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Operated by the Los Alamos National Security, LLC for the DOE/NNSA
IMPACT ProjectDrag coefficients of Low Earth Orbit satellites computed
with the Direct Simulation Monte Carlo method
Andrew Walker, ISR-1
LA-UR 12-24986
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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 gas-surface 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
Knudsen Number, Kn = λ/L0 0.01 0.1 1
10 100 ∞
EulerEqns.
Navier-StokesEqns.
Boltzmann Equation / Direct Simulation Monte Carlo
Inviscid Limit
Free MolecularLimit
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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
CollisionsMovement
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
Freestream Boundary
Sphere = 300 K
Free
stre
am B
ound
ary
Freestream Boundary
Freestream B
oundary, ,
DAC Flowfield
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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)
– Sphere–
Speed ratio, Most Probable speed,
= magnitude of velocity = Boltzmann’s constant = atmospheric temperature
= surface temperature = angle of attack
= normal momentum accommodation coefficient
= tangential momentum accommodation coefficient Closed-form solutions from Schaaf and Chambre (1958) and Sentman (1961)
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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)
– Sphere
• Gas-surface interactions are controlled by the accommodation coefficient(s). Generally, CD is most sensitive to the accommodation coefficient(s).
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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
Specular Reflection
Incident Velocity, Vi
Reflected Velocity, Vr
𝜃𝑖 𝜃𝑟
=
Diffuse Reflection
𝜃𝑖
=R(0,1)
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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:
𝜃𝑖
𝛼=1.0
𝜃𝑖
𝛼=0 .5
𝜃𝑖
𝛼=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– , 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, = 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:
Flat Plate
Cube Cuboid
Sphere
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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, 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