Comments on

VORTICITY AND VORTEX DYNAMICS By:

Jie-Zhi Wu, Hui-Yang Ma, and Ming-De Zhou Springer Verlag, 2006

By: Mohsen Hassan vand

Wednesday, September 24, 2008

1

Notations Pg x : Page x P x : Paragraph number x L x : Line number x LB x : Line number x from bottom : is suggested to be changed to

1. From an aesthetic point of view, the title of book in Pg III

Because in this way:

a) we keep the same format of the book title on the hardcover, b) the combined term vortex dynamics which is also the main topic of the book

locates on a single line.

2. Pg V, item 3, L 5: the word late in Late Sir is written with capital letter L, but in Pg 2, P3, late Prof. Shi-Jia Lu is written with small letter l. Therefore, L l or l L.

3. Pg V(Preface), item 2

the dissipation of the smallest structures into heat. the dissipation of the smallest structures turbulent kinetic energy into heat.

4. Pg VI, P 1 , L1: the material has been the materials have been

5. Pg VI, P 2 ,

L 3: Xuesong Wu Xue-Song Wu L 9: I am in doubt here whether the Susan is an English name or a Chinese name

(this name exists in both English and Chinese). If it is a Chinese name here, it should be written as Su-San.

2

L 9: In standard Chinese, there isnt any character or word spelling Au as in Au-Kui.

Pg VI, P3, L3: Jain-Ming Wu Jian-Ming Wu

6. Pg 1, P2, L8, From evolution, instability, and decay of vorticity and vortices we can deduce these two concepts:

instability of vortices and

instability of vorticity instability of vortices is meaningful, but what is the meaning of instability of vorticity?

7. Pg 2, last P, last L: dissipate into heat decay due to dissipation

8. Pg 3, Sect. 1.2, L6: In vortex motions. , . is used but in many cases in the text[e.g.,

L15, in observed. (Lighthill 1956).], . is used. It is suggested to use a uniform format in the whole text and all . . or . . Also: , , or , ,

Sect. 1.2, L 13: hydraulic engineers hydraulics engineers . Compare (1932, first edition: 1879) in L17 with

(1858, English translation 1867) in L 5. Therefore: (1858, English translation 1867) (1858, English translation: 1867) or

(1932, first edition: 1879) (1932, first edition 1879) . footnote 3: 1995 (1995)

9. Pg 4,

P3, L3-4: techniques technologies at the bottom: Big whirls have Big whorls have

10. Pg6,

L1: biofluiddynamics bio-fluid dynamics Sect. 1.3, P 2, L 12: Lugt (1983) (Lugt 1983)

11. Pg 8, P 3, last sentence: The relation between .. are also

The relation between .. is also

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12. Pg 14, P 1, L 4: differenting (2.2) differentiating (2.2)

13. Pg 16,

P1, L1: Some developments of fluid mechanics Some developments in fluid mechanics P1, L5:

by adding to the Eulerian description condition (2.10) by adding condition (2.10) to the Eulerian description

last P, L5: Sect. 3.2.3 3.3.1 last P, L5: well-ordered ordered

Because, well-ordered is a terminological term in mathematics which is not the concern here. Also, in Chinese version of the book, the term is used which according to Chinese dictionary of ciba( PowerWord), it means ordered. LB1, Figure. 2.1 Figure 2.1

Note: There are many cases in the text where Figure. needs to be changed to Figure . Please search and change them.

14. Pg 17, Fig.2.1.: The notations (a), (b) and (c) are not necessary, because they are not being

referred in the text. Therefore they can be deleted.

15. Pg 19, (2.19b): ijk - ijk

16. Pg 20, L1 under (2.22): D measures the change D measures the time rate of change

17. Pg 21, Fig. 2.4. The deformation of small ..is shown The deformation of a small ..is shown.

18. Pg 24, L1 under (2.33): The font of S should be the same with that in first line of the Page

S last line: If V(t) is a material volume F

If V(t) is a material volume The point at the end of relation (2.35a) should be deleted.

19. Pg 25, P1, L 7 : dynamic and thermodynamic theorems

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dynamics and thermodynamics theorems

20. Pg 26, L1 under (2.42): The sentence First, let is suggested to be started at the beginning

of a new paragraph (such as the sentence Second, observe at the beginning of a new paragraph in six lines under (2.42)).

L2 under (2.42): Dividing both sides of (2.42) by l2 and let Dividing both sides of (2.42) by l2 and letting P1 above (2.43), linear form of the ith component of n

linear combination of the components of n A general comment: in the whole text, the notation for the material volume element,

dv, is different from that of control volume element, dV. But there is no such distinction between material surface element and control surface element (i.e., dS is used in both cases). For example, see (2.42) and (2.35b).

21. Pg 27, item 2 of Remarks, V is the deviator of T

V is proportional to the deviator of T In L1 above (2.51), point x0: point x0: last L, total moment acted to total moment acted on

22. Pg 28:

It is better to swap (2.52) with (2.53), because (2.53) is derived from (2.44) first and then (2.52) is derived by integrating (2.53).

L1 under (2.54): work is transferred work is transformed Regarding L1 under (2.56), heat flux through the boundary of a fluid body

(entering the fluid.): The application of the relation is not limited to fluids only, or only through boundaries. The diffusion heat transfer also occurs within the bulk of bodies wherever there is a temperature gradient. Therefore, it is suggested the sentence to be changed to:

heat flux and . Tq nn

=

is the heat flow rate across a unit area with unit outward

normal n. footnote 4: continuous mechanics continuum mechanics or mechanics of

continuous media [see Pg 742: Landau L.D.,. Mechanics of Continuous Media]. footnote 4, their form their forms

23. Pg 29:

5

In (2.59), the subscript of specific heat v is better to be written with non-italic v font because in L1under the relation it is written with non-italic font as cv

Note: In (2.63), p in cp is also non-italic. Relation (2.63) is better to be written in differential form, i.e., dh= cpdT = de+d(p/)

[see relation (2.59)] . last line: d-operation d-operator. Regarding de=cvdT and dh=cpdT:

generally, for pure materials,

( ) ( ) ( )v T v T

u u udu dT dv c dT dvT v v

= + = +

,

( ) ( ) ( )p T p T

h h hdh dT dp c dT dpT p p

= + = +

, and in most fluids, the second terms are not zero or negligible. Note: in (2.59) density is not constant. Regarding (2.54) and D:D= DijDji (,Dij=Dji) 0 , to satisfy 0 , it requires

0 and 0 , but in the second line of the formulas under (2.62),

? 24. Pg 30, L 1: But by (2.46) But by (2.47).

25. Pg 31, Fig.2.5: The relative vorticitytangent components..

26. Pg 32,

L2 before (2.73), Reynolds transfer theorem Reynolds transport theorem L2 before (2.71), acted to acted on

27. Pg 33 , Sect. 2.2.5,

P 1, L6: their gradient their gradients P 2, L 3: O(Re-1) O(Re)

28. Pg 34,

L2: infinitely large peaks of and/or infinitely large peaks of

and

. Because, the vorticity in a vortex sheet (or dilatation in a shock wave) is not necessarily infinite large.

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In L1 under (2.78), we have total-enthalpy equation but in L1before (2.65) there is no hyphen between total and enthalpy [i.e. total enthalpy equation].

P2: Pr is not always of O(1). For gases Pr 0.7 at ordinary conditions. In some specific liquids and conditions, Pr can be very big or small.[Ref: Unit Operations of Chemical Engineering, 7th edition, Pg 1109]. What can be said here is: the viscosity and thermal conductivity have the same physical root, and therefore whenever one of them is negligible, the other one is also negligible.[ Ref: Fluid Mechanics , by :Joseph H. Spurk , 1997, Spring-Verlag- Pg 83, the paragraph under (3.9): ignoring the friction stresses(or viscosity)implies that we should in general ignore the heat conduction(or conductivity).]

L1 under (2.81d): When .should be added. When .should be taken into account. Because the surface tension effect appears in the form of a pressure jump across the surface

1 1

x y

pR R

= + ,

which is implicitly included in [[pn]] in (2.81b). (2.82a), [[fg]] 1/2[[fg]]

29. Pg 35:

Regarding the Fig. 2.6. a,b: In the spherical fluid element shown (which is infinitesimal in size, but however it would certainly have a non-zero size), the upper hemisphere moves faster (on average ) than the lower hemisphere and consequently the element deforms and loses its spherical shape after elapse of a finite time. (In the figure, the element is shown at some finite time intervals). The vorticity is a property, and therefore we cannot trace a fluid particle in space and time and then expecting it really rotates. In Fig. b, the overall motion of the particle should be shown such that it does not rotate (for example, the vorticity makes the point on the top of element in the figure to rotate clockwise, but simultaneously because of the dilatation effect, it will be restored to its initial position and therefore moves on a straight line parallel to the vortex sheet), and different layers within the element slide over each other and element does not rotate.. Fig. 2.6. , Fig. 2.6. (a) Tangent discontinuity in strictly inviscid flow. (b) Vortex sheet

Fig. 2.6. Tangent discontinuity in (a) strictly inviscid flow, (b) vortex sheet

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30. Pg 38, relation (2.92): d3x is suggested to be corrected to dx3. I do know that in most books and texts, d3x is used (and generally accepted!) for denoting volume differential elements, but mathematically and dimensionally is a common mistake.

31. Pg 40, P3, with 2 = 0 ,that is, 2 = 0

32. Pg 41: In (2.102), and throughout the text, all log ln. Note: Except for log10. in (2.104b), dV dV

33. Pg 42, P1 under (2.105b)

The formulas not only showbut also, via (2.103), tells The formulas not only showbut also, via (2.103), tell

34. Pg45, in (2.119), RHS of second relation, u(k, t) ku(k, t) in (2.120), 2 is always equal to 1. It seems 2 k2

in (2.122), it seems h(k) h (k) < h(k) h (k)> in (2.123), both indices are subscript, but in (2.127), one of them is subscript and the

other one is superscript.

35. Pg46, a =M +N a =M +N. Also it seems better to change a to

36. Pg47, in (2.127), should be lower index of , like previous pages.

37. Pg 48, L2, divergenceless divergence-free.

Note: divergence-free has been used throughout the text.

in (2.132), n T^(x) T^(x) . n

38. Pg 50, L1 under (2.138): delete of dS .

39. Pg 52, L1 under (2.147), BB It is suggested to use a uniform format(with or without{}) for Dij in (2.146) and upper

part of Pg 22. Fig. 2.8.:

8

-It is suggested that the angles and 3 to be shown on the figure (because they are being referred in the text). - 3 is the angle between p3 and axes (=135 or = - 45). But on the figure, the angle between p1 and is shown. - on the figure, the boundary is denoted by B , but in the text B has been used for referring the boundary.

40. Pg 53, in (2.149), n T^ T^. n

41. Pg 54,

L1under (2.154):

2=.is called the enstrophy 1/2

2=1/2. is called enstrophy. The definitive article the should be omitted because this is for the first time that the word enstrophy is being defined and used in the book. After definition, the the could come before the word enstrophy. P1 above (2.156a):

Therefore, the work done by ts.always directly dissipated into heat. The work done by ts, that is, ws, is equal to the last term of (2.155). As it is mentioned in the same page, this term can be either positive or negative. If it is positive, according to (2.155), it directly dissipates into heat. But if it is negative, we come to an interesting result, i.e., some part of the dissipation heat resulted from the compressing and shearing processes (see, 2.155) is converted into work( not in the form of increase in kinetic energy, but by changing the surface strain rate tensor ). Note that, while the energy which is dissipated into heat is lost in the case of incompressible flow and can therefore no longer be transformed into mechanical

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energy, the energy transformed into heat is still usable in compressible flow. For adiabatic incompressible flow, De/Dt = , all the dissipated energy flows into the increase of the internal energy, which does not depend on the density, since density is a constant rather than a state variable in incompressible flow.

42. Pg 54, if the comment No. 22, first bullet, is acceptable, it should also be applied to (2.156a)

and (2.156b).

43. Pg 55, L1 under (2.158): imcompressible incompressible.

44. Pg 58, L 1,

s

dpcd

=

s

pc

= .

L4 under (2.170), vortex sound vorticity sound

45. Pg 59, L1 under (2.172b): This pair.contain This pair.contains. P 2, L 7: throughor through throughor .

46. Pg 60,

L 4 under (2.176): the total ..acting to but in lines 2-3 under (2.177) the total ..acting on. Therefore to on or on to.

LB 4, [0- , t] t = (0

-, 0

+).

0- is not a fixed or definite number as 0, so we should use (0

- or [0 as lower limits of

the intervals. The font used for in [0-, t] is different (bold font) from that of in its next line O(t).

at the bottom of the Page,

0

t

0

0

+

.

47. Pg 63, item 1 in Summary, L 4, boundaries is boundaries are

48. Pg 64, item 5 , L 8, transferred transformed or dissipated

49. Pg 65, L 5: No real flow no real flow or no real flow

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50. Pg 68, Fig.3.2: Comparing with its initial position, the vertical component of the end point of

the arrow in the rotated ( and deformed, as well) fluid particle should not be changed and should be put on the point with maximum vertical distance from the solid surface.

Because inside the boundary layers on flat plates with parallel flows, there is no downward component for the velocity in the boundary layer but rather there may exist upward component for the velocity in semi-infinite flat plates.

51. Pg 69,

L 3 under (3.3): q=0 q 0, such that q = constant,. Fig. 3.3. It seems that in Fig. (a), the particle has spin with respect to its center of mass,

therefore this should be shown by bowed arrows. In Fig.(b), the flow is irrotational and the particle has no spin. Therefore the small arrows should be removed from the figure.

Note: if we put the images of fluid element at three positions in (b) together,

we see no rotation in the element.

52. Pg 70, L2 before (3.8), mass center center Please recheck (3.8).

53. Pg 71: Regarding the sentence just before the section 3.2, This is why., it would be interesting if we view the vorticity as the curl of a fluid particle linear momentum (not

angular momentum) per unit mass: .( )dmVVdm

= . In this way, the vorticity which is a

11

kinematic property is linked to a dynamic property, momentum. Compare this interpretation with the content of item 1 in page 127.

54. Pg 73, LB6 Fig. 3.5c is Figure3.5c is P 2, L 7: and climb up, but in Fig.3.5.(b) the direction of the arrows shows climb

down. If the cup is started to rotate, during the transient process of spinning up, the flow near the wall climbs up, but if the cup is fixed and inside water is stirred by a spoon, the flow near wall climbs down.

55. Pg 75, Fig.3.6: the titles (a) and (b) of the two figures are missing. L 5 under (3.15), Helmholtzs first theorem Helmholtz first theorem

56. Pg 76: the second term on the R.H.S of the first equality in the formula before (3.17a),

i jl kx x ...i jl kx x

57. Pg 77: the equation before (3.22), 0v v . In (3.22), H(t) v H(t).

Also, in LHS of equation (3.23) ( , )x t ( , )v x t .

58. Pg 78, L 1, the righthand side the right-hand side

59. Pg 79,

60. Pg 80, L3 under (3.28), B = 1/2(n 1).B =

12

61. Pg 82: L3 under (3.34): arclength arc length Please apply this throughout the text.

62. Pg 83,

L 2 before (3.38), O( O( Please apply this throughout the text, wherever O( means order of magnitude Note: O is italic.

Regarding (3.38), what is ? Is scalar?

Because, in this way, the 0th, 1th and nth terms in

0 , 0 , 0

1( ) ( ) ( ) ...2i i i j ij

G x x G x G x x G = + +

takes the form, , and respectively.

63. Pg 84, in (3.42), x x

64. Pg 85, L1 before (3.48): orthogonal orthonormal L1 after (3.48): In two dimensions with In two dimensions,

65. Pg 87, P1 under (3.58),

for steady incompressible flow the continuity equation reduces to (u) = 0

for incompressible flow the continuity equation reduces to u = 0 Or

for steady flow the continuity equation reduces to (u) = 0. Note: See 2.39 L 2 under (3.58) , the velocity and vorticity field

the velocity and vorticity fields 66. Pg 88, Line 2,

any incompressible Beltrami flow any homogeneous Beltrami flow.

Because in incompressible flows u=0 not (u)=0.

13

67. Pg 89:

L4, 0 results in =(t) , not =constant ( Trkalian flow). L2, L4 0 0 at the bottom of the page

( )

y x

( )

y x

. 68. Pg 90,

in RHS of (3.68) ,

and in L1under (3.68), Indicating indicating and in (3.67): = = = .

69. Pg 91, P 3, L 3: Conceptually,

tangledness of vorticity lines tangledness of vortex filaments Note: in both Fig. 3.9. and Fig. 3.10. , the term vortex filament has been used.

70. Pg 92: P 3, unkotted unknotted

71. Pg 93, Please recheck the formula before (3.76) 72. Pg 96,

L1: convergence property of I and L are poor

convergence property of I and L is poor.

22

2 R zC

Rx n uds e =

221

2 2 zC

Rx n uds e = .

73. Pg 97, in Fig.3.13., lines in hachured triangular surface element should be parallel to

vector tds, or deleted. Also, the closed loop C should have no thickness.

14

Regarding the contents of the paragraph under (3.88), we know that 0dS = for any closed surface.

1 2

0S S

dS dS+ = , 1 2S S

dS dS= S c= for any open vector surface

spanned by any closed loop. Now, imagine a vortex ring like 1 2 3 1 2 3 in following figure, in which the distance between the two parallel lines 1 1 and 3 3

is small enough to have 3

0S . Therefore, 1 2 1 23

S S S S S S= + + + . But for this

shape, 1 2

S S= , and 0S = . That is , in this special example, 0S = minimum

surface spanned by the loop. In this figure, Minimum surface spanned by the loop is shown by the shadowed area. I personally did the experiment with a wire frame, shaped like1 2 3 1 2 3 , and soap-water solution. After putting the wire frame inside the solution and drawing it out, a soap film formed , bounded by frame, with a shape very close to the shadowed surface shown in the figure.

last line: (3.74) (3.71).

74. Pg 98:

in (3.94) 1

2 12

.

in (3.95a) 1

8

1

8 .

15

75. Pg100, L3, have no local nor global have neither local nor global

76. Pg 104,

L1 under (3.117a): where k=n-1 and n=2,3 is where k=n-1; and n=2,3 is .

In equation (3.117b), ,t denotest

, but in equation(3.178), tu

denotesut

. Therefore it is suggested ,t t . The sentence under (3.116)

Finally, the rate of..are closely related to Finally, the rates of..are closely related to last sentence:

We than use. We then use. . last L :

( denotes that[that the] Lamb vector) should be deleted because the Lamb vector has already been defined in previous lines( in paragraph after (3.116) , L 3).

77. Pg 105, L2 under (3.121),

the second term of (3.120) the right hand side term of (3.120) Last sentence in paragraph under (3.121).

Hence, the vortical impulse and angular impulseis time invariant. Hence, the vortical impulse and angular impulseare time invariant. . L1 before (3.123) (and also Pg 108, P 2),

Write = t , Write = t,, because throughout the text(or at least, most often), the magnitudes of the components of vectors stands before the unit vectors.

78. Pg 106: LB 3, and bi-normal vectors and binormal vectors.

Please see the last line of the Pg 68, .., and the binormal.... In the end of P2,

, where the last three points imply an exponential growing. , implying an exponential growing. Because the curve is inherently exponential and all points on the curve show an exponential trend.

16

LB 2, and torsion of -line and torsion of neighboring -vector lines. This is physically (not solely mathematically), or at least from vortex dynamics point of view, more sensible and meaningful. Please see the sentence under the relation (3.3).

79. Pg 107: In Fig. 3.15. , the cubes are not necessary to be shown, and the xyz-axes are enough. Anyway, it is better the relative distance between the indices x,y, z letters and cubes to be the same in three figures.

80. Pg 108: P 2 , (Constantin (1994)) (Constantin 1994). See bottom of the Pg 3,

(Lighthill 1956) and (Birkhoff 1960), or Pg 120, (Salmon 1988). last line of the P 3, and if and r is neither..

and if and r are neither...

81. Pg 109, 3.6.1 Local and Integral Conservation

3.6.1 Local and Global Conservation or 3.6.1 Differential and Integral Conservation in P1 under (3.130), L11-12,

vorticity conservation circulation conservation or total vorticity conservation Note: From (3.114) ,C= cte but from(3.109) we cannot conclude that is constant . There is no vorticity creation, but the vorticity of each fluid element, , can be changing(with the constraint of total vorticity conservation) solely because of kinematic issues, like a spinning acrobat who opens his arms...

82. Pg 115, L 3,L 4 under (3.154a,b),

g = constant and h = constant g = const. and h = const. Please apply = constant = const. throughout the text. P 2 under (3.154),

into the acceleration formula (2.11), i.e., a = u/t + u

into a = u/t + u u 83. Pg 116: last P, L 2, It seems that

This is so for incompressible flow without density stratification.

17

This is so for flows without density stratification.

84. Pg 117, L 1: The phrase has a potential is not necessary and can be omitted. L1 under (3.159) , transfer transform

Sect. 3.6.3, L 2, We now seek... we now seek.... Sect. 3.6.3, L 6, Salmon (1998) Salmon (1988).

85. Pg 118, L 2 after (3.164): (Salmon (1998)) (Salmon 1988). L 3 under (3.163):

+e as the sum of potential and internal energies,.. +h as the sum of potential and enthalpy,..

because in this way, from 1dh Tds dp

= + , under isentropic condition (2.78) we get

1dh dp

= or 1h p

= , which has been used to derive the formula after (3.164).

But from 1( )de Tds pd

= , under isentropic condition (2.78) we get 1( )de pd

=

and therefore, the formula after (3.164) cannot be derived. in the formula under (3.164),

2x/2+ . Also, it is better d3x dv or d3x d3X

86. Pg 119, L 2: now simply reads. now simply reads

87. Pg 120, L 9 under (3.173c): energy conservation entropy conservation

88. Pg 121, Last formula

and in L2 under it, (3.134) (3.133b) In LHS of the equation after (3.175) , .

89. Pg 123: In (3.176),

18

v

v

In (3.179b),

. 0 = . 0 = 90. Pg 124:

In (3.182),

2( )O

3( )O L 3 under (3.185),u u

91. Pg 125: Regarding the sentences under (3.190) , the simple shear flow exemplified

in Sect.2.1.2, Cij = 0; D1 ,D2= 0 and D3= -k , therefore 0 = .

92. Pg 126, L1, Minimum Enstrophy Theorem Minimum Total Enstrophy Theorem

93. Pg 127, L 1:

circulating-preserving flow circulation-preserving flow

94. Pg 128, bottom of the item 2: mathematic definition mathematical definition

95. Pg 131, in the first line under sect. 4.1.1, Consider an incompressible flow with uniform density Consider a flow with constant density .

96. Pg 133, last P, regarding the term the rotational part of the Lorentz force creates, suppresses, and propagates a vorticity field it seems that the rotational part of the Lorentz force creates and suppresses a vorticity field but it does not cause the propagation a vorticity field.

97. Pg 135: The second sentence under the (4.10b) seems to be incorrect: In (2.61), Ds is caused by viscous dissipation or heat transfer, and Ds is for "one" fluid particle. Here in 4.10b , ds does not mean entropy increase or decrease of one fluid particle, it is just the difference of entropy between "two" adjacent fluid particles on a material loop C due to their pressure difference( not due to viscous dissipation).

98. Pg 136,

19

in the paragraph under the first formula, wake vortices wakes

P2, L7 under the formula entropy increase entropy creation

In Fig.4.3., T-s diagram, the abscissa, S s. From perspective of view, the red line is necessary.

Fig. 4.3., Vortex layer behind a curved shock wave. Vortical flow region behind a curved shock wave.

99. Pg 137, the formula under (4.13) :

( . ) . . .U n U n U n U K = + =

( . ) . . .U n U n U n U K = + = L1 before (4.14): due to 0U = due to 0U =

Regarding the contents of L1-L3 under (4.14): as well as the density ratio across it, but completely independent of the theromodynamic (to be corrected to thermodynamic) process inside the shock layer , it is to be noted that the density ratio across the shock layer itself is determined by the thermodynamic process inside the shock layer(that is, it is dependent on thermodynamic process inside the shock layer).

100. Pg 138: Regarding the title (and content) of 4.1.3., it seems more logical that the title to be

rearranged as Viscosity Creation, Diffusion, and Dissipation of Vorticity at Boundaries .

P1 of sect. 4.1.3,

20

For incompressible flow For homogeneous flow Regarding the sentence under (4.15) ,The first effect., one may ask why the

creation of vorticity is not the first effect of viscosity. In Pg 141, P 2 from bottom, L 4, This is the third effect of viscosity, the creation of vorticity is called the third effect of viscosity. I think there is no priority among viscosity effects. Sometimes the vorticity is created by other effects such as baroclinicity but dissipated by viscosity. Please see L1 of page 201 if the first milestone by no-slip condition. No-slip condition is one of the effect of viscosity.

Therefore it is suggested: The first effect. One of the effects. and

This is the third effect of viscosity This is another effect of viscosity. In (4.17), it seems necessary

.vn vn =

.vn v

n =

Please refer to,

1994 71 (3.5.6).

101. Pg 139, L2-3. Vorticity is a point function. So, this term how much vorticity is diffused is meaningless for the same reason that the term how much temperature is diffused is meaningless. Instead, e.g., we can say how much is diffused or how much heat is diffused. Fig.4.4.

Intrinsic frame e1,e1,n Intrinsic frame e1,e2,n L3,

vorticity will be diffused vorticity will diffuse P 2, L 4,

(1989, 1990)) (1989, 1990) P2, L1,

vorticity diffusive flux vorticity diffusion flux See same page, L1. It is better to change all vorticity diffusive flux vorticity diffusion flux throughout the text. In most cases in the text, vorticity diffusion flux has been used.

102. Pg 140: In (4.24a), a Bn a = a n a =

103. Pg 141: In (4.26a), pa p

21

104. Pg 142, last L: they become the root of they become the roots of

105. Pg 143, L1 before (4.29): (4.24b) and (2.172b) (4.24b) and (2.172a,b).

Fig. 4.6 Schematic . vorticity generation by pressure gradient Fig. 4.6 Schematic . vorticity creation by tangent pressure gradient . Note : in the title of this section and most parts of the book, the creation has been used. In Pg 144, the lines under Sect. 4.1.4, the term generation has been used. Also, note that in Pg 126, last P, entropy production has been used while in Pg 8, P 3 ; entropy generation has been used. To keep uniformity throughout the book, it is suggested to change all generations or productions and their derivatives to creation and its derivatives, if applied and possible.

Comparing with(4.29b), it is better to swap the LHS of (4.29a) with its RHS. That is /n = p/s p/s =/n

106. Pg 144,

Considering the paragraph above (4.30), the definition of B+ and B-,

L1 under (4.34): which is evidently independent of viscosity which is apparently independent of viscosity.

Because is implicitly a function of viscosity.

107. Pg 146, P1 under (4.44), L 4, L5:

at y [0, 1] and flow 2 (e.g., the air) at y = (1,) at y [0, 1] and flow 2 (e.g., the air) at y (1,)

P 2 under(4.44), fluid 2 flow 2. See P 1 under(4.44), L4 , Both flow 1 (e.g., the water) at y [0, 1] and flow 2

108. Pg 147, Fig.4.7., In (a), the small letter t denotes the time[but in (b), (c), the

capital letter T ]. It is better to keep the same format in three figures.

22

On the Fig.(a), t=10 T =10 See the horizontal axis of Fig (b),(c), which is T.

109. Pg 148: Last P, L 2, example 3 third example.

110. Pg 149, Fig. 4.8. Soundvortex interaction

Fig. 4.8. Soundvorticity interaction The title of this subsection in Pg 148 is better to be changed. L 2 under (4.51b), In example 3 In third example,.

111. Pg 152:

In the equation before (4.57a),

4C

2

C

In L1 before (4.57a), it is suggested to delete with density . In L1 before (4.58a),

From (4.57a) From (4.57a) and incompressibility condition,. It may be necessary to put a note under (4.59), notifying that the

characteristic length in Re here ( and (4.67),,(4.71)), is radius a , not diameter. Regarding the sentence under (4.59),

the total kinetic energy K is time-invariant it seems that the total kinetic energy is infinite because at the far field , the velocity is assumed to be constant (according to(4.56a)). Therefore the integral of kinetic energy over an infinite domain is infinite.

112. Pg 153: L2 of P 1 under section 4.2.2,

was ignored. was ignored at infinity. In L5 of P 1 under section 4.2.2,

the inertial force is of 3( )O R . On the other hand, 1 2( )O R = .

the inertial force is of 2( )O R . On the other hand,. 1 3( )O R = .

113. Pg 154: In (4.65),

(1 cos )kR (1 cos )

2R

In L1 before (4.66),

(1)R O+ (1)R O=

23

In (4.66),

(1 )

2R+

(1 cos )

2R +

In (4.67),

2 36 (1 Re)

8D U a = +

36 (1 Re)8

D Ua= +

114. Pg 155: the sentence under (4.68),

One sees that 0 = not only on the sphere but also along the revolutionary surface One sees that 0 = not only on the sphere but also along the line 0 = and the revolutionary surface. Fig. 4.10., the small circle in the center of the sphere is to be deleted

115. Pg 156:

On the ordinate of the diagram in Fig. 4.11., R Re. Because, in the solid curve for the model Two-term Stokes expansion for R=infinity, the R in for R=infinity stands for the dimensionless radius and it may be mistaken with Reynolds number R on the ordinate. On the diagram:

Experimant Experiment In (4.71b),

132

136

and

3

32

.

Last P ,

?( up to here, the notation has not been used). 116. Pg 157:

P 1, L 3,

? P 2 , L 5,

the velocity and vorticity profiles the velocity profile and the magnitude of vorticity. Because the vorticity vectors here in the figure for 2D are perpendicular to and pointing outward of the page, and they cannot be shown as vectors, like velocity vectors in Fig.4.12. (a), which are from left to right.

24

In Fig. 4.12.(b), the vertical axis is y not . The horizontal axis is , which is overlapped with x -axis.

P 1 , L 6, take opposite takes opposite. Fig. 4.12.:

Sketch of the profiles of velocity (a) and vorticity (b), and the variation of fluxes of vorticity (c) and enstrophy (d)..

Sketch of the profiles of (a) velocity and (b) vorticity; and the variation of fluxes of (c) vorticity and (d) enstrophy ... Because, up to here, in all figures, the trend was like(a) velocity not velocity (a).

117. In following places the scalar 0 should be changed to vectorial 0. [Pg 35, L 8],[Pg 39, L 5 under (2.97)],[Pg 59, P 2, L 10],[Pg 75, footnote 4],[Pg 77: third relation of (3.20b)], [Pg 89, Ls 2, 4], [Pg 100: In L1 before (3.100)], [Pg 116, last P, L 3],[Pg 126: In L1just before P 2], [Pg 185,P2 ,L6],[Pg 185, L6],.

118. Pg 160: in and under the (4.79b),

Ei Ei. See (12.108) and (12.110). Also, Ei is to be introduced here(not under (12.108)), that is to add , where Ei(x) is the Eiry function, after (4.79b)

119. Pg 161, Fig. 4.13, the title of vertical axis

6Ua 6Ua L1

homotopy analysis method homology analysis method See homology analysis method in Pg 164, L6 under(4.87)

120. Pg 164,L4 under (4.87) has to be used, and the computed has to be used. The computed

121. Pg 165, L2: the 35th order in computational and physical world of continuum mechanics

seems to be meaningless! See Pg 246, L2, too. L1 under(4.90)

y-integral Y-integral 122. Pg 166, L1 under (4.95a,b)

25

the kinetic-energy dissipation the mechanical-energy dissipation

123. Pg 168, L1 under (4.102), surface tensor surface tension

124. Pg 169,

L1 before (4.105) U = (Us, Un) (4.104b) U = (Us, Un), (4.104b) In (4.106)

= 2U, = 2Us,

125. Pg 170, in (4.109), W We

126. Pg 171, Fig. 4.16.

or

127. Pg 172, top of the fig.4.17., Surface vorticity check to be deleted . On the same figure

10.0 10 Can (4.116b) be integrated to

? Compare this with (4.116a).

128. Pg 176,

considering

in L2 before (4.131a), it is better that in (4.131a)

26

in P1 above (4.132a),

.11. .11

129. Pg 177, RHS of (4.138), (4.139), first integrals may need a minus coefficients.

130. Pg 178, Fig. 4.20 (b, c) two-dimensional flow (b), (c) two-dimensional flow

131. Pg 179, L1

Then, let n = e2 e1 point from the side to the + side, Then, let n = e2 e1 point from the +side to the -side, 0r Then, let n = e1 e2 point from the side to the + side, In (4.141b),

u22 u12 in L4 under (4.141b),

u2e2 u2-e2

In (4.142),

Note : the functional form of Z is the same for and .

132. Pg 180, in (4.143),

L9 under (4.145), considering the horizontal axis of fig.4.21,

t = x/U. t = z/U. (4.146), considering the horizontal axis of fig.4.21,

y x (4.146), considering the (4.147), A 0. If so, in L1above (4.147)

A 0. L12 under (4.145), (see Fig. 4.21 later). (see Fig. 4.21).

133. Pg 181, in the top curve of fig. 4.21,

27

0 t= 0 134. Pg 183, L5 under(4.153)

x1(x, t) + ix2(s, t) x1(s, t) + ix2(s, t)

135. Pg 184, Fig. 4.22, for the symmetry reasoning it is better to modify the figures as follows:

It seems

136. Pg 185,

28

L1, The term-to-temathrmcorrespondence The term-to-term correspondence L2 under (4.155),

fomathrmof ??

considering the comment on (2.82a), the formula under the (4.155) needs a 1/2 factor too. And (4.156) should be modified.

137. Pg 187,

in (4.158b), V B P1, L12,

he uses they use

138. Pg 189, L2, fomathrma ??

139. Pg 190, under (4.168a,b), where suffix B means values on B. is not necessary and can be deleted.

140. Pg 191, in (4.170) and (4.171), to keep the same format throughout the book :

, 141. Pg 192,

L3 under (4.173b), the inviscid equations for u and the inviscid equations for ue and e in (4.175),

Because it is obvious that the integration is taken over B.

142. Pg 193, L1 under (179), tt L1 before (179), fomathrmis ??

143. Pg 194, P3, L5, confimathrmthe?? last line,

unable obtaining unable in obtaining

144. Pg 195, in top figure of Fig. 4.24., t=4.0 can be deleted because it is already been

29

quoted in the text under the figure.

145. Pg 197, Fig. 4.28

The title of the horizontal axis: Cl at different angles of attack angle of attack Cd at different angles of attack angle of attack As a general comment, in many figures of the text, such as Fig. 4.28a, the divisions on

the axis(here the vertical axis) are too fine, and need coarsening.

146. Pg 198, P1, L6 , fomathrma ?? P1, L2-3 under (4.183),

The boundary conditions there is The boundary conditions there, are

147. Pg 202, L4 under (5.1b), semiseddle semisaddle

148. Pg 203, L2, to keep the general format of the book,

lh lh. See hlu, at the same line L4, to keep the general format of the book,

h/(2) h/2. See h2l/2 in L5. 149. Pg 204, Sect. 5.2, L1,

hureristic heuristic

150. Pg 205, (5.4), in right formula, it is better . L3 under (5.4),

The vector fields ( p, ) or equivalently (,) on B forms The vector fields ( p, ) or equivalently (p,) on B form In next paragraph , (,) (p,).

151. Pg 206, Fig. 5.4. Orthogonal and -lines

Fig. 5.4. Orthogonal and p-lines in footnote3,

the right-hand side of (5.6) and (5.7) become

30

the right-hand sides of (5.6) and (5.7) become

152. Pg207, under (5.9),

Note vectorial zero

153. Pg 209, L1 before (5.19a),

see (4.24b) and (4.28) see (4.23) , (4.24b) and (4.28) L1under (5.17)

(5.15) (5.12) In (5.18a)

x' iP x' iP the formula under (5.20)

33 32

154. Pg 210, L1 under (5.23), u3P 3 P

155. Pg 211, please recheck (5.27a,b,c),

156. Pg 213, Fig. 5.6. (a)Local Fig. 5.6. (a) Local

157. Pg 214,

in(5.28) Also , apply this in (5.29), (5.34) L1 under(5.28) :

y = x y = x See L2 under 5.2.4 in Pg215

31

P2, L5,

we may again be satisfied with assuming it may again be satisfied by assuming

in(5.30), y yy

158. Pg215, in (5.31), A zA , zC(x, z), z 2C(x, z), and in (5.32) dr rdr L2 under 5.2.4 and L1 of Pg 216 and L8 in P1 under (5.37)

= =

159. Pg 216, considering (5.36), in (5.37)

LB3,

do not givenor reveal neither give nor reveal

160. Pg 217, Fig. 5.7. and 2 Fig. 5.7. 2 and

161. Pg 218,

in (5.40)

L2 after (5.40)

by (5.40a). by (4.29a).

162. Pg 219, in Fig. 5.8., considering the related text, it is better to denote the origin with xs instead of 0.

32

163. Pg 222, L2 after title Main deck, it seems

164. Pg 223, please recheck (5.51a,b)

165. Pg225, (5.55)

166. Pg 227, L9, Helmholtzs Helmholtz L10, L11

In this flow model the separated vortex sheet appears as a free streamline along which q = |u| and p are constant, In this flow model the separated vortex sheet appears as a free streamline along which p is constant, and then according to Bernoulli equation q = |u| is constant too in(5.61)

L1 of the last P,

The earlier dilemma The above-mentioned dilemma P1 under (5.62), it is better

Now, by (5.60) and (5.61), k > 0.(Fig. 5.10). k < 0 should Now, by (5.60), .(Fig. 5.10). By (5.61), k < 0 should.

167. Pg 228,

sentence 1 before (5.63), O(Re1/16) O(Re5/16) considering (5.63), P2 after (5.63),last line,

P3, L3

, P3 under (5.63), L3

33

found by the calculation. The computed variation A, P, and found by calculation. The computed A, p, and LB5,

full-NavierStokes full NavierStokes

168. Pg229, Fig. 5.11. A, pressure P, Fig. 5.11. A, pressure p,

169. Pg 230, in (5.66b,c)

, P1 under (5.66d), last line,

see the sketch of Fig. 5.13. see Fig. 5.13. regarding the other formulas in (5.69)

170. Pg 231, P2, L3,

Sect. 5.2.2. Sect. 5.2.3.

171. Pg 232, P1 under(5.70), L7,

Condition (5.71) Condition (5.70)

P1 under(5.70), L10, as tends to as -lines tends to

172. Pg 235, footnote 9, Here and below hereafter or henceforth

173. Pg 238, L1 under(5.72a,b)

This criterion is known as the MRS criterion. This is known as the MRS criterion.

34

in (5.72a,b)

s = uy(s) = 0, us = xt(s). s = u,y(s) = 0, us = x,t(s).

174. Pg 239, last L, Evidently, (5.77a,b) is precisely (5.72). Evidently, (5.77)is precisely (5.72). or Evidently, (5.77a,b) is precisely (5.72a,b). in (5.75b), -

L2 under (5.75b)

term (xs x)1/4: term (xs x)1 / 2 :

175. Pg241, Fig. 5.18.

where where

J = xy yx = 1 (5.81a) J = xy yx (5.81a) is the Jacobian for incompressible flow, is the Jacobian, which for incompressible flow J =1 ,

176. Pg 242, L1 under (5.87),

35

if u and u is if u and u are in (5.88),

0 0 177. Pg 243,

P1 under (5.90), L10, (normalized by wall vorticity) can be deleted because the same phrase has been stated in the text for Fig. 5.19.

L1 before (5.90),

(a dot denotes d/dt) (a dot denotes /t) Because, otherwise we can directly write:

P1 under (5.90), L9. considering the text under Fig. 5.19 :

On top of Fig. 5.19 are the profiles of velocity and vorticity (normalized by wall vorticity) close to separation, from which it is evident

From Fig. 5.19,it is evident

178. Pg 245, P2, L9,

a upwelling an upwelling In this page, gradient operator is not

179. Pg 247, in L2 in the text for Fig. 5.22., it seems (5.109) (5.111)

180. Pg 249, LB2,

of (5.106) up to quadratic of (5.102) up to quadratic

181. Pg 250, L1 under(5.111) into (5.108) into (5.106)

182. Pg 251, item 2, L1 A general flow-separation can be studies A general flow-separation can be studied

183. Pg 256, LB2,

An inspection of (6.1.5) An inspection of (6.2) 184. Pg 257,

36

L1, This kind of vortices are called pure vortices, These kinds of vortices are called pure vortices,

L4: This kind of vortices are called This kind of vortices is called

185. Pg 258, it seems that the LHS of (6.12b) u (/r ) u (/r )

186. Pg 259, P1 under(6.15a,b), This set of equations are. and has been This set of equations is .and has been

187. Pg 260,

under the (6.20), outer radii R1 and R2 and angular velocity 1 and 2, outer radii R1 and R2 and angular velocities 1 and 2,

(5.19) If

Then

Note : in (5.19b), second one, +and _ are swapped. (6.21)

188. Pg261, it seems,

189. Pg 262,

L1, The mode n = 0 First, the mode n = 0...

See L1 before (6.26a): Second, the mode n = 1... L4 under (6.25b),

37

LB2, f = constant f = const.

It is suggested all = constant = const.

P2 under (6.25b), L1

Note: Dimensionally, is incorrect.

(6.26a)

(6.26b)

where represents the total angular momentum about the axis where represents the total angular momentum per unit length and about the z-axis

(6.27)

190. Pg 263, in Fig. 6.1.

similarity variables (c). In (c) similarity variables (c). In (c) Note the font of second (c)

in Fig. 6.1.(a) : t=0.5 0.5 and in Fig. 6.1. b: t=1, t=2, t=8 1, 2, 8

It is suggested to add the corresponding values of the vertical variables to the vertical axes.

191. Pg 264, L1 under (6.32)

have similar behavior have similar behaviors Note that when we talk about similarity, we are talking about at least two objects.

38

(6.33a)

and (6.33b)

L2 under (6.29),

a uniform stretching. an exponential stretching. L3 under (6.32) :

and faster rotation time, and faster rotation, 192. Pg 265, L4 under (6.34b)

OseenLamb vortex Burgers vortex

193. Pg 266, L2 of P2 under (6.38)

can be made disappear can be made disappeared L1 under(6.37b)

nonstretched vortices, nonstretching vortices, in(6.38), array 3x3 of D, it seems 2 2 L2 under(6.37b)

the stretching as it should. the stretching as it should be.

194. Pg 267, P1

comparing (6.43) with (6.44c), in (6.43) d d

Because = /4has already been defined in (6.33a).Also = /4 is the definition of and cannot be written after the term if and only if.

195. Pg268, The sentence in L1 needs to be revised.

39

L1 after 6.2.3, the coupling of are .. rather than derived the coupling of is rather than being derived

P2, L3 Thus, some behaviors of these vortices are lack of physical background. Thus, some behaviors of these vortices are due to lack of physical background.

L1 after (6.46), quasi-cylindric quasi-cylindrical

196. Pg 269, Fig. 6.2., last L,

Note: See the title of the horizontal axis

197. Pg 270, L4 under (6.47)

F = 2(1 x2)f, F = 2(1 x2)f and K=1/2, (6.48a)

G(x) K2G(x) and In (6.48c), 4 2

198. Pg271, LB2,

and axial angular momentum is infinity and axial angular momentum are infinity

199. Pg272 , under 6.3, L7, and Leonard 1992 and and Leonard (1992) ; and

200. Pg 273, Fig. 6.4.

last sentence

40

Then (6.52) or (6.53) Then (6.52) and (6.53)

201. Pg 274, L10

asymptotic thing ring theory asymptotic thin ring theory (6.57a)

202. Pg 275,

(6.60) 2/E(k) 2 E(k)/ k

L2 under (6.62)

203. Pg 276,to keep the harmony with the other parts of the text, in this page and

Pg 277, all

except (6.66), which

Note: See (6.113),(5.98), bottom of Pg 152,

204. Pg 277, L1 under 6.3.2

Where is (6.3.6)??

205. Pg 278, last L, adnvection advection Pease recheck (6.70). L1 under (6.71)

(6.55) (6.54) 206. Pg 280

41

L1 under (6.75a,b) (6.3.26a) (6.75a)

L1 above (6.76) (6.3.26b) (6.75b)

L2 before (6.77), equation (6.1.14) equation (6.14)

207. Pg 281, L2 above (6.80),

L3 under 6.3.3.

(thing rings) (thin rings)

L1 under (6.80) uniform vorticity constant vorticity See L4 of P2 under 6.3.3.( = 0 is constant,)

208. Pg 282, (6.82)

R r0 L4 under (6.81)

streamfunction stream function Please apply this throughout the text.

L1 of P2 under (6.82), it seems that (6.63) is wrong.

209. Pg 283, L1 under (6.84)

thing-ring thin-ring and

(6.86),

42

210. Pg 285, L2 under (6.91)

= 0 = a/c cosh0 = a/c sinh 0 = b/c cosh 0 = b/c

211. Pg 287, last line,

neutually stable neutrally stable,

(6.104b)

212. Pg 290, L1

Chaplykin Chaplygin Note: Please apply Chaplykin Chaplygin throughout the text.

In (6.110) and (6.111)

213. Pg291,

L2 they must be it must be

L1 of P2 under 6.4.3,

See in (6.114) 214. Pg 292

Fig. 6.12. Single vortex row (a) and associated streamlines (b) Fig. 6.12. (a) Single vortex row , and (b) associated streamlines

43

Last line

buff body bluff body

215. Pg 293, in (6.119),

216. Pg 294,

in (6.121),

L2 above (6.124)

2(t 2) =0 t 2=0 217. Pg 295,

Fig. 6.14., x x /a and y y /b [if so, range of vertical axis needs to be changed as -1 to 1]

A slight modification to the figure makes it more meaningful(and beautiful)

218. Pg 296, L4,

Recall that the outer solution there is inviscid Recall that the outer solution there, is inviscid

219. Pg 297, L1 after (6.131) ,

Considering L2 of Pg 297 u U

in (6.133), At the vortex center we require

At the vortex center we require

44

or We require

Please recheck (6.135).

220. Pg 299, please recheck (6.147)

in (6.144b)

221. Pg 300, L2 under (6.150),

222. Pg 301 Fig. 6.15. Axial strain (a) and biaxial strain (b). Fig. 6.15. (a) Axial strain , and (b) biaxial strain .

223. Pg 302, L1, to keep the uniformity of the text , it is better:

in (6.156a), it seems that 2 - 2

224. Pg 303, last L,

velocity V = /4L speed V = /4L

or velocity V = /4L.ey

225. Pg 304, (6.166) =

45

226. Pg 305, Please recheck (6.168)

227. Pg 306, in (6.174) , it seems that the exponent ,

Also see (6.175),(6.178) under (6.174)

LB3,

.strain field has been .strain field have been

228. Pg 307, in(6.177) and (6.178)

in (6.176),

?? 229. Pg 308,

Fig. 6.16. strain vortex Fig. 6.16. strained vortex Note that throughout the text strained vortex has been used.

230. Pg 316, under (6.202),

assumed condition (6.193) yields an inequality equation assumed condition, (6.193) yields an inequality

L1 under (6.203b), between the Q-,-, and 2-criteria. among the Q-,-, and 2-criteria.

L3 under (6.203b) streamlines merge into or diverge out streamlines converge into or diverge out

231. Pg 317 , L1 (6.179) (6.182)

232. Pg 318, LB5,

46

does not forms does not form 233. Pg 319, Fig. 6.20

the qualitative feature of remain the qualitative feature of remains

234. Pg 321, last L, So far the definition problem still remains an open issue. So far the definition problem remains an open issue.

235. Pg 323, LB8 a the vector field a vector field LB6 property properties

236. Pg 324, footnote1: considering , the names such as lugt should also be written

as: . 237. Pg 325,

L1 of P2, A three-dimensional The three-dimensional

In L3 of P2, easier to examined easier to examine or easier to be examined.

In (7.3) and in L1under it, P2 P and Q2 Q

In L5 of the paragraph under (7.4), summerized summarized

238. Pg 327, In Fig. 7.2.

239. Pg 328, in L4 of the text for Fig.7.3. In S4 is In S4, is

240. Pg 330, Fig. 7.5.

47

241. Pg336, L11, They a lso They also

242. Pg337, L1, spatial dimensions of the flow is spatial dimensions of the flow are

243. Pg339, Fig. 7.12. Supercritical bifurcation (a) and subcritical bifurcation (b) Fig. 7.12. (a) Supercritical bifurcation and (b ) subcritical bifurcation

244. Pg 341, L2 before (7.15),

(3.643.67) (3.64)(3.67) In L3 under (7.16), it is better to delete:

, with unit vectors (t,n) being tangent and normal to the line,

245. Pg 342, Fig.7.13.

246. Pg 343, L1 under (7.21b),

(7.19a,bb) and (7.19a,ba) (7.19b) and (7.20a)

247. Pg345, in the last formula at the bottom,

248. Pg 346, L2 before(7.31)

48

see L1 under (7.30d) 249. Pg347, L6 and L9 under (7.34)

y = o(x). y = O(x2). q = o(x) q = O(x2) and o(x2). O(x2).

250. Pg348, P2, L11

in instability 3 3? Last Line,

A (s = 0) A (s = sA) 251. Page 349

In L1 after (7.37c), and in (7.37d),

L1 under (7.36)

, 252. Pg 350, L5 of last P,

Re = UD/ (based on diameterD) Re = UD/ 253. Pg 351,

in (7.38), it seems,

In L1 after(7.39)

Secondly, Second, because at the same page , P3, we have First not Firstly L2 of P3,

Sadovski Sadovskii

254. Pg 352, L1 after (7.41) Thirdly, Third,

255. Pg353,L1 after (7.45), it seems

49

256. Pg355, at the bottom r cosd r cotd

257. Pg356, L3 before (7.55), may be via (7.52), via (7.51 ),

258. Pg 357, in the first and second formulas under (7.56), g/x dg/dx

259. Pg362,P2, L7 has 4 degrees has four degrees

260. Pg 364, P1, L5,

a symmetric a symmetric Note: font a is changed. In L3 after (7.68),

the vorticity the vorticity

261. Pg 367, in L3 before and L2 after (7.69),( Also, in Pg 377, L1 )) CPb Cpb

262. Pg 368, Fig. 7.25. , the upper and lower figures should be named as (a) and (b).

263. Pg 369, Fig.7.26,

264. Pg 370,

L5 Then for Re in 230260 Then for 230

50

L2, buff cylinders bluff cylinders P1, L3, Re 2105 Re= 2105

267. Pg 378, L3 before (7.73)

Averaging (7.70) over T/2 yields time averaging over T and spatial averaging over D of (7.70) yields

L1 before (7.73) by U2/2 by 2 U2

268. Pg 379, L1, nor?? L1 under (7.76)

balance of work rate and kinetic energy balance rate of work and kinetic energy

269. Pg 380,LB6

(7.80) (7.81a)

270. Pg383, L2 Logically this phrase: the formation of a vortex is the rolling-up of vortex layer. is not correct. Because, the formation of a vortex cannot be expressed in terms of a vortex. Intuitively it seems, as far as there is no rolling up in a vortex layer, a vortex layer is not a vortex(it is a misnomer ) and it is just a shear layer with small thickness.

271. Pg384, L1

we may divide problem we may divide the problem L2 before (8.1)

= ez = ez

272. Pg385, in (8.5)

See (8.4)

51

273. Pg 386, (8.13)

It is more beautiful! And more similar to (8.12).

(8.11)

274. Pg388,

(8.21)

L2 before (8.21),

as r there is p = p, as r there is p p, (8.23)

275. Pg389,

All p - p0 pa - p0 L7

causes an axial-flow deficit, . an axial-flow increment indicate that this axial-flow increment causes an axial-flow deficit, . an axial-flow excess indicate that this axial-flow excess Note: Excess is a better antonym for deficit. Increment is the antonym of decrement. Also, please see Pg 392, L3 (deficit at the trailing-vortex) and L9 (axial-flow excess)

in (8.25)

52

276. Pg 390,

L1 before (8.30), initial condition upstream condition because in (8.28a), time t is not explicitly an independent variable. Also, see Pg 391,L1 and L2 under (8.33). L4 and L5 under (8.27)

dw 2/dz dwa2/dz 277. Pg 391, (8.32)

Also, see (8.33).

L1 before (8.33) the leading terms of (8.32) is the leading term of (8.32) is

L1 after (8.33) The solutions (8.30), (8.31a), and (8.32) have been independent of The solutions (8.30), (8.31a), and (8.32) are independent of

all ei() Ei() In L2 before (8.34),

by (8.30) by (8.30b)

278. Pg 393, LB2, (and to keep the same format, throughout the text)

quasi-cylindrical quasicylindrical

Compare in P1 under (8.40), L5, with

in Fig. 8.2. the formula after (8.38)

53

279. Pg394,

(8.41d)

in (8.43)

280. Pg 396,

Fig. 8.3. Left and right polarized Fig. 8.3. Left and righthanded polarized

281. Pg 397,L3 of the text for Fig. 8.4., vortex lines vorticity lines See L3 of the Pg397.

282. Pg 398, Fig. 8.5. Evolution of .. The letters on the left-hand side of figure show the different time of evolution. Fig. 8.5. Evolution of

283. Pg 401, L3 under (8.50)

under (8.51)

54

284. Pg402, L2 before(8.55)

L6 under (8.52)

localized induction approximation local induction approximation See the title of 8.2.1

285. Pg 403, (8.58)

L3 under (8.59)

and phase and phase or (total torsion) Note: is not phase.

286. Pg 404, L3 under (8.61)

SerretFrenet FrenetSerret

In (8.62a) c c

In (8.62), s Please note = s ct in L4 before (8.62a).

287. Pg 405, L2 under (8.65)

rigid rotation rigid-body rotation under (8.66)

= 0 = 0 and V = 0 V = 0 288. Pg 407,

L1 under (8.69),.

L8 under 8.2.2, the Hasomoto the Hasimoto

L3 under (8.70), K-dV equation Korteweg-de Vries equation(K-dV equation)

55

Because here, it is the first time that K-dV is introduced in the text. 289. Pg 408,

in Fig. 8.9.,

Fig. 8.9. Three-dimensional curvilinear coordinates systems. Fig. 8.9. Three-dimensional curvilinear coordinate systems. Note that except here, throughout the text coordinate system has been used (not coordinates systems). And

and

Because the changes shown in the figure are finite (not differential).

290. Pg 409, L4 under (8.74),

= Re1/2 =O( Re1/2) (8.72),

And h = 1cos h = 1rcos

291. Pg 411, at the bottom,

56

Now, regarding to the evolution of the core-structure function C(t), Klein and Ting (1995) have shown . Regarding the evolution of the core-structure function C(t), Klein and Ting (1995) have shown

292. Pg 412, in(8.86)

Question: in (8.86b),

LB2

Ludgren Lundgren Note: See L1 under (8.86b) please recheck (8.85b)

293. Pg413,

8.2.3 Nonlocal Effects of Self-Stretch 8.2.3 Nonlocal Effects of Self-Stretching In most cases in the text this term Self-Stretching has been used.

under (8.90)

Last line of the Pg:

294. Pg 415, at the bottom,

57

and

295. Pg 416, L3, it is better

growing solutions growing effects

296. Pg418, according to the related text, Fig. 8.13. The hairpin creation. Fig. 8.13. The hairpin formation.

297. Pg 419, In L4,

circulations kwith singular vorticity field strengthskwith singular vorticity density field

it seems that in (8.102) and(8.118) (8.101)

In (8.103) and (8.105), (8.125a)

Under (8.104)

298. Pg420, L3 under(8.108) the latter are the latters are

299. Pg421, (8.109)

58

LB4

For unbounded fluid there is Nc = 3. For unbounded fluid , Nc = 3.

300. Pg 423, (8.119)

301. Pg 424,

Regarding L1 under (8.122) The triangles at t = 1 and 20 are plotted.. But, the triangle collapses at t = 32 < 20? Therefore:

The triangles at t = 1 and 20 are plotted in Fig. 8.16. The triangles at time-step= 1 and 20 are plotted in Fig. 8.16.

the formula above (8.22),

302. Pg425,

under (8.123)

(8.125a)

303. Pg 428

Fig. 8.20. Stability boundaries for Karman vortex street consisting of point vortices (a), and finite-core vortex patches (b). Fig. 8.20. (a) Stability boundaries for Karman vortex street consisting of point vortices, and (b) finite-core vortex patches .

L2,

59

the wavenumber the wave number Note: Please apply wavenumber wave number throughout the text

304. Pg 429,P3 L4,

axisymmetrinization ?? axisymmetrization

305. Pg 433

306. Pg 434

L7 under (8.129),

is Trkalian flows, which is a Beltrami flow is Trkalian flows, which is a Beltramian flow

307. Pg 435, L3

most have been or to be addressed most have been or are to be addressed L7

Vertical flow Vortical flow

in L7 under 8.4.1, at hight at height

in L9 under 8.4.1, Re = /2, but in some places such as Pg 400, 497, Re = /. 308. Pg 437, LB2

sheds out of sheds off 309. Pg438, LB6 during the interaction evolves as time. during the interaction evolution. 310. Pg 439, in Fig. 8.28.

contour contours

60

311. Pg 441, last L,

vorticity antisymmetric vorticity symmetric Note: See Saffman, Pg86

312. Pg 442, all

Under (8.134)

Above (8.135)

313. Pg 443, L1 above (8.137) Apply Applying

314. General comment: The title of the book covers mainly the contents of parts I and II, while parts III, IV constitute a substantial body of the book. This is a comprehensive book! It is suggested to add a phrase to the title of the book to reflect the matter.

315. General comment: It is suggested to change the title of chap 9 Vortical-Flow Stability and Vortex Breakdown Vortical-Flow Instability and Vortex Breakdown In this chapter, the count of the word instability is much more than instability and more discussions are about the causes of instability not stability.

316. Pg 444, L1 before and L3 after(8.138)

317. Pg 446, item 3, L8 Hasomoto Hasimoto

318. Pg454, the second integral in(9.5) should have lower and upper limits.

61

(9.6) F(k , n, ;Re, ...) D (k , n, ;Re, ...) Note: This is the dispersion relation. See (9.10)

In (9.6) u(x,t) u(x,t) Note: Font x is changed.

In (9.8) kix and krx kiz and krz Note : see (9.5) and (9.7)

319. Pg455, Fig. 9.2.

Note: it is better to delete the negative times.

LB2 Monkwwitz Monkwitz Note: ww= 2 double u= 4 u (just as fun!)

320. Pg456,LB5 -plane (i.e. some -plane (i.e., some

321. Pg 458, it seems that the convention of the book, the vectors and matrices to be shown with bold font is not regarded completely in 9.1.2 . Please see (9.23) and (9.32).

322. Pg 459,

(9.15a,b) (9.15a,b,c) L6,

L, the later L, the latter

62

323. Pg 460, Why E/t not dE/dt?

L1 under (9.18)

A +A A + A At the same Pg

324. Pg461,Fig. 9.5.

Note: u1 and u2 to be shown with directions.

325. Pg464, in this page some u, 0 are to be written in bold font.

326. Pg 465, L2 above (9.34)

zero, i.e. zero, i.e., L1 under (9.34)

( ) and ( ), and few lines under(9.34)

327. Pg 466, the time sequence of the figures in Fig. 9.6.(b) should be shown with

numbers(such as fig (a))

328. Pg468,

329. Pg 470

63

In (9.52), (9.51) Los LOS Lsq LSq

330. Pg 471 The Fjortoft theorem. The Fjortoft Theorem.

LB7,

The case (e), (f) are The cases (e), (f) are

331. Pg472 Fig. 9.7. Velocity and vorticity profiles for the parallel shear flows: (a) stable; (b)

stable; (c) stable according to Fjrtoft; (d) unstable according to Fjrtoft; (e) stable according to Fjrtoft; (f ) unstable according to Fjrtoft

Fig. 9.7. Velocity and vorticity profiles for the parallel shear flows: (a) and (b) stable; and (c) stable (d) unstable (e) stable (f ) unstable according to Fjrtoft

In LB2 an Reynolds a Reynolds

332. Pg 473, in (9.57):

In (9.58):

(z) (y) and : for Note: see (9.57)

333. Pg 474, Fig. 9.8.

64

334. Pg 475, LB3, (9.23)?! under 9.2.3

Reynolds was the first to realize that transition is caused by boundary-layer instability. Reynolds was the first to realize that transition is caused by flow instability. Note: Historically, the concept of boundary-layer was found by Prandtl, years after Reynolds

335. Pg 477, (9.66)

336. Pg 478, in (9.67) Los LOS

Lsq LSq And

337. Pg 479,

65

338. Pg 480, P2, L1-L2, its velocity profiles take its velocity profile takes

339. Pg481,L7 i.e. the i.e., the

340. Pg 483, L1- L2 after (9.75) In consistent (9.75), Khorrami . lead to In consistent (9.74), Khorrami . leads to

341. Pg 485, in Fig. 9.14. vs axial vs. axial Note : Except here, throughout the text vs. has been used.

342. Pg486, L5 Monkwwitz Monkwitz

343. Pg 486, Fig. 9.15. In (a),(b),(c) and (d) W,U,U,U V, W, W, W, respectively.

344. Pg489, In Fig. 9.18.

345. Pg 490, Fig. 9.19.

66

346. Pg491,L3 after(9.78)

The second terms The second term 347. Pg 492,L4

coexistence. coexist. 348. Pg 493, in (9.82)

Note: font of x is changed. 349. Pg 494,

in (9.89)

and one formula before it,

See the second formula before (9.89).

350. Pg495, L5 Note that c and 2 are the natural and forcing frequencies of (9.89), Note that 2 and c are the natural and forcing frequencies of (9.89),

351. Pg, 496, Fig. 9.22. Horizontal axis : strain strain rate Vertical axis : growth rate maximum growth rate

352. Pg 497, (9.94),

(9.96a)

67

353. Pg 498, L4 under (9.98)

n + 2-wave (n + 2)-wave In this page, and Pg499 all k

In Fig. 9.23., horizontal axes, K kc

And

k remains in the unstable band around k only

k remains in the unstable band around kc only

354. Pg499, L7, L8

P2 under 9.4.3, L6,

cutoff approximation (see Sect. 8.2.1; the cutoff cut-off approximation (see Sect. 8.2.1; the cut-off

355. Pg 500, Fig. 9.24. a/h and kh a/b and kb Note: in Saffman (Pg 237), h is the distance between two vortices. But here, in this text (Pg 499, L14 from bottom), this distance is shown by b.

356. Pg 501,Last P, L1 (a/b)2

68

358. Pg504,

LB4, attack. The axial deceleration attack, the axial deceleration

359. Pg 505, (9.99)

In (9.100)

Because, circulation =(), and note L3 under (9.100)

In L4 of P2 under (9.100),

in turn induced in turn induce in Fig. 9.28., it is better

and The generation the axial gradient The generation of axial gradient

360. Pg 506,

L1, to make the latter makes the latter

P1,L5, the external fluid to move outward like passing a bluff body the surrounding fluid to move outward like by- passing a bluff body

69

LB2 the linearly stability theory the linear stability theory

361. Pg 509, please recheck (9.103) Fig. 9.30. s > 0 (a) s> s0

362. Pg 510, Fig. 9.31. Compare the title of vertical axis

with (9.101),

. On the figure : (present throry) (present theory)

Benjamin (1962) (Benjamin 1962) LB2