Andrew Walker (IMPACT): Drag coefficients for low earth
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UNCLASSIFIED IMPACT Project Drag coefficients of Low Earth Orbit satellites computed with the Direct Simulation Monte Carlo method Andrew Walker, ISR-1 UNCLASSIFIED 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 UNCLASSIFIED 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 UNCLASSIFIED 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 UNCLASSIFIED 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 UNCLASSIFIED 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 π, π, π∞ Sphere ππ = 300 K πΌπ = 1.0 ππ‘ = 1.0 Freestream Boundary UNCLASSIFIED 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) – πΆπ·,πΉπ = 2 ππ 2 2 − ππ sin2 π + ππ‘ cos 2 π π −π + UNCLASSIFIED 2 − ππ 2sin2 π + 1 π 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) – πΆπ·,πΉπ = 2 ππ 2 2 − ππ sin2 π + ππ‘ cos 2 π π −π + UNCLASSIFIED 2 − ππ 2sin2 π + 1 π 2 + Gas-surface interaction models • Maxwell’s Model – A fraction of molecules, π, are specularly reflected. The remainder, 1−π, 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 ππ Incident Velocity, Vi ππ ππ Reflected Velocity, Vr ππ = ππ cos(ππ )=R(0,1) Diffuse Reflection Specular Reflection UNCLASSIFIED 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, πΌ – The energy accommodation coefficient is defined as: πΌ = πΈπ −πΈπ πΈπ −πΈπ – For example, if π∞ > ππ then the angular distribution may look like: ππ ππ ππ πΌ = 0.5 πΌ = 1.0 πΌ = 0.0 πΌ increases, molecules are closer to thermal equilibrium with surface UNCLASSIFIED Gas-surface interaction models • Cercignani-Lampis-Lord (CLL) Model – Reemission from a surface is controlled by two accommodation coefficients: – ππ‘ , tangential momentum accommodation coefficient – απ , 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) UNCLASSIFIED 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, π = 7500 m/s Satellite surface temperature, ππ = 300 K Atmospheric translational temperature, π∞ = 1100 K Atmospheric number density, π = 7.5 x 1014 m-3 Normal energy accommodation coefficient, απ = 1.0 Tangential momentum accommodation coefficient, ππ‘ =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 UNCLASSIFIED Geometries Investigated • Four geometries have been investigated thus far: Sphere Flat Plate Cube Cuboid UNCLASSIFIED Sensitivity Analysis – Satellite Velocity • Flat Plate and Sphere are relatively insensitive to changes in π – CD ~2.1 – 2.2 over range of π • Cuboid is most sensitive to π – Lower U increases shear on “long” sides – CD ~2.65 – 3.15 over range of π UNCLASSIFIED Sensitivity Analysis – Surface Temperature • All geometries are relatively insensitive to ππ • For each geometry, CD changes by ~0.1 over entire range of ππ • Dependence of sphere is slightly different – Cube and cuboid solutions are the superposition of several flat plates UNCLASSIFIED Sensitivity Analysis – Atm. Temperature • Flat plate and sphere are relatively insensitive to π∞ – CD ~2.1 – 2.15 over range of π∞ • Cuboid is most sensitive to π∞ – Higher π∞ increases shear on “long” sides – CD ~2.45 – 3.1 over range of π∞ • Cube is moderately sensitive to π∞ UNCLASSIFIED 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 UNCLASSIFIED Sensitivity Analysis – Tang. Acc. Coefficient • The flat plate is independent of ππ‘ – 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 ππ‘ due to geometry UNCLASSIFIED Sensitivity Analysis – Norm. Acc. Coefficient • The DAC CLL solution does not agree with closed-form solution – Closed-form solution is defined in terms of ππ whereas DAC CLL is in terms of πΌπ – There is no relation between ππ and πΌπ – Agrees at πΌπ = 0 and 1 – Error grows with increasing πΌπ • Can be made to agree by modifying the gas-surface interaction term in the closed-form solution UNCLASSIFIED 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 • πΌπ is the most sensitive parameter of those investigated for each geometry UNCLASSIFIED 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, πΏ ≈ 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 π∞ and π which can lead to increased shear UNCLASSIFIED 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 UNCLASSIFIED
