# Hypersonic Aerodynamics and Heat Transfer

## Description

Results of theoretical and experimental studies of the problems concerned super/hypersonic flows around the models of real space vehicle configurations such as "Buran"-orbiter, winged cone (airplane-like body), Clipper space vehicle, Martian planetary probe are presented. Theoretical analysis is based on solution 2/3D Navier- Stokes equations and their simplified asymptotic models with account for equilibrium and nonequilibrium chemical reactions taking place at the background of relaxation of excited internal energy modes of the particles in the shock layer and on the vehicle surface.

Exact and simple relationships for the transport of species mass and heat are derived from rigorous kinetic theory for multicomponent mixtures of gases and plasma with different diffusion characteristics of the species (governing relations - thermodynamic parameter gradients expressed through the fluxes) as well as velocity slip, temperature and species concentration jumps boundary conditions for the surfaces of finite catalycity in multicomponent chemically and thermally nonequilibrium gas flow.

Phenomenological and kinetic models are developed for heterogeneous catalytic reactions on the thermal protection materials of the space vehicles entering Earth and Mars atmospheres along gliding trajectories.

The problem of multicomponent thermally and chemically nonequilibrium air flow in inductively coupled plasma torch and jet flow around the models installed in the facility work section in considered. The possibility to specify catalytic properties of real heat shield coatings are demonstrated using numerical and test values of heat fluxes or surface equilibrium radiation temperatures.

Original iterative/marching method to solve viscous shock layer equations is presented. The method is based on splitting of marching component of the pressure gradient onto hyperbolic part and a part that minimizes the elliptic part to a maximum degree that are then subjected to the global iterations procedure. Using the splitting of this kind allows numerical obtaining the drug and heat transfer coefficients within single or couple global iterations with reasonable for practical purposes accuracy.

The knowledge area connected to development of large computer software for conjugate problem of ballistics, aerodynamics, vehicle heat transfer and thermal strength of constructions of variable mass and shape is developed for all stages of the modern rocket-space vehicles designing.

383 pages,
© 2013

## Table of Contents:

Chapter 1: Asymptotically Simplified Gas-Dynamic Models of Supersonic and Hypersonic Aerodynamics and Heat Transfer

1.2: Theory of the Boundary Layer in the Second Approximation

1.3: Composite Systems of Equations for a Viscous Fluid: Equations of the Viscous Shock Layer, and Parabolized Navier-Stokes Equations

1.4: Approximation of the Thin Viscous Shock Layer (TVSL)

1.5: Equations of the Viscous Shock Layer

1.6: Parabolized Navier-Stokes Equations

1.8: Navier-Stokes Equations

1.9: Numerical Solution of the Simplified NS Equations

Chapter 2: Generalized Equations of the Viscous Shock Layer with Slip Conditions on the Surface and Bow Shock Wave

2.2: Two-Dimensional Navier-Stokes Equations in the Natural Coordinate System Fitted to the Body Surface

2.4: Drag and Heat-Transfer Coefficients

2.5: Generalized Viscous Shock Layer Equations at Low, Moderate, and High Reynolds Numbers

2.6: Generalized Boundary Conditions on the Bow Shock Wave

2.7: Navier-Stokes Equations and Boundary Conditions in Dorodnitsyn-Lees Variables

2.8: Boundary Conditions in the Variables ξ and η

2.9: Estimation of the Order of the Coefficients of the NS Equations in the Variables ξ and η

2.10: Drag and Heat-Transfer Coefficients in the Variables ξ and η

2.11: Generalized Viscous Shock Layer Equations with Slip and Temperature Jump Conditions on the Body Surface and with Generalized Rankine-Hugoniot Conditions on the Bow Shock Wave

Chapter 3: New Form of 'Forces via Fluxes' Transport Relations for Multicomponent Gas and Plasma Mixtures with Exact Transport Coefficients and Applications

3.2: Classical (Old) Form of Transport Relations in the Form of “Fluxes via Thermodynamic Forces”

3.3: New Exact Form of the Transport Relations for the Component Mass and Heat in the Mixture, Resolved with Respect to the Gradients of Hydrodynamic Variables via Fluxes (Force via Fluxes): Exact Stefan–Maxwell Relations

3.4.1 Hydrodynamic Equations for Thermochemically Equilibrium Flows of a Multicomponent Plasma

3.4.2 Effect of Separation of Chemical Elements

3.4.3 Further Applications of the New Form of the Transport Equations

Chapter 4: Slip Boundary Conditions on a CatalyticWall in a Chemically Reacting Multicomponent Many-Temperature Gas Flow with Excited Internal Degrees of Freedom of Particles

4.2: Kinetic Justification of Gas-Dynamic Equations in the Case of Relaxation of Internal Degrees of Freedom for Chemically Reacting Multicomponent Mixtures of Gases

4.2.1 Kinetic Equations and Zeroth Approximation

4.2.2 Hydrodynamic Equations in the Zeroth Approximation

4.2.3 First Approximation

4.2.4 Navier–Stokes Equations and Transport Coefficients

4.3: Boundary Conditions for a Chemically Reacting Gas with Different Vibrational Temperatures of the Components

4.3.1 Kinetic Boundary Conditions

4.3.2 Asymptotic Equation and Zeroth Approximation of the Internal Problem

4.3.3 Formulation of the Problem for the First Approximation

4.3.4 Maxwell–Loyalka Method and Boundary Conditions

4.3.5 Application of the Slip Boundary Conditions

Chapter 5: Physicochemical Models of Hypersonic Flows

5.2: Equations of Conservation (Balance) of Mass, Momentum, and Energy for Gas Mixtures

5.3: Internal Energy of the Species

5.4: Equation of Conservation (Balance) of the Energy of Internal Degrees of Freedom of Particles

5.5: Equation of Conservation (Balance) of the Energy of the Electron Component

5.6: Molecular Transport Relations and Transport Coefficients

5.7: Fluxes of the Energy of Internal Degrees of Freedom

5.8: Kinetics of Dissociation and Ionization Reactions for a One-Temperature Mixture of Gases and Plasma

5.9: Thermally and Chemically Nonequilibrium Regimes of Hypersonic Flow

5.10: Exchange Terms in Equations of Energy Balance of Internal Degrees of Freedom: Effect of Vibrational-Dissociation-Vibrational Interaction (VDVI)

5.11: Heterogeneous Recombination and Heterogeneous Deactivation of Internal Degrees of Freedom of Particles

5.12: Interrelation of Gas-Phase Exchange Reactions and Heterogeneous Recombination Reactions of Atoms in Dissociated Air

5.13: Formation of Excited Particles in the Flow and on the Body Surface

5.14: Effect of Electron-Excited Particles on Gas-Phase Kinetics

5.15: Locally Thermochemically Equilibrium Flows of Gas Mixtures with Different Diffusion Properties of the Species

Chapter 6: Modeling of Catalytic Properties of Thermal Protection Coatings of Space Vehicles

6.2: Experimental Methods and Facilities

6.3: Laboratory and In-Flight Data

6.4: Theoretical Models of Heterogeneous Catalysis during Reentry into the Earth’s Atmosphere

6.5: Heterogeneous Catalytic Processes in Entering the Martian Atmosphere

6.6: Modeling of Catalytic Properties of Thermal Protection Coatings on Space Vehicles on the Basis of Quantum Mechanics and Molecular Dynamics

Chapter 7: Navier–Stokes-Based Numerical Simulation of Flows of a Chemically and Thermally Nonequilibrium Air Plasma in the Discharge Channel and Underexpanded Jets of the IPG-4 Induction Plasmatron

7.3: Thermochemical Model

7.4: Navier–Stokes Equations in the Integral Form

7.5: Calculation of the Induction Plasma Flow in the Discharge Channel and in Underexpanded Jets Escaping from the Sonic Nozzle of the Plasmatron

Chapter 8: Numerical Study of Specific Features of Heat Transfer in a Hypersonic Flow Around a Blunted Cone Lying on a Triangular Plate with Blunted Edges

8.2: Thermodynamic Properties

8.3: Chemical and Transport Models of the Gas Medium

8.4: Geometry of the Body Surface

8.5: Method of the Solution

8.6: Construction of the Computational Grid

8.7: Free-Stream Parameters and Data for Computations

Chapter 9: Model of Partial Chemical Equilibrium for Solving Problems of a Hypersonic Viscous Gas Flow around Bodies

9.2: Formulation of the Problem

9.3: Model of Partial Chemical Equilibrium

9.4: Comparative Analysis of the Problem Solutions within the Framework of NS and FVSL Equations

9.5: Results of Solving the Problem by Using the Partial Chemical Equilibrium Model

9.6: Application of the Partial Chemical Equilibrium Model in the Martian Atmosphere

Chapter 10: Numerical Modeling of Heat Transfer of Hypersonic Flying Vehicles Flying in the Earth’s Atmosphere

10.2: Calculation Methods

10.3: Physicochemical Model of Air

10.4: Boundary Conditions

10.5: Thin Triangular Plate with a Blunted Nose in a Viscous Hypersonic Flow

10.6: Experimental Study of Heat Transfer on a Model of a Reentry Vehicle

10.7: Flow Around a Midwing Hypersonic Vehicle

10.8: Flow around a Low-Wing Hypersonic Vehicle

Chapter 11: Heat Transfer and Flow Structure near the Planetary Probe Surface

11.2: Examined Configurations

11.3: Wind Tunnels and Flow Parameters

11.5: Method of Numerical Simulation of Laminar and Turbulent Flows

11.5.1 Formulation of the Problem

11.5.1.1 Differential Navier–Stokes Equations

11.5.1.2 Boundary and Initial Conditions

11.5.2 Reynolds-Averaged Navier–Stokes Equations

11.5.3 Modeling of Real Gas Flows

11.5.4 Approximation of Equations

11.5.5 Solution of Nonlinear Grid Equations

11.5.6 Solution of Systems of Linear Algebraic Equations

11.5.6.1 Direct Method of Solving a System of Linear Algebraic Equations

11.5.6.2 Iterative Method of Solving a System of Linear Algebraic Equations

11.5.6.3 Convergence Acceleration by Means of Preconditioning

11.5.7 Efficiency of the Numerical Solution of Grid Equations

11.5.8 Construction of the Computational Grid

11.5.9 Development of a Set of Universal Codes

11.5.10 Study of Convergence of the Calculated Data

11.6: Flow Structure and Heat Transfer near the Surface of Model No. 1

11.6.2 Transitional and Turbulent Flows

11.7: Flow Structure and Heat Transfer near the Surface of Model No. 2

11.7.2 Separation Region Length

11.8: Comparison of Models No. 1 and No. 2

Chapter 12: Methodology of Formation of the Windward Surface of a Winged Reentry Vehicle with Reduced Thermal Intensity

Chapter 13: Numerical Modeling of Hypersonic Heat Transfer on the Windward Side of the Buran Reentry Vehicle

13.2: Model of the Medium

13.2.1 Basic Parameters of the Medium: Equation of State

13.2.2 Chemical Reactions in the Gas Phase

13.2.3 Thermodynamic Functions

13.2.4 Model of Molecular Transport Processes

13.2.5 Model of Heterogeneous Processes

13.3: Constitutive Equations and Method of Solving the Problem

13.3.2 Constitutive Equations

13.3.3 Boundary Conditions

13.3.4 Discretization of the Computational Domain

13.3.5 System of Difference Equations

13.3.6 Metric Coefficients

13.3.7 Regularization of Difference Equations

13.3.8 Organization of Flow Field Calculation

13.3.9 Solution of Block Equations

13.4: Calculation of the Flow around the Buran Reentry Vehicle

13.4.1 Description of the Windward Side of the Buran Reentry Vehicle

13.4.2 Coordinate System and Difference Grid

13.4.3 Presentation of Calculated Data

13.4.4 Analysis of Heat Transfer Calculations on the Windward Surface of the Buran Reentry Vehicle

Chapter 14: Mathematical Modeling of Heat and Mass Transfer during Aerothermochemical Destruction of Thermal Protection Materials

14.2: Object of Research: Thermal Destruction of the Binder

14.3: Heterogeneous Chemical Interaction between Silica and Carbon in Internal Layers of the Material

14.4: Melting Ablation of Silica

14.5: Mechanical Ablation of Carbon and Gaseous Products of Material Destruction

14.6: Energy Conservation Equation

14.7: Sublimation of Condensed Components of the Material from the Wall

14.8: Heterogeneous Chemical Reactions Proceeding on the Wall

14.9: Ablation of Silica from the Wall

14.10: Ablation of Condensed Carbon from the Wall

14.11: Correlation Dependence between Rates of Destruction of Solid Species of the Material

14.12: System of Boundary Conditions on the Wall

14.13: System of Boundary Conditions on the Front of Primary Pyrolysis of the Binder

Chapter 15: Modeling of Turbulent Compressible Near-Wall Flows

15.2: Specific Features of the Structure of Turbulent Compressible Near-Wall Flows. Algebraic Models

15.2.1 Structure of the Turbulent Boundary Layer

15.2.2 Variants of the Prandtl Model

15.2.3 Models of Effective Transport Coefficients

15.3: One-Equation Differential Turbulence Models

15.3.1 Models with an Equation for Turbulent Viscosity

15.3.2 Model with the Equation for the Turbulent Kinetic Energy

15.4: Two-Equation Models

15.4.1 Two-Equation K–L Models

15.4.2 Two-Equation K–ε Models

15.4.3 Relations for Linear K–ε Models

15.4.4 Two-Equation K–ω Models

15.5: Three-Equation K–F–R Models

15.6: Models Based on Equations for the Reynolds Stress

15.6.1 Differential Models for the Reynolds Stress

15.6.2 Models Based on Algebraic Relations for the Reynolds Stress

15.6.3 Allowance for Compressibility Effects in High-Order Closure Models

Chapter 16: Numerical Method of Solving Viscous Shock Layer Equations in a Wide Range of Reynolds Numbers

16.2: Formulation of the Problem

16.3: Characteristic Analysis of the System of VSL Equations and the Model of a Parabolic-Hyperbolic Viscous Shock Layer

16.4: Splitting of the Streamwise Pressure Gradient and Method of Global Iterations

16.5: Marching Method of Solving the Cauchy Problem with a Transonic Bifurcation

16.5.1 Numerical Solution of a Model Problem of the One-Dimensional Theory of the Laval Nozzle

16.5.2 Marching Method of Solving the System of PHVSL Equations

16.6: Convergence of Global Iterations

16.6.1 Inviscid Shock Layer

16.6.2 Viscous Shock Layer

Chapter 17: Analytical Method of Solving Thin Viscous Shock Layer Equations at Low Reynolds Numbers

17.2: Thin Viscous Shock Layer Model at Low Reynolds Numbers Re. Two-Dimensional Flows

17.3: TVSL Model in the Vicinity of the Stagnation Line in a Three-Dimensional Flow

17.4: TVSL Model in the Vicinity of the Plane of Symmetry in a Three-Dimensional Flow

17.5: Regimes and Parameters of the Hypersonic Rarefied Gas Flows

17.6: Asymptotic Solution of TVSL Equations

17.7: Estimation of the Accuracy and Applicability Area of the Analytical Solution

Chapter 18: Near-Wall Domain Decomposition in Turbulence Modeling

18.2: Non-Overlapping Linear Domain Decomposition

18.3: Nonoverlapping Nonlinear Domain Decomposition

18.4: Interface Boundary Conditions for RANS Equations

18.4.1 One-Dimensional Domain Decomposition

18.4.1.1 Exact Domain Decomposition

18.4.1.2 Approximate Domain Decomposition

18.5: Interface Boundary Conditions for LR RANS Equations

18.5.2 Comparison Against Analytical and Numerical Wall Functions

18.5.3 Extension to Multidimensional Problems

18.5.4 Preconditioning Technique Based on Domain Decomposition