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Hypersonic Non-Equilibrium Flow and Its Thermodynamic Relations

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Hypersonic Non-Equilibrium Flow and Its Thermodynamic Relations c UARI Research Report No. 30 HYPERSONIC NON-EQUILIBRIUM FLOW AND ITS THERMODYNAMIC RELATIONS by Rudolf Hermann '0- (ACCESSION NUMBER* (THRUI eLL (CObEI e (CATEGORY1 Invited lecture presented at the 4. SPACE SYMPOSIUM at the University of Goettingen, Germany October 18-22, 1965 The preparation of this lecture was supported by the National Aeronautics and Space Administration under research grant NsG-381 UNIVERSITY OF ALABAMA RESEARCH INSTITUTE Huntsville, Alabama November 1965 I.\* UARl Research Report No. 30 HYPERSONIC NON -EQUI LlBRlUM FLOW AND ITS THERMODYNAMIC RELATIONS Rudolf Herma nn Invited lecture presented at the 4. SPACE SYMPOSIUM at the University of Goettingen, Germany October 18-22, 1965 The preparation of this lecture was supported by the National Aeronautics and Space Administration under research grant NsG-381 UNIVERSITY OF ALABAMA RESEARCH INSTITUTE Huntsville, Alabama November 1965 I . I F OREW ORD The preparation of this lecture was supported by the National Aeronautics and Space Administration under research grant NsG-381. The visit to the Symposium was sponsored by a cooperation of West-Germany's Aerospace Research Centers. The author acknowledges the efforts of Mr. Manfred J. Loh, Dip1 .-lng., for valuable assistance in the preparation of this manuscript, and of Mr. Kenneth 0. Thompson, Research Associate, for critical comments. This lecture of research in hypersonic non-equil ibrium flow presents the state of the art until September 1965. Co-workers of the author during the past four years in the subject matter being discussed here were, in chronological order, Kenneth 0. Thompson, Janardanarao Yalamanchil i, Walter Melnik, R. Douglas Archer, and. Jurgen Thoenes. Their substantial contributions are gratefully ac know Iedged. .. II TABLE OF C ONT 'NTS Page F OREW OR D I .. TABLE OF CONTENTS II LIST OF TABLES AND FIGURES IV 1. INTRODUCTION 1 2. NON-EQUILIBRIUM FLOW, HYPERSONIC FLOW FIELDS, AND FLIGHT REGIONS 2 2.1 Description of Equilibrium Flow, Non-Equilibrium Flow, and Frozen Flow 2 2.2 Hypersonic Flow Fields, Flow Conditions, and Flight Regions during Re-Entry 4 3. THERMODYNAMIC RELATl ONS INCLUDING REAL GAS EFFECTS 7 3.1 The Simplified Air Model 7 3.2 Dissociation of Oxygen and Nitrogen in Equilibrium 7 3.2.1 Process of Dissociation and Recombination 7 3.2.2 Definition of Degrees of Dissociation a, p 8 3.2.3 Law of Mass Action 0 3.2.4 The Pressure Equilibrium Constants K and KPP 10 Pa 3.2.5 Degree of Oxygen and Nitrogen Dissociation as Function of Pressure and Temperature 11 3.3 Thermal Equation of State for Oxygen Dissociation Only 13 3.3.1 Simplified Air Model With Restriction to Oxygen Dissociation 13 3.3.2 Derivation of Thermal Equation of State 14 3.4 Equations for Internal Energy and Enthalpy for Oxygen Dissociation On1 y 15 3.4.1 Energy States of a Gas Particle; and Partition Functions 15 11 3.4.2 Internal Energy Equation IW 3.4.3 Enthalpy Equation 18 3.5 Dissociation and Recombination Rate Equations for Oxygen Dissoc iation 18 3.5.1 Departure From Equilibrium 18 3.5.2 Dissociation and Recombination Rate Equation 19 3.5.3 Net Production and Source Function 21 3.5.4 Reaction Rate Constants 22 3.6 Equilibrium Constants for Oxygen Dissociation Calculated From Stat istical Thermodynamics 23 3.6.1 Concentration Equilibrium Constant K, 23 3.6.2 Pressure Equilibrium Constant K 23 P 3.6.3 Equilibrium Flow Relation 24 ...Ill Page 4. NON-EQUILIBRIUM FLOW WITH OXYGEN DISSOCIATION THROUGH A HYPERSONIC NOZZLE 25 4.1 Assumptions and Basic Equations 25 4.2 Flow Calculations 26 4.3 Numerical Results of Hypersonic Nozzle Flow 28 5. NON-EQUILIBRIUM FLOW AROUND BLUNT BODIES WITHOUT AND WITH DISSOCIATION IN THE FREE STREAM 30 5.1 Direct and Inverse Methods for Flow Calculation 30 5.2 The Integral Method of Dorodnitsyn and Its Application to the Inviscid Flow Around a Circular Cylinder 31 5.2.1 Basic Equations 31 5.2.2 Boundary Conditions 32 5.2.3 Application of the One-Strip Integral Method 33 5.3 Numerical Calculation of the Parameters in the Flow Field Between Shock and Cylinder Surface 35 5.3.1 Calculation of the Flow Along the Stagnation Streamline and the Stagnation Point Condition 35 5.3.2 Calculation of the Flow Variables Along the Cylinder Surface and Behind the Shock 36 5.4 Discussion of the Results of the Flow Around a Cylinder 37 5.4.1 Flow Along the Stagnation Streamline 37 5.4.2 Flow Around Circular Cylinder 37 5.5 Comparison With Flow Around a Sphere 39 6. NON-EQUILIBRIUM FLOW AROUND A POINTED CONE WITHOUT AND WITH DISSOCIATION IN THE FREE STREAM 40 6.1 Review of Cone Flow Calculations for Real Gases 40 6.2 Basic Equations for the Flow Around a Sharp Biconvex Twodimensional and a Pointed Axisymmetric Body 41 6.3 Application of the One-Strip Integral Method for the Flow Calculation Around a Pointed Cone 43 V.U.T, ’I C----nI a_--.. F!nw Around a Cone 44 .I 6.3.2 Equilibrium Flow Around a Cone 40 6.3.3 The General Case of Non-Equilibrium Flow Around a Cone 46 6.4 Presentation of the Results 48 6.4.1 Numerical Technique 48 6.4.2 Frozen Flow Results 49 6.4.3 Equilibrium Flow Results 49 6.4.4 Non-Equilibrium Flow Results 50 7. SUMMARY 52 8. LIST OF SYMBOLS 54 9. REFERENCES 57 10. TABLES 61 -.I.,mrr ii. tIbUKCJ 62 IV LIST OF TABLES AND FIGURES Page TABLE I: Air Model and Physical Constants 61 TABLE 11: Atomic and Molecular Constants 61 FIGURE 1: Blunt Body Configuration and Flow Field 62 FIGURE 2: Hypersonic Flow Regions for Apollo Vehicle 62 FIGURE 3: Flight Region of Manned Re-Entry Vehicles from Circular Orbit and Lunar Return. Also Equilibrium Conditions Behind a Normal Shock, Presented in an Altitude-Velocity Diagram 63 FIGURE 4: Temperature Ts and Compressibility Factor Zs for Equili- brium Conditions Behind a Normal Shock. Also Flight Region of Manned Re-Entry Vehicles from Circular Orbit and Lunar Return, Presented in a Pressure-Enthalpy Diagram 63 FIGURE 5: Oxygen and Nitrogen Equilibrium Constants Kpa and KPp as Functions of Temperature 64 FIGURE 6: Degree of Oxygen Dissociation a and Nitrogen Dissociation p as Functions of Temperature and Pressure 64 FIGURE 7: Oxygen Dissociation Rate Constant for Various Colliding Bodies as Function of Temperature 65 FIGURE 8: Degree of Oxygen Dissociation Along Hypersonic Nozzle in Equilibrium and Non-Equilibrium Flow 66 FIGURE 9: . Temperature Distribution Along the Hypersonic Nozzle in Equilibrium and Non-Equilibrium Flow for Pt = 50 atm 66 FIGURE 10: Mach Number bisrri'uui ;UII A!GZS !-!>~er:cn!c Nnizle in Equilibrium and Non-Equilibrium Flow 67 FIGURE 11: Coordinate System and Shock Wave Geometry on Circular Cylinder 67 FIGURE 12: Degree of Oxygen Dissociation Along Stagnation Streamline for Various Body Radii 68 FI GURE 13: Temperature Along Stagnation Streamline for Various Body Radii 68 FIGURE 14: Calculated Shock Shapes for Circular Cylinder 69 FIGURE 15: Stagnation Shock Detachment Distance on Circular Cylinder 69 V Page FIGURE 16: Velocity Distribution Along Body Surface 70 FIGURE 17: Degree of Oxygen Dissociation Along Body Surface 70 FI GURE 18: Temperature Distribution Along Body Surface 71 l- FIGURE 19: Pressure Distribution Along Body Surface 71 FIGURE 20: Shock Waves in Front of a Cylinder and a Sphere for Non- '- Equilibrium Flow 72 FIGURE 21: Coordinate System and Shock Geometry on Arbitrary Circular I Pointed Body 72 FIGURE 22: Shock Layer Thickness as Function of Cone Semivertex Angle for Chemically and Vibrationally Frozen Flow Around a Circular Cone. Included is Free Stream Oxygen Dissociation at 3 Mach Numbers 73 FIGURE 23: Pressure on Body Surface as Function of Cone Length for Chemical Non-Equilibrium Flow Around a Circular Cone 73 FIGURE 24: Temperature and Density on Body Surface as Function of Cone Length for Chemical Non-Equilibrium Flow Around a Circular Cone 74 FIGURE 25: Oxygen Dissociation on Body Surface as Function of Cone Length for Chemical Non-Equilibrium Flow Around a Circular Cone 74 FIGURE 26: Shock Layer Thickness as Function of Cone Length for Frozen, Non-Equilibrium, and Equilibrium Flow Around a Circular Cone 75 1 1. INTRODUCTION The return of manned and unmanned satellites and space vehicles to the earth's surface with their structure intact is one of the most important problems in astronautics today. During re-entry of a space vehicle through the atmosphere, extremely high velocities are encountered. For instance, at return from a lunar mission nearly para- bolic velocity (1 1.3 kmlsec), corresponding to about Mach number 35, is reached. Strong heating, starting behind the shockwave, occurs particular1y in the stagnation point region of the re-entering space vehicle (in case of parabolic velocity up to a maximum of 1l,00O0K at 60 km altitude). Already at much smaller velocities, namely for flight above Mach number 3, air no longer behaves as a perfect gas, and with increasing Mach number, molecular vibration, dissociation, electronic excitation, ionization, and ultimately complete plasma formation progressively take pLace. These phenomena will change considerably the chemical composition of the air and this change will extend along the body. Since in most hypersonic applications, the flow has insufficient time to obtain ther- modynamic equilibrium, there will generally be non-equilibrium flow in the shock layer. There also exist two I imiting cases. Depending on the specific free stream velocity, density, temperature, atmospheric composition, and the absolute size of the body, the flow may be either almost frozen, or the flow may reach very nearly equilibrium.
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