Compressibility is a property inherent in any material, but it does not always manifest itself. Experience suggests that it affects the medium motion only at velocities comparable to the speed of sound.
Why do we study compressibility? It turns out that in order to calculate the aircraft streamlining or the internal flow in its engine, or the shell muzzle velocity, or the dynamic load of a shock wave from an accidental blast on a structural element, and in many other cases it is necessary to know and understand the laws of the Dynamics of Compressible Media (DCM) and be able to apply them in practice. This textbook is designed to help readers achieve this goal and learn the basics of DCM.
This field of knowledge is high-tech and always focuses on the future: modern developments of hypersonic aircraft, designing more advanced structural elements for airplanes and helicopters, calculating the car aerodynamics, etc.
Paradoxes have always given impetus to the search for new technological devices. Unusual effects in DCM include the flow chocking in supersonic outflow from reservoirs (Sect.2.2); the shock wave formation inside an initially smooth flow (Sect.5.3); the generation of a "spallation saucer" of armor inside a tank when a shell hits it (Sect.5.5); the dog-leg of a plane discontinuity surface at shockwave reflection from a rigid wall (Sec.8.1).
The way to understand these and other effects is through the creation of quantitative models of a moving compressible fluid.
Author(s): Oleksandr Girin
Publisher: Springer
Year: 2022
Language: English
Pages: 315
City: Cham
Preface
Contents
About the Author
Introduction
1. Scope of the Dynamics of Compressible Fluids
2. The Subject Matter of Dynamics of Compressible Fluids
1 General Equations of Gas Motion
1.1 The Thermodynamic Model of a Perfect Gas; Adiabatic Formulae
1.1.1 Internal State of a Gas Particle; Thermodynamic Variables
1.1.2 Perfect Gas Model; Polytropic Gas
1.1.3 Adiabatic Formulae
1.2 Governing Equations of Gas Motion; Mathematical Model …
1.3 Speed of Propagation of Small Disturbances in Ideal Gas; Sound Speed
1.4 Thermodynamics of a Moving Gas
1.4.1 Bernoulli—Saint-Venant Equation; Enthalpy
1.4.2 Stagnation Gas State; Isentropic Formulae
1.4.3 Laval's Number; Other Characteristic States of a Moving Gas
References
2 Continuous Flows
2.1 Equations of One-Dimensional Steady Gas Flow; Rule of a Stream Reversal
2.2 Gas Outflow from Reservoir; Saint-Venant—Vantzel Formula
2.3 Supersonic Outflow Mode; Laval's Nozzle
References
3 Discontinuity in a Gas Flow
3.1 Conservation Laws at a Strong Discontinuity Surface
3.2 Classification of Strong Discontinuities; Shocks
3.3 Normal Shock Theory
3.4 Normal Shock Regularities
3.4.1 Velocity Jump
3.4.2 Pressure Jump
3.4.3 Density Jump
3.4.4 Entropy Jump
3.5 Shock Adiabatic Curve and Its Properties
3.5.1 Equation of Shock Adiabatic Curve
3.5.2 ``Asterisk'' Property
3.5.3 Limiting Degree of Gas Compression in Shock Waves
3.5.4 Approximation of Strong Shocks
3.5.5 Approximation of Weak Shocks
References
4 Governing Equations and Initial-Boundary-Value Problems
4.1 Geometry of One-Dimensional Flows
4.2 Equations of Motion in Euler's Form; Initial and Boundary Conditions
4.2.1 Euler's Equations of Motion
4.2.2 Initial Conditions
4.2.3 Boundary Conditions
4.3 Equations of Motion in Lagrange's Form
4.4 Equations of Motion in Characteristic Form; the Characteristic …
4.5 The Method of Characteristics
4.6 Generalized Cauchy Problem (Type I Problem) …
4.7 The Goursat Problem (Type II Problem)
4.8 Combined Problem of a Special Type (Type III Problem)
4.9 Characteristics as Trajectories of a Possible Weak Discontinuity of a Solution
4.9.1 Relationships Along the Weak Discontinuity Trajectory
4.9.2 Breakup of Arbitrary Weak Discontinuity
References
5 Isentropic Gas Flows with Plane Waves
5.1 Riemann Method
5.1.1 Riemann Invariants
5.1.2 Riemann Variables; Riemann Method
5.1.3 The Euler–Poisson Equation
5.1.4 The Remarkable Case γ= 3
5.2 The Riemann Waves
5.2.1 Simple Waves
5.2.2 Adjoining Theorem
5.2.3 Simple Wave Equations
5.2.4 Properties of Simple Waves
5.3 Gradient Catastrophe
5.4 The Piston Problem
5.4.1 Case When the Piston Is Pulled Out from Gas
5.4.2 Case of Piston Moving with Constant Velocity
5.4.3 Gas Outflow into the Vacuum
5.4.4 Piston Moves into Gas; Shock Wave Induction Time
5.5 Interaction of Simple Wave with a Contact Surface …
5.5.1 Analysis of the Flow Structure
5.5.2 Qualitative Analysis of the Interaction
5.5.3 The Limit Cases
References
6 Methods of Wave Interaction Analysis
6.1 Method of (u, p)-Diagrams
6.1.1 ( u,p ) -Diagrams of Simple Waves
6.1.2 ( u,p ) -Diagrams of Shock Waves
6.2 Breakup of Arbitrary Strong Discontinuity (Riemann's Problem)
6.2.1 The Problem Formulation
6.2.2 Lemma About the Disturbances
6.2.3 Existence and Uniqueness of the Solution
6.2.4 Acoustic Approximation
References
7 Shock—Wave Flows
7.1 Shock Tube Performance
7.1.1 The Device Description
7.1.2 The Problem Formulation
7.1.3 Shock Tube Solution
7.2 Piston Moving with a Constant Velocity
7.2.1 Piston Moves into the Gas
7.2.2 Piston Moves Out from the Gas
7.3 Shock Wave Reflection from Rigid Wall; Amplification Factor
7.3.1 The Problem Formulation
7.3.2 The Problem Solution
7.3.3 Shock Wave Percussive Ability
7.4 Interaction of Shock Wave with Contact Surface
7.4.1 The Problem Formulation
7.4.2 Qualitative Analysis of the Flow
7.5 Interaction of Two Shock Waves
7.5.1 The Problem Formulation
7.5.2 Qualitative Analysis
7.6 Interaction of Shock Wave with Simple Wave; Entropy Trace
7.6.1 The Problem Formulation
7.6.2 Qualitative Analysis of the Flow
7.7 The Problem of the Internal Ballistics (Lagrange's Problem)
7.7.1 The Main Assumptions
7.7.2 The Problem Formulation
7.7.3 Solution in the Domain of Simple Wave
7.8 Strong Point Blast in Gas
7.8.1 Explosion Phenomenon
7.8.2 The Problem Formulation
7.8.3 Self-similarity of the Solution
7.8.4 Regularities of Gas Motion at Strong Point Blast
7.9 Long-Range Asymptotic Behavior of Shock Waves
References
8 Steady Plane Irrotational Flows
8.1 Theory of an Oblique Shock
8.1.1 Interaction of Supersonic Flow with a Wedge; Velocity Triangle
8.1.2 The Properties of Shock Polar
8.1.3 Oblique Reflection of a Plane Shock from a Rigid Wall
8.2 Equations of Steady Plane Irrotational Gas Motion
8.2.1 Equations and Methods
8.2.2 The Characteristics of Equations of Plane Irrotational Steady Flow
8.2.3 Simple Waves
8.3 Supersonic Flow Around a Convex Corner; Prandtl–Meyer Flow
8.4 Plane Supersonic Outflow from a Slit
8.5 Elements of the Theory of Thin Aerodynamic Profile
8.5.1 The Main Concepts
8.5.2 Linearization of Equations of Motion
8.5.3 Thin Profile in a Subsonic Stream; The Prandtl–Glauert Rule
8.5.4 Thin Profile in a Supersonic Stream; Akkeret's Formula; Wave Drag
References
Appendix A Numerical Method of Characteristics for the 1-D Unsteady Flows (Massau's Scheme)
A.1 General Features of the Method
A.2 Algorithms of the Numerical Method of Characteristics
A.2.1 Governing Equations of 1-D Unsteady Gas Flow in Characteristic Form
A.2.2 Calculations in the Internal Node
A.2.3 Implementation of Boundary Conditions
A.2.3.1 ``Rigid Wall''
A.2.3.2 ``Piston''
A.2.3.3 ``Shock Front''
A.2.3.4 ``Contact Surface''
A.3 Reverse Method of Characteristics (Hartree Scheme)
A.4 Scheme for Isentropic Flows with Plane Waves
Appendix B Godunov's Method for the Calculations of 1-D Unsteady Flows
B.1 General Properties of the Method
B.2 Scheme of the Method
B.2.1 Initial Data Processing
B.2.2 Development of the Difference Scheme
B.2.3 Searching for uk ,pk and Flow Configuration
B.2.4 Determination of R,U,P
B.2.5 Determination of the Slopes Wleft ,W'left ,Wk , W'right ,Wright of the Sectors' Borders
B.2.6 Finding the Relevant Sector
B.3 Approximate Solution of the Discontinuity Breakup Problem
B.3.1 The Acoustic Approximation
B.3.2 Isentropic Approximation
B.4 Algorithms of Boundary Conditions' Fulfillment
B.4.1 ``Rigid Wall''
B.4.2 ``Piston''
B.4.3 ``Shock Front''
B.4.4 ``Contact Surface''
B.5 Determination of a Stable Time-Step
B.6 Example Structure and Flowchart of Program Code for Godunov's Method
Appendix C Numerical Methods for Two-Dimensional Flows
C.1 Method of Characteristics for 2-D Steady Supersonic Flows
C.1.1 The Characteristic form of Equations of Gas Motion in Ehlers' Variables
C.1.2 Calculation Scheme for an Internal Node
C.1.3 Calculation Scheme at the Symmetry Axis
C.1.4 Calculation of the Node at the Rigid Wall
C.1.5 Calculation of a Node at Free Surface
C.2 Breakup-Based Scheme of the Predictor—Corrector Type for 2-D Steady Supersonic Flows
C.2.1 Governing Equations
C.2.2 Approximation of the Computational Domain
C.2.3 The Corrector Stage: the Finite-Difference Scheme
C.2.4 The Predictor Stage: Determination of R,U,V,P
C.2.5 Boundary Condition Fulfillment
C.2.6 Choice of Time Step; Use of Auxiliary Variables
C.3 Godunov's Scheme for 2-D Unsteady Flows
C.3.1 The Case of Plane-Parallel Flow
C.3.1.1 The ``Corrector'' Stage
C.3.1.2 The Stage ``Predictor''
C.3.2 The Case of a Fixed Rectangular Grid
References
Index
