FUNDAMENTAL OF COMPRESSIBLE FLUID DYNAMIC

CONTENTS
Stagnation effects
Nozzle
Normal Shock
Isothermal Flow
Fanno Flow
Rayleigh Flow
Evacuation and filling semi rigid Chambers
Evacuating and filling chambers under external forces
Oblique Shock
Prandtl–Meyer
Transient problem
1 Introduction 1
1.1 What is Compressible Flow ?
1.2 Why Compressible Flow is Important?
1.3 Historical Background
1.3.1 Early Developments
1.3.2 The shock wave puzzle
1.3.3 Choking Flow
1.3.4 External flow
1.3.5 Filling and Evacuating Gaseous Chambers
1.3.6 Biographies of Major Figures

2 Fundamentals of Basic Fluid Mechanics
2.1 Introduction
2.2 Fluid Properties
2.3 Control Volume
2.4 Reynold’s Transport Theorem

3 Speed of Sound
3.1 Motivation
3.2 Introduction
3.3 Speed of sound in ideal and perfect gases
3.4 Speed of Sound in Real Gas
3.5 Speed of Sound in Almost Incompressible Liquid
3.6 Speed of Sound in Solids
3.7 Sound Speed in Two Phase Medium

4 Isentropic Flow
4.1 Stagnation State for Ideal Gas Model
4.1.1 General Relationship
4.1.2 Relationships for Small Mach Number
4.2 Isentropic Converging-Diverging Flow in Cross Section
4.2.1 The Properties in the Adiabatic Nozzle
4.2.2 Isentropic Flow Examples

CONTENTS v
4.2.3 Mass Flow Rate (Number)
4.3 Isentropic Tables
4.3.1 Isentropic Isothermal Flow Nozzle
4.3.2 General Relationship
4.4 The Impulse Function
4.4.1 Impulse in Isentropic Adiabatic Nozzle
4.4.2 The Impulse Function in Isothermal Nozzle
4.5 Isothermal Table
4.6 The effects of Real Gases

5 Normal Shock
5.1 Solution of the Governing Equations
5.1.1 Informal Model
5.1.2 Formal Model
5.1.3 Prandtl’s Condition
5.2 Operating Equations and Analysis
5.2.1 The Limitations of the Shock Wave
5.2.2 Small Perturbation Solution
5.2.3 Shock Thickness
5.2.4 Shock or Wave Drag
5.3 The Moving Shocks
5.3.1 Shock or Wave Drag Result from a Moving Shock
5.3.2 Shock Result from a Sudden and Complete Stop
5.3.3 Moving Shock into Stationary Medium (Suddenly Open Valve)
5.3.4 Partially Open Valve
5.3.5 Partially Closed Valve
5.3.6 Worked–out Examples for Shock Dynamics
5.4 Shock Tube
5.5 Shock with Real Gases
5.6 Shock in Wet Steam
5.7 Normal Shock in Ducts
5.8 More Examples for Moving Shocks
5.9 Tables of Normal Shocks, k = 1.4 Ideal Gas

6 Normal Shock in Variable Duct Areas
6.1 Nozzle efficiency
6.2 Diffuser Efficiency
7 Nozzle Flow With External Forces
7.1 Isentropic Nozzle (Q = 0)
7.2 Isothermal Nozzle (T = constant)

vi CONTENTS
8 Isothermal Flow 143
8.1 The Control Volume Analysis/Governing equations
8.2 Dimensionless Representation
8.3 The Entrance Limitation of Supersonic Branch
8.4 Comparison with Incompressible Flow
8.5 Supersonic Branch
8.6 Figures and Tables
8.7 Isothermal Flow Examples
8.8 Unchoked situations in Fanno Flow

9 Fanno Flow
9.1 Introduction
9.2 Model
9.3 Non–Dimensionalization of the Equations
9.4 The Mechanics and Why the Flow is Choked?
9.5 The working equations
9.6 Examples of Fanno Flow
9.7 Supersonic Branch
9.8 Maximum Length for the Supersonic Flow
9.9 Working Conditions
9.9.1 Variations of The Tube Length ( 4fLD ) Effects
9.9.2 The Pressure Ratio, P2,P1 , effects
9.9.3 Entrance Mach number, M1, effects
9.10 Practical Examples for Subsonic Flow
9.10.1 Subsonic Fanno Flow for Given 4fLD and Pressure Ratio
9.10.2 Subsonic Fanno Flow for a Given M1 and Pressure Ratio
9.11 The Approximation of the Fanno Flow by Isothermal Flow
9.12 More Examples of Fanno Flow
9.13 The Table for Fanno Flow

10 Rayleigh Flow
10.1 Introduction
10.2 Governing Equation
10.3 Rayleigh Flow Tables
10.4 Examples For Rayleigh Flow

11 Evacuating SemiRigid Chambers
11.1 Governing Equations and Assumptions
11.2 General Model and Non-dimensioned
11.2.1 Isentropic Process
11.2.2 Isothermal Process in The Chamber
11.2.3 A Note on the Entrance Mach number
11.3 Rigid Tank with Nozzle
11.3.1 Adiabatic Isentropic Nozzle Attached
11.3.2 Isothermal Nozzle Attached
CONTENTS vii
11.4 Rapid evacuating of a rigid tank
11.4.1 With Fanno Flow
11.4.2 Filling Process
11.4.3 The Isothermal Process
11.4.4 Simple Semi Rigid Chamber
11.4.5 The “Simple” General Case
11.5 Advance Topics

12 Evacuating under External Volume Control
12.1 General Model
12.1.1 Rapid Process
12.1.2 Examples
12.1.3 Direct Connection
12.2 Summary

13 Oblique Shock
13.1 Preface to Oblique Shock
13.2 Introduction
13.2.1 Introduction to Oblique Shock
13.2.2 Introduction to Prandtl–Meyer Function
13.2.3 Introduction to Zero Inclination
13.3 Oblique Shock
13.4 Solution of Mach Angle
13.4.1 Upstream Mach Number, M1, and Deflection Angle,
13.4.2 When No Oblique Shock Exist or When D > 0
13.4.3 Upstream Mach Number, M1, and Shock Angle, µ
13.4.4 Given Two Angles, ± and µ
13.4.5 Flow in a Semi–2D Shape
13.4.6 Small ± “Weak Oblique shock”
13.4.7 Close and Far Views of the Oblique Shock
13.4.8 Maximum Value of Oblique shock
13.5 Detached Shock
13.5.1 Issues Related to the Maximum Deflection Angle
13.5.2 Oblique Shock Examples .
13.5.3 Application of Oblique Shock
13.5.4 Optimization of Suction Section Design
13.5.5 Retouch of Shock or Wave Drag
13.6 Summary
13.7 Appendix: Oblique Shock Stability Analysis

14 Prandtl-Meyer Function
14.1 Introduction
14.2 Geometrical Explanation
14.2.1 Alternative Approach to Governing Equations
14.2.2 Comparison And Limitations between the Two Approaches
viii CONTENTS
14.3 The Maximum Turning Angle
14.4 The Working Equations for the Prandtl-Meyer Function
14.5 d’Alembert’s Paradox
14.6 Flat Body with an Angle of Attack
14.7 Examples For Prandtl–Meyer Function
14.8 Combination of the Oblique Shock and Isentropic Expansion

A Computer Program
A.1 About the Program
A.2 Usage
A.3 Program listings

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