__3.1 Classiﬁcation of Fluid Flows__

__3.1 Classiﬁcation of Fluid Flows__

Fluid mechanics is a subject in which many rather complicated phenomena are encountered, so it is important that we understand some of the descriptions and

simpliﬁcations of several special ﬂuid ﬂows. Such special ﬂows will be studied in detail in later chapters. Here we will attempt to classify them in as much detail as possible.

__UNIFORM, ONE-, TWO-, AND THREE-DIMENSIONAL FLOWS__

__UNIFORM, ONE-, TWO-, AND THREE-DIMENSIONAL FLOWS__

A dependent variable in our study of ﬂuids depends, in general, on the three space coordinates and time, i.e., **V**(*x*, *y*, *z*, *t*). The ﬂow that depends on three space coordinates is a *three-dimensional ﬂow*; it could be a steady ﬂow if time is not involved, such as would be the case in the ﬂow near the intersection of a wing and the fuselage of an aircraft ﬂying at a constant speed. The ﬂow in a washing machine would be an unsteady, three-dimensional ﬂow.

Certain ﬂows can be approximated as two-dimensional ﬂows; ﬂows over a wide weir, in the entrance region of a pipe, and around a sphere are examples that are of special interest. In such *two-dimensional ﬂows *the dependent variables depend on only two space variables, i.e., *p*(*r,**q*) or **V**(*x**, y, t*). If the space coordinates are *x *and *y*, we refer to the ﬂow as a *plane ﬂow*.

*One-dimensional ﬂows *are ﬂows in which the velocity depends on only one space variable. They are of special interest in our introductory study since they include the ﬂows in pipes and channels, the two most studied ﬂows in an introductory course. For ﬂow in a long pipe, the velocity depends on the radius *r*, and in a wide channel (parallel plates) it depends on *y*, as shown in Fig. 3.4.

The ﬂows shown in Fig. 3.4 are also referred to as *developed ﬂows*; the velocity proﬁles do not change with respect to the downstream coordinate. This demands that the pipe ﬂow shown is many diameters downstream of any change in geometry, such as an entrance, a valve, an elbow, or a contraction or expansion. If the ﬂow has not developed, the velocity ﬁeld depends on more than one space coordinate, as is the case near a geometry change. The developed ﬂow may be unsteady, i.e., it may depend on time, such as when a valve is being opened or closed.

Finally, there is the *uniform ﬂow*, as shown in Fig. 3.5; the velocity proﬁle, and other properties such as pressure, is uniform across the section of pipe. This proﬁle

**F****i****g****u****re 3.4 **One-dimensional ﬂow: (*a*) ﬂow in a pipe and (*b*) ﬂow in a wide channel.

**F****i****g****u****re 3.5 **A uniform ﬂow in a pipe.

is often assumed in pipe and channel ﬂow problems since it approximates the more common turbulent ﬂow so well. We will make this assumption in many of the problems throughout this book.

__VISCOUS AND INVISCID FLOWS__

__VISCOUS AND INVISCID FLOWS__

In an *inviscid ﬂow *the effects of viscosity can be completely neglected with no signiﬁcant effects on the solution to a problem. All ﬂuids have viscosity and if the viscous effects cannot be neglected, it is a *viscous ﬂow*. Viscous effects are very important in pipe ﬂows and many other kinds of ﬂows inside conduits; they lead to losses and require pumps in long pipelines. But, there are ﬂows in which we can neglect the inﬂuence of viscosity. Consider an *external ﬂow*, ﬂow external to a body, such as the ﬂow around an airfoil or a hydrofoil, as shown in Fig. 3.6. If the airfoil is moving relatively fast (faster than about 1 m/s), the ﬂow away from a thin layer near the boundary, a *boundary layer*, can be assumed to have zero viscosity with no signiﬁcant effect on the solution to the ﬂow ﬁeld (the velocity, pressure, temperature ﬁelds). All the viscous effects are concentrated inside the boundary layer and cause the velocity to be zero at the surface of the airfoil, the *no-slip condition*. Since inviscid ﬂows are easier to solve than viscous ﬂows, the recognition that the viscosity can be ignored in the ﬂow away from the surface leads to much simpler solutions. This will be demonstrated in Chap. 8 (External Flows).

**F****i****g****u****re 3.6 **Flow around an airfoil.

__LAMINAR AND TURBULENT FLOWS__

__LAMINAR AND TURBULENT FLOWS__

A viscous ﬂow is either a laminar ﬂow or a turbulent ﬂow. In a *turbulent ﬂow *there is mixing of ﬂuid particles so that the motion of a given particle is random and very irregular; statistical averages are used to specify the velocity, the pressure, and other quantities of interest. Such an average may be “steady” in that it is independent of time, or the average may be unsteady and depend on time. Figure 3.7 shows steady and unsteady turbulent ﬂows. Notice the noisy turbulent ﬂow from a faucet when you get a drink of water.

In a *laminar ﬂow *there is negligible mixing of ﬂuid particles; the motion is smooth and noiseless, like the slow ﬂow of water from a faucet. If a dye is injected into a laminar ﬂow, it remains distinct for a relatively long period of time. The dye would be immediately diffused if the ﬂow were turbulent. Figure 3.8 shows a steady and an unsteady laminar ﬂow. A laminar ﬂow could be made to appear turbulent by randomly controlling a valve in the ﬂow of honey in a pipe so as to make the velocity appear as in Fig. 3.7. Yet, it would be a laminar ﬂow since there would be no mixing of ﬂuid particles. So, a simple display of *V*(*t*) is not sufﬁcient to decide if a particular ﬂow is laminar or turbulent. To be turbulent, the motion has to be random, as in Fig. 3.7, but it also has to have mixing of ﬂuid particles.

As a ﬂow starts from rest, as in a pipe when a valve is slightly opened, the ﬂow starts out laminar, but as the average velocity increases, the laminar ﬂow becomes unstable and turbulent ﬂow ensues. In some cases, as in the ﬂow between rotating cylinders, the unstable laminar ﬂow develops into a secondary laminar ﬂow of vortices, and then a third laminar ﬂow, and ﬁnally a turbulent ﬂow at higher speeds.

There is a quantity, called the *Reynolds number*, that is used to determine if a ﬂow is laminar or turbulent. It is

where *V *is a characteristic velocity (the average velocity in a pipe or the speed of an airfoil), *L *is a characteristic length (the diameter of a pipe or the distance from the leading edge of a ﬂat plate), and *n *is the kinematic viscosity. If the Reynolds number is larger than a critical Reynolds number, the ﬂow is turbulent; if it is lower than a critical Reynolds number, the ﬂow is laminar. For ﬂow in a pipe, assuming the typically rough pipe wall, the critical Reynolds number is usually taken to be 2000; if the wall is smooth and free of vibrations, and the entering ﬂow is free of disturbances, the critical Reynolds number can be as high as 40 000. The critical Reynolds number is different for each geometry. For ﬂow between parallel plates, it is taken as 1500 using the average velocity and the distance between the plates. For a boundary layer on a ﬂat plate, it is between 3 × 105 and 106, using *L *as the distance from the leading edge.

We do not refer to an inviscid ﬂow as laminar or turbulent. In an external ﬂow, the inviscid ﬂow is called a *free-stream ﬂow*. A free stream has disturbances but the disturbances are not accompanied by shear stresses, another requirement of both laminar and turbulent ﬂows. The free stream can also be irrotational or it can possess vorticity.

A *boundary layer *is a thin layer of ﬂuid that develops on a body due to the viscosity causing the ﬂuid to stick to the boundary; it causes the velocity to be zero at the wall. The viscous effects in such a layer can actually incinerate a satellite on reentry. Figure 3.9 shows the typical boundary layer on a ﬂat plate. It is laminar near the leading edge and undergoes transition to a turbulent ﬂow with sufﬁcient

**F****i****g****u****re 3.9 **Boundary-layer ﬂow on a ﬂat plate.

length. For a smooth rigid plate with low free-stream ﬂuctuation level, a laminar layer can exist up to Re = 106, *L *being the length along the plate; for a rough plate, or a vibrating plate, or high free-stream ﬂuctuations, a laminar ﬂow exists up to about Re = 3 × 105.

__INCOMPRESSIBLE AND COMPRESSIBLE FLOWS__

__INCOMPRESSIBLE AND COMPRESSIBLE FLOWS__

Liquid ﬂows are assumed to be incompressible in most situations (water hammer1 is an exception). In such *incompressible ﬂows *the density of a ﬂuid particle as it moves along is assumed to be constant, i.e.,

This does not demand that the density of all the ﬂuid particles be the same. For example, salt could be added to a water ﬂow at some point in a pipe so that down- stream of the point the density would be greater than at some upstream point. Atmo spheric air at low speeds is incompressible but the density decreases with increased elevation, i.e., *r *= *r*(*z*), where *z *is vertical. We usually assume a ﬂuid to have constant density when we make the assumption of incompressibility. Constant density requires

The ﬂow of air can be assumed to be incompressible if the velocity is sufﬁciently low. Air ﬂow in conduits, around automobiles and small aircraft, and the takeoff and landing of commercial aircraft are all examples of incompressible airﬂows. The *Mach number *M where

is used to determine if a ﬂow is compressible; *V *is the characteristic velocity and *c *

is the speed of sound. If M < 0.3, we assume the ﬂow to be incompressible. For air near sea level this is about 100 m/s (200 mph) so many air ﬂows can be assumed to be incompressible. Compressibility effects are considered in some detail in Chap. 9 (Compressible Flows).

__EXAMPLE 3.3__

A river ﬂowing through campus appears quite placid. A leaf ﬂoats by and we estimate that it ﬂoats about 2 m in 10 s. We wade in the water and estimate the depth to be about 60 cm. Is the ﬂow laminar or turbulent?

**Solution**

We estimate the Reynolds number to be, assuming *T *= 20°C and using Table C.1,

The ﬂow is highly turbulent at this Reynolds number, contrary to our observation of the placid ﬂow. Most internal ﬂows are turbulent, as observed when we drink from a drinking fountain. Laminar ﬂows are of minimal importance to engineers when compared to turbulent ﬂows; a lubrication problem is one exception.

*3.2 Bernoulli’s Equation*

*3.2 Bernoulli’s Equation*

Bernoulli’s equation may be the most often used equation from ﬂuid mechanics, but it is also the most often misused equation. In this section it will be derived and the restrictions required for its derivation will be highlighted. But, before the equation is derived, consider the ﬁve assumptions required: negligible viscous effects (no shear stresses), constant density, steady ﬂow, the ﬂow is along a streamline, and the velocity is measured in an inertial reference frame.

We apply Newton’s second law to a cylindrical particle that is moving on a streamline, as shown in Fig. 3.10. A summation of inﬁnitesimal forces acting on the particle is

where *a *is the *s*-component of the acceleration vector. It is given by Eq. (3.9)

where we think of the *x*-direction being in the *s*-direction so that *u = **V*:

where *∂ **V*/*∂ **t *= 0, assuming a steady ﬂow. (This leads to the same acceleration expression as presented in physics or dynamics where *a**x *= *VdV*/*dx*, providing an

**F****i****g****u****re 3.10 **A particle moving along a streamline.

inertial reference frame is used in which no Coriolis or other acceleration components are present.) Next, we observe that

Now, divide Eq. (3.19) by *dsdA *and use the above expressions for *a**s *and cos*q *and earrange. There results

If we assume that the density *r *is constant (this is more restrictive than incompressibility) so it can be moved after the partial derivative, and we recognize that *V**∂ **V*/*∂ **s *= *∂ *(*V *2/2)/*∂ **s*, we can write our equation as

This means that along a streamline the quantity in parentheses is constant, i.e.,

where the constant may change from one streamline to the next; along a given streamline the sum of the three terms is constant. This is often written, referring to two points on the same streamline, as

Either of the two forms above is the famous *Bernoulli’s equation*. Let’s once again state the assumptions required to use Bernoulli’s equation:

• Inviscid ﬂow (no shear stresses)

• Constant density

• Steady ﬂow

• Along a streamline

• Applied in an inertial reference frame

The ﬁrst three listed are primarily ones usually considered. There are special applications where the last two must be taken into account; but those special applications will not be presented in this book. Also, we often refer to a constant-density ﬂow as an incompressible ﬂow, even though constant density is more restrictive [refer to the comments after Eq. (3.16)]. This is because incompressible ﬂows, in which the density changes from one streamline to the next, such as in atmospheric ﬂows, are not encountered in an introductory course.

Note that the units on all the terms in Eq. (3.26) are meters. Consequently, *V *2/2g is called the *velocity head*, *p*/*r**g *is the *pressure head*, and *h *is simply the *head*. The sum of the three terms is often referred to as the *total head*. The pressure *p *is the *static pressure *and the sum *p *+ *r**V *2/2 is the *total pressure *or *stagnation pressure *since it is the pressure at a *stagnation point*, a point where the ﬂuid is brought to rest along a streamline.

The difference in the pressures can be observed by considering the measuring probes shown in Fig. 3.11. The probe in Fig. 3.11*a *is a *piezometer*; it measures the

**F****i****g****u****re 3.11 **Pressure probes: (*a*) the piezometer, (*b*) a pitot tube, and (*c*) a pitot-static tube.

static pressure, or simply, the pressure at point 1. The *pitot tube *in Fig. 3.11*b *measures the total pressure, the pressure at a point where the velocity is zero, as at point 2. And, the *pitot-static tube*, which has a small opening the side of the probe as shown in Fig. 3.11*c*, is used to measure the difference between the total pressure and the static pressure, i.e., *r**V *2/2; this is used to calculate the velocity. The expression for velocity is

where point 2 must be a stagnation point with *V*2 = 0. So, if only the velocity is desired, we simply use the pitot-static probe shown in Fig. 3.11*c*.

Bernoulli’s equation is used in numerous ﬂuid ﬂows. It can be used in an internal ﬂow over short reaches if the viscous effects can be neglected; such is the case in the well-rounded entrance to a pipe (see Fig. 3.12) or in a rather sudden contraction of a pipe. The velocity for such an entrance is approximated by Bernoulli’s equation to be

Another common application of the Bernoulli’s equation is from the free stream to the front area of a round object such as a sphere or a cylinder or an airfoil. A sketch

**F****i****g****u****re 3.12 **Flow from a reservoir through a pipe.

is helpful, as shown in Fig. 3.13. For many ﬂow situations the ﬂow separates from the surface, resulting in a separated ﬂow, as shown. If the ﬂow approaching the object is uniform, the constant in Eq. (3.25) will be the same for all the streamlines. Bernoulli’s equation can then be applied from the free stream to the stagnation point at the front of the object, and to points along the surface of the object up to the separation region.

We often solve problems involving a pipe exiting to the atmosphere. For such a situation the pressure just inside the pipe exit is the same as the atmospheric pressure just outside the pipe exit, since the streamlines exiting the pipe are straight near the exit (see Fig. 3.12). This is quite different from the entrance ﬂow of Fig. 3.12 where the streamlines near the entrance are extremely curved.

To approximate the pressure variation normal to a curved streamline, consider the particle of Fig. 3.10 to be a parallelepiped with thickness *dn *normal to the

streamline, with area *dA**s *of the side, and with length *ds*. Use

**F****i****g****u****re 3.13 **Flow around a sphere or a long cylinder.

where we have used the acceleration to be *V *2/*R*, *R *being the radius of curvature in the assumed plane ﬂow. If we assume that the effect of gravity is small compared to the acceleration term, this equation simpliﬁes to

Since we will use this equation to make estimations of pressure changes normal to

a streamline, we approximate ∂*p*/∂*n *= Δ*p*/Δ*n *and arrive at the relationship

Hence, we see that the pressure decreases as we move toward the center of the curved streamlines; this is experienced in a tornado where the pressure can be extremely low in the tornado’s “eye.” This reduced pressure is also used to measure the intensity of a hurricane; that is, lower the pressure in the hurricane’s center, larger the velocity at its outer edges.

**EXAMPLE 3.4**

The wind in a hurricane reaches 200 km/h. Estimate the force of the wind on a window facing the wind in a high-rise building if the window measures 1 m by 2 m. Use the density of the air to be 1.2 kg/m^{3}.

__Solution__

Use Bernoulli’s equation to estimate the pressure on the window:

where the velocity must have units of m/s. To check the units, use kg = N · s2/m. Assume the pressure to be essentially constant over the window so that the force is then

This force is large enough to break many windows, especially if they are not properly designed.

**EXAMPLE 3.5**

A piezometer is used to measure the pressure in a pipe to be 20 cm of water. A pitot tube measures the total pressure to be 33 cm of water at the same general location. Estimate the velocity of the water in the pipe.

**Solution**

The velocity is found using Eq. (3.27):

where we used the pressure relationship *p *= ρ*g**h*.

__Quiz No. 1__

__Quiz No. 1__

1. A velocity ﬁeld in a plane ﬂow is given by **V **= 2*yt***i **+ *x***j **m/s. The magnitude of the acceleration at the point (4, 2 m) at *t *= 3 s is

(A) 52.5 m/s2

(B) 48.5 m/s2

(C) 30.5 m/s2

(D) 24.5 m/s2

2. A velocity ﬁeld in a plane ﬂow is given by **V **= 2*xy***i **+ *yt***j **m/s. The vorticity of the ﬂuid at the point (0, 4 m) at *t *= 3 s is

(A) −4**k **rad/s

(B) −3**j **rad/s

(C) −2**k **rad/s

(D) −3**i **rad/s

3. The parabolic velocity distribution in a channel ﬂow is given by *u*(*y*) = 0.2(1 − *y*2) m/s, with *y *measured in centimeters. What is the acceleration of a ﬂuid particle at a location where *y *= 0.5 cm?

(A) 0

(B) 2 m/s2

(C) 4 m/s2

(D) 5 m/s2

4. The equation of the streamline that passes through the point (2, −1) if the velocity ﬁeld is given by **V **= 2*xy***i **+ *y*2 **j **m/s is

(A) *x*^{2} = 4*y*^{2}

(B) *x *= −2*y*

(C) *x*^{2} = −4*y*

(D) *x *= 2*y*^{2}

5. A drinking fountain has a 2-mm-diameter outlet. If the water is to be laminar, what is the maximum speed that the water should have?

(A) 0.5 m/s

(B) 1 m/s

(C) 2 m/s

(D) 4 m/s

6. Which of the following ﬂows could be modeled as inviscid ﬂows?

(a) Developed ﬂow in a pipe

(b) Flow of water over a long weir

(c) Flow in a long, straight canal

(d) The ﬂow of exhaust gases exiting a rocket

(e) Flow of blood in an artery

(f) Flow of air around a bullet

(g) Flow of air in a tornado

(A) d, e

(B) d, g

(C) b, e

(D) b, g

7. Salt is being added to fresh water in a pipe at a certain location. In the vicinity of that location the term *D**r*/*Dt *is nonzero. Which term in the expression for *D**r*/*Dt *is nonzero if the *x*-axis is along the pipe axis? Assume uniform conditions.

(A) *u*∂ρ /∂ *y*

(B) ρ∂*u */∂ *x*

(C) *u*∂ρ /∂ *x*

(D) ρ∂*u */∂ *y*

8. A pitot probe measures 10 cm of water on a small airplane ﬂying where the temperature is 20°C. The speed of the airplane is nearest

(A) 40 m/s

(B) 50 m/s

(C) 60 m/s

(D) 70 m/s

9. The ﬂuid in the pipe is water and *h *= 10 cm of mercury. The velocity *V *is nearest

(A) 7 m/s

(B) 8 m/s

(C) 5 m/s

(D) 4 m/s

10. Select the false statement for Bernoulli’s equation

(A) It can be applied to an inertial coordinate system.

(B) It can be applied to an unsteady ﬂow.

(C) It can be applied in an inviscid ﬂow.

(D) It can be applied between two points along a streamline.

11. Water ﬂows through a long-sweep elbow on a 2-cm-diameter pipe at an average velocity of 20 m/s. Estimate the increase in pressure from the inside of the pipe to the outside of the pipe midway through the elbow if the radius of curvature of the elbow averages 12 cm at the midway section.

(A) 60 kPa

(B) 66.7 kPa

(C) 75 kPa

(D) 90 kPa

__Quiz No. 2__

__Quiz No. 2__

1. Find the rate of change of the density in a stratiﬁed ﬂow where

*r *= 1000(1− 0.2*z*) and the velocity is **V **= 10(*z *– *z*^{2})**i**.

2. A velocity ﬁeld is given in cylindrical coordinates as

What are the three acceleration components at the point (3 m, 90°)?

3. The trafﬁc in a large city is to be studied. Explain how it would be done using (a) the Lagrangian approach and (b) the Eulerian approach.

4. Find the unit vector normal to the streamline at the point (2, 1) when *t *= 2 s if the velocity ﬁeld is given by **V **= 2*xy***i **+ *y*2*t***j **m/s.

5. A leaf is ﬂoating in a river seemingly quite slowly. It is timed to move 6 m in 40 s. If the river is about 1.2 m deep, determine if the placidly ﬂowing river is laminar or turbulent.

6. Which of the following ﬂows would deﬁnitely be modeled as a turbulent ﬂow?

(a) Developed ﬂow in a pipe

(b) Flow of water over a long weir

(c) Flow in a long, straight canal

(d) The ﬂow of exhaust gases exiting a rocket

(e) Flow of blood in an artery

(f) Flow of air around a bullet

(g) Flow of air in a tornado

7. Air ﬂows over and parallel to a 10-m-long ﬂat plate at 2 m/s. How long is the laminar portion of the boundary layer if the air temperature is 30°C. Assume a high-ﬂuctuation level on a smooth rigid plate.

8. The pitot and piezometer probes read the total and static pressures as shown. Calculate the velocity *V*.

9. Determine the velocity *V *in the pipe if water is ﬂowing and *h *= 20 cm of mercury.

10. A car is travelling at 120 km/h. Approximate the force of the air on the 20-cm-diameter ﬂat lens on the headlight.