Fluid Mechanics - Lecture notes - Chapters 1 - 14 PDF

Preview

Summary

chapters 1-14...

Description

Chapter 1 Introduction A fluid is usually defined as a material in which movement occurs continuously under the application of a tangential shear stress. A simple example is shown in Figure 1.1, in which a timber board floats on a reservoir of water.

Figure 1.1 Use of a floating board to apply shear stress to a reservoir surface. If a force, F,is applied to one end of the board, then the board transmits a tangential shear stress, -, to the reservoir surface. The board and the water beneath will continue to move as long as F and -are nonzero, which means that water satisfies the definition of a fluid. Air is another fluid that is commonly encountered in civil engineering applications, but many liquids and gases are obviously included in this definition as well. You are studying fluid mechanics because fluids are an important part of many problems that a civil engineer considers. Examples include water resource engineering, in which water must be delivered to consumers and disposed of after use, water power engineering, in which water is used to generate electric power, flood control and drainage, in which flooding and excess water are controlled to protect lives and property, structural engineering, in which wind and water create forces on structures, and environmental engineering, in which an understanding of fluid motion is a prerequisite for the control and solution of water and air pollution problems. Any serious study of fluid motion uses mathematics to model the fluid. Invariably there are numerous approximations that are made in this process. One of the most fundamental of these approximations is the assumption of a continuum. We will let fluid and flow properties such as mass density, pressure and velocity be continuous functions of the spacial coordinates. This makes it possible for us to differentiate and integrate these functions. However an actual fluid is composed of discrete molecules and, therefore, is not a continuum. Thus, we can only expect good agreement between theory and experiment when the experiment has linear dimensions that are very large compared to the spacing between molecules. In upper portions of the atmosphere, where air molecules are relatively far apart, this approximation can place serious limitations on the use of mathematical models. Another example, more relevant to civil engineering, concerns the use of rain gauges for measuring the depth of rain falling on a catchment. A gauge can give an accurate estimate only if its diameter is very large compared to the horizontal spacing between rain droplets. Furthermore, at a much larger scale, the spacing between rain gauges must be small compared to the spacing between rain clouds. Fortunately, the assumption of a continuum is not usually a serious limitation in most civil engineering problems.

1.2

Chapter 1 — Introduction

Fluid Properties The mass density, ', is the fluid mass per unit volume and has units of kg/m3. Mass density is a function of both temperature and the particular fluid under consideration. Most applications considered herein will assume that ' is constant. However, incompressible fluid motion can occur in which ' changes throughout a flow. For example, in a problem involving both fresh and salt water, a fluid element will retain the same constant value for ' as it moves with the flow. However, different fluid elements with different proportions of fresh and salt water will have different values for ', and ' will not have the same constant value throughout the flow. Values of 'for some different fluids and temperatures are given in the appendix. The dynamic viscosity, µ , has units of kg / ( m  s )  N  s / m 2 * and is the constant of proportionality between a shear stress and a rate of deformation. In a Newtonian fluid, µ is a function only of the temperature and the particular fluid under consideration. The problem of relating viscous stresses to rates of fluid deformation is relatively difficult, and this is one of the few places where we will substitute a bit of hand waving for mathematical and physical logic. If the fluid velocity, u , depends only upon a single coordinate, y, measured normal to u , as shown in Figure 1.2, then the shear stress acting on a plane normal to the direction of y is given by -  µ

du dy

(1.1)

Later in the course it will be shown that the velocity in the water beneath the board in Figure 1.1 varies linearly from a value of zero on the reservoir bottom to the board velocity where the water is in contact with the board. Together with Equation (1.1) these considerations show that the shear stress, -, in the fluid (and on the board surface) is a constant that is directly proportional to the board velocity and inversely proportional to the reservoir depth. The constant of proportionality is µ . In many problems it is more convenient to use the definition of kinematic viscosity   µ /'

(1.2)

Figure 1.2 A velocity field in which u changes only with the in which the kinematic viscosity, , has units of m2/s. coordinate measured normal to Values of µ and  for some different fluids and the direction of u . temperatures are given in the appendix.

*

A Newton, N, is a derived unit that is related to a kg through Newton's second law, F Thus, N kg m /s 2 .

ma .

Chapter 1 — Introduction

1.3

Surface tension, ), has units of N / m  kg / s 2 and is a force per unit arc length created on an interface between two immiscible fluids as a result of molecular attraction. For example, at an air-water interface the greater mass of water molecules causes water molecules near and on the interface to be attracted toward each other with greater forces than the forces of attraction between water and air molecules. The result is that any curved portion of the interface acts as though it is covered with a thin membrane that has a tensile stress ). Surface tension allows a needle to be floated on a free surface of water or an insect to land on a water surface without getting wet. For an example, if we equate horizontal pressure and surface tension forces on half of the spherical rain droplet shown in Figure 1.3, we obtain  p %r 2  )2%r

(1.3)

in which  p = pressure difference across the interface. This gives the following result for the pressure difference: Figure 1.3 Horizontal pressure and surface tension force acting on half of a spherical rain droplet.

p 

2) r

(1.4)

If instead we consider an interface with the shape of a half circular cylinder, which would occur under a needle floating on a free surface or at a meniscus that forms when two parallel plates of glass are inserted into a reservoir of liquid, the corresponding force balance becomes  p 2r  2)

(1.5)

which gives a pressure difference of p 

) r

(1.6)

A more general relationship between  p and ) is given by p  )

1 1  r1 r2

(1.7)

in which r1 and r2 are the two principal radii of curvature of the interface. Thus, (1.4) has r1  r2  r while (1.6) has r1  r and r2   . From these examples we conclude that (a) pressure differences increase as the interface radius of curvature decreases and (b) pressures are always greatest on the concave side of the curved interface. Thus, since water in a capillary tube has the concave side facing upward, water pressures beneath the meniscus are below atmospheric pressure. Values of ) for some different liquids are given in the appendix. Finally, although it is not a fluid property, we will mention the “gravitational constant” or “gravitational acceleration”, g , which has units of m/s2. Both these terms are misnomers because

1.4

Chapter 1 — Introduction

g is not a constant and it is a particle acceleration only if gravitational attraction is the sole force acting on the particle. (Add a drag force, for example, and the particle acceleration is no longer g . ) The definition of g states that it is the proportionality factor between the mass, M , and weight, W, of an object in the earth's gravitational field. W  Mg

(1.8)

Since the mass remains constant and W decreases as distance between the object and the centre of the earth increases, we see from (1.8) that g must also decrease with increasing distance from the earth's centre. At sea level g is given approximately by g  9.81 m/ s 2

(1.9)

which is sufficiently accurate for most civil engineering applications.

Flow Properties Pressure, p , is a normal stress or force per unit area. If fluid is at rest or moves as a rigid body with no relative motion between fluid particles, then pressure is the only normal stress that exists in the fluid. If fluid particles move relative to each other, then the total normal stress is the sum of the pressure and normal viscous stresses. In this case pressure is the normal stress that would exist in the flow if the fluid had a zero viscosity. If the fluid motion is incompressible, the pressure is also the average value of the normal stresses on the three coordinate planes. Pressure has units of N /m 2  Pa , and in fluid mechanics a positive pressure is defined to be a compressive stress. This sign convention is opposite to the one used in solid mechanics, where a tensile stress is defined to be positive. The reason for this convention is that most fluid pressures are compressive. However it is important to realize that tensile pressures can and do occur in fluids. For example, tensile stresses occur in a water column within a small diameter capillary tube as a result of surface tension. There is, however, a limit to the magnitude of negative pressure that a liquid can support without vaporizing. The vaporization pressure of a given liquid depends upon temperature, a fact that becomes apparent when it is realized that water vaporizes at atmospheric pressure when its temperature is raised to the boiling point. Pressure are always measured relative to some fixed datum. For example, absolute pressures are measured relative to the lowest pressure that can exist in a gas, which is the pressure in a perfect vacuum. Gage pressures are measured relative to atmospheric pressure at the location under consideration, a process which is implemented by setting atmospheric pressure equal to zero. Civil engineering problems almost always deal with pressure differences. In these cases, since adding or subtracting the same constant value to pressures does not change a pressure difference, the particular reference value that is used for pressure becomes immaterial. For this reason we will almost always use gage pressures.*

*

One exception occurs in the appendix, where water vapour pressures are given in kPa absolute. They could, however, be referenced to atmospheric pressure at sea level simply by subtracting from each pressure the vapour pressure for a temperature of 100 C (101.3 kPa).

Chapter 1 — Introduction

1.5

If no shear stresses occur in a fluid, either because the fluid has no relative motion between particles or because the viscosity is zero, then it is a simple exercise to show that the normal stress acting on a surface does not change as the orientation of the surface changes. Consider, for example, an application of Newton's second law to the two-dimensional triangular element of fluid shown in Figure 1.4, in which the normal stresses )x , )y and )n have all been assumed to have different magnitudes. Thus (Fx  ma x gives )x  y  )n  x 2   y 2 cos 

'  x  y ax 2

(1.10)

in which ax  acceleration component in the x direction. Since the triangle geometry gives y

cos 

(1.11)

x 2  y 2

we obtain after inserting (1.11) in (1.10) for cos and dividing by y )x  )n 

' x ax 2

(1.12)

Thus, letting x  0 gives )x  )n

(1.13)

A similar application of Newton's law in the y direction gives Figure 1.4 Normal stress forces acting on a two-dimensional triangular fluid element.

)y  )n

(1.14)

Therefore, if no shear stresses occur, the normal stress acting on a surface does not change as the surface orientation changes. This result is not true for a viscous fluid motion that has finite tangential stresses. In this case, as stated previously, the pressure in an incompressible fluid equals the average value of the normal stresses on the three coordinate planes.

1.6

Chapter 1 — Introduction

Figure 1.5 The position vector, r, and pathline of a fluid particle. Let t = time and r (t )  x (t ) i  y (t ) j  x (t ) k be the position vector of a moving fluid particle, as shown in Figure 1.5. Then the particle velocity is V 

dr dy dz dx k i  j   dt dt dt dt

(1.15)

If we define the velocity components to be V  ui  vj  wk

(1.16)

then (1.15) and (1.16) give u 

dx dt

v 

dy dt

w 

dz dt

(1.17 a, b, c)

If e t = unit tangent to the particle pathline, then the geometry shown in Figure 1.6 allows us to calculate V 

s et r (t   t )  r (t ) dr   V et  dt t t

(1.18)

in which s = arc length along the pathline and V  ds /dt   V   particle speed. Thus, the velocity vector is tangent to the pathline as the particle moves through space.

1.7

Chapter 1 — Introduction

Figure 1.6 Relationship between the position vector, arc length and unit tangent along a pathline.

It is frequently helpful to view, at a particular value of t , the velocity vector field for a collection of fluid particles, as shown in Figure 1.7.

Figure 1.7 The velocity field for a collection of fluid particles at one instant in time. In Figure 1.7 the lengths of the directed line segments are proportional to  V   V , and the line segments are tangent to the pathlines of each fluid particle at the instant shown. A streamline is a continuous curved line that, at each point, is tangent to the velocity vector V at a fixed value of t .The dashed line ABis a streamline, and, if d r = incremental displacement vector along AB , then V   dr

(1.19)

in which d r  dx i  dy j  dz k and  is the scalar proportionality factor between  V and d r . [Multiplying the vector d r by the scalar  does not change the direction of d r , and (1.19) merely requires that V and d r have the same direction. Thus,  will generally be a function of position along the streamline.] Equating corresponding vector components in (1.19) gives a set of differential equations that can be integrated to calculate streamlines.

1.8

Chapter 1 — Introduction

dx dz dy   w u v



1 

(1.20)

There is no reason to calculate the parameter  in applications of (1.20). Time, t , is treated as a constant in the integrations. Steady flow is flow in which the entire vector velocity field does not change with time. Then the streamline pattern will not change with time, and the pathline of any fluid particle coincides with the streamline passing through the particle. In other words, streamlines and pathlines coincide in steady flow. This will not be true for unsteady flow. The acceleration of a fluid particle is the first derivative of the velocity vector. a 

dV dt

(1.21)

When V changes both its magnitude and direction along a curved path, it will have components both tangential and normal to the pathline. This result is easily seen by differentiating (1.18) to obtain a 

det dV et  V dt dt

(1.22)

The geometry in Figure 1.8 shows that de t dt



e t t   t  e t (t ) t



V 1 s en  e R t R n

(1.23)

in which R radius of curvature of the pathline and e n  unit normal to the pathline (directed through the centre of curvature). Thus, (1.22) and (1.23) give a 

dV V2 e et  R n dt

(1.24)

Equation (1.24) shows that a has a tangential component with a magnitude equal to dV /dt and a normal component, V 2 /R , that is directed through the centre of pathline curvature.

Chapter 1 — Introduction

1.9

Figure 1.8 Unit tangent geometry along a pathline.  e t (t   t )    e t (t )   1 so that  e t t   t  e t (t )   1  

Review of Vector Calculus When a scalar or vector function depends upon only one independent variable, say t , then a derivative has the following definition: dF (t ) F (t   t )  F (t ) as  t  0  t dt

(1.25)

However, in almost all fluid mechanics problems p and V depend upon more than one independent variable, say x , y , z and t . [ x , y , z and tare independent if we can change the value of any one of these variables without affecting the value of the remaining variables.] In this case, the limiting process can involve only one independent variable, and the remaining independent variables are treated as constants. This process is shown by using the following notation and definition for a partial derivative: 0F (x , y, z , t ) F (x , y  y, z , t )  F (x , y , z , t ) as y  0  0y y

(1.26)

In practice, this means that we calculate a partial derivative with respect to y by differentiating with respect to y while treating x , z and t as constants. The above definition has at least two important implications. First, the order in which two partial derivatives are calculated will not matter. 0 2F 0 2F  0y0x 0x0y

(1.27)

1.10

Chapter 1 — Introduction

Second, integration of a partial derivative 0 F (x , y , z , t )  G (x , y , z , t ) 0y

(1.28)

in which G is a specified or given function is carried out by integrating with respect to y while treating x , z and t as constants. However, the integration “constant”, C, may be a function of the variables that are held constant in the integration process. For example, integration of (1.28) would give F (x , y , z , t ) 

P

G (x , y , z , t ) dy  C (x , z , t )

(1.29)

in which integration of the known function G is carried out by holding x , z and t constant, and C (x , z , t ) is an unknown function that must be determined from additional equations. There is a very useful definition of a differential operator known as del: /  i

0 0 0  k  j 0y 0x 0z

(1.30)

Despite the notation, del / is not a vector because it fails to satisfy all of the rules for vector algebra. Thus, operations such as dot and cross products cannot be derived from (1.30) but must be defined for each case. The operation known as the gradient is defined as /1  i

01 01 01  k  j 0y 0x 0z

(1.31)

in which 1 is any scalar function. The gradient has several very useful properties that are easily proved with use of one form of a very general theorem known as the divergence theorem P

/1 d~ 

P

1 e n dS

S

in which ~ is a volume enclosed by the surface S with an outward normal e n .

(1.32)

1.11

Chapter 1 — Introduction

Figure 1.9 Sketch used for derivation of Equation (1.32).

A derivation of (1.32) is easily carried out for the rectangular prism shown in Figure 1.9. x2

01 01 i i 1 x2 , y, z dy dz dx dy dz  d~  i PP PP P 0 x P 0x S x 

(1.33)

2

1

i 1 x1 , y, z dy dz PP S1

Since i is the outward normal on S2...