andrew walker, isr-1

U N C L A S S I F I E D


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


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


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


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


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


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


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)


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).


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)


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


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)


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


Geometries Investigated• Four geometries have been investigated thus far:

Flat Plate

Cube Cuboid

Sphere


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


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


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


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


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


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


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


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


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

andrew walker, isr-1

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