Basics of Fluid Mechanics, 2014a

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11.5. NORMAL SHOCK 417 is different in the present case (as oppose to the stationary shock), one side has increase the momentum of the control volume. This increase momentum in the control volume causes the shock drag. In way, it can be view as continuous acceleration of the gas around the body from zero. Note this drag is only applicable to a moving shock (unsteady shock). The moving shock is either results from a body that moves in gas or from a sudden imposed boundary like close or open valve 9 . In the first case, the forces or energies flow from body to gas and therefor there is a need for large force to accelerate the gas over extremely short distance (shock thickness). In the second case, the gas contains the energy (as high pressure, for example in the open valve case) and the energy potential is lost in the shock process (like shock drag). For some strange reasons, this topic has several misconceptions that even appear in many popular and good textbooks 10 . Consider the following example taken from such a book. Fig. -11.16. The diagram for the common explanation for shock or wave drag effect a shock. Please notice the strange notations (e.g. V and not U) and they result from a verbatim copy. Example 11.13: A book (see Figure 11.16) explains the shock drag is based on the following rational: The body is moving in a stationary frictionless fluid under one–dimensional flow. The left plane is moving with body at the same speed. The second plane is located “downstream from the body where the gas has expanded isotropically (after the shock wave) to the upstream static pressure”. the bottom and upper stream line close the control volume. Since the pressure is the same on the both planes there is no unbalanced pressure forces. However, there is a change in the momentum in the flow direction because U 1 >U 2 . The force is acting on the body. There several mistakes in this explanation including 9 According to my son, the difference between these two cases is the direction of the information. Both case there essentially bodies, however, in one the information flows from inside the field to the boundary while the other case it is the opposite. 10 Similar situation exist in the surface tension area.
418 CHAPTER 11. COMPRESSIBLE FLOW ONE DIMENSIONAL the drawing. Explain what is wrong in this description (do not describe the error results from oblique shock). Solution Neglecting the mistake around the contact of the stream lines with the oblique shock(see for retouch in the oblique chapter), the control volume suggested is stretched with time. However, the common explanation fall to notice that when the isentropic explanation occurs the width of the area change. Thus, the simple explanation in a change only in momentum (velocity) is not appropriate. Moreover, in an expanding control volume this simple explanation is not appropriate. Notice that the relative velocity at the front of the control volume U 1 is actually zero. Hence, the claim of U 1 >U 2 is actually the opposite, U 1
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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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Page 430 and 431: 10.3. POTENTIAL FLOW FUNCTIONS INVE
Page 432 and 433: 10.4. CONFORMING MAPPING 369 flow d
Page 434 and 435: 10.4. CONFORMING MAPPING 371 The un
Page 436 and 437: 10.5. UNSTEADY STATE BERNOULLI IN A
Page 438 and 439: 10.6. QUESTIONS 375 Table -10.2. Di
Page 440 and 441: CHAPTER 11 Compressible Flow One Di
Page 442 and 443: 11.3. SPEED OF SOUND 379 to be cont
Page 444 and 445: 11.3. SPEED OF SOUND 381 Solution T
Page 446 and 447: 11.3. SPEED OF SOUND 383 consider a
Page 448 and 449: 11.4. ISENTROPIC FLOW 385 Now, subs
Page 450 and 451: 11.4. ISENTROPIC FLOW 387 Static Pr
Page 452 and 453: 11.4. ISENTROPIC FLOW 389 that the
Page 454 and 455: 11.4. ISENTROPIC FLOW 391 Using the
Page 456 and 457: 11.4. ISENTROPIC FLOW 393 Solution
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Page 460 and 461: 11.4. ISENTROPIC FLOW 397 The tempe
Page 462 and 463: 11.4. ISENTROPIC FLOW 399 Table -11
Page 468 and 469: 11.4. ISENTROPIC FLOW 405 Example 1
Page 470 and 471: 11.5. NORMAL SHOCK 407 In a shock w
Page 472 and 473: 11.5. NORMAL SHOCK 409 Energy equat
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Page 476 and 477: 11.5. NORMAL SHOCK 413 or in a dime
Page 478 and 479: 11.5. NORMAL SHOCK 415 which differ
Page 482 and 483: 11.5. NORMAL SHOCK 419 Table -11.3.
Page 484 and 485: 11.6. ISOTHERMAL FLOW 421 Table -11
Page 486 and 487: 11.6. ISOTHERMAL FLOW 423 Rearrangi
Page 488 and 489: 11.6. ISOTHERMAL FLOW 425 dT 0 T 0
Page 490 and 491: 11.6. ISOTHERMAL FLOW 427 10 2 0 1
Page 492 and 493: 11.6. ISOTHERMAL FLOW 429 The fact
Page 494 and 495: 11.6. ISOTHERMAL FLOW 431 Calculati
Page 496 and 497: 11.6. ISOTHERMAL FLOW 433 maximum M
Page 498 and 499: 11.6. ISOTHERMAL FLOW 435 From the
Page 500 and 501: 11.7. FANNO FLOW 437 The energy con
Page 502 and 503: 11.7. FANNO FLOW 439 Differentiatin
Page 504 and 505: 11.7. FANNO FLOW 441 11.7.3 The Mec
Page 506 and 507: 11.7. FANNO FLOW 443 pressure on th
Page 508 and 509: 11.7. FANNO FLOW 445 10 2 0 1 2 3 4
Page 510 and 511: 11.7. FANNO FLOW 447 End Solution A
Page 512 and 513: 11.7. FANNO FLOW 449 M x M y T y T
Page 514 and 515: 11.7. FANNO FLOW 451 11.7.5.1 Maxim
Page 516 and 517: 11.7. FANNO FLOW 453 At the startin
Page 518 and 519: 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.
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11.7. FANNO FLOW 467 3.0 2.5 2.0 Co
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11.8. THE TABLE FOR FANNO FLOW 469
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11.9. RAYLEIGH FLOW 471 Table -11.6
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11.10. INTRODUCTION 473 or in simpl
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11.10. INTRODUCTION 475 or explicit
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11.10. INTRODUCTION 477 Table -11.7
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11.10. INTRODUCTION 479 Example 11.
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11.10. INTRODUCTION 481 Example 11.
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11.10. INTRODUCTION 483 what the ma
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CHAPTER 12 Compressible Flow 2-Dime
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12.2. OBLIQUE SHOCK 487 12.2 Obliqu
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12.2. OBLIQUE SHOCK 489 The relatio
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12.2. OBLIQUE SHOCK 491 Where S = 3
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12.2. OBLIQUE SHOCK 493 by evaluati
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12.2. OBLIQUE SHOCK 495 Figure (12.
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12.2. OBLIQUE SHOCK 497 be made. Th
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12.2. OBLIQUE SHOCK 499 demonstrate
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12.2. OBLIQUE SHOCK 501 or explicit
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12.2. OBLIQUE SHOCK 503 Fig. -12.9.
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12.2. OBLIQUE SHOCK 505 Table -12.1
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12.2. OBLIQUE SHOCK 507 Example 12.
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12.2. OBLIQUE SHOCK 509 In the obli
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12.2. OBLIQUE SHOCK 511 M x M ys M
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12.2. OBLIQUE SHOCK 513 M x M y T y
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12.2. OBLIQUE SHOCK 515 M x M yw θ
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12.2. OBLIQUE SHOCK 517 The pressur
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12.2. OBLIQUE SHOCK 519 From these
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12.3. PRANDTL-MEYER FUNCTION 521 as
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12.3. PRANDTL-MEYER FUNCTION 523 Th
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12.3. PRANDTL-MEYER FUNCTION 525 No
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1 2 12.7. FLAT BODY WITH AN ANGLE O
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12.8. EXAMPLES FOR PRANDTL-MEYER FU
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12.9. COMBINATION OF THE OBLIQUE SH
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12.9. COMBINATION OF THE OBLIQUE SH
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CHAPTER 13 Multi-Phase Flow 13.1 In
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13.4. KIND OF MULTI-PHASE FLOW 537
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13.5. CLASSIFICATION OF LIQUID-LIQU
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13.5. CLASSIFICATION OF LIQUID-LIQU
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13.6. MULTI-PHASE FLOW VARIABLES DE
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13.6. MULTI-PHASE FLOW VARIABLES DE
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13.7. HOMOGENEOUS MODELS 547 13.7 H
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13.7. HOMOGENEOUS MODELS 549 There
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13.8. SOLID-LIQUID FLOW 551 where
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13.8. SOLID-LIQUID FLOW 553 So far
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13.9. COUNTER-CURRENT FLOW 555 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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