Atmospheric Regimes on Entry Basic fluid parameters Definition of Mean Free Path Rarified gas Newtonian flow Continuum Newtonian flow (hypersonics) 2014 David L. Akin - All rights reserved http://spacecraft.ssl.umd.edu 1
Basic Fluids Parameters M Mach Number = v a a speed of sound = p RT R = < m 1 ordered energy random energy = 2 mv2 1 2 m v2 g 2 = v2 3RT = 3 Re Reynold s number = Re = ṁv A = Av2 µ v L A = vl µ v 2 a 2 = 3 M 2 inertial force viscous force
Random vs. Ordered Energy 3
More Fluid Parameters K Knudsen number K = number of collisions with body number of collisions with other molecules K = L mean free path L vehicle characteristic length 4
Estimating Mean Free Path Assume: All molecules are perfect rigid spheres Each has diameter σ, mass m, and velocity Consider a cube with side length L containing N molecules N/6 molecules are traveling in each direction ±X ±Y ±Z v 5
Consider Collisions in +Z Direction v v 2 v () Area = 2 number of potential +Z collisions n(+z) = 1 6 N 2 L L 3 = 1 6 N 2 L 2 frequency of +Z collisions f(+z) = n(+z) t f(+z) = = 2 v 3m 6 N 2 L 2 L 2 v 6
Consider Collisions in +X Direction v v () 2 v frequency of +X collisions f(+x) = n(+x) p 2 2 v = t 6m f( X) =f(+y )=f( Y )=f(+x) f( Z) =0 Total frequency of collisions f = 3 (1 + 2p 2) 2 v m 7
Mean Free Path = v f = m/ 2 3 (1 + 2p 2) / 1 at sea level: at 100 km: =6.7 10 8 m =0.3 m 1 ft 8
Five Basic Flow Regimes Free molecular regime Near-free molecular regime Transition regime Viscous merged boundary layer Continuous regime 9
Entry Flow Regimes ref: Frank J. Regan, Reentry Vehicle Dynamics AIAA Education Series, NY, NY 1984 10
Flow Regime Definitions Knudsen number in rarified flow K = RN Mean free path after collision c = 4 p Tw T 1 1 2 1 M 1 If T w T 1 c = 1.9 1 M 1 11
Free Molecular Regime Orbital flight ` Molecule encountering a boundary (e.g., surface of vehicle) attains the state of the boundary a er a single collision K c 10 or K 1 > 5.24M 1 12
Newtonian Flow Mean free path of particles much larger than spacecra --> no appreciable interaction of air molecules Model vehicle/ atmosphere interactions as independent perfectly elastic collisions V V α α 13
Newtonian Analysis mass flux = (density)(swept area)(velocity) V ρ A sin(α) A dm dt =(ρ)(a sin α)(v ) α 14
Momentum Transfer Momentum perpendicular to wall is reversed at impact Bounce momentum is transferred to vehicle Momentum parallel to wall is unchanged Vsin(α) V V F F = dm dt V = ρv A sin α(2v sin α) =2ρV 2 A sin 2 α 15
Lift and Drag L = F cos α =2ρV 2 A sin 2 α cos α D = F sin α =2ρV 2 A sin 3 α c L = c D = L 1 2 ρv 2 A = 4 sin2 α cos α D 1 2 ρv 2 A = 4 sin3 α 16 V α L D = cos α sin α = cot α D F L
Flat Plate Newtonian Aerodynamics 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 0 20 40 60 80 100 Angle of Attack (deg) Lift coeff. Drag coeff. L/D 17
Example of Newtonian Flow Calculations Consider a cylinder of length l, entering atmosphere transverse to flow da = rdθdl dṁ = ρda cos θv = ρv cos θrdθdl df = dṁ V =2ρV 2 cos 2 θrdθdl dd = df cos θ =2ρV 2 cos 3 θrdθdl dl = df sin =2 V 2 cos 2 sin rd d` V df r θ dl dd 18
Integration to Find Drag Coefficient Integrate from θ = π 2 π 2 D = + π 2 π 2 By definition, l 0 =2 V 2 r` dd =2ρV 2 r Z + 2 2 D = 1 2 ρv 2 Ac D + π 2 π 2 l 0 and, for a cylinder cos 3 θdθdl cos 3 d = 8 3 V 2 r` A =2rl V 2 r`c D = 8 3 V 2 r` =) c D = 8 3 19
Near Free Molecular Flow Regime Also known as slip region Gas molecule only attains state of moving boundary a er several collisions Molecules near the wall will have a different velocity from the wall Temperature will be nearly discontinuous function of separation from wall 10 apple K c apple 1 3 or 5.4M 1 apple K 1 apple 0.175M 1 20
Transition Region Very difficult to treat analytically For engineering purposes, usually treated as interpolation between slip and viscous flow 0.175M 1 apple K 1 apple 1 21
Viscous Merged Layer Regime Viscous effects in forming shock and boundary layer must be treated in a unified manner Boundary layer on the wall alters the conditions for the forming shock wave Large pressure gradients across the shock wave significantly alter the boundary layer Neither shocks nor boundary layers can be treated as discontinuities 1 apple K 1 apple 0.1 s / 1 22
Continuous Regime Classical fluid mechanics of high Reynolds number Shock waves and boundary layer treated as discontinuities K 1 > 0.1 s / 1 Subdivided based on Mach number Incompressible (subsonic) (M apple 0.8) Transonic ( 0.8 apple M apple 1.3) Supersonic ( 1.3 apple M apple 5) Hypersonic ( 5 apple M) 23
Continuum Newtonian Flow (Hypersonics) Air molecules predominately interact with shock waves Effect of shock wave passage is to decelerate flow and turn it parallel to vehicle surface V α Shock wave 24
Continuum Newtonian Flow (Hypersonics) Treat hypersonic aerodynamics in manner similar to previous Newtonian flow analysis All momentum perpendicular to wall is absorbed by the wall V α 25
Mass Flux (unchanged) mass flux = (density)(swept area)(velocity) V ρ A sin(α) A dm dt =(ρ)(a sin α)(v ) α 26
Momentum Transfer Momentum perpendicular to wall is absorbed at impact and transferred to vehicle Vsin(α) V Vcos(α) F Momentum parallel to wall is unchanged F = dm dt V = ρv A sin α(v sin α) =ρv 2 A sin 2 α 27
Lift and Drag L = F cos α = ρv 2 A sin 2 α cos α D = F sin α = ρv 2 A sin 3 α c L = c D = V L 1 2 ρv 2 A = 2 sin2 α cos α D 1 2 ρv 2 A = 2 sin3 α 28 α L D = cos α sin α = cot α D F L
Modified Newtonian Flow Coefficient of pressure in classical Newtonian flow c p =2sin 2 ( ) Coefficient of pressure in modified Newtonian flow c p = c pmax sin 2 ( ) Cp(max) is the pressure coefficient behind a normal shock at flight conditions c pmax = P shock P 1 1 2 1v 2 1 29
Maximum Coefficient of Pressure c pmax = 2 M 2 1 ( apple ( + 1) 2 M 2 1 4 M 2 1 2( 1) 1 apple 1 +2 M 2 1 +1 1 ) as M! 1 c pmax! apple ( + 1) 2 4 1 apple 4 +1 c pmax! 1.839 for =1.4 c pmax! 2 for =1 30