Basics of Fluid Mechanics, 2014a

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12.2. OBLIQUE SHOCK 507 Example 12.4: A cone shown in Figure (12.11) is exposed to supersonic flow and create an oblique shock. Is the shock shown in the photo weak or strong shock? Explain. Using the geometry provided in the photo, predict at which Mach number was the photo taken based on the assumption that the cone is a wedge. Solution The measurement shows that cone angle is 14.43 ◦ and the shock angle is 30.099 ◦ . With given two angles the solution can be obtained by utilizing equation (12.59) or the Potto-GDC. M 1 M ys M yw θ s θ w δ P 0y P 0x 3.2318 0.56543 2.4522 71.0143 30.0990 14.4300 0.88737 Because the flow is around the cone it must be a weak shock. Even if the cone was a wedge, the shock would be weak because the maximum (transition to a strong shock) occurs at about 60 ◦ . Note that the Mach number is larger than the one predicted by the wedge. End Solution Oblique Shock 3 90 k = 1 4 2.5 80 70 2 60 M y θ δ 1.5 50 40 1 30 0.5 20 10 0 0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 M x Thu Jun 30 15:14:53 2005 Fig. -12.12. Maximum values of the properties in an oblique shock.
508 CHAPTER 12. COMPRESSIBLE FLOW 2–DIMENSIONAL 12.2.3 Application of Oblique Shock α 1 α 2 α 3 normal shock oblique shock Fig. -12.13. Two variations of inlet suction for supersonic flow. One of the practical applications of the oblique shock is the design of an inlet suction for a supersonic flow. It is suggested that a series of weak shocks should replace one normal shock to increase the efficiency (see Figure (12.13)) 17 . Clearly, with a proper design, the flow can be brought to a subsonic flow just below M =1. In such a case, there is less entropy production (less pressure loss). To illustrate the design significance of the oblique shock, the following example is provided. Example 12.5: The Section described in Figure 12.13 and 12.14 air is flowing into a suction section at M = 2.0, P = 1.0[bar], and T =17 ◦ C. Compare the different conditions in the two different configurations. Assume that only a weak shock occurs. α 1 7 ◦ ÓÐÕÙ ×Ó× 1 2 3 4 « ¾ 7 ◦ neglect the detached distance Normal shock Solution Fig. -12.14. Schematic for Example (12.5). The first configuration is of a normal shock for which the results 18 are M x M y T y T x ρ y ρ x P y P x P 0y P 0x 2.0000 0.57735 1.6875 2.6667 4.5000 0.72087 17 In fact, there is general proof that regardless to the equation of state (any kind of gas), the entropy is to be minimized through a series of oblique shocks rather than through a single normal shock. For details see Henderson and Menikoff “Triple Shock Entropy Theorem,” Journal of Fluid Mechanics 366, (1998) pp. 179–210. 18 The results in this example are obtained using the graphical interface of POTTO–GDC thus, no input explanation is given. In the past the input file was given but the graphical interface it is no longer needed.
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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
Page 520 and 521: 11.7. FANNO FLOW 457 11.7.7.1 Choki
Page 522 and 523: 11.7. FANNO FLOW 459 back pressure
Page 524 and 525: 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.
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Page 548 and 549: CHAPTER 12 Compressible Flow 2-Dime
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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 566 and 567: 12.2. OBLIQUE SHOCK 503 Fig. -12.9.
Page 568 and 569: 12.2. OBLIQUE SHOCK 505 Table -12.1
Page 572 and 573: 12.2. OBLIQUE SHOCK 509 In the obli
Page 574 and 575: 12.2. OBLIQUE SHOCK 511 M x M ys M
Page 576 and 577: 12.2. OBLIQUE SHOCK 513 M x M y T y
Page 578 and 579: 12.2. OBLIQUE SHOCK 515 M x M yw θ
Page 580 and 581: 12.2. OBLIQUE SHOCK 517 The pressur
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Page 584 and 585: 12.3. PRANDTL-MEYER FUNCTION 521 as
Page 586 and 587: 12.3. PRANDTL-MEYER FUNCTION 523 Th
Page 588 and 589: 12.3. PRANDTL-MEYER FUNCTION 525 No
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Page 598 and 599: CHAPTER 13 Multi-Phase Flow 13.1 In
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Page 610 and 611: 13.7. HOMOGENEOUS MODELS 547 13.7 H
Page 612 and 613: 13.7. HOMOGENEOUS MODELS 549 There
Page 614 and 615: 13.8. SOLID-LIQUID FLOW 551 where
Page 616 and 617: 13.8. SOLID-LIQUID FLOW 553 So far
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13.9. COUNTER-CURRENT FLOW 557 The
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13.9. COUNTER-CURRENT FLOW 561 For
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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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