Andrew Walker (IMPACT): Drag coefficients for low earth

Report
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IMPACT Project
Drag coefficients of Low Earth Orbit satellites computed
with the Direct Simulation Monte Carlo method
Andrew Walker, ISR-1
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Operated by the Los Alamos National Security, LLC for the DOE/NNSA
LA-UR 12-24986
Outline
• Motivation
• Direct Simulation Monte Carlo (DSMC) method
• Closed-form solutions for drag coefficients
• Gas-surface interaction models
– Maxwell’s model
– Diffuse reflection with incomplete accommodation
– Cercignani-Lampis-Lord (CLL) model
• Fitting DSMC simulations with closed-form solutions
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Motivation
• Many empirical atmospheric models infer the atmospheric
density from satellite drag
– Some models assume a constant value of 2.2 for all satellites
– The drag coefficient can vary a great deal from the assumed value of
2.2 depending on the satellite geometry, atmospheric and surface
temperatures, speed of the satellite, surface composition, and gassurface interaction
• Without physically realistic drag coefficients, the forward
propagation of LEO satellites is inaccurate
– Inaccurate tracking of LEO satellites can lead to large uncertainties in
the probability of collisions between satellites
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Direct Simulation Monte Carlo (DSMC)
• DSMC is a stochastic particle method that can solve gas
dynamics from continuum to free molecular conditions
– DSMC is especially useful for solving rarefied gas dynamic problems
where the Navier-Stokes equations break down and solving the
Boltzmann equation can be expensive
– DSMC is valid throughout the continuum regime but becomes
prohibitively expensive compared to the Navier-Stokes equations
Boltzmann Equation / Direct Simulation Monte Carlo
Euler
Eqns.
0
Inviscid
Limit
Navier-Stokes
Eqns.
0.01
0.1
1
10
Knudsen Number, Kn = λ/L
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100
∞
Free
Molecular
Limit
Direct Simulation Monte Carlo (DSMC)
• Particle movement and collisions are decoupled based on
the dilute gas approximation
– Movement is performed by applying F=ma
– Collisions are allowed to occur between molecules in the same cell
Movement
Collisions
Possible Collision
Partners
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Direct Simulation Monte Carlo (DSMC)
• These drag coefficient calculations utilize NASA’s DSMC
Analysis Code (DAC)
– Parallel
– 3-dimensional
– Adaptive timestep and spatial grid
DAC Flowfield
Freestream Boundary
, , ∞
Sphere
 = 300 K
 = 1.0
 = 1.0
Freestream Boundary
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Closed-form Solutions
• Closed-form solutions for the drag coefficient, CD, have been
derived for a variety of simple geometries:
– Flat Plate (both sides exposed to the flow)
– , =
2

2
2 −  sin2  +  cos 2   − +
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2 − 
2sin2  +
1
2
+
Closed-form Solutions
• The key term in each of these expressions is the last term
which accounts for the reemission of molecules from the
surface (e.g. the gas-surface interaction):
– Flat Plate (both sides exposed to the flow)
– , =
2

2
2 −  sin2  +  cos 2   − +
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2 − 
2sin2  +
1
2
+
Gas-surface interaction models
• Maxwell’s Model
– A fraction of molecules, , are specularly reflected. The remainder,
1−, are diffusely reflected.
– Momentum and energy accommodation are coupled (e.g. if a
molecule is diffusely reflected, it is also fully accommodated).
– Intuitive and simple to implement
– Unable to reproduce molecular beam experiments

Incident
Velocity, Vi


Reflected
Velocity, Vr
 = 
cos( )=R(0,1)
Diffuse Reflection
Specular Reflection
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Gas-surface interaction models
• Incomplete Energy Accommodation with Diffuse Reflection
– All molecules are diffusely reflected but may lose energy to the
surface depending on the energy accommodation coefficient, 
– The energy accommodation coefficient is defined as:  =
 −
 −
– For example, if ∞ >  then the angular distribution may look like:



 = 0.5
 = 1.0
 = 0.0
 increases, molecules are closer to thermal equilibrium with surface
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Gas-surface interaction models
• Cercignani-Lampis-Lord (CLL) Model
– Reemission from a surface is controlled by
two accommodation coefficients:
–  , tangential momentum accommodation
coefficient
– α , normal energy accommodation
coefficient
– Normal and tangential components are
independent but tangential momentum and
energy are coupled.
– Able to reproduce molecular beam
experiments (as shown in the figure to the
right)
Figure from Cercignani
and Lampis (1971)
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Local Sensitivity Analysis
• Drag coefficients are computed with the DAC CLL model as
well as with the closed-form solution for that geometry
• Each parameter is varied independently with the nominal
parameters defined as:
–
–
–
–
–
–
Satellite velocity relative to atmosphere,  = 7500 m/s
Satellite surface temperature,  = 300 K
Atmospheric translational temperature, ∞ = 1100 K
Atmospheric number density,  = 7.5 x 1014 m-3
Normal energy accommodation coefficient, α = 1.0
Tangential momentum accommodation coefficient,  =1.0
• CD are compared between the DAC CLL model and the
closed-form solutions by computing the local percent error at
each data point
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Geometries Investigated
• Four geometries have been investigated thus far:
Sphere
Flat Plate
Cube
Cuboid
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Sensitivity Analysis – Satellite Velocity
• Flat Plate and Sphere
are relatively
insensitive to changes
in 
– CD ~2.1 – 2.2 over
range of 
• Cuboid is most
sensitive to 
– Lower U increases
shear on “long” sides
– CD ~2.65 – 3.15 over
range of 
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Sensitivity Analysis – Surface Temperature
• All geometries are
relatively insensitive to

• For each geometry, CD
changes by ~0.1 over
entire range of 
• Dependence of sphere
is slightly different
– Cube and cuboid
solutions are the
superposition of several
flat plates
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Sensitivity Analysis – Atm. Temperature
• Flat plate and sphere
are relatively
insensitive to ∞
– CD ~2.1 – 2.15 over
range of ∞
• Cuboid is most
sensitive to ∞
– Higher ∞ increases
shear on “long” sides
– CD ~2.45 – 3.1 over
range of ∞
• Cube is moderately
sensitive to ∞
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Sensitivity Analysis – Number Density
• The closed-form
solutions assume free
molecular flow
• DAC CLL simulations
show this assumption
breaks down across all
geometries for number
densities above ~1016
m-3 (with a 1 m satellite
length scale)
• This corresponds to an
altitude of ~200 km or
above
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Sensitivity Analysis – Tang. Acc. Coefficient
• The flat plate is
independent of 
– The flat plate is
infinitesimally thin and
therefore there is no
shear at this angle of
attack
• For the cube, cuboid,
and sphere, the
dependence is linear
– Sphere is most
sensitive to  due to
geometry
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Sensitivity Analysis – Norm. Acc. Coefficient
• The DAC CLL solution
does not agree with
closed-form solution
– Closed-form solution is
defined in terms of 
whereas DAC CLL is in
terms of 
– There is no relation
between  and 
– Agrees at  = 0 and 1
– Error grows with
increasing 
• Can be made to agree by
modifying the gas-surface
interaction term in the
closed-form solution
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Sensitivity Analysis – Norm. Acc. Coefficient
• Modified closed-form
solutions agree with
DAC CLL model
– Used least squares
error method to find
best fit
– Modified closed-form
solution isn’t perfect but
is within ~0.5% percent
error
•  is the most
sensitive parameter of
those investigated for
each geometry
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Conclusions
• Closed-form solutions, which assume free molecular flow,
are valid above ~200 km where the density is below
~1016 m-3 assuming a satellite length scale,  ≈ 1 m
• DAC CLL simulations agree well with the closed-form
solution except in terms of the normal energy
accommodation coefficient
– This is because closed-form solutions are cast in terms of the normal
momentum accommodation coefficient
– Can modify closed-form solutions to agree with DAC CLL model
• CD is most sensitive to:
– Geometry
– Normal energy accommodation coefficient
– “Long” bodies such as the cuboid are also sensitive to ∞ and 
which can lead to increased shear
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Future Work
• Thus far, only simple geometries where the closed-form
solution is known have been investigated
– Allows for verification of the DAC CLL model vs. closed-form solution
• Use DAC CLL model to find empirical closed-form fits to
realistic and complicated satellite geometries (e.g. CHAMP)
• Recreate Langmuir isotherm fit for normal energy
accommodation coefficient (Pilinski et al. 2010) with the
GITM physics-based atmospheric model
• Perform global sensitivity analysis with Latin Hypercube
sampling
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