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AERODYNAMICS: SOME FUNDAMENTAL PRINCIPLES AND EQUATIONS 141

The stream function ~ defined above applies to both compressible and incompressible flow. Now consider the case of incompressible flow only, where p = constant. Equation (2.134) can be written as

 

v=a(Ji/p)

(2.140)

 

an

 

 

We define a new stream function, for incompressible flow only, as 1/1 == Ji/ p.

Then Eq. (2.140) becomes

 

 

a1/1

 

 

v=-

 

 

an

 

and Eqs. (2.138) and (2.139) become

 

 

 

 

 

a1/1

(2.141a)

 

u= -

 

ay

 

 

a1/1

(2.141b)

 

v= --

 

ax

 

 

 

 

 

 

 

1 al/l

(2.142a)

 

 

V= --

 

 

r

r ao

 

and

 

a1/1

(2.142b)

vo = --

 

 

 

ar

 

 

 

 

 

 

The incompressible stream function 1/1 has characteristics analogous to its more general compressible counterpart Ji. For example, since Ji(x, y) = c is the equation of a streamline, and since p is a constant for incompressible flow, then I/I(x, y) == Ji/ p = constant is also the equation for a streamline (for incompressible flow only). In addition, since I1Ji is mass flow between two streamlines (per unit depth perpendicular to the page), and since p is mass per unit volume, then physically 111/1 = I1Ji/ p represents the volume flow (per unit depth) between two streamlines. In SI units, 111/1 is expressed as cubic meters per second per meter perpendicular to the page, or simply m2 / s.

In summary, the concept of the stream function is a powerful tool in aerodynamics, for two primary reasons. Assuming that Ji(x, y) [or I/I(x, y)] is known through the two-dimensional flow field, then:

1. Ji = constant (or 1/1 = constant) gives the equation of a streamline.

2.The flow velocity can be obtained by differentiating Ji (or 1/1), as given by Eqs. (2.138) and (2.139) for compressible flow and Eqs. (2.141) and (2.142) for incompressible flow.

142 FUNDAMENTALS OF AERODYNAMICS

We have not yet discussed how Jj(x,y) [or «f!(x,y)] can be obtained in the first place; we are assuming that it is known. The actual determination of the stream function for various problems is discussed in Chap 3.

2.1S VELOCITY POTENTIAL

Recall from Sec, 2.12 that an irrotational flow is defined as a flow where the vorticity is zero at every point. From Eq. (2.120), for an irrotational flow,

~=VxV=O

(2.143)

Consider the following vector identity: if 4> is a scalar function, then

v x (V4» = 0

(2.144)

i.e., the curl of the gradient of a scalar function is identically zero. Comparing Eqs. (2.143) and (2.144), we see that

I

V=V4> I

(2.145)

 

Equation (2.145) states that for an irrotational flow, there exists a scalar function 4> such that the velocity is given by the gradient of 4>. We denote 4> as the velocity potential. 4> is a function of the spatial coordinates; i.e., 4> = 4> (x, y, z), or 4> = 4>(r, fJ, z), or 4> = 4>(r, fJ, <1». From the definition of the gradient in cartesian coordinates given by Eq. (2.16), we have, from Eq. (2.145),

••

a4>

a4>.

a4>

(2.146)

Ul+ vJ+ Wk=-l+- J+- k

 

ax

ay

az

 

The coefficients of like unit vectors must be the same on both sides of Eq. (2.146). Thus, in cartesian coordinates,

a4>

a4>

a4>

 

(2.147)

u= -

v=-

w=-

ax

ay

az

 

 

 

 

 

 

 

In a similar fashion, from the definition of the gradient in cylindrical and spherical coordinates given by Eqs. (2.17) and (2.18), we have, in cylindrical coordinates,

 

v = a4>

1 a4>

a4>

(2.148)

 

V ---

V= -

 

r ar

e - r afJ

Z az

 

and in spherical coordinates,

 

 

 

 

a4>

1 a4>

1 a4>

(2.149)

V= -

V ---

V<j>=---

r

ar

e - r afJ

r sin fJ a<1>

 

f

AERODYNAMICS: SOME FUNDAMENTAL PRINCIPLES AND EQUATIONS 143

The velocity potential is analogous to the stream function in the sense that derivatives of ¢ yield the flow-field velocities. However, there are distinct differences between ¢ and .f (or t/I):

1.The flow-field velocities are obtained by differentiating ¢ in the same direction as the velocities [see Eqs. (2.147) to (2.149)], whereas.f (or t/I) is differentiated normal to the velocity direction [see Eqs. (2.138) and (2.139), or Eqs. (2.141) and (2.142)].

2.The velocity potential is defined for irrotational flow only. In contrast, the stream function can be used in either rotational or irrotational flows.

3.The velocity potential applies to three-dimensional flows, whereas the stream function is defined for two-dimensional flows only.t

When a flow field is irrotational, hence allowing a velocity potential to be defined, there is a tremendous simplification. Instead of dealing with the velocity components (say, u, v, and w) as unknowns, hence requiring three equations for these three unknowns, we can deal with the velocity potential as one unknown, therefore requiring the solution of only one equation for the flow field. Once ¢ is known for a given problem, the velocities are obtained directly from Eqs. (2.147) to (2.149). This is why, in theoretical aerodynamics, we make a distinction between irrotational and rotational flows and why the analysis of irrotational flows is simpler than that of rotational flows.

Because irrotational flows can be described by the velocity potential ¢, such flows are called potential flows.

In this section, we have not yet discussed how ¢ can be obtained in the first place; we are assuming that it is known. The actual determination of ¢ for various problems is discussed in Chaps. 3, 6, 11, and 12.

2.16 RELATIONSHIP BETWEEN THE STREAM FUNCTION AND VELOCITY POTENTIAL

In Sec. 2.15, we demonstrated that for an irrotational flow, V=V¢. At this stage, take a moment and review some of the nomenclature introduced in Sec. 2.2.5 for the gradient of a scalar field. We see that a line of constant ¢ is an isoline of ¢; since ¢ is the velocity potential, we give this isoline a specific name, equipotential line. In addition, a line drawn in space such that V¢ is tangent at every point is defined as a gradient line; however, since V¢ = V, this gradient line is a streamline. In turn, from Sec. 2.14, a streamline is a line of constant .f (for a two-dimensional floW). Because gradient lines and isolines are perpendicular (see Sec. 2.2.5,

t Ji (or"') can be defined for axisymmetric flows, such as the flow over a cone at zero degrees angle of attack. However, for such flows, only two spatial coordinates are needed to describe the flow field (see Chap. 6).

144 FUNDAMENTALS OF AERODYNAMICS

Gradient of a Scalar Field), then equipotential lines = constant) and streamlines (Iii = constant) are mutually perpendicular.

To illustrate this result more clearly, consider a two-dimensional, irrotational, incompressible flow in cartesian coordinates. For a streamline, I/J(x, y) = constant. Hence, the differential of I/J along the streamline is zero; i.e.,

aI/J

aI/J

(2.150)

dI/J = -

dx +- dy = 0

ax

ay

 

From Eqs. (2.141a and b), Eq. (2.150) can be written as

 

dI/J = - v dx + u dy = 0

(2.151)

Solve Eq. (2.151) for dy/ dx, which is the slope of the I/J = constant line, i.e., the slope of the streamline:

(:~)~~const

u

 

 

 

 

v

 

(2.152)

 

 

 

 

Similarly, for an equipotential line, ¢(x, y) = constant. Along this line,

 

0

(2.153)

= -

dx +- dy =

ax

ay

 

 

 

From Eq. (2.147), Eq. (2.153) can be written as

 

 

= u dx + v dy = 0

 

(2.154)

Solving Eq. (2.154) for dy/ dx, which is the slope of the

¢ = constant line, i.e.,

the slope of the equipotential line, we obtain

 

 

 

 

 

u

 

(2.155)

 

 

v

 

 

 

 

 

Combining Eqs. (2.152) and (2.155), we have

 

 

 

(dY)

 

1

 

(2.156)

dx ~~const

(dy/ dX)",~const

 

Equation (2.156) shows that the slope of a I/J = constant line is the negative reciprocal of the slope of a ¢ = constant line, i.e., streamlines and equipotential lines are mutually perpendicular.

2.17SUMMARY

Return to the road map for this chapter, as given in Fig. 2.1. We have now covered both the left and right branches of this map and are ready to launch into the solution of practical aerodynamic problems in subsequent chapters. Look at each block in Fig. 2.1; let your mind flash over the important equations and concepts represented by each block. If the flashes are dim, return to the appropriate sections of this chapter and review the material until you feel comfortable with these aerodynamic tools.

AERODYNAMICS: SOME FUNDAMENTAL PRINCIPLES AND EQUATIONS 145

For your convenience, the most important results are summarized below:

Basic Flow Equations

Continuity equation

 

:t 1# p d'V + #p V . dS = 0

(2.39)

 

 

 

 

'v

S

 

or

ap

 

(2.43)

-+V·(pV)=O

 

at

 

 

or

Dp

 

(2.99)

-+pV ·v=o

Dt

Momentum equation

:t 1# pV d'V+# (pV·dS)V= -# P ds+1# pf d'V+ Fviscous (2.55)

'V S S

or

a(pu)

 

_

ap

 

 

m:

--+ V .

(pu V) - --+ pj,

+ (.'1'Jviscous

 

at

 

 

ax"

 

 

 

a(pv)

 

_

ap

 

 

m:

 

--+ V . (pvV) -

--+ piv

 

+ (~\.Liscous

 

at

 

 

ay .

 

 

 

a(pw)

 

 

ap

 

 

m:

 

--+ V .

(pwV) = --+ p/z

+ (~z)viscous

 

at

 

 

az

 

 

 

or

Du

 

ap

 

 

 

 

-- --+p.l" + (gi )

viscous

 

P Dt -

ax

'Jx

x

 

Dw ap

p Dt = - az + p/z + (giz )viscous

Energy equation

= #.f qp d'V + Qviscous - #p V . dS

T S

+ 1# p(f· V) d'V + "'viscous

(2.61a)

(2.61b)

(2.61c)

(2.104a)

(2.104b)

(2.104c)

(2.86)

·V

146 FUNDAMENTALS OF AERODYNAMICS

 

 

 

(2.87)

or

pq - V . (pV) + p(r· V) + Q~iscous+ W~iSCOUS

(2.105)

 

 

 

 

 

 

 

 

Substantial derivative

 

 

 

 

D

a

(2.95)

 

-==-+(V·V)

 

Dt

at

 

local convective derivative derivative

A streamline is a curve whose tangent at any point is in the direction of the velocity vector at that point. The equation of a streamline is given by

dsxV=O

(2.106)

or, in cartesian coordinates,

 

wdyvdz =0

(2.108a)

u dzwdx=O

(2.108b)

vdx- u dy =0

(2.108c)

 

 

The vorticity ~ at any given point is equal to twice the angular velocity

of a fluid element w, and both are related to the velocity field by

 

~=2w=VxV

(2.120)

When V x V ~ 0, the flow is rotational. When V x V = 0, the flow is irrotational.

Circulation r is related to lift and is defined as

 

(2.127)

Circulation is also related to vorticity via

 

r== - tv. ds = - ff(V x V) . dS

(2.128)

s

AERODYNAMICS: SOME FUNDAMENTAL PRINCIPLES AND EQUATIONS 147

df

(V xV)·n==--

(2.129) I

dS

The stream function Ijj is defined such that Ijj(x, y) = constant is the equation of a streamline, and the difference in the stream function between two streamlines, !1Ijj, is equal to the mass flow between the streamlines. As a consequence of this definition, in cartesian coordinates,

aIj;

pu=-

(2.138a)

ay

 

aIj;

(2.138b)

pv= --

ax

 

and in cylindrical coordinates,

 

1 aIjj

 

pVr =; ae

(2.139a)

 

aIj;

(2.13,9b)

pVe=--

ar

 

For incompressible flow, Ij; == Ijj/ p is defined such that

Ij;(x, y) = constant

denotes a streamline and !1Ij; between two streamlines is equal to the volume flow between these streamlines. As a consequence of this definition, in cartesian coordinates,

aIj;

u= -

(2.141a)

 

ay

 

 

aIj;

(2.141b)

v= --

 

ax

 

and in cylindrical coordinates,

 

 

 

1 aIj;

(2.142a)

V= --

r

rae

 

 

aIj;

(2.142b)

Ve = --

 

ar

 

The stream function is valid for both rotational and irrotational flows, but it is restricted to two-dimensional flows only.

The velocity potential cP is defined for irrotational flows only, such that

V=VcP

(2.145)

148 FUNDAMENTALS OF AERODYNAMICS

In cartesian coordinates,

 

 

acp

acp

 

 

acp

 

u= -

v= -

w=-

 

 

ax

ay

 

 

az

In cylindrical coordinates,

 

 

 

 

 

 

v = acp

 

v =acp

 

r

ar

 

 

Z

az

In spherical coordinates,

 

 

 

 

 

 

acp

 

1 acp

V<1>-

 

1_ acp

v=-

V ---

r sin (} a<l>

r

ar

e -

r a(}

 

An irrotational flow is called a potential flow.

(2.147)

(2.148)

(2.149)

A line of constant cp is an equipotential line. Equipotential lines are perpendicular to streamlines (for two-dimensional irrotational flows).

PROBLEMS

2.1.Consider a body of arbitrary shape. If the pressure distribution over the surface of the body is constant, prove that the resultant pressure force on the body is zero. [Recall that this fact was used in Eq. (2.68).]

2.2.Consider an airfoil in a wind tunnel (Le., a wing that spans the entire test section).

Prove that the lift per unit span can be obtained from the pressure distributions on the top and bottom walls of the wind tunnel (i.e., from the pressure distributions on the walls above and below the airfoil).

2.3.Consider a velocity field where the x and y components of velocity are given by u=CX/(X2+y2) and v=cY/(X2+y2), where c is a constant. Obtain the equations

of the streamlines.

2.4. Consider a velocity field where the x and y components of velocity are given by u = cy/(x 2+ y2) and v = -cx/(x2+ y2), where c is a constant. Obtain the equations of the streamlines.

2.5.Consider a velocity field where the radial and tangential components of velocity are Vr = 0 and Ve = cr, respectively, where c is a constant. Obtain the equations of the

streamlines.

2.6. Consider a velocity field where the x and y components of velocity are given by u = ex and v = -cy, where c is a constant. Obtain the equations of the streamlines.

2.7.The velocity field given in Prob. 2.3 is called source flow. For source flow, calculate:

(a)The time rate of change of the volume of a fluid element per unit volume

(b)The vorticity

Hint: It is simpler to convert the velocity components to polar coordinates and deal with a polar coordinate system.

AERODYNAMICS: SOME FUNDAMENTAL PRINCIPLES AND EQUATIONS 149

2.8.The velocity field given in Proh. 2.4 is called vortex flow. For vortex flow, calculate:

(a)The time rate of change of the volume of a fluid element per unit volume

(b)The vorticity

Hint: Again, for convenience use polar coordinates.

2.9.Is the flow field given in Prob. 2.5 irrotational? Prove your answer.

2.10.Consider a flow field in polar coordinates, where the stream function is given as t/I = t/I(r, 8). Starting with the concept of mass flow between two streamlines, derive Eqs. (2.139a and b).

2.11.Assuming the velocity field given in Prob. 2.6 pertains to an incompressible flow, calculate the stream function and velocity potential. Using your results, show that lines of constant 4> are perpendicular to lines of constant t/I.