Aero-thermal Demise of Reentry Debris: A Computational Model

Transcription

1 Aero-thermal Demise of Reentry Debris: A Computational Model by Troy M. Owens Bachelor of Science In Aerospace Engineering Florida Institute of Technology 2013 A thesis submitted to the College of Mechanical and Aerospace Engineering at Florida Institute of Technology in partial fulfillment of the requirements for the degree of Master of Science in Aerospace Engineering Melbourne, Florida August, 2014

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3 All rights reserved. Copyright 2014 by T. M. Owens. No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the author. For information address: T. M. Owens. PRINTED IN THE UNITED STATES OF AMERICA

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5 We the undersigned committee hereby approve the attached thesis Aero-Thermal Demise of Reentry Debris: A Computational Model by Troy M. Owens Dr. Daniel Kirk Professor, Mechanical and Aerospace Engineering Associate Dean for Research Major Advisor Dr. David Fleming Professor, Mechanical and Aerospace Engineering Committee Member Dr. Ronnal Reichard Professor, Marine and Environmental Systems Director of Laboratories Coordinator; Senior Design Committee Member Dr. Hamid Hefazi Professor, Mechanical and Aerospace Engineering Department Head

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7 Abstract Title: Aero-thermal Demise of Reentry Debris: A Computational Model Author: Troy Owens Major Advisor: Daniel R. Kirk, Ph.D. The modeling of fragment debris impact is an important part of any space mission. Planned debris or failure at launch and reentry need to be modeled to understand the hazards to property and populations. With more accurate impact predictions, a greater confidence can be used to close areas for protection and generate destruct criteria for space vehicles. One aspect of impact prediction that is especially difficult to simulate in a simple yet accurate way is the aero-thermal demise of reentry debris. This thesis will attempt to address the problem by using a simple set of inputs and combining models for the earth, atmosphere, impact integration and stagnation-point heating. Current tools for analyzing reentry demise are either too simplistic or too complex for use in range safety analysis. NASA s Debris Assessment Software 2.0 (DAS 2.0) has simple inputs that a range safety analyst would understand, but only gives the demise altitude as output and no ability to specify breakup conditions. Object Reentry Survival Analysis Tool (ORSAT), the standard for reentry demise analysis, requires inputs that only the vehicle manufacturer knows and a trained operator. The output from ORSAT gives a full range of fragment properties and for numerous breakup conditions. This thesis details a computational model with simple inputs like DAS 2.0, but an output closer to that of ORSAT, that will be useful in many mission risk analysis scenarios. This is achieved by using 1) WGS 84, a fourth order spherical harmonic model of the earth s surface and gravity; 2) the 1976 U.S. Standard Atmosphere; 3) an impact integrator for a spherical rotating earth; and 4) a stagnation-point heating correlation based on the Fay-Riddell theory. iii

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9 Contents Abstract... iii Contents... v List of Keywords and Abbreviations... ix List of Exhibits... xi List of Symbols... xiii Symbols for Impact Integration... xiii Symbols for Fay-Riddell Stagnation Point Heating... xv Symbols for Aero-thermal Demise... xvii 1 Introduction DAS 2.0: Debris Assessment Software ORSAT: Orbital Reentry Survival Analysis Tool Aerospace Survivability Tables SCARAB: Spacecraft Atmospheric Re-entry and Aerothermal Break-up Computation Model Risk Analysis Impact Integration Equations of Motion Relative Angular Motion Equations for Flight Over a Rotating Spherical Earth Stagnation-Point Heating Fay and Riddell Theory Laminar Boundary-Layer in Dissociated Gas Boundary Layer Ordinary Differential Equations Heat Transfer Rate...24 v

10 3.1.4 Equilibrium Boundary Layer Detra, Kemp and Riddell Correlation Radiation Heat Balance Algorithm Earth Model Zonal Harmonic Gravity Vector Atmospheric Model Lower Atmosphere Upper Atmosphere Impact Integrator Aero-thermal Demise Fragment Properties Material Properties Shape Assumptions Stagnation Point Heating Liquid Fraction Fragment Tables Results Understanding Aero-heating Reentry Trajectory, Heat Flux, and Bulk Temperature Varying Breakup Altitude Varying Initial Temperature of Debris Fragment Varying Initial Velocity Varying Flight Path Angle...67 vi

11 5.1.6 Varying Materials of Debris Fragment Varying Mass of Debris Fragment Model Comparisons DAS Aerospace Survivability Tables Input and Output Debris Fragment Catalog Conclusions Practical Application Validation Performance Possible Future Work...89 References...91 Appendix...94 Appendix A: Material Properties...94 Appendix B: Supplemental Algorithms...98 Alternate Correlations...98 Trajectory Site Direction Cosines ECEF Coordinates to XYZ Coordinates ECEF Coordinates to Aeronautical Coordinates Appendix C: MATLAB Code... Error! Bookmark not defined. demiseutility.m... Error! Bookmark not defined. demise.m... Error! Bookmark not defined. glidederivatives.m... Error! Bookmark not defined. Atmosphere1976.m... Error! Bookmark not defined. createearth.m... Error! Bookmark not defined. vii

12 getgravity.m... Error! Bookmark not defined. fragmentexporter.m... Error! Bookmark not defined. fragmentimporter.m... Error! Bookmark not defined. viii

13 List of Keywords and Abbreviations Keyword/Abbreviation AST Boeing X-37 OTV CAIB Casualty Area CFD CSV Definition Office of Commercial Space Transportation Orbital Test Vehicle. Unmanned spacecraft, launches as rocket lands as a space plane Columbia Accident Investigation Board Area where a debris fragment will cause human casualty, same as hazard area Computational fluid dynamics Comma-sepperated values, a data file format DAS 2.0 Debris Assessment Software 2.0 Ec ECEF EFG ESA FAA Hazard Area JARSS MP MATLAB Estimated or expected casualties, usually measured in micro-casualties Earth Centered Earth Fixed, same as EFG Earth Fixed Geocentric, same as ECEF European Space Agency Federal Aviation Adminstration Area where a debris fragment will cause a human casualty, same as casualty area Joint Advance Range Safety System: Mission Planning Programing language and integrated development environment (IDE) developed by MathWorks ix

14 Micro-casualties Mission Analyst Mollier diagram NASA ODE ORSAT Pi Planned Debris Risk Analyst SCARAB Risk of human casualty due to debris hazard. Equal to 1*10-6 casualty per event, see Ec Someone who performs a flight safety analysis for a mission, see Risk Analyst Enthalpy-entropy chart, h-s chart. Plots the total heat against entropy National Aeronautics and Space Administration Ordinary differential equation Orbital Reentry Survival Analysis Tool Probability of Impact Debris from staging and other planned events Someone who performs a flight safety analysis for a mission, see Mission Analyst Spacecraft Atmospheric Re-entry and Aerothermal Break-up WGS 84 World geodetic Survey of 1984 x

15 List of Exhibits Figure 1: Casualty Risk from Reentry Debris...3 Figure 2: Aluminum 2024-T8xx Tumbling Hollow Sphere Survivability Table (Ref. 25)...6 Figure 3: Coupling and Dependence for Disintegration Prediction (Ref. 12)...7 Figure 4: Thermal Fragmentation and Mechanical Breakup (Ref. 12)...8 Figure 5: Relative Angular Motion (Ref. 26)...13 Figure 6: Kinematics of Rotation (Ref. 26)...13 Figure 7: Coordinate Systems (Ref. 26)...16 Figure 8: Aerodynamic and Propulsive Forces (Ref. 26)...17 Figure 9: Stagnation Point Heat Transfer Rate vs. Flight Velocity (Ref. 5)...29 Figure 10: Corrected (red) and Uncorrected (blue) Cd Impacts...52 Figure 11: Single Fragment, Altitude vs. Time...55 Figure 12: Single Fragment, Altitude vs. Range...55 Figure 13: Single Fragment, Heat Flux vs. Time...56 Figure 14: Single Fragment, Temperature vs. Time...57 Figure 15: Varying Breakup Altitude, Altitude vs. Time...58 Figure 16: Varying Breakup Altitude, Altitude vs. Range...58 Figure 17: Varying Breakup Altitude, Heat Flux vs. Time...59 Figure 18: Varying Breakup Altitude, Temperature vs. Time...60 Figure 19: Varying Breakup Altitude, Liquid Fraction vs. Time...61 Figure 20: Varying Temperature, Temperature vs. Time...62 Figure 21: Varying Temperature, Liquid Fraction vs. Time...62 Figure 22: Varying Temperature, Heat Flux vs. Time...63 Figure 23: Varying Velocity, Altitude vs. Time...64 Figure 24: Varying Velocity, Altitude vs. Range...64 Figure 25: Varying Velocity, Heat Flux vs. Time...65 xi

16 Figure 26: Varying Velocity, Temperature vs. Time...66 Figure 27: Varying Velocity, Liquid Fraction vs. Time...66 Figure 28: Varying Flight Path Angle, Altitude vs. Time...67 Figure 29: Varying Flight Path Angle, Altitude vs. Range...68 Figure 30: Varying Flight Path Angle, Heat Flux vs Time...69 Figure 31: Varying Flight Path Angle, Temperature vs. Time...69 Figure 32: Varying Flight Path Angle, Liquid Fraction vs. Time...70 Figure 33: Varying Materials, Heat Flux vs. Time...71 Figure 34: Varying Materials, Temperature vs. Time...71 Figure 35: Varying Materials, Liquid Fraction vs. Time...72 Figure 36: Varying Mass, Altitude vs Time...73 Figure 37: Varying Mass, Altitude vs. Range...74 Figure 38: Varying Mass, Heat Flux vs. Time...75 Figure 39: Varying Mass, Temperature vs. Time...75 Figure 40: Varying Mass, Liquid Fraction vs. Time...76 Table 1: WGS84 Ellipsoid Derived Geometric Constants...31 Table 2: Local Arrays (1976 Std. Atmosphere)...34 Table 3: Atmosphere Fit Parameters (1976 Std. Atmosphere)...36 Table 4: Debris Fragment Shape Assumptions...44 Table 5: Example Fragment...54 Table 6: DAS 2.0 Debris Fragments...77 Table 7: Computational Model Debris Fragments, Compared to DAS Table 8: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder...80 Table 9: Survivability Table Liquid Fraction Comparison, Aluminum Cylinder...81 Table 10: Demise Utility Input...82 Table 11: Demise Utility Output, Adjusted Fragment Tables...83 xii

17 List of Symbols Symbols for Impact Integration Variable Value Units Description a m s Local speed of sound a m Earth s semi-major axis β 1 m Atmospheric decay parameter, scale height C L Coefficient of lift C L Coefficient of lift at the smallest glide angle C D Coefficient of drag C D Coefficient of drag at the smallest glide angle C D 0 Zero lift coefficient of drag F N N Force normal to the flight path F T N Force tangential to the flight path g m s 2 Local gravitational acceleration g 0 m s 2 Gravitational acceleration at Earth s surface γ rad Flight path angle, glide angle of vehicle K Induced drag factor, function of Mach m kg Mass of the vehicle xiii

18 n Wing loading factor n s Structural loading limit ω rad s Angular velocity of Earth φ rad Glide turn roll angle of vehicle φ rad Latitude of the vehicle θ rad Longitude of vehicle ψ rad Heading of vehicle r m EFG magnitude radius to vehicle R m Mean radius of Earth S m 2 Plane area of the vehicle V y m s Vehicle velocity Trajectory aerodynamic coordinate state xiv

19 Symbols for Fay-Riddell Stagnation Point Heating Variable Value Units Description c i Mass fraction of component i c p J kgk Specific heat per unit mass at constant pressure D m 2 s Molecular diffusion coefficient D T m 2 sk Thermal diffusion coefficient e e i J kg Internal energy per unit mass, of component i h J kg Enthalpy per unit mass of mixture h i J kg Perfect gas enthalpy per unit mass of component i h i 0 h A 0 k J kg J kg W K Heat of formation of component i at 0 K per unit mass Dissociation energy per unit mass of atomic products Thermal conductivity l m Characteristic length L L i T Lewis Number: D i ρc p k ratio of the rate of thermal diffusivity to the mass diffusivity, thermal Lewis number μ Pa s Absolute viscosity Nu Nusselt Number: qxc pw k w (h s h w ) ratio of convective to conductive heat transfer ν Pa s Kinematic viscosity p Pa Pressure xv

20 Pr 0.71 Prandtl Number: c pμ k ratio of momentum diffusivity to thermal diffusivity, value for air Φ Dissipation function q W m 2 Heat flux q q i m s Vector mass velocity, vector diffusion velocity r m Cylindrical radius of body R h m Body nose radius, radius of heating Re ρ kg m 3 Reynolds Number: u e x v w ratio of the inertial to viscous forces Mass density T K Absolute temperature u v w i x y m s m s kg m 3 s m s m s x component of velocity y component of velocity Mass rate of formation of component i per unit volume and time Distance along meridian profile Distance normal to the surface xvi

21 Symbols for Aero-thermal Demise Variable Value Units Description a m s Local speed of sound A w m 2 Wetted area β hc C Dhc kg m 2 Hypersonic continuum ballistic coefficient Hypersonic continuum coefficient of drag C Dβ Coefficient of drag corrected for ballistic coefficient C DMach Coefficient of drag from fragment drag tables C pb C p J kgk J kgk Mean Specific heat capacity of the fragment Specific heat capacity of air δ m Recession length, flat plate ε b ε w Surface emissivity of the fragment g m s 2 Standard gravitational acceleration γ radians Flight angle of the fragment h h i m Fragment height and interior height, box h f J kg Heat of fusion H r H r 0 k m Hazard radius & user defined hazard radius Area-averaging factor (a less conservative 0.8 for composites) l l i m LF Fragment length and interior length, cylinder, flat plate and box Liquid fraction of the fragment m b kg Thermal mass of the fragment xvii

22 m i kg Mass of the interior of the fragment were it solid m LF kg Thermal mass adjust by liquid fraction of the fragment σ sbc W m 2 K 4 Stefan-Boltzmann constant q s W m 2 Stagnation heat flux q rad W m 2 Radiation heat flux Q W s Net heat flow Q 0 W Heat of initial temperature Q melt W Heat of melting Q a W Heat of ablation Q W Heat content of the fragment r r i m Radius and interior radius of fragment, sphere and cylinder R m Mean radius of Earth R h m Radius of heating R 287 ρ b ρ J K kg m 3 kg m 3 Gas constant for air Density of the fragment material Free stream air density S m 2 Aerodynamic reference area of the fragment t m Fragment thickness, flat plate T 0 K Initial body bulk temperature of the fragment T b K Body bulk temperature of the fragment T melt K Melting temperature of the fragment T ref 300 K Reference temperature xviii

23 T s K Adiabatic stagnation temperature T w K Hot wall temperature of the fragment U ref U m s m s Reference velocity, ft/s Free stream velocity w w i m Fragment width and interior width, flat plate and box z m Altitude position of the fragment xix

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25 T. M. Owens 1 1 Introduction In the past few years there has been a steady rise in the number of space launches as well as the privatization of America s launch capabilities. With the increase in space missions, there is an increase in demand for range safety analysis to address possible risk to the population. Tools used to evaluate this risk rely on accurate vehicle debris fragment models to estimate the probability of a human casualty. By developing simple to use and accurate models for the aero-thermal demise of reentry debris, better predictions of the probability of impact (Pi) and estimated casualties (Ec) can be made. The computational model detailed in this thesis combines several well established algorithms for modeling earth geometry, gravitational acceleration, atmospheric properties, impact propagation and aero-thermal demise to model the aerothermal demise of reentry debris. With simple inputs the model generates a debris catalog that can be used by other risk analysis tools. There is also the possibility that the impact integrator could be used within an existing tool in order to consider demise of an existing debris catalog. To understand what the work in this thesis is attempting to accomplish, it helps to understand how current tools work to estimate reentry debris survivability and casualty risk. The Debris Assessment Software 2.0 (DAS 2.0) and Orbital Reentry Survival Analysis Tool (ORSAT) are used by National Aeronautics and Space Adminstration (NASA) and other American launch providers Ref. 17, Ref. 2. Aerospace Survivability Tables were developed for the Federal Aviation Authority (FAA) and Office of Commercial Space Transportation (AST) Ref. 24. The Spacecraft Atmospheric Re-entry and Aerothermal Break-up (SCARAB) tool is used by the European Space Agency (ESA) Ref. 11. DAS 2.0 is a simplistic first order solution, like this thesis, whereas ORSAT and SCARAB are pseudo-cfd programs with much

26 2 Aero-thermal Demise greater complexity. The computational model is based in part on the algorithm used to build the survivability table. 1.1 DAS 2.0: Debris Assessment Software 2.0 The Debris Assessment Software is a NASA utility built to perform a variety of orbital debris assessments according to the NASA Technical Standard , Process for Limiting Orbital Debris Ref. 13. The technical standards are a set of mission requirements which can govern the acceptance of a mission for launch. The reentry-survivability model, which checks requirement from the technical standard, is the portion relevant to this thesis. The safety guideline is in the NASA Safety Standard , Guidelines and Assessment Procedures for Limiting Orbital Debris, and it states that "the total debris casualty area for components and structural fragments will not exceed 8 m 2." This equates to 100 micro-casualties per reentry event or 1:10,000 Ref. 16. Figure 1 shows the output from DAS2.0 s assessment of requirement for the example missions. The top portion summarizes the inputs. The mission LEO1 has a number of sub-components or debris fragments with a variety of different materials and shapes. The output shows that the mission LEO1 is non-compliant. Several of the debris fragments survive to impact giving a total casualty area of m 2, just over the limit of 8 m 2. The components each have a demise altitude, casualty area and impact energy. The demise altitude is when the debris fragment is fully ablated. If the demise altitude is 0, the fragment has survived to impact and has a casualty area and impact energy. As the simplest of the models, DAS 2.0 has an advantage in that it does not require the user to have a detailed knowledge of the spacecraft s geometry. Just the overall shape (sphere, flat plate, cylinder or box), rough dimensions and material for each of the fragment classes are required. However, DAS 2.0 is limited

27 T. M. Owens 3 in that its output does not allow for the creation of a demise modified debris catalog because the output only has the impact mass and energy. Also DAS 2.0 is not particularly useful for missions with planned reentry as it can only have a single failure event at an altitude of 78 km. Its ease of use serves as a benchmark for the computational model developed in this thesis. Figure 1: Casualty Risk from Reentry Debris

28 4 Aero-thermal Demise 1.2 ORSAT: Orbital Reentry Survival Analysis Tool The Orbital Reentry Survival Analysis Tool (ORSAT) is a much more complex and higher fidelity tool for analyzing the thermal breakup of spacecraft during reentry Ref. 2. Like DAS 2.0, it is was developed to meet the requirements of NASA standards, specifically NASA Technical Standard , A Process for Limiting Orbital Debris. Like DAS 2.0, the casualty risk from all reentry debris should be less than 1:10,000. ORSAT uses integrated trajectory, atmospheric, aerodynamic, aerothermodynamic and ablation algorithms to find the impact risk. It is able to use three different atmospheres, 1976 U.S. standard, MSISe-90 and GRAM-99 atmosphere. It can model either spinning or tumbling modes for the fragment debris. The drag coefficients are found from the kinetic energy at impact. The stagnation point continuum heating rates are found for spheres and correlated for other geometries and flight regimes. To find the surface temperature, it is able to use both a lumped mass and 1-D conduction. Demise is assumed once the net heat absorbed reaches the heat of ablation for the material. Unfortunately, ORSAT is only available to the Orbital Debris Program Office at Johnson Space Center so a true comparison cannot be made in this thesis. However, there are some capabilities that ORSAT obviously has that this computational model will not. Probably the most significant is ORSAT s ability to define more complex geometries and predict aerodynamic breakup. 1.3 Aerospace Survivability Tables The Aerospace Survivability Tables is a set of tabular data on the demise of various fragments that is part of a larger tool to estimate the total casualty expectation made by The Aerospace Corporation Ref. 24. The model has been validated against

29 T. M. Owens 5 the Columbia Accident Investigation Board (CAIB) report for casualty expectation and impact probability and The Aerospace Corporation s higher fidelity model Atmospheric Heating and Breakup tool (AHaB) for survivability of debris. Their model does not change the fragment properties as they impact nor does it account for the wall gradient temperature. It uses the Detra-Kemp-Riddell stagnation point heating correlation with a radiation heat balance to determine the amount of ablation for the fragments. The algorithm sits nicely between simple tools like DAS 2.0 and pseudo CFD tools like ORSAT which is why it was chosen as a starting point for the computational model outlined in this thesis. The tables cover three materials, aluminum 2024-T8xx, stainless steel and titanium (6 Al-4 V), and three hollow shapes, spheres, cylinders and flat plates. There is the choice between 1541 R and 540 R as breakup temperature of the debris. The tables also vary the breakup flight conditions. It covers from 46 to 30 Nmi in altitude with the flight path angles of -0.5, -3.5 and -5.5 degrees. The velocities at each altitude are based on what is to be expected from a reentry trajectory. As an example at 42 Nmi there is a choice between 25,000, 23,000 and 21,000 ft/s. Figure 2 shows an example table from the Aerospace Survivability Tables for an aluminum sphere. The rows are for radius in feet and the columns for weight in pounds. The values in the tables are liquid fractions, how much of the mass of the fragment has ablated, with one being fully demised and zero meaning the debris fragment has survived intact to impact. The s in the table indicates that the fragment has skipped off the atmosphere.

30 6 Aero-thermal Demise Figure 2: Aluminum 2024-T8xx Tumbling Hollow Sphere Survivability Table (Ref. 24) The obvious disadvantage to the tables is the limitation of having to choose the fragment and breakup conditions that best fit for the survivability analysis instead of computing it. The accuracy only to the first decimal is not a major concern as the Detra-Kemp-Riddell stagnation point heating correlation is only accurate to 10% of the heating rate, at best Ref SCARAB: Spacecraft Atmospheric Re-entry and Aerothermal Break-up The SCARAB tool is very similar to ORSAT. It was developed primarily for use by the European Space Agency and partners Ref. 11. The program is broken into five disciplines with different dependencies and couplings, flight dynamics,

31 T. M. Owens 7 aerodynamics, aerothermodynamics, structural analysis, thermal analysis and deformation/disintegration/melting as seen in Figure 3. A spacecraft is defined by geometric modeling, materials and physical modeling. SCARAB can use a variety of panelized geometric primitives to construct more complex shapes and volume elements. Figure 3: Coupling and Dependence for Disintegration Prediction (Ref. 11) SCARAB uses a materials database with 20 parameters that can be extended to a three phase model (gas, liquid and solid). The aerodynamic and aero-heating analysis is broken into free molecular, hypersonic continuum and rarefied transitional flow regimes. The thermal analysis uses thin and thick thermal heating which allows layered melting of solids with low heat conductivity. The latest versions of SCARAB can also perform a finite element analysis to find the stress resulting from inertial and aerodynamic forces. An example of the thermal fragmentation and mechanical breakup as computed by SCARAB can be seen in Figure 4. As with ORSAT the main difference between this tool and the computational model developed for this thesis is the complexity of geometry and the ability to predict structural failure.

32 8 Aero-thermal Demise Figure 4: Thermal Fragmentation and Mechanical Breakup (Ref. 11) 1.5 Computation Model There is an obvious gap in complexity level between the computational model developed in this thesis and those of tools like ORSAT and SCARAB. However most of these deficiencies are not critical to performing a flight risk analysis. The model in this thesis only takes into consideration the aero-thermal analysis of the hypersonic continuum flow regime. This is adequate for approximating demise as the reentry flight speeds are typically on the order of Mach 10 and maximum aero-heating occurs from 80 to 50 kms of altitude. The ORSAT and SCARAB tools are also designed for impact prediction of spacecraft and not designed for landing reentry, whereas the model in this thesis is mainly concerned with the failure of launch and reentry vehicles. Therefore, the very high altitude flight regimes are of little importance because minimal heat transfer takes place at supersonic and

33 T. M. Owens 9 subsonic flight speeds; there is no real value added to include them in this analysis. While the inability to predict a breakup event may seem like a major weakness of the code developed for this thesis, it is of minimal importance at the stage of risk analysis for this computation model and its expected use. Typically a debris catalog will be provided to the mission analysis that may have several failure modes such as an intact crash, partial breakup and full breakup. The probability of each of these outcomes is determined through some other analysis, perhaps by the vehicle manufacture itself. Thus, there is no way a complex geometry could be constructed from the debris catalog provided. The tools that the mission analyst will use to predict risk typically assume failure at every point in the trajectory at the failure rates from the probability of outcomes. Therefore, knowing precisely when a part will fail is not as important as knowing where it will land if it failed at that trajectory time and what sort of casualty risk can be expected. The advantage this utility will have over some other reentry demise analysis tools is that it will take in a standard debris catalog and return the standard debris catalog with values adjusted for aero-thermal demise. This allows a risk analyst to use the existing workflow and simply run the computational model developed in this thesis before other risk tools to account for the demise casualty reduction. 1.6 Risk Analysis Understanding the desire for a tool that can predict aero-thermal demise requires some understanding of how a mission risk analysis is performed. The typical main risk criterion is 1:10,000 or 100 micro-casualties. To determine this, the casualty expectation is found by summing the probability of every possible event and the casualty consequences at each mission event. The general form of the casualty expectation equation for n possible events is as follows Ref. 1,

34 10 Aero-thermal Demise n E c = P i A ci D pi i=1 Eq. 1 Where the P i is the impact probability, A ci is the casualty or hazard area of the debris and D pi is the population density of the area at risk for the i th event. The population in an area is partially under the control of the launch provider as they can close portions of the launch area. This, however, is restrictive to other work and may not be possible in public areas. The probability of impact could possibly be changed by increasing reliability of the trajectory; however, this is generally not an option open to the mission analyst. Thus, the only real way to find a reduction in the expected casualties is to change the hazard area of the debris. One possible method is to introduce the effects of sheltering. Sheltering is the effect that buildings will have on the risk of human casualty. This is dependent on the time of week and day as well as the mass and speed of the impacting debris fragment. It does not always reduce the expected casualties however. Some large debris is considered to cause building collapses which will cause more casualties than if the debris were to impact an open area. Therefore, the aero-thermal demise of reentry debris should be considered. Partially demising fragments will not greatly reduce the casualty area; however, the mass loss can mean greater benefit from the effects of sheltering. Fully demising debris has no casualty area. As a result, there will be a clear reduction in expected casualties. This and the accurate prediction of where the debris will impact due to the changing ballistic coefficient give a clear benefit to performing demise analysis along with risk analysis.

35 T. M. Owens 11 2 Impact Integration The impact integration relies on the derivation of the equations of motion over a rotating spherical earth. Because of the very high initial speeds and altitudes of the debris fragments, many terms that would be otherwise ignored for impact or ballistic calculations must be included. The probably of impact is not explored. This is a separate and distinct problem that involves creating a bivariate normal distribution of impact probability defined by an impact covariance. The covariance should take into account factors such as explosion velocity, position, velocity, wind and drag uncertainties. The probability distribution can be built through a Monte Carlo set of impact propagations, typically on the order of 10,000 or more. Other approximations of Gaussian distributions such as Jacobian-based techniques or a Julier-Ulhmann method can be used for much faster propagation Ref Equations of Motion The most important component of the impact integration is the derivation of force equations. The following is a derivation of the equations of motion that will result in three force equations for velocity, heading and flight path angle from Ch. 2 of Vinh Ref. 25. These general equations of flight over a spherical, rotating earth allow for use with high performance reentry vehicles like the Boeing X-37, capsules like the Apollo Command Module or a piece of reentry debris. Consider a body with a point mass defined by a position vector, r(t), velocity vector, V(t), and mass, m(t). The total force, F, at each instance is a sum of the gravitational, m g, aerodynamic, A, and propulsion thrust forces, T. F = T + A + mg Eq. 2

36 12 Aero-thermal Demise For reentry debris fragments, the propulsive force is zero, and in a vacuum, the aerodynamic forces are zero. The kinematic vector equation for the inertial system defined by the position, velocity and mass is, dr dt = V Eq. 3 Newton s second law gives the force equation. m dv dt = F Eq. 4 Because Newton s second law requires a fixed system, and the earth s system is rotating, a transformation is required Relative Angular Motion Consider a fixed inertial reference frame system and a rotating system, OX 1 Y 1 Z 1 and Oxyz respectively. The system Oxyz rotates with respect to the fixed system OX 1 Y 1 Z 1 with angular velocity ω. Let A be an arbitrary vector in the rotating system as seen in Figure 5, A = A x i + A y j + A z k Eq. 5

37 T. M. Owens 13 Figure 5: Relative Angular Motion (Ref. 25) The i, j and k unit vectors in the rotating reference system, Oxyz, are functions of time. Therefore, the time derivative of A with respect to the fixed system OX 1 Y 1 Z 1 is, da dt = (da x dt i + da y dt j + da z dt k ) di + (A x dt + A dj y dt + A dk z dt ) Eq. 6 At point P, the linear velocity in a fixed system rotating with angular velocity ω at position vector r as seen in Figure 6 is, V = dr dt = ω r Eq. 7 Figure 6: Kinematics of Rotation (Ref. 25)

38 14 Aero-thermal Demise Then, according to Poisson s formulas, di = ω i dt dj = ω j dt Eq. 8 dk dt = ω k Using this with the definition of vector A in the equation of the time derivative of A the following equation is found, di A x dt + A dj y dt + A dk z = ω A Eq. 9 dt Then in the rotating system Oxyz the time derivative of A is, δa δt = da x dt i + da y dt j + da z dt k Eq. 10 The equation for the time derivative with respect to the fixed system, Eq. 6, can be rewritten as, da dt = δa δt + ω A Eq Equations for Flight Over a Rotating Spherical Earth OX 1 Y 1 Z 1, is the inertial reference frame, with the origin at the center of a spherical earth s gravitation field. The plane OX 1 Y 1 is the equatorial plane. The reference frame OXYZ is fixed with respect to earth with OZ and OZ 1 coincident. The atmosphere is assumed to rotate at the same constant angular acceleration, ω.

39 T. M. Owens 15 The equation for absolution acceleration is found by setting the position vector A = r and taking the time derivative of Eq. 11 in the earth fixed frame OXYZ. dv dt = δ2 r δr + 2ω + ω (ω r) δt2 Eq. 12 δt The equation Eq. 4 can then be put in the earth-fixed system, m δ2 r δr = F 2mω mω (ω r) δt2 Eq. 13 δt Or, m dv dt = F 2mω V mω (ω r) Eq. 14 The velocity, V, is the velocity relative to the earth-fixed system. From this there are two acceleration forces as the earth rotates. They are the Coriolis acceleration, 2ω V, and the transport acceleration, ω (ω r). The Coriolis acceleration is zero when the flight path angle is parallel to the earth s pole and reaches a maximum of 2ωV when the flight path angle is perpendicular to the polar axis. The transport acceleration is zero when the body is at the poles and at its maximum, ω 2 r, when the body is on the equatorial plane. The fixed coordinate system, OXYZ, and rotating coordinate system, Oxyz, can be seen in Figure 7. The longitude is angle θ and latitude is φ.the angle γ is the flight path angle and Ψ, the heading. The flight path angle is positive for a launch trajectory and negative for a reentry trajectory. The heading is the angle between the local parallel of the latitude and the projection of the velocity vector on earth s surface with right hand positive about the x axis.

40 16 Aero-thermal Demise Figure 7: Coordinate Systems (Ref. 25) Thereby the velocity vector in the rotating system Oxyz is, V = V sin γ i + V cos γ cos Ψ j + V cos γ sin Ψ k Eq. 15 The angular velocity in the Oxy plane is, ω = ω sin φ i + ω cos φ k Eq. 16 This can then be used to find the Coriolis and transport accelerations in terms of the unit vectors, ω V = ωv cos γ cos φ cos Ψ i + ωv(sin γ cos φ cos γ sin Ψ)j + ωv cos γ sin φ cos Ψ k Eq. 17 ω (ω r) = ω 2 cos 2 φ i + ω 2 r sin φ cos φ k Eq. 18 The force of gravitational acceleration in the total force F is,

41 T. M. Owens 17 mg = mg(r)i Eq. 19 The aerodynamic forces, lift and drag, can be put into terms of tangential and normal forces to the flight plane. The angle between thrust and the velocity vector is angle ε as seen in Figure 8. The propulsive and aerodynamic forces can be grouped, F T = T cos ε D F N = T sin ε + L Eq. 20 Figure 8: Aerodynamic and Propulsive Forces (Ref. 25) In the unit vector form, the normal force can be defined as, F T = F T sin γ i + F T cos γ cos Ψ j + F T cos γ sin Ψ k Eq. 21 The normal force in vector form requires a coordinate transformation between the rotating reference frame and the flight plane, as well as accounting for the roll angle σ. This results in the equation, F N = F N cos σ cos γ i F N (cos σ sin γ cos Ψ + sin σ sin Ψ)j + F T (cos σ sin γ sin Ψ sin σ cos Ψ)k Eq. 22 Then, the equation Eq. 8 can be put in terms of the latitude and longitude.

42 18 Aero-thermal Demise di dt dθ dφ = cos φ j + dt dt k dθ i + sin φ dt k dj dθ = cos φ dt dt dk dt = dφ dt i sin φ dθ dt j Eq. 23 Then, using the equation Eq. 23 in Eq. 15, The derivative of velocity is, dv dt V = dr dt dv dγ = [sin γ + V cos γ dt dt V2 r cos2 γ] i dθ dφ i + r cos φ j + r dt dt k Eq [cos γ cos Ψ dv dγ dψ V sin γ cos Ψ V cos γ sin Ψ dt dt dt + V2 r cos γ cos Ψ (sin γ cos γ sin Ψ tan φ)] j Eq [cos γ sin Ψ dv dγ dψ V sin γ sin Ψ + V cos γ cos Ψ dt dt dt + V2 r cos γ (sin γ sin Ψ cos γ cos2 Ψ tan φ)] k Next, by substituting equation Eq. 25 into Eq. 14, the scalar equations of motion are found to be, sin γ dv dγ + V cos γ dt dt V2 r cos2 γ = F T m sin γ + F N cos σ cos γ m g + 2ωV cos γ cos Ψ cos φ + ω 2 r cos φ Eq. 26

43 T. M. Owens 19 sin γ dv dγ + V cos γ dt dt V2 r cos2 γ = F T m sin γ + F N cos σ cos γ m g + 2ωV cos γ cos Ψ cos φ + ω 2 r cos φ Eq. 27 sin γ dv dγ + V cos γ dt dt V2 r cos2 γ = F T m sin γ + F N cos σ cos γ m g + 2ωV cos γ cos Ψ cos φ + ω 2 r cos φ Eq. 28 Solving for the derivatives dv, dγ dt dt, and dψ dt, dv dt = F T m g sin γ + ω2 r cos φ (sin γ cos φ cos γ sin ψ sin φ) Eq. 29 V dγ dt = F N V2 cos φ g cos γ + cos γ m r + 2ωV cos ψ cos φ + ω 2 r cos φ (cos γ cos φ sin γ sin ψ sin φ) Eq. 30 V dψ dt = F N sin φ m cos γ V2 cos γ cos ψ tan φ + 2ωV(tan γ sin ψ cos φ sin φ) r ω2 r cos ψ sin φ cos φ cos γ Eq. 31 The ω 2 r term is the transport acceleration and the 2ωV term is the Coriolis acceleration. If the speeds are much less than orbital, then the equations could be simplified further to not include the Coriolis and transport accelerations; however, for the purposes of this thesis, they are necessary terms.

44 20 Aero-thermal Demise 3 Stagnation-Point Heating Stagnation point heating is the main mode of heat transfer for bodies reentering an atmosphere. Convective heat transfer depends on the properties of the atmosphere, planet and reentering bodies. Radiative heat transfer balances the convective heat transfer in the net heat flux. 3.1 Fay and Riddell Theory The Theory of Stagnation Point Heat Transfer in Dissociated Air by Fay and Riddell Ref. 6 is probably the seminal work on stagnation point heating theory. Many of the correlations for stagnation point heating find their roots in Fay and Riddell s theory and the work of others at Avco Research Laboratory in the 1950 s. The Fay- Riddell theory reduces a set of general boundary-layer equations for stagnation point heating into nonlinear ordinary differential equations for a broad flight regime. The derivation starts with the equation for the heat flux in a quiescent dissociated gas where h A 0 is the dissociation energy per unit mass, D is the diffusion coefficient and c A is the atomic mass fraction. q = k T + h A 0Dρ c A Eq. 32 The first term on the right of equation Eq. 32 is the transport of kinetic energy and the second term is the potential recombination energy of the dissociated gas. This can be simplified by neglecting the process of dissociation and recombination as well as substituting for the temperature gradient and assuming a Lewis number of unity.

45 T. M. Owens 21 q = k c p (h + c A h A 0) Eq Laminar Boundary-Layer in Dissociated Gas Stagnation point heat transfer is a combination of thermal and aerodynamic effects. Thereafter, the partial differential equations associated with the boundary layer need to be found. The general continuity equation for the mass rate of formation of the species i per unit volume and time is, [ρ(q + q i )c i ] = w i Eq. 34 The mass average velocity, q i,of the species i can be found by, q i = D i c i c i D i T T T Eq. 35 The first term on the right hand side is the concentration diffusion, the second is the thermal diffusion and the pressure diffusion is assumed negligible. The continuity equation is summed for all species so it takes the form, (ρq) = 0 Eq. 36 In addition, the energy equation for a fluid element is required, ρq c i e i = (k T) ( ρq i c i h i ) + w i h i 0 + p q + Φ Eq. 37 With Φ being the dissipation function, the steady-state energy equation can be rewritten using the idea gas assumption, conservation of mass, continuity equation and relationship of enthalpy to internal energy.

46 22 Aero-thermal Demise ρq c i (h i h i 0) = [k T ρq i c i (h i h i 0)] + q p + Φ Eq. 38 Taking into account the boundary layer assumptions, the equations Eq. 34, Eq. 36, and Eq. 38 can be rewritten as partial differentials. The centrifugal forces are neglected assuming the boundary-layer thickness is much less than the radius of curvature of the body. The x is tangential and y is normal to the surface with u and v being the velocity components respectively. (ρru) x + (ρrv) y = 0 Eq. 39 ρuc ix + ρvc iy = (D i ρc iy + D i Tρc i T y T ) y + w i Eq. 40 ρuh x + ρvh y = (kt y ) y + up x + μu y 2 + [ D i ρ(h i h i 0)c iy + D i Tρc i (h i h i 0) T y T ] y Eq. 41 The equation of motion is, ρuu x + ρvu y = p x + (μu y ) y Eq. 42 The equation Eq. 41 can be rewritten in terms of temperature instead of enthalpy for simpler use with transport coefficients. c p(ρut x + ρvt y ) = (kt y ) y + up x + μu y 2 + w i (h i h i 0) + [ D i ρc iy + D i Tρc i T y T ] y Eq. 43 Similarly, equation Eq. 41 can be rewritten to simply be in terms of the enthalpy,

47 T. M. Owens 23 ρu (h + u2 2 ) + ρv (h + u2 2 ) = [ k (h + u2 x y c p 2 ) ] + up x y y + μu 2 y + [ (D i ρ k ) (h i h i 0)c iy + D i Tρc i (h i h i 0) T y c p T ] y Eq. 44 The equations Eq. 39, Eq. 40, Eq. 42 and Eq. 43 or Eq. 61 form a system of partial differential equations that must be solved Boundary Layer Ordinary Differential Equations In order to simplify the solution, the partial differential equations are reduced to ordinary differential equations. An exact solution can only exist when the boundary layer is considered to be frozen or in thermodynamic equilibrium. The first step is to set transformations of the independent variables and dimensionless independent variables. y η ( ru 2ξ ) ρdy 0 Eq. 45 x ξ ρ w μ w u r 2 dx 0 Eq. 46 f η u ; u η f = f η 0 dη Eq. 47 g = (h + u2 2 ) h s Eq. 48 θ = T T Eq. 49

48 24 Aero-thermal Demise s i = c i c i Eq. 50 The subscript is the free stream condition and w is for the condition at the wall. At the stagnation point, the equations for f, g, θ and s i are functions of η as ξ increases. Also, assuming a thermodynamic equilibrium, the equations Eq. 39, Eq. 40, Eq. 42 and Eq. 43 or Eq. 61 can be written as, ρv = [( 2ξf ξ + f 2ξ ) ξ x + 2ξf y η x ] r Eq. 51 [ l Pr (L is iη + LiTs i T η )] + fs T iη + w i [(2 du 1 η ρc is dx ) ] = 0 Eq. 52 s (lf ηη ) η + ff ηη (ρ s ρ f η 2 ) = 0 Eq. 53 ( c p c pw l Pr ) η + c p fθ η + [(2 du 1 c pw dx ) ] w i (h i h i 0) s ρ c pwt s + c pi c is l c pw Pr (L is iη + LiTs i θ η θ ) θ η = 0 η Eq. 54 ( l Pr g + fg η + { l η)η Pr [c is (h i h i 0) h s ] [(L i 1)s iη + L its i θ η θ ]} η = 0 Eq Heat Transfer Rate The local heat transfer rate, which is a sum of the conduction and diffusion transports at the wall, is given by the equation, q = (k T y ) c i + [ ρ(h i h i 0) (D i y + D i T c i T T y )] y=0 y=0 Eq. 56

49 T. M. Owens 25 The dimensionless terms from the previous section can be used to get the equation, q = ( rk wρ w u T 2ξ ) [θ η + c i (h i h i 0) c pt (L i s iη + LiTs i θ η )] θ η=0 Eq. 57 At the stagnation point, rρ w u 2ξ = 2 ν w ( du dx ) s Eq. 58 Thereby allowing the heat transfer equation to be rewritten as, q = Nu Re ρ wu w ( du dx ) (h s h w ) s Pr Eq Equilibrium Boundary Layer The equilibrium boundary layer is found through a numerical solution of equations Eq. 52, Eq. 53, Eq. 54 and Eq. 55, as explained in Fay-Riddell Ref. 6. Because a catalytic wall is assumed it is not necessary to find the frozen heat transfer rate. For a Lewis number of unity the heat transfer parameter relies only on the variation of ρμ across the boundary-layer giving the equation, Nu Re = 0.67 ( ρ 0.4 sμ s ) ρ w μ w Eq. 60 A further simplification can be made if only a single species air is considered with an average heat of formation from atomic oxygen and hydrogen found by,

50 26 Aero-thermal Demise h A 0 = c is ( h i 0) c is atoms atoms Eq. 61 Also, the numerical solution effect of the Lewis number can be taken into account by the equation, Nu Re ( Nu Re ) = 1 + (L ) h D Eq. 62 h L=1 s From equations Eq. 60 and Eq. 62, with the Prandtl number set to 0.71, the stagnation point heat transfer rate equation Eq. 59 is found to be, q = 0.94(ρ s μ s ) 0.1 (ρ sl μ sl ) 0.4 [1 + (L ) h D h s ] ( du dx ) s Eq. 63 The velocity gradient as defined by a modified Newtonian flow is, ( du dx ) = 1 s R 2(p s p ) Eq. 64 ρ s So, with these various correlations, the stagnation point heat transfer rate can be developed.

51 T. M. Owens Detra, Kemp and Riddell Correlation The correlation for stagnation point heating in a continuum flow developed by Detra, Kemp and Riddell Ref. 4 is an exact formulation that takes into account the high temperature dissociation phenomena. It starts with the formula from the Fay-Riddell theory equation, q s = 0.94 (1 h s ) (ρ h s μ s ) 0.1 (ρ sl μ sl ) 0.4 h sl ( du sl dx ) [1 s (L 1) h D h sl ] Eq. 65 The viscosity is extrapolated using Sutherland s law, the Prandtl number is made a constant 0.71, and the Lewis number is also taken as a constant. The assumption is that there is a thermodynamic equilibrium; however, the correlation can be used with a nonequalibrium boundary layer as long as the surface is catalytic. This formula can be reduced to a function of density and velocity. This is done using a Mollier diagram of the National Bureau of Standards data Ref. 10. The solution of the shock wave equations is found through iteration, and the inviscid flow properties are used to find the stagnation point velocity gradient. Assuming a Newtonian pressure distribution gives, ( du e dx ) = 1 2p sl Eq. 66 s R h ρ sl The variation of the stagnation point heating should vary with respect to density and velocity by approximately ρ u 3 (this can be seen in many of the other similarly derived correlations examples in Appendix B: Supplemental Algorithms). By using this velocity distribution and correlating the equation Eq. 65 to

52 28 Aero-thermal Demise experimental data from hypersonic shock tubes, the equation for the stagnation point heating flux is Ref. 4, q s = ρ ( U 3.15 ) ( h s h b ) Eq. 67 R h ρ sl U ref h sl h ref This is for units of Btu/ft 2 -sec which can be converted by multiplying by a factor of 11,364 to W/m 2. The reference enthalpy is the enthalpy at 300 K. The equation is accurate ± 10 % over a range of 7,000 to 25,000 fps from sea level to 250,000 ft (2,134 to 7,620 m/s from sea level to 76,200 m). A plot of experimental data versus the correlation can be seen in Figure 9. Equation Eq. 67 is Eq. (2) in the plot. These are the results from a shock tube experiment using air by Avoco Research Laboratory to measure the stagnation point heat transfer rate. The experiments simulated three flight altitudes of roughly 111,00 to 127,000 ft; 64,000 to 80,000 ft; and 11,000 to 31,000 ft (33,833 to 38,710 m; 19,510 to m; and 3,353 to 9,449 m) Ref. 4.

53 T. M. Owens 29 Figure 9: Stagnation Point Heat Transfer Rate vs. Flight Velocity (Ref. 4) Some similarly derived correlations from the Fay-Riddell theory can be found in Appendix B: Supplemental Algorithms, Alternate Correlations. The reason for choosing this particular correlation over the others is that it is directly relates to the work performed by Fay and Riddell, as well as that used in the Reentry Hazard Analysis Handbook Ref. 24, upon which the stagnation point heating algorithm is in part based Radiation Heat Balance The other major source of heating is radiation through emission. The radiative cooling is accounted for by the Stefan-Boltzmann law assuming a lumped-mass

54 30 Aero-thermal Demise node. Like the stagnation point heating flux, the radiation energy flux is in units of W/m 2. q rad = εσ sbc T 4 Eq. 68 The cold-wall heat flux is averaged over the surface of the debris fragment by the fraction of instantaneous cold-wall flux at the stagnation point as seen in equation Eq. 69 Ref. 8. This fraction, the area averaging factor (0 < k 2 < 1), is assumed to have a value of 0.12 for a reasonable match to past data of tumbling reentry debris Ref. 24. For composites like graphite reinforced epoxy, this value can be set to 0.8 for a more accurate, though less conservative, mass loss rate Ref. 8. q rad = k 2 q s Eq. 69 This can then be put into the heat energy balance or net heat flow equation. The heat input less the heat output is equal to the heat absorbed. Q = (k 2 q s ε b σ sbc T b 4 )A w Eq. 70 The wall temperature, because of the lumped-mass assumption, is the temperature of the body. This is found by using the following, T b = Q m b C p b, for Q < m b C p b T melt Eq. 71 { T melt, for Q m b C p b T melt

55 T. M. Owens 31 4 Algorithm The following algorithm has been implemented in MATLAB code. See Error! Reference source not found.. Coordinate transforms and other supplemental algorithms can be found in Appendix B: Supplemental Algorithms. Coordinate transformations are modified from the function libraries outlined in the Joint Advanced Range Safety System Mathematics and Algorithms document Ref Earth Model The model of earth employed was developed from the Department of Defense (DoD) World Geodetic System 1984 (WGS84). The WGS84 earth model was created by the National Imagery and Mapping Agency (NIMA) in order to define a common, simple and accessible 3-dimensional coordinate system. The WGS84 also has a method for finding gravity using ellipsoidal zonal harmonics Ref. 14. Table 1 is a collection of the derived geometric constants. The values were obtained through precise GPS ephemeris estimation process. The method can potentially be used for other planetary bodies if the geometric constants are known. These are set in the createearth function. Table 1: WGS84 Ellipsoid Derived Geometric Constants Variable Value Units Description a m Semi-major axis b m Semi-minor axis f 1/ Flattening e First Eccentricity 2 e First Eccentricity Squared