Andrew Walker, ISR-1

1. IMPACT ProjectDrag coefficients of Low Earth Orbit satellites computed with the Direct Simulation Monte Carlo method Andrew Walker, ISR-1 LA-UR 12-24986

2. 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

3. 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

4. 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. Navier-Stokes Eqns. 0 0.01 0.1 1 10 100 ∞ Free MolecularLimit Inviscid Limit Knudsen Number, Kn= λ/L

5. 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

6. 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 , , Freestream Boundary Freestream Boundary Freestream Boundary

7. 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)

8. 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).

9. 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 Reflected Velocity, Vr Incident Velocity, Vi = =R(0,1) Specular Reflection Diffuse Reflection

10. 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: increases, molecules are closer to thermal equilibrium with surface

11. 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)

12. 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

13. Geometries Investigated • Four geometries have been investigated thus far: Flat Plate Sphere Cube Cuboid

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. Conclusions • Closed-form solutions, which assume free molecular flow, are valid above ~200 km where the density is below ~1016m-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

22. 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 (Pilinskiet al. 2010) with the GITM physics-based atmospheric model • Perform global sensitivity analysis with Latin Hypercube sampling