Stability and Suppression of Turbulence in Relaxing Molecular Gas Flows (Fluid Mechanics and Its Applications (117))

This book presents an in-depth systematic investigation of a dissipative effect which manifests itself as the growth of hydrodynamic stability and suppression of turbulence in relaxing molecular gas flows. The work describes the theoretical foundations of a new way to control stability and laminar turbulent transitions in aerodynamic flows. It develops hydrodynamic models for describing thermal nonequilibrium gas flows which allow the consideration of suppression of inviscid acoustic waves in 2D shear flows. Then, nonlinear evolution of large-scale vortices and Kelvin-Helmholtz waves in relaxing shear flows are studied. Critical Reynolds numbers in supersonic Couette flows are calculated analytically and numerically within the framework of both linear and nonlinear classical energy hydrodynamic stability theories. The calculations clearly show that the relaxation process can appreciably delay the laminar-turbulent transition. The aim of the book is to show the new dissipative effect, which can be used for flow control and laminarization. This volume will be of interest and useful to mechanical engineers, physicists, and mathematicians who specialize in hydrodynamic stability theory, turbulence, and laminarization of flows. Preface 6 Acknowledgements 9 Contents 10 About the Authors 13 List of Figures 14 List of Tables 20 Introduction 21 Thermal Relaxation and Stability of Molecular Gas Flows 21 References 21 1 Physico-Mathematical Models of Relaxing Molecular Gas Flows 29 1.1 Elements of Physical Kinetics 30 1.2 Systems of Equations of Relaxation Gas Dynamics 33 1.2.1 One-Temperature Models of the Flow 34 1.2.2 Two-Temperature Models of Relaxing Flows 37 1.2.3 Landau--Teller Equation 41 1.3 Parameters of Thermal Relaxation in Diatomic Gases 43 1.3.1 Bulk Viscosity Phenomenon 44 1.3.2 Rotational Relaxation 48 1.3.3 Vibrational Relaxation 56 1.4 Absorption of Acoustic Waves in the Relaxation Process 58 References 60 2 Linear Stability of Inviscid Plane-Parallel Flows of Vibrationally Excited Diatomic Gases 63 2.1 Equations of the Linear Stability Theory 64 2.2 Some General Necessary Conditions of Instability Growth 65 2.3 Growth Rates and Eigenfunctions of Unstable Inviscid Modes in a Free Shear Flow 71 2.3.1 Formulation of the Problem 71 2.3.2 Numerical Method and Results 73 References 77 3 Linear Stability of Supersonic Plane Couette Flow of Vibrationally Excited Gas 78 3.1 Statement of Problem and Basic Equations 80 3.2 Inviscid Stability Problem 85 3.2.1 Linear Equations for Inviscid Disturbances 85 3.2.2 Necessary Instability Conditions of Inviscid Modes 86 3.2.3 Numerical Collocation Method for Spectral Problem 91 3.2.4 Effect of Vibrational Relaxation on Growth of Second Acoustic Mode 94 3.3 Linear Stability of Supersonic Couette Flow at Finite Reynolds Numbers 98 3.3.1 Numerical Calculations of Spectral Problem 98 3.3.2 Structure of Spectra of Viscous Disturbances 100 3.3.3 Neutral Stability Contours and Critical Reynolds Numbers 104 References 111 4 Asymptotic Theory of Neutral Linear Stability Contours in Plane Shear Flows of a Vibrationally Excited Gas 112 4.1 Asymptotic Solutions of Linear Stability Equations 114 4.1.1 Asymptotics of Inviscid Solutions in Neighborhood of a Singular Point 115 4.1.2 Asymptotics of Viscous Solutions at High Reynolds Numbers 117 4.2 Asymptotics of a Neutral Stability Curve of the Supersonic Couette Flow ƒ 123 4.2.1 Secular Equation and Its Solution 123 4.2.2 Asymptotics of the Critical Reynolds Numbers and Branches of the Neutral Stability Curve 130 4.2.3 Numerical Calculations of Secular Equation 133 References 136 5 Energy Theory of Nonlinear Stability of Plane Shear Flows of Thermally Nonequilibrium Gas 137 5.1 Energy Stability Analysis of a Plane Compressible Flow. Effect of a Bulk Viscosity 139 5.1.1 Basic Equations and Functionals 139 5.1.2 Variational Problem. Quality Properties and Asymptotics of Low Critical Reynolds Numbers 144 5.1.3 Results of Numerical Calculation of the Spectral Problem 153 5.2 Energy Stability Analysis of a Plane Vibrationally Excited Flow. Effect of a Vibrational Relaxation 157 5.2.1 Energy Balance Equation of Total Disturbances 157 5.2.2 Asymptotics of Low Critical Reynolds Numbers 162 5.2.3 Numerical Calculation of Low Critical Reynolds Numbers 170 References 177 6 Evolution of a Large-Scale Vortex in Shear Flow of a Relaxing Molecular Gas 178 6.1 Navier--Stokes Model Flow. Effect of Bulk Viscosity 179 6.1.1 Parametrization of a Model Flow 179 6.1.2 Basic Equations and Initial-Boundary Conditions 181 6.1.3 Numerical Calculations of a Model Flow 185 6.2 Effect of a Vibrational Relaxation on Damping Vortex Structure 189 6.2.1 Basic Equations and Initial-Boundary Conditions 189 6.2.2 Numerical Scheme and Results of Calculations 191 References 194 7 Dissipation of the Kelvin--Helmholts Waves in a Relaxing Molecular Gas 195 7.1 Nonlinear Evolution of the Kelvin--Helmholtz Instability ƒ 196 7.1.1 Formulation of the Problem 196 7.1.2 Calculation of Initial Perturbations 198 7.1.3 Numerical Calculations of the Evolution of Perturbations 200 7.1.4 Effect of Bulk Viscosity on Vorticity Kinematics 201 7.1.5 Dissipation of the Kinetic Energy of Disturbances 206 7.2 Effect of a Vibrational Relaxation on the Kelvin--Helmholtz Instability 213 7.2.1 Formulation of Problem 213 7.2.2 Evolution of Disturbances in a Vibrationally Nonequilibrium Diatomic Gas 215 References 222 Index 223 Front Matter....Pages i-xxxii Physico-Mathematical Models of Relaxing Molecular Gas Flows....Pages 1-34 Linear Stability of Inviscid Plane-Parallel Flows of Vibrationally Excited Diatomic Gases....Pages 35-49 Linear Stability of Supersonic Plane Couette Flow of Vibrationally Excited Gas....Pages 51-84 Asymptotic Theory of Neutral Linear Stability Contours in Plane Shear Flows of a Vibrationally Excited Gas....Pages 85-109 Energy Theory of Nonlinear Stability of Plane Shear Flows of Thermally Nonequilibrium Gas....Pages 111-151 Evolution of a Large-Scale Vortex in Shear Flow of a Relaxing Molecular Gas....Pages 153-169 Dissipation of the Kelvin–Helmholts Waves in a Relaxing Molecular Gas ....Pages 171-198 Back Matter....Pages 199-201