Basics of Fluid Mechanics, 2014a

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12.2. OBLIQUE SHOCK 513 M x M y T y T x ρ y ρ x P y P x P 0y P 0x 1.6100 0.66545 1.3949 2.0485 2.8575 0.89145 The temperature ratio combined upstream temperature yield T 2 =1.3949 × 300 ∼ 418.5K and the same for the pressure P 2 =2.8575 × 3=8.57[bar] And the velocity √ √ U n2 = M yw kRT =2.128 1.4 × 287 × 418.5 = 872.6[m/sec] End Solution Example 12.9: For Mach number 2.5 and wedge with a total angle of 22 ◦ , calculate the ratio of the stagnation pressure. Solution Utilizing GDC for Mach number 2.5 and the angle of 11 ◦ results in M x M ys M yw θ s θ w δ P 0y P 0x 2.5000 0.53431 2.0443 85.0995 32.8124 11.0000 0.96873 End Solution Example 12.10: What is the maximum pressure ratio that can be obtained on wedge when the gas is flowing in 2.5 Mach without any close boundaries? Would it make any difference if the wedge was flowing into the air? If so, what is the difference? Solution It has to be recognized that without any other boundary condition, the shock is weak shock. For a weak shock the maximum pressure ratio is obtained at the deflection point because it is closest to a normal shock. To obtain the maximum point for 2.5 Mach number, either use the Maximum Deflection Mach number’s equation or the Potto–GDC M x M ymax θ max δ P y P x T y T x P 0y P 0x 2.5000 0.94021 64.7822 29.7974 4.3573 2.6854 0.60027
514 CHAPTER 12. COMPRESSIBLE FLOW 2–DIMENSIONAL In these calculations, Maximum Deflection Mach’s equation was used to calculate the normal component of the upstream, then the Mach angle was calculated using the geometrical relationship of θ = sin −1 M 1n /M 1 . With these two quantities, utilizing equation (12.12) the deflection angle, δ, is obtained. End Solution Example 12.11: Consider the schematic shown in the following figure. Assume that the upstream Mach number is 4 and the deflection angle is δ = 15 ◦ . Compute the pressure ratio and the temperature ratio after the second shock (sometimes referred to as the reflective shock while the first shock is called the incidental shock). stream line M 1 = 4 3 2 1 θ δ Fig. -12.17. Schematic for Example (12.11). Solution This kind of problem is essentially two wedges placed in a certain geometry. It is clear that the flow must be parallel to the wall. For the first shock, the upstream Mach number is known together with deflection angle. Utilizing the table or the Potto–GDC, the following can be obtained: M x M ys M yw θ s θ w δ P 0y P 0x 4.0000 0.46152 2.9290 85.5851 27.0629 15.0000 0.80382 And the additional information by using minimal information ratio button in Potto–GDC is M x M yw θ w δ P y P x T y T x P 0y P 0x 4.0000 2.9290 27.0629 15.0000 1.7985 1.7344 0.80382 With a Mach number of M =2.929, the second deflection angle is also 15 ◦ . With these values the following can be obtained: M x M ys M yw θ s θ w δ P 0y P 0x 2.9290 0.51367 2.2028 84.2808 32.7822 15.0000 0.90041 and the additional information is
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Basics of Fluid Mechanics Genick Ba
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iii ‘We are like dwarfs sitting o
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CONTENTS Nomenclature xxiii GNU Fre
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CONTENTS vii 4 Fluids Statics 69 4.
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CONTENTS ix 9.2.1 Construction of t
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CONTENTS xi 12.5 The Working Equati
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LIST OF FIGURES 1.1 Diagram to expl
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LIST OF FIGURES xv 4.32 Polynomial
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LIST OF FIGURES xvii (a) Streamline
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LIST OF FIGURES xix 12.22The angles
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LIST OF TABLES 1 Books Under Potto
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NOMENCLATURE ¯R Universal gas cons
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LIST OF TABLES xxv w Work per unit
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The Book Change Log Version 0.3.4.0
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LIST OF TABLES xxix ˆ Add discussi
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LIST OF TABLES xxxi ˆ Add the open
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Notice of Copyright For This Docume
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GNU FREE DOCUMENTATION LICENSE xxxv
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GNU FREE DOCUMENTATION LICENSE xxxv
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GNU FREE DOCUMENTATION LICENSE 7. A
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CONTRIBUTOR LIST How to contribute
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CREDITS Typo corrections and other
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About This Author Genick Bar-Meir i
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Prologue For The POTTO Project This
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CREDITS xlix process. Someone has t
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CREDITS li have a new version every
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Prologue For This Book Version 0.3.
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VERSION 0.1 APRIL 22, 2008 lv and t
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How This Book Was Written This book
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Preface "In the beginning, the POTT
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To Do List and Road Map This book i
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CHAPTER 1 Introduction to Fluid Mec
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1.2. BRIEF HISTORY 3 lay is one of
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1.3. KINDS OF FLUIDS 5 results. The
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1.4. SHEAR STRESS 7 For cases where
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1.5. VISCOSITYVISCOSITY 9 The veloc
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1.5. VISCOSITYVISCOSITY 11 always p
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2.5 2.0 1.5 1.0 0.5 0 100 200 300 4
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1.5. VISCOSITYVISCOSITY 15 Chemical
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1.5. VISCOSITYVISCOSITY 17 2 Reduce
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1.5. VISCOSITYVISCOSITY 19 In very
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1.6. FLUID PROPERTIES 21 The total
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1.6. FLUID PROPERTIES 23 straight.
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1.6. FLUID PROPERTIES 25 Table -1.5
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1.6. FLUID PROPERTIES 27 Two layers
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1.6. FLUID PROPERTIES 29 1.6.2.1 Bu
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1.7. SURFACE TENSION 31 terials. Th
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1.7. SURFACE TENSION 33 or Or after
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1.7. SURFACE TENSION 35 The maximum
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1.7. SURFACE TENSION 37 kipenii,”
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1.7. SURFACE TENSION 39 Equation (1
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1.7. SURFACE TENSION 41 Example 1.1
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1.7. SURFACE TENSION 43 Table -1.7.
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CHAPTER 2 Review of Thermodynamics
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2.1. BASIC DEFINITIONS 47 Since the
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2.1. BASIC DEFINITIONS 49 when the
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2.1. BASIC DEFINITIONS 51 Utilizing
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CHAPTER 3 Review of Mechanics This
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3.2. CENTER OF MASS 55 3.2 Center o
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3.3. MOMENT OF INERTIA 57 The momen
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3.3. MOMENT OF INERTIA 59 Equation
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3.3. MOMENT OF INERTIA 61 Example 3
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3.3. MOMENT OF INERTIA 63 or x (1 a
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3.5. ANGULAR MOMENTUM AND TORQUE 65
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3.5. ANGULAR MOMENTUM AND TORQUE 67
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CHAPTER 4 Fluids Statics 4.1 Introd
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4.3. PRESSURE AND DENSITY IN A GRAV
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4.4. FLUID IN A ACCELERATED SYSTEM
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4.5. FLUID FORCES ON SURFACES 101 T
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4.5. FLUID FORCES ON SURFACES 103 x
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4.5. FLUID FORCES ON SURFACES 107 I
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4.5. FLUID FORCES ON SURFACES 111 C
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4.5. FLUID FORCES ON SURFACES 113 A
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4.5. FLUID FORCES ON SURFACES 115 F
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4.6. BUOYANCY AND STABILITY 117 In
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4.6. BUOYANCY AND STABILITY 119 ρ
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4.6. BUOYANCY AND STABILITY 121 ( )
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4.6. BUOYANCY AND STABILITY 123 The
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4.6. BUOYANCY AND STABILITY 125 on
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4.6. BUOYANCY AND STABILITY 127 A w
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4.6. BUOYANCY AND STABILITY 129 It
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4.6. BUOYANCY AND STABILITY 131 Thu
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4.6. BUOYANCY AND STABILITY 133 The
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4.6. BUOYANCY AND STABILITY 135 A n
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4.6. BUOYANCY AND STABILITY 137 of
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4.7. RAYLEIGH-TAYLOR INSTABILITY 13
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4.7. RAYLEIGH-TAYLOR INSTABILITY 14
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4.8. QUALITATIVE QUESTIONS 143 can
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Part I Integral Analysis 145
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148 CHAPTER 5. MASS CONSERVATION di
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150 CHAPTER 5. MASS CONSERVATION It
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152 CHAPTER 5. MASS CONSERVATION 5.
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154 CHAPTER 5. MASS CONSERVATION Th
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156 CHAPTER 5. MASS CONSERVATION Wh
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158 CHAPTER 5. MASS CONSERVATION Th
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162 CHAPTER 5. MASS CONSERVATION ri
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166 CHAPTER 5. MASS CONSERVATION En
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168 CHAPTER 5. MASS CONSERVATION Ex
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170 CHAPTER 5. MASS CONSERVATION So
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172 CHAPTER 5. MASS CONSERVATION In
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174 CHAPTER 6. MOMENTUM CONSERVATIO
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198 CHAPTER 7. ENERGY CONSERVATION
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CHAPTER 8 Differential Analysis 8.1
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8.2. MASS CONSERVATION 229 The firs
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8.2. MASS CONSERVATION 231 Combinin
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8.2. MASS CONSERVATION 233 Equation
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8.2. MASS CONSERVATION 235 The dens
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8.2. MASS CONSERVATION 237 Example
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8.3. CONSERVATION OF GENERAL QUANTI
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8.4. MOMENTUM CONSERVATION 241 8.4
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8.4. MOMENTUM CONSERVATION 243 The
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8.5. DERIVATIONS OF THE MOMENTUM EQ
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8.6. BOUNDARY CONDITIONS AND DRIVIN
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8.6. BOUNDARY CONDITIONS AND DRIVIN
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8.7. EXAMPLES FOR DIFFERENTIAL EQUA
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CHAPTER 9 Dimensional Analysis This
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9.1. INTRODUCTORY REMARKS 275 the r
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9.1. INTRODUCTORY REMARKS 277 End S
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9.1. INTRODUCTORY REMARKS 279 Deriv
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9.2. BUCKINGHAM-π-THEOREM 281 para
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9.2. BUCKINGHAM-π-THEOREM 283 magn
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9.2. BUCKINGHAM-π-THEOREM 285 Tabl
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9.2. BUCKINGHAM-π-THEOREM 287 End
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9.2. BUCKINGHAM-π-THEOREM 289 Equa
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9.2. BUCKINGHAM-π-THEOREM 291 the
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9.2. BUCKINGHAM-π-THEOREM 293 than
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9.2. BUCKINGHAM-π-THEOREM 295 Unde
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9.2. BUCKINGHAM-π-THEOREM 297 Solu
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9.3. NUSSELT’S TECHNIQUE 299 The
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9.3. NUSSELT’S TECHNIQUE 301 boun
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9.3. NUSSELT’S TECHNIQUE 303 Noti
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9.3. NUSSELT’S TECHNIQUE 305 In t
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9.3. NUSSELT’S TECHNIQUE 307 Wher
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9.4. SUMMARY OF DIMENSIONLESS NUMBE
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9.5. SUMMARY 319 It can be noticed
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9.6. APPENDIX SUMMARY OF DIMENSIONL
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BIBLIOGRAPHY [1] Buckingham, E. On
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CHAPTER 10 Inviscid Flow or Potenti
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10.1. INTRODUCTION 327 where in thi
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10.1. INTRODUCTION 329 With the ide
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10.1. INTRODUCTION 331 The partial
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10.2. POTENTIAL FLOW FUNCTION 333 T
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10.2. POTENTIAL FLOW FUNCTION 335 t
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10.2. POTENTIAL FLOW FUNCTION 341 9
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10.3. POTENTIAL FLOW FUNCTIONS INVE
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10.4. CONFORMING MAPPING 369 flow d
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10.4. CONFORMING MAPPING 371 The un
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10.5. UNSTEADY STATE BERNOULLI IN A
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10.6. QUESTIONS 375 Table -10.2. Di
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CHAPTER 11 Compressible Flow One Di
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11.3. SPEED OF SOUND 379 to be cont
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11.3. SPEED OF SOUND 381 Solution T
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11.3. SPEED OF SOUND 383 consider a
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11.4. ISENTROPIC FLOW 385 Now, subs
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11.4. ISENTROPIC FLOW 387 Static Pr
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11.4. ISENTROPIC FLOW 389 that the
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11.4. ISENTROPIC FLOW 391 Using the
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11.4. ISENTROPIC FLOW 393 Solution
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11.4. ISENTROPIC FLOW 395 Expressin
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11.4. ISENTROPIC FLOW 397 The tempe
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11.4. ISENTROPIC FLOW 399 Table -11
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11.4. ISENTROPIC FLOW 405 Example 1
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11.5. NORMAL SHOCK 407 In a shock w
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11.5. NORMAL SHOCK 409 Energy equat
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11.5. NORMAL SHOCK 411 The density
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11.5. NORMAL SHOCK 413 or in a dime
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11.5. NORMAL SHOCK 415 which differ
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11.5. NORMAL SHOCK 417 is different
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11.5. NORMAL SHOCK 419 Table -11.3.
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11.6. ISOTHERMAL FLOW 421 Table -11
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11.6. ISOTHERMAL FLOW 423 Rearrangi
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11.6. ISOTHERMAL FLOW 425 dT 0 T 0
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11.6. ISOTHERMAL FLOW 427 10 2 0 1
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11.6. ISOTHERMAL FLOW 429 The fact
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11.6. ISOTHERMAL FLOW 431 Calculati
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11.6. ISOTHERMAL FLOW 433 maximum M
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11.6. ISOTHERMAL FLOW 435 From the
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11.7. FANNO FLOW 437 The energy con
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11.7. FANNO FLOW 439 Differentiatin
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11.7. FANNO FLOW 441 11.7.3 The Mec
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11.7. FANNO FLOW 443 pressure on th
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11.7. FANNO FLOW 445 10 2 0 1 2 3 4
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11.7. FANNO FLOW 447 End Solution A
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11.7. FANNO FLOW 449 M x M y T y T
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11.7. FANNO FLOW 451 11.7.5.1 Maxim
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11.7. FANNO FLOW 453 At the startin
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11.7. FANNO FLOW 455 Fanno Flow 5 4
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11.7. FANNO FLOW 457 11.7.7.1 Choki
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11.7. FANNO FLOW 459 back pressure
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11.7. FANNO FLOW 461 The solution i
Page 526 and 527: 11.7. FANNO FLOW 463 11.7.8 The Pra
Page 528 and 529: 11.7. FANNO FLOW 465 0.9 0.8 0.7 0.
Page 530 and 531: 11.7. FANNO FLOW 467 3.0 2.5 2.0 Co
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Page 534 and 535: 11.9. RAYLEIGH FLOW 471 Table -11.6
Page 536 and 537: 11.10. INTRODUCTION 473 or in simpl
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Page 540 and 541: 11.10. INTRODUCTION 477 Table -11.7
Page 542 and 543: 11.10. INTRODUCTION 479 Example 11.
Page 544 and 545: 11.10. INTRODUCTION 481 Example 11.
Page 546 and 547: 11.10. INTRODUCTION 483 what the ma
Page 548 and 549: CHAPTER 12 Compressible Flow 2-Dime
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Page 552 and 553: 12.2. OBLIQUE SHOCK 489 The relatio
Page 554 and 555: 12.2. OBLIQUE SHOCK 491 Where S = 3
Page 556 and 557: 12.2. OBLIQUE SHOCK 493 by evaluati
Page 558 and 559: 12.2. OBLIQUE SHOCK 495 Figure (12.
Page 560 and 561: 12.2. OBLIQUE SHOCK 497 be made. Th
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Page 564 and 565: 12.2. OBLIQUE SHOCK 501 or explicit
Page 566 and 567: 12.2. OBLIQUE SHOCK 503 Fig. -12.9.
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Page 570 and 571: 12.2. OBLIQUE SHOCK 507 Example 12.
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Page 584 and 585: 12.3. PRANDTL-MEYER FUNCTION 521 as
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Page 610 and 611: 13.7. HOMOGENEOUS MODELS 547 13.7 H
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Page 614 and 615: 13.8. SOLID-LIQUID FLOW 551 where
Page 616 and 617: 13.8. SOLID-LIQUID FLOW 553 So far
Page 618 and 619: 13.9. COUNTER-CURRENT FLOW 555 The
Page 624 and 625: 13.9. COUNTER-CURRENT FLOW 561 For
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13.9. COUNTER-CURRENT FLOW 563 The
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13.10. MULTI-PHASE CONCLUSION 565 1
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APPENDIX A The Mathematics Backgrou
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A.1. VECTORS 569 where θ is the an
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A.1. VECTORS 571 Divergence The sam
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A.1. VECTORS 573 R · S =(xî + y 2
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A.1. VECTORS 575 The curl is writte
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A.1. VECTORS 577 The divergence of
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A.2. ORDINARY DIFFERENTIAL EQUATION
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A.3. PARTIAL DIFFERENTIAL EQUATIONS
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A.4. TRIGONOMETRY 595 A.4 Trigonome
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SUBJECTS INDEX 597 Subjects Index S
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SUBJECTS INDEX 599 Ideal gas, 91 Re
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SUBJECTS INDEX 601 Slip condition r
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AUTHORS INDEX 603 Authors Index B B
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