Chapter 6

Since the doublet described in Section 6.5.4 was developed by letting a source–sink pair approach one another, it might be expected that a uniform flow in the positive x direction combin

Ch˜Ïng 6: Dòng th∏ - L¸c nâng & L¸c c£n

Bài gi£ng cıa TS Nguyπn QuËc fi, nguyenquocy@hcmut.edu.vn

Ngày 17 tháng 11 n´m 2015 NÎi dung c¶n n≠m:

Hàm dòng, hàm th∏ và các tính

chßt

Các dòng th∏ cÏ b£n

Bài toán chÁng nh™p

Khái niªm l¸c nâng và l¸c c£n

∞c tr˜ng cıa dòng bao quanh v™t

- LÓp biên L¸c c£n: do ma sát và do áp sußt L¸c nâng: phân bË áp sußt b∑ m∞t

và dòng ch£y xoáy

1 / 40

Dòng l˛ t˜ng vs Dòng th¸c

6.6 Superposition of Basic, Plane Potential Flows 313

more blunt shape is obtained Downstream from the point of maximum body width, the surface pressure increases with distance along the surface This condition 1called an adverse pressure gradient2 typically leads to separation of the flow from the surface, resulting in a large low-pressure wake on the downstream side of the body Separation is not predicted by potential theory 1which simply indicates

a symmetrical flow 2 This is illustrated by the figure in the margin for an extreme blunt shape, showing the differences between potential and viscous flow Therefore, the potential solution for the Rankine ovals will give a reasonable approximation of the velocity outside the thin, viscous boundary layer and the pressure distribution on the front part of the body only.

6.6.3 Flow around a Circular Cylinder

As was noted in the previous section, when the distance between the source–sink pair approaches zero, the shape of the Rankine oval becomes more blunt and in fact approaches a circular shape Since the doublet described in Section 6.5.4 was developed by letting a source–sink pair approach one another,

it might be expected that a uniform flow in the positive x direction combined with a doublet could be

used to represent flow around a circular cylinder This combination gives for the stream function

which indicates that the doublet strength, K, must be equal to Thus, the stream function for flow around a circular cylinder can be expressed as

(6.112)

and the corresponding velocity potential is

(6.113)

A sketch of the streamlines for this flow field is shown in Fig 6.26.

The velocity components can be obtained from either Eq 6.112 or 6.113 as

(6.114)

and

(6.115)

On the surface of the cylinder it follows from Eq 6.114 and 6.115 that and

As shown by the figure in the margin, the maximum velocity occurs at the top and bottom of the cylinder and has a magnitude of twice the upstream velocity, U As we move

away from the cylinder along the ray 1u ! #p " 2 2 u ! p " 2 the velocity varies, as is illustrated in Fig 6.26.

V6.7 Ellipse

Potential Flow

Viscous Flow

A doublet combined with a uniform flow can be used to rep- resent flow around

a circular cylinder.

0 1 2

more blunt shape is obtained Downstream from the point of maximum body width, the surface pressure

leads to separation of the flow from the surface, resulting in a large low-pressure wake on the

the differences between potential and viscous flow Therefore, the potential solution for the Rankine ovals will give a reasonable approximation of the velocity outside the thin, viscous boundary layer and the pressure distribution on the front part of the body only.

6.6.3 Flow around a Circular Cylinder

As was noted in the previous section, when the distance between the source–sink pair approaches zero, the shape of the Rankine oval becomes more blunt and in fact approaches a circular shape Since the doublet described in Section 6.5.4 was developed by letting a source–sink pair approach one another,

it might be expected that a uniform flow in the positive x direction combined with a doublet could be

used to represent flow around a circular cylinder This combination gives for the stream function

(6.110)

and for the velocity potential

(6.111)

In order for the stream function to represent flow around a circular cylinder, it is necessary that

flow around a circular cylinder can be expressed as

(6.112)

and the corresponding velocity potential is

(6.113)

A sketch of the streamlines for this flow field is shown in Fig 6.26.

The velocity components can be obtained from either Eq 6.112 or 6.113 as

(6.114)

and

(6.115)

As shown by the figure in the margin, the maximum velocity occurs at the top and bottom of

V6.7 Ellipse

Potential Flow

Viscous Flow

A doublet combined with a uniform flow can be used to rep- resent flow around

a circular cylinder.

0 1 2

Dòng l˛ t˜ng vs Dòng th¸c

more blunt shape is obtained Downstream from the point of maximum body width, the surface pressure increases with distance along the surface This condition 1called an adverse pressure gradient2 typically leads to separation of the flow from the surface, resulting in a large low-pressure wake on the downstream side of the body Separation is not predicted by potential theory 1which simply indicates

a symmetrical flow 2 This is illustrated by the figure in the margin for an extreme blunt shape, showing the differences between potential and viscous flow Therefore, the potential solution for the Rankine ovals will give a reasonable approximation of the velocity outside the thin, viscous boundary layer and the pressure distribution on the front part of the body only.

6.6.3 Flow around a Circular Cylinder

As was noted in the previous section, when the distance between the source–sink pair approaches zero, the shape of the Rankine oval becomes more blunt and in fact approaches a circular shape Since the doublet described in Section 6.5.4 was developed by letting a source–sink pair approach one another,

it might be expected that a uniform flow in the positive x direction combined with a doublet could be

used to represent flow around a circular cylinder This combination gives for the stream function

it follows that for if

which indicates that the doublet strength, K, must be equal to Thus, the stream function for flow around a circular cylinder can be expressed as

(6.112)

and the corresponding velocity potential is

(6.113)

A sketch of the streamlines for this flow field is shown in Fig 6.26.

The velocity components can be obtained from either Eq 6.112 or 6.113 as

(6.114)

and

(6.115)

On the surface of the cylinder it follows from Eq 6.114 and 6.115 that and

As shown by the figure in the margin, the maximum velocity occurs at the top and bottom of the cylinder and has a magnitude of twice the upstream velocity, U As we move

away from the cylinder along the ray 1u ! #p " 2 2 u ! p " 2 the velocity varies, as is illustrated in Fig 6.26.

V6.7 Ellipse

Potential Flow

Viscous Flow

A doublet combined with a uniform flow can be used to rep- resent flow around

a circular cylinder.

0 0 1 2

The pressure distribution on the cylinder surface is obtained from the Bernoulli equation

pressure can be expressed as

(6.116)

A comparison of this theoretical, symmetrical pressure distribution expressed in dimensionless form with a typical measured distribution is shown in Fig 6.27 This figure clearly reveals that only on the upstream part of the cylinder is there approximate agreement between the potential flow and the experimental results Because of the viscous boundary layer that develops on the cylinder, the main flow separates from the surface of the cylinder, leading to the large difference between the theoretical, frictionless fluid solution and the experimental results on the downstream side of the cylinder 1see Chapter 92.

The resultant force 1per unit length2 developed on the cylinder can be determined by integrating the pressure over the surface From Fig 6.28 it can be seen that

■ Figure 6.26 The flow around a circular cylinder.

a

r U

θ

■ Figure 6.27 A comparison of ical (inviscid) pressure distribution on the surface of a circular cylinder with typical

Experimental

Theoretical (inviscid)

V6.8 Circular cylinder with separation

V6.9 Forces on suspended ball

c06DifferentialAnalysisofFluidFlow.qxd 2/17/12 4:42 PM Page 314

3 / 40

Dòng th∏ 2D:

Hàm dòng

~

V u, v Xét dòng không quay, PT liên tˆc cho i∑u kiªn b£o toàn th∫ tích:

u x

V™y, có th∫ mô t£ 2 bi∏n u x, y , v x, y b¨ng 1 hàm x, y

hàm vô h˜Óng x, y ˜Òc gÂi là hàm dòng

4 / 40

Tìm hàm vô h˜Óng x, y :

u x v y

v™y có th∫ mô t£ hai bi∏n u x, y , v x, y b¨ng 1 hàm x, y

x, y ˜Òc gÂi là th∏ v™n tËc

5 / 40

Dòng th∏ 2D:

Dòng không nén ˜Òc + không quay

u

x v

y u x

Dòng th∏ 2D:

PT Bernoulli cho dòng th∏ trong m∞t phØng n¨m ngang

Nh˜ v™y, ã có u và v khi bi∏t ho∞c ,

2 r ~ V V cho dòng th∏ ~ Dòng Ín ‡nh V ~

t 0, vì v™y 1

d2 < d

V2 > V

V2

V1V

9 / 40

closely spaced streamlines, shown in Fig 6.8a The lower streamline is designated and the upper

one Let dq represent the volume rate of flow 1per unit width perpendicular to the x–y

plane2 passing between the two streamlines Note that flow never crosses streamlines, since bydefinition the velocity is tangent to the streamline From conservation of mass we know that the

inflow, dq, crossing the arbitrary surface AC of Fig 6.8a must equal the net outflow through surfaces

Thus, the volume rate of flow, q, between two streamlines such as and of Fig 6.8b can be

determined by integrating Eq 6.39 to yield

as shown by the figure in the margin

Substitution of these expressions for the velocity components into Eq 6.41 shows that thecontinuity equation is identically satisfied The stream function concept can be extended toaxisymmetric flows, such as flow in pipes or flow around bodies of revolution, and to two-dimensional compressible flows However, the concept is not applicable to general three-dimensionalflows using a single stream function

vr!1

r

0c0u vu! "0c

■Figure 6.8 The flow between two streamlines.

The change in the

value of the stream

c c

c

c c

closely spaced streamlines, shown in Fig 6.8a The lower streamline is designated and the upper

one Let dq represent the volume rate of flow 1per unit width perpendicular to the x–y

plane 2 passing between the two streamlines Note that flow never crosses streamlines, since by definition the velocity is tangent to the streamline From conservation of mass we know that the

inflow, dq, crossing the arbitrary surface AC of Fig 6.8a must equal the net outflow through surfaces

Thus, the volume rate of flow, q, between two streamlines such as and of Fig 6.8b can be

determined by integrating Eq 6.39 to yield

as shown by the figure in the margin.

Substitution of these expressions for the velocity components into Eq 6.41 shows that the continuity equation is identically satisfied The stream function concept can be extended to axisymmetric flows, such as flow in pipes or flow around bodies of revolution, and to two- dimensional compressible flows However, the concept is not applicable to general three-dimensional flows using a single stream function.

vr! 1

r

0c 0u vu! " 0c

c

■ Figure 6.8 The flow between two streamlines.

The change in the value of the stream function is related

to the volume rate

c c

c

cc

closely spaced streamlines, shown in Fig 6.8a The lower streamline is designated and the upper

one Let dq represent the volume rate of flow 1per unit width perpendicular to the x–y

plane 2 passing between the two streamlines Note that flow never crosses streamlines, since by definition the velocity is tangent to the streamline From conservation of mass we know that the

inflow, dq, crossing the arbitrary surface AC of Fig 6.8a must equal the net outflow through surfaces

Thus, the volume rate of flow, q, between two streamlines such as and of Fig 6.8b can be

determined by integrating Eq 6.39 to yield

as shown by the figure in the margin.

Substitution of these expressions for the velocity components into Eq 6.41 shows that the continuity equation is identically satisfied The stream function concept can be extended to axisymmetric flows, such as flow in pipes or flow around bodies of revolution, and to two- dimensional compressible flows However, the concept is not applicable to general three-dimensional flows using a single stream function.

vr! 1

r

0c 0u vu! " 0c

c

■ Figure 6.8 The flow between two streamlines.

The change in the value of the stream function is related

to the volume rate

c c

c

cc

We still have two variables, u and to deal with, but they must be related in a special way as indicated by Eq 6.36 This equation suggests that if we define a function called the stream

(6.37)

then the continuity equation is identically satisfied This conclusion can be verified by simply

substituting the expressions for u and into Eq 6.36 so that

Thus, whenever the velocity components are defined in terms of the stream function we know that conservation of mass will be satisfied Of course, we still do not know what is for a particular problem, but at least we have simplified the analysis by having to determine only one unknown function, rather than the two functions, and

Another particular advantage of using the stream function is related to the fact that lines

along which is constant are streamlines Recall from Section 4.1.4 that streamlines are lines in the flow field that are everywhere tangent to the velocities, as is illustrated in Fig 6.7 and the figure in the margin It follows from the definition of the streamline that the slope at any point along a streamline is given by

The change in the value of as we move from one point to a nearby point

is given by the relationship:

Along a line of constant we have so that and, therefore, along a line of constant

which is the defining equation for a streamline Thus, if we know the function we can plot lines of constant to provide the family of streamlines that are helpful in visualizing the pattern c

compo-of a stream function.

V y

x y

Dòng th∏ 2D:

i∫m nguÁn- i∫m gi∏ng

= constantφ

q 0 cho i∫m nguÁn

q 0 cho i∫m gi∏ng

1

rÁi

q 2⇡r ln r và,

q 2⇡r ✓

12 / 40

u ✓ 2⇡r

V™y,

2⇡✓ , và 2⇡ ln r

13 / 40

Dòng th∏ 2D:

L˜Ông c¸c

6.5 Some Basic, Plane Potential Flows 307

which can be rewritten as

since the tangent of an angle approaches the value of the angle for small angles.

The so-called doublet is formed by letting the source and sink approach one another while increasing the strength so that the product remains constant In this case, since Eq 6.94 reduces to

(6.95)

where K, a constant equal to is called the strength of the doublet The corresponding velocity

potential for the doublet is

(6.96)

Plots of lines of constant reveal that the streamlines for a doublet are circles through the origin

tangent to the x axis as shown in Fig 6.23 Just as sources and sinks are not physically realistic

entities, neither are doublets However, the doublet when combined with other basic potential flows

x y

A doublet is formed

by letting a source and sink approach one another.

c06DifferentialAnalysisofFluidFlow.qxd 2/17/12 4:41 PM Page 307

6.5 Some Basic, Plane Potential Flows 307

which can be rewritten as

since the tangent of an angle approaches the value of the angle for small angles

The so-called doubletis formed by letting the source and sink approach one another while increasing the strength so that the product remains constant In this case,since Eq 6.94 reduces to

(6.95)

where K, a constant equal to is called the strength of the doublet The corresponding velocity

potential for the doublet is

(6.96)

Plots of lines of constant reveal that the streamlines for a doublet are circles through the origin

tangent to the x axis as shown in Fig 6.23 Just as sources and sinks are not physically realistic

entities, neither are doublets However, the doublet when combined with other basic potential flows

2p tan#1 a2ar sin u r2#a2b

tan a#2pcm b!2ar sin u

Source

Sink

■Figure 6.23 Streamlines for a doublet.

x y

14 / 40

Dòng th∏ 2D:

ChÁng nh™p: [dòng ∑u + nguÁn= dòng qua ’n˚a v™t r≠n’]

U

Stagnation point

y

x

r

θ Source

uniform flow source @ origin

2⇡U

15 / 40

Dòng th∏ 2D:

ChÁng nh™p: [dòng ∑u + nguÁn + gi∏ng= dòng qua ’v™t r≠n’]

(b) (a)

x

y U

h h

uniform flow source @ a,0 sink @ a,0

@ i∫m d¯ng bên trái u r source U u r sink

@ i∫m d¯ng bên ph£i u r sink U u r source

0 trên b∑ m∞t v™t th∫ (tìm ˜Òc b¨ng ph˜Ïng pháp trial-error)

16 / 40

Dòng th∏ 2D:

ChÁng nh™p: [dòng ∑u + l˜Ông c¸c= dòng qua trˆ tròn tænh]

a

r U

θ

CË th∫ Rankine:

a 0, nguÁn + gi∏ng l˜Ông c¸c, oval trˆ!

a 2 q 0 U trên b∑ m∞t trˆ:

u ✓s 2U sin ✓, và,

2 ⇢U

2 1 4 sin 2 ✓ (Hãy t¸ ch˘ng minh các công th˘c ó!)

17 / 40

Dòng th∏ 2D:

ChÁng nh™p: [dòng qua trˆ tròn + xoáy t¸ do= dòng qua trˆ tròn quay ∑u]

Thành ph¶n v™n tËc ti∏p tuy∏n b∑ m∞t: u ✓s 2U sin ✓

4⇡Ua nh˜ trên hình!

18 / 40

Dòng th∏ 2D:

L¸c tác dˆng lên b∑ m∞t trˆ tròn

a

x y

F kdòng 0 F dòng 0

CT Kutta-Joukowski, hiªn t˜Òng Magnus

19 / 40

Dòng th¸c:

Khái niªm l¸c nâng - l¸c c£n

Even with a steady, uniform upstream flow, the flow in the vicinity of an object may be steady Examples of this type of behavior include the flutter that is sometimes found in the flow past airfoils 1wings2, the regular oscillation of telephone wires that “sing” in a wind, and the irreg- ular turbulent fluctuations in the wake regions behind bodies.

un-The structure of an external flow and the ease with which the flow can be described and alyzed often depend on the nature of the body in the flow Three general categories of bodies are shown in Fig 9.2 They include 1a2 two-dimensional objects 1infinitely long and of constant cross- sectional size and shape2, 1b2 axisymmetric bodies 1formed by rotating their cross-sectional shape about the axis of symmetry 2, and 1c2 three-dimensional bodies that may or may not possess a line

an-or plane of symmetry In practice there can be no truly two-dimensional bodies—nothing extends

to infinity However, many objects are sufficiently long so that the end effects are negligibly small Another classification of body shape can be made depending on whether the body is stream- lined or blunt The flow characteristics depend strongly on the amount of streamlining present In

general, streamlined bodies 1i.e., airfoils, racing cars, etc.2 have little effect on the surrounding fluid,

compared with the effect that blunt bodies 1i.e., parachutes, buildings, etc.2 have on the fluid ally, but not always, it is easier to force a streamlined body through a fluid than it is to force a similar-sized blunt body at the same velocity There are important exceptions to this basic rule.

Usu-9.1.1 Lift and Drag Concepts

When any body moves through a fluid, an interaction between the body and the fluid occurs; this effect can be given in terms of the forces at the fluid–body interface These forces can be described

in terms of the stresses—wall shear stresses on the body, due to viscous effects and normal

stresses due to the pressure, p Typical shear stress and pressure distributions are shown in Figs 9.3a and 9.3b Both and p vary in magnitude and direction along the surface.

It is often useful to know the detailed distribution of shear stress and pressure over the face of the body, although such information is difficult to obtain Many times, however, only the integrated or resultant effects of these distributions are needed The resultant force in the direction

sur-of the upstream velocity is termed the drag, d, and the resultant force normal to the upstream

y

z

y

x x

y

(c) (b)

U

(a)

p > 0

Pressuredistribution

p < 0

A body interacts

with the

surround-ing fluid through

pressure and shear

stresses.

"

(c)

9.1 General External Flow Characteristics 483

velocity is termed the lift, as is indicated in Fig 9.3c For some three-dimensional bodies there

may also be a side force that is perpendicular to the plane containing and The resultant of the shear stress and pressure distributions can be obtained by integrating the

effect of these two quantities on the body surface as is indicated in Fig 9.4 The x and y nents of the fluid force on the small area element dA are

theo-tw

u

l !! dF y! "! p sin u dA #! tw cos u dA

d !! dF x!! p cos u dA #! tw sin u dA

dF y! "1 p dA2 sin u # 1t w dA2 cos u

dF x!1 p dA2 cos u # 1t w dA2 sin u

l

dl,

■Figure 9.4 Pressure and shear forces on a small element of the surface of a body.

Pressure-sensitive paint For many years, the conventional

method for measuring surface pressure has been to use static

by hoses from the holes to a pressure measuring device

Pressure-sensitive paint (PSP) is now gaining acceptance as an alternative

a luminescent compound that is sensitive to the pressure on it and

video imaging equipment Thus, it provides a quantitative

mea-sure of the surface presmea-sure One of the biggest advantages of PSPover the entire surface, as opposed to discrete points PSP also hasstatic pressure port holes are small, they do alter the surface anddition, the use of PSP eliminates the need for a large number ofsurements to be made in less time and at a lower cost

It is seen that both the shear stress and pressure force contribute to the lift and drag, sincefor an arbitrary body is neither zero nor along the entire body The exception is a flat platealigned either parallel to the upstream flow or normal to the upstream flow as

9.1 General External Flow Characteristics 483

velocity is termed the lift, as is indicated in Fig 9.3c For some three-dimensional bodies there

may also be a side force that is perpendicular to the plane containing and The resultant of the shear stress and pressure distributions can be obtained by integrating the

effect of these two quantities on the body surface as is indicated in Fig 9.4 The x and y nents of the fluid force on the small area element dA are

tw

u

l !! dF y! "! p sin u dA #! tw cos u dA

d !! dF x!! p cos u dA #! tw sin u dA

dF y! "1 p dA2 sin u # 1tw dA2 cos u

dF x!1 p dA2 cos u # 1t w dA2 sin u

l

dl,

■Figure 9.4 Pressure and shear forces on a small element of the surface of a body.

Pressure-sensitive paint For many years, the conventional

method for measuring surface pressure has been to use static

pressure taps consisting of small holes on the surface connectedsensitive paint (PSP) is now gaining acceptance as an alternative

a luminescent compound that is sensitive to the pressure on it andvideo imaging equipment Thus, it provides a quantitative mea-

sure of the surface pressure One of the biggest advantages of PSPover the entire surface, as opposed to discrete points PSP also hasstatic pressure port holes are small, they do alter the surface anddition, the use of PSP eliminates the need for a large number ofsurements to be made in less time and at a lower cost

It is seen that both the shear stress and pressure force contribute to the lift and drag, sincefor an arbitrary body is neither zero nor along the entire body The exception is a flat platealigned either parallel to the upstream flow 1u ! 90°2or normal to the upstream flow 1u ! 02as

90°

u

c09FlowoverImmersedBodies.qxd 2/20/12 9:30 PM Page 483

20 / 40