecular collision and a characteristic flow time, or as the ratio of typical shear stress to pressure in the flow. Comparison of NavIer-Stokes calculations and direct simulation solutions of the Boltzmann equation show that the Navier-Stokes equations may be adequate for almost rarefied flow. Cheng s parameter appears to be the best correlation parameter and the best indicator of the validity of the Navier-Stokes equations for high-speed blunt body flow. For slender body flow the viscous interaction parameter based on a reference boundary layer temperature, a modified Tsein parameter similar to Chang s parameter, appears to be best. G. A. Bird Breakdown of translational and rotational equilibrium in gaseous expansions, AIAA J, 8, 1998 (1971) H. K. Cheng Hypersonic shock-layer theory of the stagnation region at low Reynolds number Proc 1961 heat Transfer and Fluid Mechanics Institute, Cornell Aero Lab Report IRN (1961) H. S. Tsein Superaerodynamics, mechanics of rarefied gases J Aerospace Sci, 13, 342 (1946)

4 Introduction Importance of mean free path λ, the average distance a typical molecule travelled between collisions. When Knudsen number λ/d is large or when the flow speed is high, gas is not in kinetic equilibrium. No single temperature characterizes the flow (T x T y T z T rot T vib ); the Navier-Stokes equations are not valid (in theory). Many different parameters have been used to characterize high speed and rarefied flow. Consider all the different correlation, or continuum breakdown parameters to show their near equivalence.

5 Show how all can be derived from the Boltzmann equation - the fundamental equation describing gas flow at the molecular level Quickly show how macroscopic fluid equations, Euler and Navier-Stokes are derived from the Boltzmann equation Comparing results of Navier-Stokes calculations with Direct Simulation Monte Carlo (DSMC) statistical solutions of the Boltzmann equation Best is a viscous interaction parameter = C M 2 /Re or Cheng s (1961) parameter or modified Tsein (1946) parameter. Modification factor C = µ T /µ T accounts for different collision rates molecular cross-sections viscosity laws for high energy collisions Will try to express everything in terms of the Mach number and Reynolds number (not the natural parameters of kinetic theory!)

6 Molecular collision rate - viscosity λ y u 2 = u 1 + du u 1 u dy λ 1 2 ρ cu 1 momentum fluxes Parallel flow, velocity gradient du/dy 1 2 ρ cu 2 Move ( or ) with thermal speed c 2RT Distance λ = mean free path Collision time τ = λ/ c, Rate ν = c/λ Shear stress = net x-momentum flux τ xy 1 2 ρ c (u 2 u 1 )= 1 du 2ρ cλ dy Viscosity µ 1 2 ρ cλ = 1 2 ρ c2 τ ρrt τ = pτ, size (cross-section) of molecules

7 Mean free path - Knudsen number Knudsen number Kn = λ/d, D = body dimension Kn rarefied (no collisions), Kn 0 continuum (collision dominated) λ = 2µ ρ c µ ρ : a state property, not a flow property T λ defined in rest frame of gas Relation to macroscopic flow parameters Kn = λ D µ ρ cd U/ c ρud/µ M Re

8 3 SPHERE DRAG DSMC calculations C D Kissel, M = 10, µ ~ T 0.5 Kissel, M = 8, µ ~ T 0.5 Kissel, M = 5, µ ~ T 0.5 Overell, M = 2.95, µ ~ T 0.74 Overell, M = 4.92, µ ~ T 0.74 Overell, M = 9.93, µ ~ T 0.74 Overell, M = 19.7, µ ~ T 0.74 Dogra, M = 10 27, air, µ ~ T Kn Typeset by FoilTEX Scaling parameters in rarefied flow

9 Free molecular limit C Df cold wall, diffuse reflection C D /C Df Kissel, M = 10, µ ~ T 0.5 Kissel, M = 8, µ ~ T 0.5 Kissel, M = 5, µ ~ T 0.5 Overell, M = 2.95, µ ~ T Overell, M = 4.92, µ ~ T 0.74 Overell, M = 9.93, µ ~ T 0.74 Overell, M = 19.7, µ ~ T 0.74 Dogra, M = 10 27, air, µ ~ T Kn

10 Tsein s parameter J. Aero. Sci Tsein proposed Kn δ = λ/δ. Flat plate length L. Boundary layer thickness δ. viscosity µ density ρ speed U ====> ~x 1/2 BOUNDARY LAYER δ LENGTH L δ L, ReL Re L = ρul µ, Re δ = ρuδ µ Re L Knudsen number based on δ: Kn δ = λ/δ = M Re δ M ReL For Re << 1, δ L, Kn δ Kn L = M/Re L

11 14 FLOW REGIMES TSEIN J AERO SCI 1946 M/Re = 10 M/Re 0.5 = 1 M/Re 0.5 = FREE MOLECULAR M/Re > 10 i.e. Kn > 10 RAREFIED M/Re 0.5 > 1 MACH NUMBER 8 6 BOLTZMANN EQN SLIP FLOW M/Re 0.5 > 0.01 BURNETT EQNS CONTINUUM M/Re 0.5 < 0.01 NAVIER STOKES REYNOLDS NUMBER

12 CYLINDER (PLANE 2D). FREEMOLECULAR Cd fm = Argon: γ = 5/3, µ = µ (T/T ) 0.72, T = 300 K, µ = 2.08e 5 M = 8, U = 2600 m/s, D = 2m C D /C Df NAVIER STOKES OK >? > SLIP FLOW > RAREFIED INVISCID C D = 1.21 DSMC NAVIER STOKES TSEIN S PARAMETER: M /Re Typeset by FoilTEX Scaling parameters in rarefied flow

13 1.1 TSEIN s PARAMETER SLIP RAREFIED FREE MOLECULAR C D /C Df Kissel, M = 10, µ ~ T 0.5 Kissel, M = 8, µ ~ T 0.5 Kissel, M = 5, µ ~ T 0.5 Overell, M = 2.95, µ ~ T Overell, M = 4.92, µ ~ T 0.74 Overell, M = 9.93, µ ~ T 0.74 Overell, M = 19.7, µ ~ T 0.74 Dogra, M = 10 27, air, µ ~ T M/Re 0.5 ~ λ/δ

14 Viscous Interaction Parameter AXIAL FORCE COEFFICIENT C A TUNNEL FLIGHT CORRELATIONS, NASA SHUTTLE M /Re 0.5 Calspan LaRC N 2 FLIGHT V 2 = C M 2 /Re C = µ T /µ T T = characteristic T of boundary layer For C 1, V 2 = Kn 2 δ i.e. Tsein s parameter Wind tunnel/flight Data Consider C later Data from Wilhite et al. AIAA Paper

15 Collision Number Number of collisions as molecule passes body ( Damköhler number ) Rarefaction parameter (collision number 1 ) P = coll. time flow time = λ/ c D/U = λ U D c Kn M = M2 Re = Kn2 δ U Λ 3 D 3 Λ Λ mean free path, body fame of reference P = τ/(d/u) = Uτ/D = Λ/D

16 Breakdown Parameter Breakdown of translational and rotational equilibrium in gaseous expansions G. A. Bird, AIAA J 74, 1998 (1970) Local non-equilibrium (i.e. multi-temperatures) when breakdown parameter B M λ ρ/ ρ > For ρ/ ρ D, B MKn M2 Re Kn2 δ Take conservative estimate B = 0.01 = Breakdown of equilibrium when TSEIN s parameter Kn 2 δ > 0.01, Kn δ > 0.10, i.e. Navier-Stokes may be valid well into slip regime

17 Boltzmann Equation Distribution function f: fdv = fraction of molecule velocities v v + dv Evoultion of f (v, r, t) : nf t + v nf r = [nf] coll r = position, t = time, n(r, t) = number density (molecules/m 3 ) Bhatnagar-Gross-Krook (BGK) approximation to collision term [nf] coll = n τ c (f e f) f e = Maxwell distribution τ c = local characteristic time = p/µ ( collision time).

18 Boltzmann Eq Macroscopic fluid equations Multiply B.E. by Q = [ m, mv, mv 2 /2 ], integrate over all velocities, integrate over finite volume V, with surface S Q nf t dvdv + Qv nf r dvdv = Q [nf] coll dvdv [ ] [ ] nfqdv dv + nfqc n dv ds = 0 (for anyf) t Conserved quantities: [ nfqdv ] = [ ρ, ρu, ρ ( u 2 /2 + C v T )] for any f Fluxes: [ nfqc n dv ], depends on form of f (on surface S) (1) f = f e Euler Fluxes (2) f = f e (1 + Φ CE ) Navier-Stokes fluxes (3) f = f e (1 + Φ CE + Φ B ) Burnett Fluxes

19 Non-dimensional Boltzmann Equation Select Lr, Ur = characteristic length, speed, (nr, T r ) = reference state τr = µr/ (nrktr), where µr = µ (Tr) = reference viscosity Non-dimensional values: ˆr = r/lr, ˆv = v/ur, ˆn = n/nr, ˆt = tur/lr, ˆf = fu 3 r ˆn ˆf ˆt + ˆv ˆn ˆf ˆr ( ) ( ) L r τ r n ( ˆf) = ˆfe Urτr τ c nr ( ) ( ) L r µ r T = Urτr µ Tr ( ) = (ζ) 1 µ r T µ Tr ( ˆn 2 ˆfe ˆf ) ( ˆn 2 ˆfe ˆf )

20 For µ/µr T/Tr, non-dimensional Boltzmann equation is ˆn ˆf ˆt ˆn ˆf ( + ˆv ˆr = ζ 1ˆn 2 ˆfe ˆf ) If (1) same non-dimensional boundary conditions and (2) µ/µ r T/T r then 1 non-dimensional solution depends only on the value of ζ ζ is a rarefaction parameter ζ could be a correlation parameter 1 consider conditions (1) and (2) in more detail later

21 Limiting Cases: ζ, ζ ζ : Collision rate zero maximum non-equilibrium ζ 0: Infinite collision rate equilibrium Non-dimensional collision time ζ ˆn ˆf ˆt + ˆv ˆn ˆf ˆr = 0 COLLISIONLESS BOLTZMANN EQUATION Typical solution is highly non-equilibrium distribution function.

22 Limiting case ζ 0 Non-dimensional collision time ζ 0. Infinite collision rate ˆf e ˆf = ζˆn 2 [ ˆf ˆt ] ˆf + ˆv ˆr 0 Result 2 f f e was built-in to BGK approximation, but same result follows from full collision term of Boltzmann equation. ζ is a rarefaction parameter: continuum: ζ = 0, free molecular ζ = 2 If ˆn 0 there may be local non-equilibrium regions embedded in continuum solution

23 Rarefaction parameter ζ P Tsein s parameter Reference quantities from the freestream Lr = D (body dimension), Ur = U, τr = µ /p λ / c ζ = U rτr Lr = U c λ D M Kn M2 Re P Kn 2 δ Note: ζ = µ p U D = µ U/D p shear stress pressure

24 Summary: Rarefaction (Breakdown) Parameter ζ ζ collision time/flow time inverse collision number (# collisions past body) ζ mean free path/boundary layer thickness Kn δ = λ/δ Tsein s parameter shear stress/pressure Regimes (for hypersonic flow) Continuum 0 < ζ < Navier-Stokes Slip 0.0 < ζ < 0.02 Navier-Stokes? Slip 0.02 < ζ < 1 Burnett Equations, DSMC Rarefied 1 < ζ < 5-20 Boltzmann Eq., DSMC Free molecular 20 < ζ < Collisionless Boltzmann

25 Non-dimensional boundary condition Assume diffuse reflection at surface Reflected with thermal speed at wall temperature cw (2RTw) 1 2 ( )1 ĉw = cw/u ref = Sw 1 T 2, where Sw = S Tw Reflected particle travels (average) distance λw before collision 3 ˆλw = λw/d == f (Kn, S, T w /T ) and the viscosity law µ = µ (T ). Similarity depends on ζ (S or M, and Re ) and Tw/T, and viscosity law. 3 See Whitfield (1971,1973) and Cathcart and Macrossan (1993)

26 Boltzmann Equation: C (µ/µr) (Tr/T ) term With reference state = freestream. Non-dimensional Boltzmann equation ˆn ˆf ˆt ˆn ˆf (ˆn + ˆv ˆr = (ζ C) 1 ˆf e ˆn ˆf ) C = (µ/µ ) (T /T ) varies throughout flow if C is same for two flows at some characteristic region of flow, then C (flow 1) C (flow 2) for all regions of flow (we hope!) e.g. Viscous interaction parameter V 2 = C M 2 /Re C ζ C Kn 2 δ C = (µ /µ ) (T /T ), T /T = ?M 2 T = characteristic temperature in boundary layer If V 2 C Kn 2 δ same for two flows, non-dimensional flows similar.

27 10 0 VISCOUS INTERACTION CORRELATIONS SHUTTLE, with C * C * = (µ * /µ )(T /T * ) AXIAL FORCE COEFFICIENT C A 10 1 T * = characteristic T in boundary layer T * = ( ?M 2 )T (BEST FIT) Calspan LaRC N 2 FLIGHT * 2 * Typeset by FoilTEX M /Re C = KnδC Scaling parameters in rarefied flow

28 Cheng s (1961) parameter K 2 c: Blunt body Thick shock - thick boundary layer: T in the merged shock/boundary layer CURVED BOW SHOCK K 2 c inverse viscous interaction parameter POST SHOCK T s REFERENCE T * = 0.5*(T s + T w ) 1/Kc 2 = C U µ p D C M2 Re Modified Tsein BLUNT BODY Kc 2 ( C Kn 2 ) 1 δ = (C ζ ) 1 WALL TEMPERATURE = T w

29 CHENG S PARAMETER (INVERSE) C D /C Df 0.8 C D /C Df Kissel, M = 10, µ ~ T 0.5 Kissel, M = 8, µ ~ T 0.5 Kissel, M = 5, µ ~ T 0.5 Overell, M = 2.95, µ ~ T Overell, M = 4.92, µ ~ T 0.74 Overell, M = 9.93, µ ~ T 0.74 Overell, M = 19.7, µ ~ T 0.74 Dogra, M = 10 27, air, µ ~ T Rarefaction parameter M 2 /Re = Kn δ Kissel, M = 10, µ ~ T 0.5 Kissel, M = 8, µ ~ T 0.5 Kissel, M = 5, µ ~ T 0.5 Overell, M = 2.95, µ ~ T Overell, M = 4.92, µ ~ T 0.74 Overell, M = 9.93, µ ~ T 0.74 Overell, M = 19.7, µ ~ T 0.74 Dogra, M = 10 27, air, µ ~ T C * Kn δ 2

30 Conclusions Rarefaction does not depend on Knudsen number alone 14 FLOW REGIMES TSEIN J AERO SCI 1946 M/Re = 10 M/Re 0.5 = 1 M/Re 0.5 = FREE MOLECULAR M/Re > 10 i.e. Kn > 10 RAREFIED M/Re 0.5 > 1 MACH NUMBER 8 6 SLIP FLOW M/Re 0.5 > 0.01 BURNETT EQNS CONTINUUM M/Re 0.5 < 0.01 NAVIER STOKES Importance of Mach number high shear 4 2 < N.S. MAY BE VALID M/R 0.5 = REYNOLDS NUMBER

31 Navier-Stokes equations can work into the slip regime CYLINDER (PLANE 2D). FREEMOLECULAR Cd fm = Argon: γ = 5/3, µ = µ (T/T ) 0.72, T = 300 K, µ = 2.08e 5 M = 8, U = 2600 m/s, D = 2m 0.9 C D /C Df NAVIER STOKES OK >? > SLIP FLOW > RAREFIED Slip wall boundary condition may improve NS solutions in slip regime - not shown here INVISCID C D = 1.21 DSMC NAVIER STOKES TSEIN S PARAMETER: M /Re 0.5

32 Rarefaction parameter ζ Near equivalence of many parameters ζ coll. time flow time Shear Pressure M2 Re V 2 ( Kn 2 ) 1 δ Breakdown Best is viscous interaction parameter = C M 2 /Re (or Cheng s parameter) Modification factor C = µ T /µ T For high energy collisions, C accounts for different collision rates collision cross-sections viscosity laws Must select (guess) characteristic temperature T for given flow configuration: B.L. temperature for flat plate, post-shock merged layer for blunt bodies

33 References [1] H. S. Tsein. Superaerodynamics, mechanics of rarefied gases. J. Aero. Sci., 13:342, [2] H. K. Cheng. Hypersonic shock-layer theory of the stagnation region at low Reynolds number. Report IRN , Cornell Aeronautical Laboratory, Proc 1961 heat Transfer and Fluid Mechanics Institute, [3] G. A. Bird. Breakdown of translational and rotational equilibrium in gaseous expansions. A.I.A.A. Journal, 8(11): , [4] D. L. Whitfield. Drag on bodies in rarefied high-speed flow. PhD thesis, The University of Tennessee, U.S.A., December 1971.

34 [5] D. L. Whitfield. Mean free path of emitted molecules and correlation of sphere drag data. A.I.A.A. Journal, 11(12): , [6] A. W. Wilhite, J. P. Arrington, and R. S. McCandless. Performance aerodynamics of aero-assisted orbital vehicles. A.I.A.A. Paper , [7] G. P. Cathcart and M. N. Macrossan. Aerodynamic drag reduction for satellites in low earth orbits. A.I.A.A. Journal, 31: , 1993.