Radiation Modeling in Shock - Tubes and Entry Flows

In the present lecture some basic problems of state and solving of radiation heat transfer equation as applied to radiation modelling in shock-tubes and entry flow are discussed. The lecture contains five parts. In the first part the radiation heat transfer equation and the general definitions of the radiation heat transfer theory are presented. The definitions introduced in the first part are widely used in other parts. The second part of the lecture (Chapter 3) presents four methods which are used for solution of radiation heat transfer problems in different aerospace applications. Well known problem of radiation heat transfer in heated gases and low-temperature plasma in view of atomic and molecular rotational lines, as well as of the vibrational molecular bands, are discussed in the third part. Some untraditional models of radiation heat transfer are presented in this part. The fourth part of the lecture is dedicated to the Monte-Carlo algorithms which are also in common use in aerospace problems, especially at prediction of emissivities of heated and radiating volumes of lightscattering gases. Several traditional and novel algorithms are presented in this part. The final part (Chapter 6) of the lecture presents examples of using of numerical simulation methods described in previous parts at solution of various radiation heat transfer problems in aerospace applications. It should be stressed that the methods of the radiation heat transfer integration and various numerical simulation results presented in the lecture were used and obtained by the author at solution of concrete problems of aerospace physical gas dynamics. The study was supported by the Russian Foundation of Basic Research (grant # 07-01-0133). 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TITLE AND SUBTITLE Radiation Modeling in Shock-Tubes and Entry Flows 5a. CONTRACT NUMBER 5b. GRANT NUMBER 5c. PROGRAM ELEMENT NUMBER 6. AUTHOR(S) 5d. PROJECT NUMBER 5e. TASK NUMBER 5f. WORK UNIT NUMBER 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Institute for Problems in Mechanics Russian Academy of Sciences 101-1, Vernadskogo prosp. Moscow, 119526 Russia 8. PERFORMING ORGANIZATION REPORT NUMBER 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S) 11. SPONSOR/MONITOR’S REPORT NUMBER(S) 12. DISTRIBUTION/AVAILABILITY STATEMENT Approved for public release, distribution unlimited 13. SUPPLEMENTARY NOTES See also ADA562449. RTO-EN-AVT-162, Non-Equilibrium Gas Dynamics From Physical Models to Hypersonic Flights (Dynamique des gaz nonequilibres Des modeles physiques jusqu’au vol hypersonique)., The original document contains color images. 14. ABSTRACT In the present lecture some basic problems of state and solving of radiation heat transfer equation as applied to radiation modelling in shock-tubes and entry flow are discussed. The lecture contains five parts. In the first part the radiation heat transfer equation and the general definitions of the radiation heat transfer theory are presented. The definitions introduced in the first part are widely used in other parts. The second part of the lecture (Chapter 3) presents four methods which are used for solution of radiation heat transfer problems in different aerospace applications. Well known problem of radiation heat transfer in heated gases and low-temperature plasma in view of atomic and molecular rotational lines, as well as of the vibrational molecular bands, are discussed in the third part. Some untraditional models of radiation heat transfer are presented in this part. The fourth part of the lecture is dedicated to the Monte-Carlo algorithms which are also in common use in aerospace problems, especially at prediction of emissivities of heated and radiating volumes of light-scattering gases. Several traditional and novel algorithms are presented in this part. The final part (Chapter 6) of the lecture presents examples of using of numerical simulation methods described in previous parts at solution of various radiation heat transfer problems in aerospace applications. It should be stressed that the methods of the radiation heat transfer integration and various numerical simulation results presented in the lecture were used and obtained by the author at solution of concrete problems of aerospace physical gas dynamics. 15. SUBJECT TERMS 16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF ABSTRACT SAR 18. NUMBER OF PAGES 102 19a. NAME OF RESPONSIBLE PERSON a. REPORT unclassified b. ABSTRACT unclassified c. THIS PAGE unclassified Standard Form 298 (Rev. 8-98) Prescribed by ANSI Std Z39-18 Radiation Modeling in Shock-Tubes and Entry Flows 10 2 RTO-EN-AVT-162 CONTENTS 1.0 Introduction ........................................................................................................................ 3 2.0 Radiation heat transfer equation and general characteristics of heat radiation transfer ..................................................................................................... 4 2.1 Classification of spectral optical models ............................................................... 4 2.2 General notations of the radiation heat transfer theory .......................................... 5 3.0 Methods of integration of radiation heat transfer equations ............................................... 9 3.1 The PN (Spherical Harmonic) approximation ........................................................ 9 3.2 Quadro-moment methods ...................................................................................... 25 3.3 Ray-Tracing method .............................................................................................. 29 3.4 Discreet ordinates method ..................................................................................... 31 4.0 Random models .................................................................................................................. 35 4.1. Formulation of Random Models for Atomic Lines ............................................... 35 4.2. Numerical Simulation Method for Calculation of Radiative Heat Transfer in Plane-Parallel Non-Uniform Layers .................................................................. 38 4.3. The Macro-Random Model for Describing of Radiative Heat Transfer with Vibrational Band Structure ............................................................................ 43 5.0 The Monte-Carlo methods .................................................................................................. 47 5.1 Line by line integration of radiation heat transfer equation on spectrum of rotational lines ................................................................................................... 47 5.2 Hybrid Statistical Method ...................................................................................... 48 5.3 Method of the smoothed coefficients ..................................................................... 53 5.4 The two-group method ........................................................................................... 53 5.5 Line-by-line integration with little number of trajectories ..................................... 54 5.6 The Monte-Carlo imitative method based on the Maximum Cross Section (MCS) method ....................................................................................................... 54 5.7 Imitative Monte-Carlo algorithm based on the quasi-random sampling of photon trajectory parameters ............................................................................. 56 5.8 The hybrid method based on the quasi-random sampling method ........................ 56 5.9 Three-dimensional simulation algorithms ............................................................. 56 5.10 Monte-Carlo Local Estimation of Directional Emissivity (MCLEDE) ................. 56 6.0 Examples of application of methods for solving radiation heat transfer equations ............ 59 6.1 P1-approximation ................................................................................................... 59 6.2 The quadro-moment method .................................................................................. 62 6.3 The ray-tracing method .......................................................................................... 63 6.4 Discrete ordinates method ..................................................................................... 68 6.5 Random models of atomic lines ............................................................................. 69 6.6 Macro-random model ............................................................................................ 73 6.7 The Monte-Carlo method ....................................................................................... 77 References ...................................................................................................................................... 95 Radiation Modeling in Shock-Tubes and Entry Flows RTO-EN-AVT-162 10 3 1.0 INTRODUCTION The term “thermal radiation” (“heat radiation”) means electromagnetic radiation of atomic and molecular, as opposed to nuclear, origin. Such radiation is emitted by matter in a state of thermal excitation, thus accounting for the designation of the radiation as thermal. The energy density of this type of radiation is given by the Planck formula for black body radiation. More generally the radiation energy distribution is described by a kinetic or transport equation, referred to historically as the equation of radiative transfer. The importance of thermal radiation in physical problems, and particularly in problems of physical gas dynamics in different aerospace applications, increases as the temperatures is raised. At moderate temperatures ( 4 5 10 10 T   K), the role of radiation is primary one of transporting energy in gases and plasmas by radiative processes. At higher temperatures ( 5 6 10 10 T   K), the energy and momentum densities of radiation field may become comparable or dominate the corresponding fluid quantities. As a rule, hydrodynamics with explicit account of the radiation energy and momentum contributions constitutes subject of investigation of “high-temperature radiation hydrodynamics”. More general definition of radiation gasdynamic implies consideration of coupled radiative and gasdynamic processes. In the partial case of the weak radiation-gasdynamic interaction there is possibility to study radiative and gasdynamic phenomena separately. For example, such a case is realized for spacecraft at entry velocities up to 6 8  km/s. Radiative gas dynamics (RadGD) is the directions of physical gas dynamics which is connected to such challenging sciences as: astrophysics, physics of stars and Sun, research of a structure of substance (atomic and molecular spectroscopy), interaction of laser radiation and high-energy beams with materials, plasma generators, rocket engines (of chemical, plasma, electric, nuclear or laser types), spacecraft's thermal protection, heat exchange in steam boiler, in aircraft and rocket engines, in working volumes of various power installations (including nuclear). Figure 1.1 shows hierarchical division of the Radiative gas dynamics. In order to solve any RadGD problem there is a necessity to create a radiative model of a gas or plasma. The radiative model is defined as the set of optical model and radiation transfer model. The optical model includes spectral, group and integral absorption, emission and scattering coefficients, which are in turn based on cross-sections (or probabilities) of elementary radiative processes predicted by quantum mechanics and quantum chemistry. The absorption, emission and scattering coefficients can be determined only with use data on distribution functions for atomic and molecular particles, and also on their energy states. Thermodynamics and statistical physics provide all necessary information for these purposes. The radiation transfer model, composing the second part of the radiative model, is based on the thermodynamics and statistical physics, and is designed for prediction such characteristics of a radiation field as the radiation energy density U, the radiation flux W , and the divergence of the radiation flux div rad Q W  . A spectral region of the electromagnetic radiation, which is of practical interest for various aerospace applications ranges from 0.05m until 20m. This spectral region is divided on the following sub regions:  0.05 0.4    m is the ultraviolet region;  0.4 0.7    m is the visible region;  0.7 20.0    m is the infrared region, where  is the wavelength interval. A radiation emitted by matter in this spectral region is called as heat radiation. Radiation Modeling in Shock-Tubes and Entry Flows 10 4 RTO-EN-AVT-162 Figure 1.1: Basic scheme of Radiative Gas Dynamics 2.0 RADIATION HEAT TRANSFER EQUATION AND GENERAL CHARACTERISTICS OF HEAT RADIATION TRANSFER 2.1 Classification of spectral optical models Classification of the spectral optical models was introduced in [1]: 1) Optical model   , k T   is spectral, quasi-spectral, multigroup and combined models of absorption coefficients representing in the table, analytical, graphical or in a computer module form. Here  is the Wavenumber; T is the temperature;  is set of the parameters (pressure or density, concentration of the chemical components). In the case of a nonequilibrium medium instead of T we should use set of effective temperatures (electronic temperature, vibrational temperature, rotational temperature). 2) Spectral (line-by-line) model   , k T   is the function with continuous and lines structure without any smoothing. 3) Multigroup model is the smoothing spectral model in a set of spectral ranges i  . We suppose in the each spectral range i  the absorption coefficient is independent of wavenumber. 4) Quasi-spectral (quasi-continuum) model is the smeared rotational line model for a set of spectral ranges. The value i  in this model must be larger than maximal values of the rotational lines widths (that is not less than 25 ÷ 50 cm 1 ). As a rule, quasi-spectral model is included to multigroup model. Radiative Gas Dynamics Radiation Heat Transfer Optical model: Absorption coefficients, emission coefficients, scattering coefficients and scattering indicatrix Cross-sections of the elementary radiative processes Thermodynamics and Statistical Physics Gas Dynamics Chemical Physics Radiative Model Radiation Transfer Model Methods for solving RHT problems Quantum mechanics and quantum chemistry Radiation Modeling in Shock-Tubes and Entry Flows RTO-EN-AVT-162 10 5 5) Total absorption model is the Planck mean absorption coefficient, or (and) the Rosseland mean absorption coefficient, or (and) the Chandrasekhar mean absorption coefficient. 6) Combined model is the sum of the spectral line absorption and the multigroup absorption model (as a rule the selective atomic lines absorption on the continuum or quasi-continuum background). 7) Radiative heat transfer model is the set of:  conditions (the thermodynamic conditions in a medium, radiation heat transfer boundary conditions, the spectral resolution of numerical calculations);  equation of radiation heat transfer;  mathematical method using for solving this equation. 8) Radiative model is the set of the optical model and the radiation heat transfer model. 9) Optimum radiative model is the radiative model containing minimum number of the spectral groups and in the same time, permitting to obtain physical adequately results. Figure 2.1 shows example of air low-temperature plasma. Wavenumber, 1/cm 5000