Fluid Mechanics and Its Applications

Volume 117

Series editor

André Thess, German Aerospace Center, Institute of EngineeringThermodynamics, Stuttgart, Germany

Founding Editor

René Moreau, Ecole Nationale Supérieure d’Hydraulique de Grenoble,Saint Martin d’Hères Cedex, France

Aims and Scope of the Series

The purpose of this series is to focus on subjects in which fluid mechanics plays afundamental role.

As well as the more traditional applications of aeronautics, hydraulics, heat andmass transfer etc., books will be published dealing with topics which are currentlyin a state of rapid development, such as turbulence, suspensions and multiphasefluids, super and hypersonic flows and numerical modeling techniques.

It is a widely held view that it is the interdisciplinary subjects that will receiveintense scientific attention, bringing them to the forefront of technologicaladvancement. Fluids have the ability to transport matter and its properties as wellas to transmit force, therefore fluid mechanics is a subject that is particularly open tocross fertilization with other sciences and disciplines of engineering. The subject offluid mechanics will be highly relevant in domains such as chemical, metallurgical,biological and ecological engineering. This series is particularly open to such newmultidisciplinary domains.

The median level of presentation is the first year graduate student. Some texts aremonographs defining the current state of a field; others are accessible to final yearundergraduates; but essentially the emphasis is on readability and clarity.

More information about this series at http://www.springer.com/series/5980

Yurii N. Grigoryev • Igor V. Ershov

Stability and Suppressionof Turbulence in RelaxingMolecular Gas Flows

123

Yurii N. GrigoryevInstitute of Computational TechnologiesRussian Academy of SciencesNovosibirskRussia

Igor V. ErshovInstitute of Computational TechnologiesRussian Academy of SciencesNovosibirskRussia

ISSN 0926-5112 ISSN 2215-0056 (electronic)Fluid Mechanics and Its ApplicationsISBN 978-3-319-55359-7 ISBN 978-3-319-55360-3 (eBook)DOI 10.1007/978-3-319-55360-3

Library of Congress Control Number: 2017934871

© Springer International Publishing AG 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This monograph describes the results of systematic investigations of the authors inthe field of stability and initial stages of the laminar-turbulent transition in shearflows of thermally nonequilibrium molecular gases. The dissipative effect arising insuch flows due to relaxation of internal degrees of freedom of polyatomic moleculeshas been recently considered as a tool for increasing stability of these flows and fordelaying flow turbulization. Linear stability of plane-parallel shear flows of vibra-tionally excited gases is studied in the monograph in the general statement. Detailedresults on linear and nonlinear stability of a plane Couette flow are presented,including analytical estimates and numerical calculations of the critical Reynoldsnumbers. Nonlinear evolution of large-scale (coherent) vortex structures and thetotal cycle of the development of the Kelvin–Helmholtz instability in a thermallyexcited carrier shear flow are considered.

The mathematical model of flows with relaxation at moderate levels of excitationis based on the full Navier–Stokes equations for a viscous heat-conducting gas withallowance for bulk viscosity. The case of a strongly nonequilibrium vibrationallyexcited gas is described by the full system of equations of two-temperature aero-gasdynamics, where relaxation of vibrational modes is simulated by the Landau–Teller equation for vibrational temperature. The monograph will be useful foraerodynamicists, physicists, mathematicians, and students performing research inthe field of hydrodynamic stability theory, turbulence, and flow laminarization.

The book contains an Introduction and seven chapters. The modern status ofinvestigations of the influence of relaxation processes on hydrodynamic stabilityand turbulence suppression is reviewed in the Introduction. In particular experi-ments on application of the dissipative effect for control of the laminar-turbulenttransition in real hypersonic flows are described.

Chapter 1 has an introductory character and provides some auxiliary material togive an idea of notions and results of physical kinetics, kinetic theory, and acousticsof molecular gases, which are used in the book. The main goal of this chapter is todemonstrate the feasibility and adequacy of mathematical models used in theauthors research. In particular the evolution of the concept of bulk viscosity inmechanics and kinetic theory of gases is briefly described, because this

v

phenomenon is still disputable in aerodynamics. Qualitative properties of theLandau–Teller relaxation equation for the vibrational mode energy, which plays akey role in subsequent considerations, are discussed. The physical mechanism ofdissipation of acoustic waves on the background of the relaxation process in athermally nonequilibrium molecular gas is described.

Chapter 2 is devoted to investigations of linear stability of plane-parallel flows ofan inviscid nonheat-conducting vibrationally excited gas. Some classical resultsof the theory of linear stability of ideal gas flows, in particular the first and secondRayleigh’s theorems and Howard’s theorem, are generalized. An equation of theenergy balance of disturbances is derived, which shows that vibrational relaxationgenerates an additional dissipative factor, which enhances flow stability.Calculations of the most unstable inviscid modes with the maximum growth rates ina free shear layer are described. It is shown that enhancement of excitation ofvibrational modes leads to reduction of the growth rates of inviscid disturbances.

Chapter 3 describes the results of numerical and analytical studies of linearstability of a supersonic Couette flow of a vibrationally excited gas. Even and oddinviscid modes of disturbances are analyzed as functions of the Mach number,depth of excitation of vibrational levels, and characteristic relaxation time. Thegeneral structure of the spectrum of plane perturbations is studied for finiteReynolds numbers. Two most unstable acoustic viscous modes are identified.Results calculated using the constant viscosity model and Sutherland’s law arecompared. Neutral stability curves are calculated, which show that the dissipativeeffect of vibrational mode excitation is inherent in both models of viscosity. Therelative increase in the critical Reynolds number caused by excitation is approxi-mately 12%.

An asymptotic theory of the neutral stability curve for a supersonic planeCouette flow of a vibrationally excited gas is developed in Chap. 4. The initialmathematical model consists of equations of two-temperature viscous gas dynam-ics, which are used to derive a spectral problem for a linear system of eighth-orderordinary differential equations. Unified transformations of the system for all shearflows are performed in accordance with the classical scheme. The spectral problemwith two boundary conditions, which was not considered previously in availablepublications, is reduced to an algebraic secular equation with separation into the“inviscid” and “viscous” parts. The properties of the generalized Airy functions areused for asymptotic estimates of “viscous” solutions. The neutral stability curvesobtained on the basis of the numerical solution of the secular equation agree wellwith the previously obtained results of the direct numerical solution of the originalspectral problem.

The energy stability theory extended by the authors to the case of compressibleflows of a vibrationally excited molecular gas is used in Chap. 5 to study stability ofa subsonic Couette flow. In particular a universal approach is proposed forderivation of equations of the energy balance of disturbances for energy functionalsthat adequately reflect the evolution of the total energy of oscillations for anarbitrary level of thermal excitation. Based on these equations variational problemsare posed for determining the critical Reynolds number of the possible beginning

vi Preface

of the laminar-turbulent transition. Their asymptotic solutions are obtained in thelimit of long-wave disturbances and yield an explicit dependence of Recr on thebulk viscosity coefficient, Mach number, and vibrational relaxation time.

Neutral stability curves are calculated for arbitrary wavenumbers on the basisof the numerical solution of eigenvalue problems. It is shown that the minimumcritical Reynolds numbers in realistic (for diatomic gases) ranges of flow parametersare reached on modes of streamwise disturbances and increase with increasing bulkviscosity coefficient, Mach number, vibrational relaxation time, and degree ofexcitation of vibrational modes. The results obtained in the study qualitativelyconfirm the asymptotic estimates for Recr.

Chapter 6 contains the results of the numerical study of a model problem forestimating the influence of thermal relaxation on the turbulized flow outside thelimits of the laminar-turbulent transition. Nonlinear evolution of a large-scalevortex structure in a plane shear flow of a molecular gas is considered. Suchstructures are inevitable attributes of the final stage of the laminar-turbulent tran-sition and turbulence generation in plane wakes, mixing layers, and submerged jets.The results of numerical simulations reported in this chapter lead to a conclusionabout a noticeable damping effect of thermal relaxation on nonlinear dynamics ofdisturbances that can be really reached in nozzle flows, underexpanded jets, orshock waves.

Chapter 7 presents the results of numerical simulations of the full cycle ofevolution of the Kelvin–Helmholtz instability, which adequately reproduce thelocal mechanism of turbulization of the free shear flow. The problem is consideredboth within the frameworks of the Navier–Stokes equations for a moderate levelof thermal nonequilibrium and using the full system of equations oftwo-temperature aerodynamics for a vibrationally excited gas. Plane waves pre-liminary calculated by numerical solution of appropriate linearized systems ofinviscid gas-dynamic equations are used as initial perturbations. The known patternof the evolution of the “cat’s-eye” large-scale vortex structure typical for theemergence and development of inertial instability is reproduced in detail. Thecalculated results show that the relative enhancement of dissipation of the kineticenergy of the structure averaged over its lifetime can reach 12–15% owing to theincrease of thermal nonequilibrium in ranges realistic for diatomic gases.

The results presented in the book clearly document the reality of the considereddissipative effect and possibility of its use in control of molecular gas flows.

Novosibirsk, Russia Yurii N. GrigoryevDecember 2016 Igor V. Ershov

Preface vii

Acknowledgements

The authors express their sincere gratitude to the Russian Foundation for BasicResearch, which provided support during the entire cycle of investigations, and tothe administration and colleagues from the Institute of Computational Technologiesof the Siberian Branch of the Russian Academy of Sciences, in frameworks ofwhich research programs the present investigations occured.

We are also grateful to Profs. Kenneth J. Haller and Mrs. E.V. Medvedeva fortheir kind assistance in preparing the English text of the manuscript. Last but notleast, we would like to thank the editors of the Springer Publishing Company fortheir kind cooperation and attention in the course of collaboration aimed at pub-lishing this book.

ix

Contents

1 Physico-Mathematical Models of Relaxing MolecularGas Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Elements of Physical Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Systems of Equations of Relaxation Gas Dynamics . . . . . . . . . . . . 5

1.2.1 One-Temperature Models of the Flow . . . . . . . . . . . . . . . . 61.2.2 Two-Temperature Models of Relaxing Flows. . . . . . . . . . . 91.2.3 Landau–Teller Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Parameters of Thermal Relaxation in Diatomic Gases . . . . . . . . . . 151.3.1 Bulk Viscosity Phenomenon. . . . . . . . . . . . . . . . . . . . . . . . 161.3.2 Rotational Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.3 Vibrational Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.4 Absorption of Acoustic Waves in the Relaxation Process . . . . . . . 30References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2 Linear Stability of Inviscid Plane-Parallel Flowsof Vibrationally Excited Diatomic Gases. . . . . . . . . . . . . . . . . . . . . . . 352.1 Equations of the Linear Stability Theory . . . . . . . . . . . . . . . . . . . . 362.2 Some General Necessary Conditions of Instability Growth . . . . . . 372.3 Growth Rates and Eigenfunctions of Unstable Inviscid Modes

in a Free Shear Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.3.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 432.3.2 Numerical Method and Results. . . . . . . . . . . . . . . . . . . . . . 45

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3 Linear Stability of Supersonic Plane Couette Flowof Vibrationally Excited Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.1 Statement of Problem and Basic Equations . . . . . . . . . . . . . . . . . . 533.2 Inviscid Stability Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.2.1 Linear Equations for Inviscid Disturbances . . . . . . . . . . . . 583.2.2 Necessary Instability Conditions of Inviscid Modes . . . . . . 59

xi

3.2.3 Numerical Collocation Method for Spectral Problem . . . . . 643.2.4 Effect of Vibrational Relaxation on Growth

of Second Acoustic Mode . . . . . . . . . . . . . . . . . . . . . . . . . 673.3 Linear Stability of Supersonic Couette Flow at Finite

Reynolds Numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.3.1 Numerical Calculations of Spectral Problem. . . . . . . . . . . . 713.3.2 Structure of Spectra of Viscous Disturbances . . . . . . . . . . . 733.3.3 Neutral Stability Contours and Critical Reynolds

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4 Asymptotic Theory of Neutral Linear Stability Contoursin Plane Shear Flows of a Vibrationally Excited Gas . . . . . . . . . . . . 854.1 Asymptotic Solutions of Linear Stability Equations . . . . . . . . . . . . 87

4.1.1 Asymptotics of Inviscid Solutions in Neighborhoodof a Singular Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.1.2 Asymptotics of Viscous Solutions at High ReynoldsNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2 Asymptotics of a Neutral Stability Curve of the SupersonicCouette Flow of a Vibrationally Excited Gas. . . . . . . . . . . . . . . . . 964.2.1 Secular Equation and Its Solution. . . . . . . . . . . . . . . . . . . . 964.2.2 Asymptotics of the Critical Reynolds Numbers and

Branches of the Neutral Stability Curve . . . . . . . . . . . . . . . 1034.2.3 Numerical Calculations of Secular Equation. . . . . . . . . . . . 106

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5 Energy Theory of Nonlinear Stability of Plane Shear Flowsof Thermally Nonequilibrium Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.1 Energy Stability Analysis of a Plane Compressible Flow.

Effect of a Bulk Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1.1 Basic Equations and Functionals . . . . . . . . . . . . . . . . . . . . 1135.1.2 Variational Problem. Quality Properties

and Asymptotics of Low Critical Reynolds Numbers . . . . . 1185.1.3 Results of Numerical Calculation of the Spectral

Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.2 Energy Stability Analysis of a Plane Vibrationally

Excited Flow. Effect of a Vibrational Relaxation . . . . . . . . . . . . . . 1315.2.1 Energy Balance Equation of Total Disturbances. . . . . . . . . 1315.2.2 Asymptotics of Low Critical Reynolds Numbers . . . . . . . . 1365.2.3 Numerical Calculation of Low Critical Reynolds

Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

xii Contents

6 Evolution of a Large-Scale Vortex in Shear Flow of a RelaxingMolecular Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.1 Navier–Stokes Model Flow. Effect of Bulk Viscosity . . . . . . . . . . 154

6.1.1 Parametrization of a Model Flow . . . . . . . . . . . . . . . . . . . . 1546.1.2 Basic Equations and Initial-Boundary Conditions. . . . . . . . 1566.1.3 Numerical Calculations of a Model Flow . . . . . . . . . . . . . . 160

6.2 Effect of a Vibrational Relaxation on DampingVortex Structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1646.2.1 Basic Equations and Initial-Boundary Conditions. . . . . . . . 1646.2.2 Numerical Scheme and Results of Calculations . . . . . . . . . 166

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7 Dissipation of the Kelvin–Helmholts Waves in a RelaxingMolecular Gas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1717.1 Nonlinear Evolution of the Kelvin–Helmholtz Instability

in the Navie–Stokes Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.1.1 Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . 1727.1.2 Calculation of Initial Perturbations . . . . . . . . . . . . . . . . . . . 1747.1.3 Numerical Calculations of the Evolution

of Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1767.1.4 Effect of Bulk Viscosity on Vorticity Kinematics. . . . . . . . 1777.1.5 Dissipation of the Kinetic Energy of Disturbances . . . . . . . 182

7.2 Effect of a Vibrational Relaxation on the Kelvin–HelmholtzInstability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.2.1 Formulation of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 1897.2.2 Evolution of Disturbances in a Vibrationally

Nonequilibrium Diatomic Gas . . . . . . . . . . . . . . . . . . . . . . 191References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Contents xiii

About the Authors

Yurii N. Grigoryev Professor, Doctor in Physics andMathematics

Place of work: Institute of ComputationalTechnologies SB RAS.

Professional Awards: The Academician Petrov’s Prizeof Russian National Committee on Theoretical andApplied Mechanics (2014).

Field of scientific interests: hydrodynamic stabilityand turbulence, kinetic theory of gases, physical andchemical processes, group methods, mathematicalmodeling, optimization.e-mail: grigor@ict.nsc.ru

Igor V. Ershov Professor, Doctor in Physics andMathematics

Place of work: Department of Information Systemsand Technologies, Novosibirsk State University ofArchitecture and Civil Engineering.

Professional Awards: The Academician Petrov’s Prizeof Russian National Committee on Theoretical andApplied Mechanics (2014).

Field of scientific interests: hydrodynamic stabilityand turbulence, kinetic theory of gases, physical andchemical processes, mathematical modeling.e-mail: i_ershov@ngs.ru; ivershov1969@gmail.comhttps://sites.google.com/site/ivershov2011eng/

xv

List of Figures

Figure 1.1 Model of rough spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Figure 1.2 Temperature dependencies of the rotational relaxation

time srt and bulk viscosity gb calculated in the roughsphere approximation for the gas pressure p ¼ 1 atmand K ¼ 2=5. a shows dependencies srtðTÞ. b showsdependencies gbðTÞ. Curves 1–3 show the data fornitrogen, oxygen, and carbon monoxide, respectively . . . . . . 22

Figure 1.3 Dependencies ZrðTÞ for nitrogen N2, oxygen O2, andcarbon monoxide CO (see the comments to this figurein the text). a shows ZrðTÞ for N2. b shows ZrðTÞ forO2. c shows ZrðTÞ for CO . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Figure 1.4 Temperature dependencies of the rotational relaxationtime srt and bulk viscosity gb. a shows dependenciessrtðTÞ. b shows dependencies gbðTÞ. Curves 1–3 showthe results for nitrogen, oxygen, and carbon monoxide,respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Figure 1.5 Dependencies a1ðTÞ. Curves 1–3 show the results fornitrogen, oxygen, and carbon monoxide, respectively . . . . . . 26

Figure 1.6 Bulk g rtb , g

vvb and shear g viscosities versus the degree

of excitation of vibrational modes of molecules hv.Curve 1 shows the dependence for the shear viscosityg. Curves 2 and 3 show the results for the bulkviscosities g rt

b and g vvb , respectively . . . . . . . . . . . . . . . . . . . . 28

Figure 1.7 Vibrational relaxation times . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 1.8 Dependencies svtðTÞ for nitrogen (1), oxygen (2),

and carbon monoxide (3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 2.1 Isolines of the growth rates aci at s ¼ 1. The dashed

and solid curves show the results for cv ¼ 0 and 0.667,respectively. Curves 1 and 2 show the growth rates forcv ¼ 0 and 0.667, respectively . . . . . . . . . . . . . . . . . . . . . . . . 46

xvii

Figure 2.2 Dependencies of the growth rates aci of the mostunstable modes against the relaxation time parameter sfor M ¼ 0:5 and cv ¼ 0 (1), 0.111 (2), 0.250 (3),and 0.667 (4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Figure 2.3 Dependencies of the real ehv; rðyÞ and imaginary ehv; iðyÞparts of the perturbation of the vibrational temperatureehv for M ¼ 0:5, s ¼ 1, and cv ¼ 0 (1), 0.250 (2),

and 0.667 (3). a shows dependencies ehv; rðyÞ.b shows dependencies ehv; iðyÞ. . . . . . . . . . . . . . . . . . . . . . . . . 48

Figure 2.4 Isolines of fluctuations of the generalized vorticity exfor M ¼ 0:5 and s ¼ 1. a is cv ¼ 0. b is cv ¼ 0:667 . . . . . . . 49

Figure 3.1 Profiles of velocity UsðyÞ and temperature TsðyÞ of themean flow for M ¼ 2 (1) and 5 (2). a shows profiles ofvelocity UsðyÞ. b shows profiles of temperature TsðyÞ.The solid curve is a constant viscosity model(g ¼ const). The dashed and dashdot curves areSutherland’s model (3.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

Figure 3.2 Dependencies crðaÞ for various of the Mach numbersM and s ¼ 1. a is M ¼ 2. b is M ¼ 5. Mode I (1 and 10), II (2 and 20), III (3 and 30), IV (4 and 40), V (5 and50), VI (6 and 60), VII (7 and 7 0), and VIII (8 and 80).The solid and dashed curves show the data for cv ¼ 0and 0.667, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Figure 3.3 Dependencies ciðaÞ for the most unstable mode II atM ¼ 1 (1 and 10), 20 (2 and 20), 10 (3 and 30), 5 (4and 40), 2 (5 and 50), 0,5 (6 and 60), and 0 (7 and 70).The solid and dashed curves show the results for cv ¼0 and 0.667, s ¼ 1, respectively. . . . . . . . . . . . . . . . . . . . . . . 69

Figure 3.4 Dependencies crðaÞ for various of the Mach numbersM at a1 ¼ 0. a is M ¼ 3. b is M ¼ 5. The solid anddashed curves are inviscid modes at cv ¼ 0 and 0.667,respectively. A is a constant viscosity model atRe ¼ 105. B is a constant viscosity model at Re ¼ 106.C is Sutherland’s model (3.10) at Re ¼ 105. D isSutherland’s model (3.10) at Re ¼ 106. Mode I(1 and 10), II (2 and 20), III (3 and 30), IV (4 and 40) . . . . . . . 75

Figure 3.5 Spectra of the eigenvalues c ¼ cr þ ici for M ¼ 3, a ¼0:1 and cv ¼ 0. a is Re ¼ 105. b is Re ¼ 106.a1 ¼ 0 (1) and 2 (2). Points I and II show the resultsfor modes I and II, respectively . . . . . . . . . . . . . . . . . . . . . . . 75

xviii List of Figures

Figure 3.6 Spectra of the eigenvalues c ¼ cr þ ici forRe ¼ 2:5� 105, M ¼ 5. a is a ¼ 0:1. b is a ¼ 1.c is a ¼ 2. d is a ¼ 3. a1 ¼ cv ¼ 0 (1), a1 ¼ 2 andcv ¼ 0:667 (2). Points I–VIII show the results formodes I–VIII, respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Figure 3.7 Dependencies x iðaÞ for constant viscosity model atvarious of the Mach numbers M and Re ¼ 5� 105.a and b are M ¼ 3. c and d are M ¼ 5. a and c aremode I. b and d are mode II. The curves show the datafor a perfect gas (solid curves), vibrationally excitedgas with a1 ¼ 2 and cv ¼ 0:667 (dashed curves),and ideal gas with cv ¼ 0 (dot-and-dashed curves) . . . . . . . . 78

Figure 3.8 Dependencies xiðaÞ for a perfect gas at Re ¼ 5� 105,M ¼ 3 (1) and 5 (2). a is mode I. b is mode II. Thesolid curves are the constant viscosity model. Thedashed curves are Sutherland’s model (3.10) . . . . . . . . . . . . . 79

Figure 3.9 Dependencies of xiðaÞ for Sutherland’s model (3.10)at Re ¼ 5� 105, M ¼ 3. a is mode I. b is mode II.The solid curve is a perfect gas. The dashed curve isthe vibrationally excited gas with a1 ¼ 2 and cv ¼ 0:667 . . . 79

Figure 3.10 Neutral stability curves xiðRe; aÞ ¼ 0 for two modelsof viscosity at M ¼ 3. a is the constant viscositymodel. b is the Sutherland’s model (3.10). The solidand dashed curves show the results for a perfect gasand for a vibrationally excited gas with a1 ¼ 2 andcv ¼ 0:667, respectively. The data for modes I and IIare marked by I and II, respectively. K1 and K 0

1 are thecritical points for mode I. K2 and K 0

2 are the criticalpoints for mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

Figure 3.11 The neutral stability curves xiðRe; aÞ ¼ 0 of the modeII for a perfect gas at M ¼ 5. The dashed and solidcurves show the results for the models of Sutherland(3.10) and constant viscosity, respectively. K2 and K 0

2are the critical points of mode II . . . . . . . . . . . . . . . . . . . . . . 80

Figure 3.12 Dependencies of RecrðMÞ and acrðMÞ for constantviscosity model (1) and Sutherland’s model (3.10) (2).a shows dependencies RecrðMÞ. b showsdependencies acrðMÞ. The solid and dashed curvesshow the results for a perfect gas and for avibrationally excited gas with a1 ¼ 2 and cv ¼ 0:667,respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

Figure 4.1 Neutral stability curves ReðaÞ for modes I and II forM ¼ 3, cv ¼ 0 (1) and cv ¼ 0:667, s ¼ 1 (2).a is mode I. b is mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

List of Figures xix

Figure 4.2 Neutral stability curves ReðaÞ for modes I and II forM ¼ 4, cv ¼ 0 (1) and cv ¼ 0:667, s ¼ 1 (2).a is mode I. b is mode II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

Figure 5.1 Spectral parameter versus the wavenumber Reða; dÞ.a is a1 ¼ 0:5. b is a1 ¼ 1:5 . . . . . . . . . . . . . . . . . . . . . . . . . . 129

Figure 5.2 Dependencies ReðaÞ for d ¼ 0 (neutral stabilitycurves) at a1 ¼ 0 (1), 0.5 (2), 1 (3), 1.5 (4), and 2 (5).The dashed curves are the asymptotic functions RecrðaÞ

(5.31). The dot-and-dashed curve is the criticalReynolds number versus the wavenumber RecrðaÞ . . . . . . . . . 129

Figure 5.3 Eigenfunctions ur, vr, ui, and vi corresponding to thecritical Reynolds numbers RecrðaÞ versus the x2coordinate for a ¼ 0 (1), 0.5 (2), 1 (3), and 2 (4).a shows eigenfunctions ur and vr . b showseigenfunctions ui and vi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Figure 5.4 Level lines of the surfaces Reða; dÞ for a1 ¼ 0 ands ¼ 2. a, b are M ¼ 3. c, d are M ¼ 5. a, c are cv ¼ 0.b, d are 0.667. The points on the line d ¼ 0 are thecritical values of the Reynolds number forcorresponding regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

Figure 5.5 Isolines of vorticity of the critical disturbancesxðx1; x2Þ at M ¼ 3, a1 ¼ 0, and s ¼ 2. a is cv ¼ 0,Recr ¼ 106:1. b is cv ¼ 0:667, Recr ¼ 201:1. Thepoints on the line x2 ¼ 0 are the maximum andminimum values of x for corresponding regime . . . . . . . . . . 147

Figure 5.6 Dependencies ReðaÞ (neutral stability curves) for thestreamwise modes of disturbances for cv ¼ 0:250(1 and 10), 0.429 (2 and 20), and 0.667 (3 and 30).a, b are M ¼ 3. c, d are M ¼ 5. a, c are a1 ¼ 0.b, d are a1 ¼ 2. The solid and dashed curvescorrespond to s ¼ 1 and 3, respectively. Thedot-and-dashed curves show the dependence of thecritical Reynolds number Recr on the wavenumber a . . . . . . . 148

Figure 5.7 Critical Reynolds number Recr versus the degree ofnonequilibrium of the vibrational mode cv at M ¼ 2(1 and 10), 3 (2 and 20), 4 (3 and 30), and 5 (4 and 40).a is a1 ¼ 0. b is a1 ¼ 2. The solid and dashed curvescorrespond to s ¼ 1 and 3, respectively . . . . . . . . . . . . . . . . . 149

Figure 6.1 Flow pattern at the initial time . . . . . . . . . . . . . . . . . . . . . . . . 156Figure 6.2 Effect of bulk viscosity disturbance evolution

(Re ¼ 100, Pr ¼ 0:74, M ¼ 0:5, b ¼ 0:2, v ¼ 3,c ¼ 1:4). a shows kinetic energy versus time fora1 ¼ 0 (1), 0.5 (2), 1 (3), 1.5 (4), and 2 (5).

xx List of Figures

b shows absolute values of the Reynolds stresses r12versus the parameter a1 at different times h . . . . . . . . . . . . . . 162

Figure 6.3 Generation of kinetic energy of disturbances. a showsversus time; the curves are constructed on the basisof the calculation results of Eq. (6.14); the points referto the calculation results of Eqs. (6.19) and (6.21).b shows versus bulk viscosity at different times h;the regime parameters and notation employed are thesame as in Fig. 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

Figure 6.4 Kinetic energy of perturbation versus time forRe ¼ 100, M ¼ 0:5, Pr ¼ 0:74, b ¼ 0:2, v ¼ 3,a1 ¼ 0�2, n ¼ 0�5, svt ¼ 0�5. a isg ¼ gb ¼ k ¼ kv ¼ 0, svt ¼ 3, n ¼ 0:5 (1), 1 (2),2 (3), 3 (4), 4 (5), and 5 (6). b is a1 ¼ 0:5, n ¼ 2,svt ¼ 0:5 (1), 1 (2), 2 (3), 3 (4), 4 (5), and 5 (6) . . . . . . . . . . 168

Figure 7.1 Vorticity field isolines x at the time t ¼ 0 forRe ¼ 100. a is M ¼ 0:2. b is M ¼ 0:5 . . . . . . . . . . . . . . . . . 178

Figure 7.2 Vorticity field isolines x at the time t ¼ 2:5 forRe ¼ 100, M ¼ 0:5. a is a1 ¼ 0. b is a1 ¼ 2. . . . . . . . . . . . . 179

Figure 7.3 Conventional area gðt; a1Þ versus time for Re ¼ 100,a1 ¼ 0 (1), 0.5 (2), 1 (3), 1.5 (4), and 2 (5).a is M ¼ 0:2. b is M ¼ 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 7.4 Dependencies xð0; x2Þj j for fixed time moments t forRe ¼ 100, M ¼ 0:5, and t ¼ 0 (1), 1 (2), 2.5 (3),and 4 (4). a is a1 ¼ 0. b is a1 ¼ 2 . . . . . . . . . . . . . . . . . . . . . 181

Figure 7.5 Time evolution of the disturbance energy E(t) forRe ¼ 100, a1 ¼ 0 (1), 0.5 (2), 1 (3), 1.5 (4),and 2 (5). a is M ¼ 0:2. b is M ¼ 0:5 . . . . . . . . . . . . . . . . . . 185

Figure 7.6 Fluctuating energy production D(t) versus time forRe ¼ 100, a1 ¼ 0 (1), 0.5 (2), 1 (3), 1.5 (4),and 2 (5). a is M ¼ 0:2. b is M ¼ 0:5 . . . . . . . . . . . . . . . . . . 186

Figure 7.7 Integral J1ðtÞ versus time for Re ¼ 100, a1 ¼ 0 (1),0.5 (2), 1 (3), 1.5 (4), and 2 (5). a is M ¼ 0:2.b is M ¼ 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Figure 7.8 Integral J3ðtÞ versus time for Re ¼ 100, a1 ¼ 0 (1),0.5 (2), 1 (3), 1.5 (4), and 2 (5). a is M ¼ 0:2.b is M ¼ 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Figure 7.9 Integral J4ðtÞ versus time for Re ¼ 100, a1 ¼ 0 (1),0.5 (2), 1 (3), 1.5 (4), and 2 (5). a is M ¼ 0:2.b is M ¼ 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

Figure 7.10 Vorticity field contours at M ¼ 0:5, a1 ¼ 0,cv ¼ 0:667, and svt ¼ 1 at point of time t ¼ 3.a is n ¼ 1. b is n ¼ 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

List of Figures xxi

Figure 7.11 Profiles of the static Tðx2Þ and vibrational Tvðx2Þtemperatures in section x1 ¼ 0 at Re ¼ 100, M ¼ 0:5,a1 ¼ 0, cv ¼ 0:667, and svt ¼ 1 at points of time t ¼ 0(1) and 3 (2). a shows profiles of the statictemperatures Tðx2Þ. b shows profiles of the vibrationaltemperatures Tvðx2Þ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

Figure 7.12 Time evolution of the static T(t) and vibrational TvðtÞtemperatures at the center of vortex structure atRe ¼ 100, M ¼ 0:5, a1 ¼ 0, and cv ¼ 0:667.a is svt ¼ 1, n ¼ 3 (1) and 5 (2). b is n ¼ 5,svt ¼ 0:5 (1) and 1 (2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Figure 7.13 Time evolution of the disturbance energy E(t) atRe ¼ 100, M ¼ 0:5, cv ¼ 0:667, svt ¼ 1, n ¼ 1 (1),3 (2), and 5 (3). a is a1 ¼ 0. b is a1 ¼ 2 . . . . . . . . . . . . . . . . 196

Figure 7.14 Time evolution of the fluctuation energy production D(t) at Re ¼ 100, M ¼ 0:5, cv ¼ 0; 667, svt ¼ 1, n ¼ 1(1), 3 (2), and 5 (3). a is a1 ¼ 0. b is a1 ¼ 2. . . . . . . . . . . . . 196

xxii List of Figures

List of Tables

Table 1.1 Bulk viscosity of some gases under standard conditionsT ¼ 273 K and p ¼ 105 Pa . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Table 2.1 Spectral characteristics and growth rates of the mostunstable inviscid modes for s ¼ 1, cv ¼ 0 and 0.667 . . . . . . . . 46

Table 2.2 Numerical absolute values of the real parts of thegeneralized vorticity ex and additive contributionsthat determine it at s ¼ 1, cv ¼ 0 and 0.667. . . . . . . . . . . . . . . 48

Table 3.1 Maximum growth rates xmax and correspondingwavenumbers a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

Table 3.2 Critical values of the Reynolds number Recrand wavenumbers acr for mode II . . . . . . . . . . . . . . . . . . . . . . 82

Table 4.1 Critical Reynolds numbers Recr and wavenumbers acrfor modes I and II (numerical calculation of thecomplete spectral problem (4.1)–(4.7)) . . . . . . . . . . . . . . . . . . 108

Table 4.2 Critical Reynolds numbers Recr and wavenumbers acrfor modes I and II (calculations by the asymptotic theory) . . . 109

Table 5.1 Critical Reynolds number Recrða1Þ as a function of thewavenumber a for d ¼ 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Table 5.2 Critical values of the Reynolds number Recr ða1; cv; s; MÞ . . . 150Table 5.3 Wavenumbers a corresponding to the critical Reynolds

numbers Recr ða1; cv; s; MÞ . . . . . . . . . . . . . . . . . . . . . . . . . . . 150Table 7.1 Wavenumbers and phase velocities of the most growing

inviscid disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175Table 7.2 Normalized deviations eSða1Þ, % for Re ¼ 100 . . . . . . . . . . . . 180Table 7.3 Normalized deviations exða1Þ, % for Re ¼ 100 . . . . . . . . . . . . 182Table 7.4 Normalized deviations eEða1Þ; % for Re ¼ 100 . . . . . . . . . . . . 185Table 7.5 Time evolution of the integral J2ða1Þ for Re ¼ 100 and

M ¼ 0:5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188Table 7.6 Relative deviations eEðnÞ, % for Re ¼ 100, svt ¼ 1,

and cv ¼ 0:667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

xxiii

Introduction

Thermal Relaxation and Stability of Molecular Gas Flows

Relaxation phenomena in molecular gases attracted researchers’ attention for thefirst time in the 1930s due to experiments performed by Kneser [1, 2], who detectedand studied the effect of anomalous absorption of ultrasound in a molecular gas(carbon dioxide). Leontovich [3, 4] proposed a clear physical interpretation of thiseffect in terms of bulk viscosity. A paper of Landau and Teller on the same topic [5]was published simultaneously. Using reduction of the level-by-level description ofexcitation of vibrational modes of a polyatomic molecule, Landau and Tellerderived a new fundamental equation of vibrational energy relaxation, which wasnamed after them. Thus was created the basis of the mathematical apparatus forstudying relaxation processes in thermally and chemically nonequilibrium gases.

The modern description of the theory of thermochemical relaxation in gases canbe found in the monographs of Clarke and McChesney [6] and of Vincenti andKruger [7]. In particular the problem of absorption of the energy of acoustic per-turbations in the thermal relaxation process in molecular gases was described indetail by Herzfeld and Litovitz [8]. The book [9] was devoted to the kinetic theoryof heat and mass transfer and energy exchange in multicomponent molecular gasesin which the generalized Chapman–Enskog method was used to derive systems ofhydrodynamic equations of different levels of closure, including the equations oftwo-temperature hydrodynamics.

The development of laser technologies stimulated investigations described in themonographs of Losev [10] and Gordiets et al. [11], where the processes of rotationaland vibrational relaxation were considered as applied to gas-dynamic laser prob-lems. The problems of stability of vibrationally excited gas flows were consideredfor the first time in publications, which were discussed in the reviews of Osipov andUvarov [12, 13]. Namely, one of the examined aspects was amplification of acousticwaves in a channel flow of a viscous nonheat-conducting gas whose nonequilibriumstate was maintained by means of permanent pumping of energy to vibrationalmodes of molecules and heat removal from translational degrees of freedom. The

xxv

system of equations of two-temperature relaxation gas dynamics was used, wherevibrational relaxation was described by the inhomogeneous Landau–Teller equationwith a source characterizing the power of vibrational energy pumping. Spatiallocalization of the energy input typical for gas-flow lasers leads to generation ofgradients of thermodynamic variables in the flow, resulting in reflection of acousticwaves. The inhomogeneous region formed in the flow may act as a resonator whereperturbations are enhanced [14, 15]. Two models of energy pumping were consid-ered, where the size of the energy input region was either substantially smaller thanthe relaxation “length” or comparable with the latter. Investigations of linear stabilityof the gas flow in the working chamber of the gas-dynamic laser showed that anincrease in the pumping region width made the gas flow less stable [15]. At the sametime, the loss of flow stability in the working chamber of gas-dynamic laser systemsdoes not mean radiation generation failure: if the vortex mode is enhanced, then theheat transfer in the arising turbulent flow becomes more intense, which ensuresbetter conditions for radiation generation [14].

Investigations of the influence of thermal relaxation on stability of typical aero-hydrodynamic flows were started in the 1990s. Nerushev and Novopashin [16]considered the influence of bulk viscosity on the laminar-turbulent transition in theflow in a circular tube. Comparative experiments on the laminar-turbulent transitionin the Hagen-Poiseuille flow were performed for nitrogen, N2, and carbon monoxide,CO. These gases are almost identical in terms of their thermodynamic and transportproperties. However, available data show that the bulk viscosity of CO calculated onthe basis of ultrasound decay data is several times greater than the correspondingvalue for N2 calculated in a similar manner. The laminar-turbulent transition eventwas fixed through the hydraulic resistance crisis. As a result, it was found that thetransition Reynolds number Ret in a “more viscous” gas (CO) is approximately 10%greater than the corresponding value for N2. However, reliability of the data obtainedin [16] was disputable from the very beginning because of the specific arrangementof those experiments and interpretation of their results. Moreover, there are alter-native data for bulk viscosity values for these gases [17–21], which were obtained byadvanced measurements of relaxation times behind shock waves in shock tubes. Itfollows from these data that the bulk viscosity values for N2 and CO are fairly closeto each other, and a small difference between these values cannot be considered asthe reason for the observed change in Ret. In contrast to the ultrasound approach, thisalternative method eliminates the influence of gas hygroscopicity, which may beresponsible for the large differences in data for CO known to be highly hygroscopic.Nerushev and Novopashin [16] were also aware of these data, but made no com-ments on the clearly visible contradiction.

Bertolotti [22] performed a pioneering theoretical study of the influence ofnonequilibrium internal degrees of freedom on the laminar-turbulent transition in aboundary layer flow. He considered the linear stability of a compressible boundarylayer on a semiinfinite flat plate for atmospheric flight conditions at an altitude H =10 km with the Mach number M = 4.5 and stagnation temperature of 1000 K, whichcorresponds to motion of real objects. There was no dissociation in the near-wallflow in the chosen regime, but internal degrees of freedom of oxygen, O2 and N2

xxvi Introduction

molecules, including vibrational ones, were excited to a sufficiently large extent. Itwas assumed in the calculations that the nonequilibrium state of rotational andvibrational degrees of freedom is created naturally. In particular, it was assumedthat the equilibrium distribution of energy over the degrees of freedom of moleculesin wind-tunnel modeling of such flight conditions is violated because of accelera-tion of the air flow in the nozzle up to a required Mach number. In this case theenergy of vibrational modes is “frozen” at the stagnation temperature level, and thestatic (translational) temperature appreciably decreases. In modeling the motion inan undisturbed atmosphere the translational temperature first increases near thestagnation point, and then the energy redistribution occurs due to acceleration of theflow behind the oblique shock wave on the blunted leading edge of the plate.Adiabatic wall conditions were imposed on the flat plate in both cases. The cal-culations based on the equations of linear stability theory show that the allowancefor bulk viscosity produces a minor stabilizing effect, reducing the amplitude of thesecond instability mode by several percent. Depending on temperature, the ratioof the coefficients of bulk viscosity gb and dynamic viscosity g varied in the intervaltypical for air: a1 ¼ gb=g ¼ 0:6 – 1.

A much more powerful and unexpectedly destabilizing effect was observed inthe case of a significant deviation from the equilibrium state, when the relaxationprocess is no longer described by the bulk viscosity model. Instead, the model oftwo-temperature relaxation hydrodynamics was used. The calculations of stabilityof a compressible boundary layer in a wind-tunnel experiment on a flat plate with asharp leading edge showed that the amplitude of low-frequency disturbances of thefirst instability mode is greater than the value estimated under the thermal equi-librium assumption approximately by a factor of 50. Because of the displacementof the upper branch of the neutral stability curve, the domain of instability of thefirst mode turned out to be significantly extended in the downstream direction. Thecalculations for a blunted flat plate moving in an undisturbed atmosphere with dueallowance for the nonequilibrium state behind the shock wave revealed that theamplitude of the first mode is twice higher than the corresponding value obtainedfor the equilibrium conditions. Bertolotti [22] explained this result by significantreduction of the static temperature of the flow because of the excess fraction ofinternal energy remaining in vibrational degrees of freedom after rapid expansion. Itwas assumed that rotational degrees of freedom instantaneously reach the equilib-rium state with translational degrees of freedom, whereas the energy in vibrationaldegrees of freedom remains frozen within the characteristic time of the flow. Thus,the change in the compressible boundary layer stability in the considered case is notdirectly induced by the relaxation process.

A series of experimental studies performed for more than a decade by Hornunggroup [23] in a unique Caltex T5 reflected shock tunnel. This setup provides theformation of a hypervelocity (both high temperature and high Mach number) flowwith the total enthalpy of 3–15 MJ/kg. The main object in those studies was a sharpcone used to study the influence of the growth of the total enthalpy H0 of theincoming hypervelocity flow of air, N2, and CO2, and also air/CO2 and N2/CO2

Introduction xxvii

mixtures on the transition Reynolds number. The laminar-turbulent transition pointon the cone was located at the place of extreme growth of the heat flux.

Intermediate results of those studies were summarized in a review paper [23]. Itshould be noted that from the very beginning, the experiments were aimed atstudying the possibility of laminar-turbulent transition delay in hypervelocity flows.The following reasons were formulated for the expected effect. It was known thatthe laminar-turbulent transition in near-wall flows on a cooled wall at high Machnumbers occurs via the second acoustic mode (cf. [22]). The accompanying pro-cesses of vibrational relaxation and dissociation intensely suppress acoustic dis-turbances. Moreover, under conditions of a hypervelocity flow, CO2 has asufficiently large acoustic coefficient of absorption in the frequency range of106–107 Hz, where the most intense amplification of the second mode is observed.The latter fact was confirmed by special calculations [24].

Unique relaxation properties of CO2 have been known to researchers since theexperiments of Kneser [1, 2]. The linear triatomic molecule has four vibrationalmodes: symmetric, asymmetric, and twice degenerate bending mode. The last modeis the most energy-consuming one and is excited at sufficiently low temperaturesof the flow T = 959 K. The influence of dissociation in pure CO2 becomesnoticeable already at temperatures of T = 1500 K.

Relaxation processes in a high-enthalpy flow of CO2 consist of two stages. Atthe first (nonstationary) stage, vibrational modes are excited to a certain level atwhich dissociation begins owing to breakdown of bonds in the course of vibrations.Then there follows the second (quasi-stationary) stage at which the temperatures ofvibrational modes remain approximately constant, and the energy flux fromtranslational degrees of freedom is mainly spent on dissociation. The temperature ofvibrational degrees of freedom lags behind the static temperature, and the relaxationprocess consists in replenishment of vibrational energy. This process combinedwith dissociation ensures absorption of the energy of acoustic disturbances, whichleads to an increase in the transition Reynolds number.

All these factors were basically confirmed in subsequent experiments. The mainresults of those experiments can be formulated as follows. It was shown that anincrease in the total enthalpy, H0, leads to a small increase in Ret

* for air and N2

and to a significant increase in Ret* for pure CO2. Here the Reynolds number was

determined on the basis of parameters related to the so-called reference temperature.This choice made the data universal in terms of the Mach number, ratio of specificheats, and wall temperature. For a fixed value of H0, the transition Reynoldsnumbers, Ret

*, for air and N2 were found to be 4–5 times smaller than the value ofRet for CO2. The laminar-turbulent transition delay was also detected for mixturesof CO2 with air and N2 . In particular, the value of Ret

* for the mixture with N2 wasmore than doubled in the case with 40% of CO2. For the mixture with air, a similarincrease was obtained already at 14%.

Thus, the main result of this test series was proof of a significant effect ofrelaxation processes in hypervelocity flows on the laminar-turbulent transition. Atthe same time, modest capabilities of the using measurement equipment did notallow reconstruction of the detailed flow pattern. The only measure

xxviii Introduction

of thermochemical nonequilibrium was value of H0 of the flow, which includes theenergies of translational, rotational, and vibrational motion of the molecules.During the process, the energy was uncontrollably distributed between differentinternal modes of molecules and was also spent on dissociation. It is not clear whichprocesses (dissociation, recombination, or energy exchange between vibrational andtranslational degrees of freedom) prevailed behind the oblique shock wave, in theboundary layer near the cold wall, and in the external flow. Thus, it was impossibleto obtain any comprehensive idea about the mechanism of laminarization in theexamined flow.

This gap was filled to a certain extent by calculations of Johnson et al. [25]performed for typical experimental conditions in the T5 wind tunnel. The flow wasdescribed by the equations of two-temperature aerodynamics supplemented with theequations of convection-diffusion of individual species with allowance for recom-bination and dissociation. A problem of linear stability of a disturbed conical flowwas considered within the framework of this model. An important issue was to takeinto account the contributions of linearization of the dependencies of the transportcoefficients, relaxation time, and reaction rate constants on temperatures and con-centrations. After detecting the beginning of growth of the most unstable mode, thecoordinate of the laminar-turbulent transition end (critical Reynolds number) wascalculated with the use of the N-factor. The calculations reproduced the increase inthe critical Reynolds number due to the increase in the total enthalpy of the flowand also the greater values of Ret

* for pure gases and mixtures with lowerthresholds of vibrational excitation and dissociation. The calculated values of theRet

* for air and N2 were higher than the corresponding experimental valuesapproximately by a factor of 1.5–2. This difference was attributed to a high noiselevel of the free stream generated in the T5 wind tunnel, which stimulated an earlierlaminar-turbulent transition in experiments. Unfortunately, the majority of theresults of [25] were obtained for less interesting flows of air and N2. A comparativecontribution of vibrational relaxation and dissociation (recombination) in differentregions of the flow and other important issues of interrelated thermochemicalprocesses were also not reported in sufficient detail.

The experiments of [23] were followed by investigations of the possibility ofcontrolling the laminar-turbulent transition and thermal protection by means of CO2

injection into the boundary layer on the cone surface. It is known that such elementsof thermal protection systems are used in various hypersonic flying vehicles. Thoseactivities were continued with recent experiments performed in the same windtunnel [26, 27]. The results of these experiments and corresponding computationswere summarized by Leyva [27]. The main problem was to heat the injected gas inthe hot boundary layer up to temperatures of the order of 2000 K without inducing apremature laminar-turbulent transition directly at the place of injection. Variousschemes of injection through holes and pores were considered. The results ofexperiments and computations turned out to be somewhat contradictory. Thus, allexperiments with holes resulted in the becoming a laminar-turbulent transitioninstead of its delaying. The same results were predicted by linear stability calcu-lations of Wagnild et al. [28] performed using a scheme similar to that of [25], but

Introduction xxix

with the use of commercial software. On the other hand, the calculations reported in[23] and performed at a sufficiently high CFD level showed that the temperatureof CO2 in the boundary layer rapidly reaches 2000 K. This is sufficient for thebeginning dissociation and vibrational relaxation, which is expected to increaseboundary layer stability. Some experiments with porous injection revealed anapproximately 15% increase in the transition Reynolds number, which then dras-tically decreased as the injection intensity was increased. However, the calculationsof Wagnild et al. [28] for porous injection predicted the laminar-turbulent transitionimmediately after the porous insert.

Comprehensive investigations on supersonic flow control with the use of initia-tion of relaxation processes are also performed now by other research teams [29–31].

It should be noted that the interest in practical application of this effect inhypersonic flying vehicles continues to increase, though the available experimentaldata and their theoretical justification are still far from implementation in realstructures.

This brief retrospective review shows that there were no systematic results onstability of flows of thermally nonequilibrium molecular gases at the early 2000s,though various applications urgently require such investigations. Our understandingof such a challenge stimulated our activities described in [32–42], which form thebasis of this book.

References

1. Kneser, H.O.: Zur Dispersionstheorie des Schalles. Ann. Phys. 403, 761–776 (1931)2. Kneser, H.O.: Die Dispersion hochfrequenter Schallwellen in Kohlensäure. Ann. Phys. 403,

777–801 (1931)3. Leontovoich, M.A.: Remarks on the theory of sound attenuation in gases [in Russian]. Zh.

Éksp. Teor. Fiz. 6, 561–576 (1936)4. Leontovoich, M.A.: Some questions in the theory of sound in absorption polyatomic gases

[in Russian]. Izv. Akad. Nauk SSSR. Ser. Fizika 5, 633–636 (1936)5. Landau, L., Teller, E.: On the theory of sound dispersion. In: Ter Haar, D. (ed.) Collected

Papers of L.D. Landau, pp. 147–153. Pergamon Press, Oxford (1965)6. Clarke, J.F., McChesney., M.: The Dynamics of Real Gases. Butterworths, London (1964)7. Vincenti, W.G., Kruger, C.H.: Introduction to Physical Gas Dynamics. Krieger, Malabar

(1986)8. Herzfeld, K.F., Litovitz, T.A.: Absorption and Dispersion of Ultrasonic Waves. Academic

Press, New York (1959)9. Nagnibeda, E.A., Kustova, E. V.: Non-equilibrium Reacting Gas Flows. Kinetic Theory of

Transport and Relaxation Processes. Springer, Berlin, Heidelberg (2009)10. Losev, S.A.: Gasdynamic Laser. Springer, Berlin, Heidelberg (1981)11. Gordiets, B.F., Osipov, A.I., Shelepin, L.A.: Kinetic Processes in Gases and Molecular Lasers.

Gordon and Breach, New York (1987)12. Osipov, A.I., Uvarov, A.V.: Kinetic and gasdynamic processes in nonequilibrium molecular

physics. Sov. Phys. Usp. 35, 903–923 (1992)13. Osipov, A.I., Uvarov, A.V.: Stability problems in a nonequilibrium gas. Phys. Usp. 39,

597–608 (1996)

xxx Introduction

14. Mukin, R.V., Osipov, A.I., Uvarov, A.V.: Interaction of small hydrodynamic perturbationswith a nonequilibrium region in a gas flow. J. Appl. Mech. Tech. Phys. 46, 818–823 (2005)

15. Mukin, R.V., Osipov, A.I., Uvarov, A.V.: Stability of an inhomogeneousvibrationally-nonequilibrium flow in a waveguide. Fluid Dyn. 42, 126–132 (2007)

16. Nerushev, A., Novopashin, S.: Rotational relaxation and transition to turbulence. Phys. Lett.A 232, 243–245 (1997)

17. Mikhailov, I.G., Soloviev, V.A., Syrnikov, Yu.P.: Fundamentals of Molecular Acoustics[in Russian]. Nauka, Moscow (1964)

18. Stupochenko, E.V., Losev, S.A., Osipov, A.I.: Relaxation Processes in Shock Waves[in Russian]. Nauka, Moscow (1965)

19. Kaye, G.W., Laby, T.H.: Tables of Physical and Chemical Constants. Longmans, Green &Co., London (1958)

20. Kikoin, I.K. (ed.): Tables of Physical Quantities, Handbook [in Russian]. Atomizdat, Moscow(1976)

21. Grigor’ev, I.S., Meilikhov, E.Z. (eds): Physical Quantities, Handbook [in Russian].Énergoatomizdat, Moscow (1991)

22. Bertolotti, F.B.: The influence of rotational and vibrational energy relaxation onboundary-layer stability. J. Fluid Mech. 372, 93–118 (1998)

23. Leyva, I.A., Laurence, S.J., Beierholm, A.W.-K., Hornung, H.G., Wagnild, R.M., Candler,G.V.: Transition delay in hypervelocity boundary layers by means of CO2/acoustic instabilityinteractions. AIAA Paper 2009-1287 (2009)

24. Fujii, K., Hornung, H.G.: Experimental investigation of high-enthalpy effects onattachment-line boundary layer transition. AIAA Journal 41, 1282–1291 (2003)

25. Johnson, H.B., Seipp, T., Candler, G.V.: Numerical study of hypersonic reacting boundarylayer transition on cones. Phys. Fluids 10, 2676–2685 (1998)

26. Beierholm, A.K.-W., Leyva, I.A., Laurence, S.J., Jewell, J.S., Hornung, H.G.: Transitiondelay in a hypervelocity boundary layer using nonequilibrium CO2 injection. GALCITTechnical Report FM 2008.001, California Institute of Technology, Pasadena (2008)

27. Leyva, I.A.: Transition delay in hypervelocity boundary layers by means of vibrationalrelaxation and acoustic instability interactions. AFRL Technical Report 2013-0054, Air ForceResearch Laboratory, Edwards AFB, California (2014)

28. Wagnild, R.M., Candler, G.V., Leyva, I.A., Jewell, J.S., Hornung, H.G.: Carbon dioxideinjection for hypervelocity boundary layer stability. AIAA Paper 2010-1244 (2010)

29. Gembarzhevskii, G.V., Generalov, N.A., Solov’ev, N.G.: Investigation of the velocityfluctuation spectrum of a vortex flow of vibrationally excited molecular gas in a glowdischarge. Fluid Dyn. 35, 222–231 (2000)

30. Vinnichenko, N.A., Nikitin, N.V., Uvarov A.V.: Karman vortex street in a vibrationallynonequilibrium gas. Fluid Dyn. 40, 762–768 (2005)

31. Kirilovskiy, S.V., Maslov, A.A., Poplavskaya, T.V., Tsyryul’nikov, I.S.: Influence ofvibrational relaxation on perturbations in a shock layer on a plate. Tech. Phys. 60, 645–655(2015)

32. Grigoryev, Yu.N., Ershov, I.V.: Relaxation-induced suppression of vortex disturbances in amolecular gas. J. Appl. Mech. Tech. Phys. 44, 471–481 (2003)

33. Grigor’ev, Yu.N., Ershov, I.V., Ershova, E.E.: Influence of vibrational relaxation on thepulsation activity in flows of an excited diatomic gas. J. Appl. Mech. Tech. Phys. 45, 321–327(2004)

34. Grigor’ev, Yu.N., Ershov, I.V.: Effect of bulk viscosity on Kelvin-Helmholtz instability.J. Appl. Mech. Tech. Phys. 49, 407–416 (2008)

35. Grigor’ev, Yu.N., Ershov, I.V.: Energy estimate of the critical Reynolds numbers in acompressible Couette flow. Effect of bulk viscosity. J. Appl. Mech. Tech. Phys. 51, 669–675(2010)

36. Grigoryev, Yu.N., Yershov, I.V.: The linear stability of inviscid shear flow of vibrationallyexcited diatomic gas. J. Appl. Math. Mech. 75, 410–418 (2011)

Introduction xxxi

37. Grigor’ev, Yu.N., Ershov, I.V.: Dissipation of vortex disturbances in a vibrationallynonequilibrium diatomic gas. Thermophys. Aeromech. 19, 183–192 (2012)

38. Grigor’ev, Yu.N., Ershov, I.V.: Critical Reynolds number of the Couette flow of avibrationally excited diatomic gas energy approach. J. Appl. Mech. Tech. Phys. 53, 517–531(2012)

39. Grigor’ev, Yu.N., Ershov, I.V.: Linear stability of the Couette flow of a vibrationally excitedgas. 1. Inviscid problem. J. Appl. Mech. Tech. Phys. 55, 258–269 (2014)

40. Grigor’ev, Yu.N., Ershov, I.V.: Linear stability of the Couette flow of a vibrationally excitedgas. 2. Viscous problem. J. Appl. Mech. Tech. Phys. 57, 247–257 (2016)

41. Grigor’ev, Yu.N., Ershov, I.V.: Asymptotic theory of neutral stability curve of the Couetteflow of a vibrationally excited gas. J. Phys.: Conf. Ser. 722, 012012 (2016)

42. Grigor’ev, Yu.N., Ershov, I.V.: Linear stability of Couette flow of vibrationallynon-equilibrium gas. AIP Conf. Proc. 1770, 030043 (2016)

xxxii Introduction