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C HAPTER 1
I NTRODUCTION TO FLUID FLOW
1.1 I NTRODUCTION
Fluid flows play a crucial role in a vast variety of natural phenomena and man- made systems. The life-cycles of stars, the creation of atmospheres, the sounds we hear, the vehicles we ride, the systems we build for flight, energy generation and propulsion all depend in an important way on the mechanics and thermody- namics of fluid flow. The purpose of this course is to introduce students in Aeronautics and Astronautics to the fundamental principles of fluid mechanics with emphasis on the development of the equations of motion as well as some of the analytical tools from calculus needed to solve practically important problems involving flows in channels along walls and over lifting bodies.
1.2 C ONSERVATION OF MASS
Mass is neither created nor destroyed. This basic principle of classical physics is one of the fundamental laws governing fluid motion and is a good departure point for our introductory discussion.
Figure 1.1 below shows an infinitesimally small stationary, rectangular control
volume through which a fluid is assumed to be moving. A control vol- ume of this type with its surface fixed in space is called an Eulerian control
volume. The fluid velocity vector has components in the
directions and the fluid density is. In a general, unsteady, com- pressible flow, all four flow variables may depend on position and time. The law of conservation of mass over this control volume is stated as
6 x 6 y 6 z
U = ( U V, ,W)
x = ( x y z, , ) l
Conservation of mass
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(1.1)
Figure 1.1 Fixed control volume in a moving fl uid. The arrows shown denote fl uxes of mass on the various faces of the control volume.
Consider a pair of faces perpendicular to the x-axis. The vector mass flux is
with units. The rate of mass flow in through the face at is the
flux in the x-direction times the area. The mass flow rate out through
Rate of mass accumulation inside the control ª« volume ®«
¨ ¬ (^) Rate of mass flow into the control ª« volume ®«
¨ ¬ (^) Rate of mass flow out of the control ª« volume ®«
x
z
y ( x y z, , )
( x+ 6 x , y+ 6 y,z+ 6 z)
6 x
6 y
lU (^) x
lU (^) x+ 6 x
6 z
U(x,y,z,t)
V(x,y,z,t)
W(x,y,z,t)
l ( x y z t , , , )
lW (^) z
lW (^) z+ 6 z
lV (^) y
lV (^) y+ 6 y
U
lU mass/area-time x lU (^) x 6 y 6 z
Conservation of mass
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. (1.6)
Note that equation (1.6) applies to both steady and unsteady incompressible flow.
1.2.2 I NDEX NOTATION AND THE EINSTEIN CONVENTION
For convenience, vector components are often written with subscripts. This is called index notation and one makes the following replacements.
(1.7)
In index notation equation (1.5) is concisely written in the form
(1.8)
where the subscript refers to the - th vector component.
Vector calculus is an essential tool for developing the equations that govern com- pressible flow and summed products such as (1.8) arise often. Notice that the sum in (1.8) involves a repeated index. The theory of Relativity is another area where such sums arise often and when Albert Einstein was developing the special and general theory he too recognized that such sums always involve an index that is repeated twice but never three times or more. In effect the presence of repeated indices implies a summation process and the summation symbol can be dropped. To save effort and space Einstein did just that and the understanding that a repeated index denotes a sum has been known as the Einstein convention ever since. Using the Einstein convention (1.8) becomes
(1.9)
,U
,x
-------^ ,V
,y
-------^ ,W
,z
( x y z, , ) A( x 1 , x 2 ,x 3 ) ( U V, ,W) A( U 1 , U 2 ,U 3 )
,l ,t
,( lU (^) i) ,x (^) i
i = 1
3
i
,l ,t
,( lU (^) i) ,x (^) i
Particle paths, streamlines and streaklines in 2-D steady flow
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Remember, the rule of thumb is that a single index denotes a vector component and a repeated index represents a sum. Three or more indices the same means that there is a mistake in the equation somewhere! The upper limit of the sum is 1, 2 or 3 depending on the number of space dimensions in the problem. In the notation of vector calculus (1.9) is written
(1.10)
and (1.6) is. Vector notation has the advantage of being concise and independent of the choice of coordinates but is somewhat abstract. The main advantage of index notation is that it expresses precisely what differentiation and summation processes are being done in a particular coordinate system. In Carte- sian coordinates, the gradient vector operator is
(1.11)
The continuity equation as well as the rest of the equations of fluid flow are given in cylindrical and spherical polar coordinates in Appendix 2.
1.3 PARTICLE PATHS , STREAMLINES AND STREAKLINES IN 2-D
STEADY FLOW
Let’s begin with a study of fluid flow in two dimensions. Figure 1.2 shows the theoretically computed flow over a planar, lifting airfoil in steady, inviscid (non- viscous) flow. The theory used to determine the flow assumes that the flow is irrotational
(1.12)
and that the flow speed is very low so that (1.6) holds.
. (1.13)
A vector field that satisfies (1.12) can always be represented as the gradient of a
scalar potential function therefore
. (1.14)
,l ,t
------+ ¢ •( lU) = 0
¢ • U = 0
,x
------^ ,
,y
------^ ,
,z
¢ × U = 0
¢ • U = 0
q = \ ( x y , )
U = ¢\
Particle paths, streamlines and streaklines in 2-D steady flow
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separation between lines increases indicating a deceleration of the fluid. Near the point of maximum wing thickness on the upper surface, the streamlines come closer together indicating a speed-up of the fluid to greater than the free stream speed. Below the airfoil the streamlines move apart indicating that the fluid slows down. Downstream of the airfoil trailing edge, the flow speed recovers to the freestream value. Notice that the largest velocity changes occur near the wing. In the upper and lower parts of the picture, far away from the wing, the flow is deflected upward by the wing but the distance between streamlines changes little and the corresponding flow speed change is relatively small.
Figure 1.2 (b) depicts streaklines in the flow over the airfoil. These are produced numerically the same way one would produce dye lines in a real flow. Fluid ele- ments that pass through a given point upstream of the airfoil are marked forming a streak. In the figure, alternating bands of fluid are marked light and dark. The flow pattern produced by the streaklines is identical to the streamline pattern but there is one added piece of information. The streakline pattern depicts the inte- grated effect of the flow velocity on the position of the fluid elements that constitute the streak.
One of the popular explanations of how an airfoil produces lift is that adjacent fluid elements on either side of the stagnation streamline ahead of the wing must meet at the trailing edge at the same time. The reasoning then goes as follows; since a particle that travels above the wing must travel farther than the one below, it must travel faster thus reducing the pressure over the wing and producing lift. Figure 1.2 (b) clearly shows that this explanation is completely erroneous. The fluid below the wing is substantially retarded compared to the fluid that passes above the wing.
The figures below show two more streamline examples in a slightly compressible situation with the effects of viscosity included. These are computations of the flow over a wing flap at a free stream speed of approximately 30% of the speed of sound. The flow conditions are the same in both cases except that in the right-hand picture the trailing edge of the main wing (visible in the upper left corner of the picture) has attached to it a small vertical flap called a Gurney flap.
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Figure 1.3 Computed streamlines over a wing fl ap.
The Gurney flap was developed by Dan Gurney in an effort to improve the speed of his new IRL race car prior to the start of the 1971 racing season in Phoenix. The car was disappointingly slow and after several days of testing, the driver Bobby Unser challenged Gurney to find a solution. Gurney decided to add a spoiler to the rear wing by riveting a length of aluminum right-angle to the trailing edge. When Unser tested the car it was just as slow as before and so everyone felt that the idea was a failure. Later Unser confided to Gurney that the reason the car was slow was that the down-force from the flap on the rear wheels was so large that the car was understeering badly. It was clear that with some adjustments to increase the down-force on the front wings the car could be made much faster. For a while the Gurney team deflected questions with the fiction that the flap was purely to increase the structural strength of the wing but eventually the competi- tion got wise and the Gurney flap was widely adopted.
While the Gurney flap increases both the lift and drag of a single wing the effect on a two-element airfoil is to cause the flow over the flap to reattach as shown in Figure 1.3 thereby increasing lift and reducing the overall drag of the wing-flap system. This allows the system to be effective at much higher flap angles.
Gurney flap
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where and are the flow velocity components in the and directions respectively.
The velocity field is assumed to be a smooth function of position. Formally, these equations are solved by integrating the velocity field forward in time.
. (1.18)
The result is a set of parametric functions for the particle coordinates and in
terms of the time, , along a particle path
. (1.19)
The solution of (1.17) can also be expressed as a family of lines derived by elim-
inating between the functions and in (1.19).
. (1.20)
Figure 1.5 Streamlines in steady fl ow. The value of a particular streamline is determined by the coordinates of a point on the streamline.
U V x y
x t( ) x 0 U x t( ( ) ,y t( )) dt 0
t
y t( ) y 0 V x t( ( ) ,y t( )) dt 0
t
x y t
x t( ) = F x( 0 , y 0 ,t) ; y t( ) =G x( 0 , y 0 ,t)
t F G
s = ^ ( x y , )
x
y
( x 0 ,y 0 )
x = F x( 0 , y 0 ,t) y = G x( 0 , y 0 ,t)
s 0 = ^ ( x 0 ,y 0 ) = ^ ( x y , ) t = 0^ U
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This is essentially how the streamlines observed in Figure 1.2(a) and Figure 1. are generated. The value of a particular streamline is determined by the initial conditions
. (1.21)
This is the situation depicted schematically in Figure 1.5.
The streaklines in Figure 1.2(b) were generated by shading a segment of fluid ele-
ments that pass through an initial point during a fixed interval in time.
Selecting a vertical line of initial points well upstream of the airfoil leads to the bands shown in the figure. The length of a segment is directly related to the velocity history of the fluid particles that make up the segment.
The stream function can also be determined as the solution of the first order ordi-
nary differential equation obtained by eliminating between the two particle path equations in (1.17),
. (1.22)
The differential of is
. (1.23)
If we use (1.17) to replace the differentials and in (1.23) the result is
. (1.24)
On a line of constant the differential and so for nonzero ,
the right hand side of (1.24) can be zero only if the expression in parentheses is zero.
Thus the stream function, , can be determined in two ways; either as the solution of a linear, first order PDE,
. (1.25)
s 0 = ^ ( x 0 ,y 0 )
( x 0 ,y 0 )
dt
dy dx
------ V x y(^ , ) U x y( , )
^ ( x y , )
d s ,x
,^
dx ,y
,^
= + dy
dx dy
d s U x y( , ) ,x
,^ (^) V x y( , ) ,y
+ ,^
= £^ ¥^ dt
s = s 0 d s = 0 dt
^ ( x y , )
U • ¢^ U x y( , ) ,x
,^ (^) V x y( , ) ,y
= + ,^ = 0
Particle paths, streamlines and streaklines in 2-D steady flow
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In order to convert equation (1.26) to a perfect differential it must be multiplied by an integrating factor. In general there is no systematic method for finding the
integrating factor and so the analytic solution of (1.26) for general functions
and remains a difficult unsolved problem in mathematics. However we shall see later that in the case of fluid flow the integrating factor can be identified using the equation for conservation of mass.
It was shown by the German mathematician Johann Pfaff in the early 1800’s that
an integrating factor for the expression always exists. That is, for
any choice of smooth functions and , there always exists a function such that,
(1.31)
and the partial derivatives are
(1.32)
A differential expression like is often called a Pfaffian form. In the language of differential geometry it is called a differential 1-form. Pfaffian forms
of higher dimension, say are often encountered in physics and
with certain restrictions on , and an integrating factor can be found. But, the existence of the integrating factor is only assured unconditionally in two dimensions.
Pfaff’s theorem can be understood by noting that the flow patterns formed by the
vector fields with local slopes and are identical whereas the local flow speeds are not since the lengths of individual vectors differ by the factor
. Since the solution of (1.17) clearly exists (we could just integrate the equations forward in time numerically) then the integrating factor must also exist.
U
V
d s = – M( x y , )V x y( , )dx +M x y( , )U x y( , )dy
,x
,^ (^) = – M( x y , )V x y( , )
, y
,^ (^) = M x y( , )U x y( , ) ®
Adx + Bdy +Cdz A x y z( , , ) B x y z( , , ) C x y z( , , )
V U ( MV) ( MU)
M x y( , )
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Perfect differentials and integrating factors are generally covered in an upper level undergraduate course in calculus. Unfortunately the presentation is often cursory and tends to get passed over fairly quickly, often without making the deep con- nection that exists to the analysis of physical problems. In this chapter we see the connection to the theory of fluid flow. In Chapter 2 we will find that the concept of a perfect differential is one of the key tools needed to develop the laws of thermodynamics.
The following sample problem is worked out in quite a bit of detail to help strengthen your understanding.
1.3.3 S AMPLE PROBLEM - THE INTEGRATING FACTOR AND RECONSTRUCTION OF A FUNC- TION FROM ITS PERFECT DIFFERENTIAL
Solve the first order ordinary differential equation
(1.33)
where is an arbitrary smooth function. Rearrange (1.33) to read
. (1.34)
This differential expression fails the cross derivative test
(1.35)
and is not a perfect differential. The integrating factor for (1.34) is known to be
(1.36)
Multiplying (1.34) by (1.36) converts it to a perfect differential.
(1.37)
We now know the partial derivatives of the solution
dy dx ------ y x =^ --H xy( )
H
, ( – yH xy( )) ,y
------------------------------ ,^ (^ x) ,x
M 1
xy +xyH xy( )
d s yH xy(^ ) xy +xyH xy( )
- -----------------------------------dx x xy +xyH xy( ) = +-----------------------------------dy
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Sometimes what appears to be a trick will work. For example you can also get to
the solution (1.42) by simply simply defining a new variable. This allows (1.33) to be solved by separation of variables. All such tricks are equivalent to finding the integrating factor.
In the general case the required integrating factor is not known. For example the equation
(1.43)
comes up in the study of viscous boundary layer flow along a flat plate. We know an integrating factor exists but it has never been found.
Fortunately when it comes to streamline patterns in two dimensions, conservation of mass can be used to determine the integrating factor.
1.3.4 I NCOMPRESSIBLE FLOW IN 2 DIMENSIONS
The flow of an incompressible fluid in 2-D is constrained by the continuity equa- tion (1.6) that in two dimensions reduces to
. (1.44)
This is exactly the condition (1.28) and so in this case the stream function and the velocity field are related by
(1.45)
and is guaranteed to be a perfect differential. The integrating factor is one. Therefore one can always write for 2-D incompressible flow
. (1.46)
If and are known functions, the stream function is determined analytically by integration of (1.46). In practice the easiest way to assign values to the stream- lines is to integrate (1.46) at some reference state well upstream of the airfoil
where the velocity is uniform.
_ = xy
dy dx
------ xy^ y^ y^
+ +^2
2x 2 – xy
, x
,U
,y
+ ,V = 0
U
,y
,^
; V
,x
,^
d s = – Vdx +Udy U V
( U V, ) = ( U ' , 0 )
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1.3.5 I NCOMPRESSIBLE, IRROTATIONAL FLOW IN TWO DIMENSIONS
If the flow is incompressible and irrotational then the velocity field is described by both a stream function (1.45) and a potential function (1.15). The two functions are related by
(1.47)
These are the well known Cauchy-Riemann equations from the theory of complex variables. Solutions of Laplace’s equation (1.16) which is the equation of motion for this class of flows can be determined using the powerful methods of complex analysis. Interestingly, the flow can be solved just from the continuity equation (1.13) without the use of the equation for conservation of momentum. The irrota- tionality condition (1.12) essentially supplants the need for the momentum equation.
1.3.6 C OMPRESSIBLE FLOW IN 2 DIMENSIONS
The continuity equation for the steady flow of a compressible fluid in two dimen- sions is
(1.48)
In this case, the required integrating factor for (1.26) is the density and we can write
(1.49)
The stream function in a compressible flow is proportional to the mass flux with
units and the convergence and divergence of lines in the flow over the flap shown in Figure 1.3 is a reflection of variations in mass flux over different parts of the flow field.
, y
,^ ,\
,x
,x
– ,^^ ,\
,y
,x
,( lU) ,y
l ( x y , )
d s = –l Vdx+ lUdy
mass/area-sec
The substantial derivative
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Remember, according to the Einstein convention, the repeated index denotes a
sum over. Use (1.50) to replace by in (1.52). The result
is called the substantial or material derivative and is usually denoted by.
. (1.53)
The substantial derivative of the velocity is
. (1.54)
The time derivative of any flow variable evaluated on a fluid element is given by a similar formula. For example the time rate of change of the density
of a given fluid particle is:
(1.55)
The substantial derivative is the time derivative of some property of a fluid ele- ment referred to a fixed frame of reference within which the fluid element moves as shown in Figure 1.6 above.
1.5.1 FRAMES OF REFERENCE
Occasionally it is necessary to transform variables between a fixed and moving set of coordinates as shown in Figure 1.7. The transformation of position and velocity is
(1.56)
k = 1 2 3, , d x (^) k dt U (^) k D Dt
D ( ) Dt
-----------^ ,^ ( )
,t
= --------- +U• ¢ ( )
DU (^) i Dt
,U (^) i ,t
--------- U (^) k
,U (^) i ,x (^) k
l ( x 1 ( )t , x 2 ( )t ,x 3 ( )t ,t)
D l Dt
--------^ ,l ,t
------ U (^) k^ ,l ,x (^) k
x' = x – X t( ) y' = y – Y t( ) z' = z – Z t( ) U' = U – X˙^ ( )t V' = V – Y˙^ ( )t W' = W – Z˙^ ( )t
The substantial derivative
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where is the displacement of the moving frame in three
coordinate directions and is the velocity of the frame. Note that in general, the frame may be accelerating.
Figure 1.7 Fixed and moving frames of reference
The substantial derivative of some property of a fluid element, (1.54), (1.55), is sometimes referred to as the “derivative moving with the fluid element”. Do not interpret this as a transformation to a coordinate system that is attached to the fluid element. If for some reason it is actually desirable to make such a transformation the velocity of the particle in that frame is zero and the substantial derivative of velocity is also zero. The substantial derivative of a variable such as the density
of the particle at the origin of the moving coordinates is.
Both the momentum and kinetic energy of a fluid particle of mass depend on the frame of reference. The momentum transforms linearly like the velocity.
(1.57)
The transformation of kinetic energy is a little more complex. In the two coordinate systems the definition of kinetic energy is the same.
(1.58)
X = ( X t( ) , Y t( ) ,Z t( )) d X dt = ( X˙^ ( )t , Y˙^ ( )t ,Z˙^ ( )t )
x
y
z
x’
y’
z’
U
U(x,y,z)
V(x,y,z)
W(x,y,z)
U’
U’(x’,y’,z’)
V’(x’,y’,z’)
W’(x’,y’,z’)
X˙^ ( )t Y˙^ ( )t
Z˙^ ( )t
D l Dt = ,l ,t
m
mU' = mU – md X dt
momentum in moving coordinates momentum in fixed coordinates
kinetic energy in moving coordinates =^1 2 ---m U( ' 2 + V' 2 +W' 2 )
