# ENGR2000 Lecture Notes - Lecture 8: John Wiley & Sons, Terminal Velocity, Stagnation Point

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Prepared by Dr Hongwei Wu Chapter 8 – Page 1 of 17

2nd – Year Fluid Mechanics, Faculty of Engineering and Computing, Curtin University

ENGR2000: FLUID MECHANICS

For Second-Year Chemical, Civil and Mechanical Engineering

FLUID MECHANICS LECTURE NOTES

CHAPTER 8 FLOW OVER IMMERSED BODY

8.1 Introduction

We have discussed pipe flow in Chapter 6, which is internal flow because the fluid is

confined within well-defined boundaries. This chapter deals with external flows, i.e. flows

over bodies immersed in the fluid. Typical examples of external flows include the flow of

water over a submarine (Figure 8-1a) or a fish (Figure 8-1b), the flow of air over an aircraft

(Figure 8-1c).

(a)

(b)

(c)

Figure 8-1 Examples; a) submarine; b) fish; c) aircraft;

The fluid forces (drag and lift) on the immersed bodies are of important considerations in

practice. In this chapter, we will learn the fundamentals of drag and lift, as well as the

methods for determining and optimising these forces in engineering applications.

8.2 Drag and lift

As shown in Figure 8-2, the forces exerted on the surface of an aerofoil by the fluid include

the pressure force (Figure 8-2a) and the viscous force (Figure 8-2b). The results of these

forces are the net drag force FD and lift force FL (Figure 8-2c). Taking a small element dA on

the aerofoil surface in Figure 8-2c, we can decompose the pressure force (PdA) and viscous

force (

dA

w

) into x-direction force dFx and y-direction force dFy, as shown in Figure 8-2d,

sin)(cos)( dAPdAdF wx

cos)(sin)( dAPdAdF wy

.

Integrating dFx and dFy along the surface, we can calculate the drag and lift forces as,

dAdAPdFFwxD

sincos

(E8-1)

dAdAPdFFwyL

cossin

. (E8-2)

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Prepared by Dr Hongwei Wu Chapter 8 – Page 2 of 17

Equations E8-1 and E8-2 reveals that 1) both shear stress and pressure contribute to the drag

and lift. In the case of drag, the former is called friction drag; the later is called pressure drag;

2) to calculate drag and lift, sufficient knowledge are required on three aspects: the body

shape as it determines the distribution of

; the distribution of pressure P and the distribution

of shear stress

w

along the surface.

Practically, it is very difficult to obtain the distribution of pressure and shear stress. Therefore,

E8-1 and E8-2 are generally not very useful. To make it easy for engineers, dimensionless

coefficients are often used instead. These dimensionless numbers are called drag coefficient

CD and lift coefficient CL, defined as

AV

F

CD

D

2

2

1

(E8-3)

AV

F

CL

L

2

2

1

(E8-4)

where A is projected (or frontal) area, i.e. the project area of the body in the flow direction.

Figure 8-2 Forces on an aerofoil [1]: a) pressure force; b) viscous force; c). resultant drag and

lift; d) pressure and viscous forces on a small surface area dA in (c), plotted in an x-y

coordination system.

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Prepared by Dr Hongwei Wu Chapter 8 – Page 3 of 17

8.3 The boundary layer

8.3.1 Hydraulician and hydrodynamicist

For a fundamental understanding on the cause of drag and lift, we must have a good

understanding on the boundary layer concept. Let’s start with some historical background on

two groups of fluid mechanicians who developed two different approaches in dealing with

problems of fluid mechanics by late 19th century.

One is the group of hydraulicians, focused on experiments and attempted to generalise useful

design equations from experimental data. This group developed the filed of experimental

hydraulics, delivering empirical solutions with little theoretical content. The other is the group

of hydrodynamicists, who focused on differential equations describing flows and tried to

apply them to practical problems. In order to solve these differential equations, the fluid was

assumed to have zero viscosity and constant density. This group developed the filed of

theoretical hydrodynamics, seeking pure theoretical solutions based on ideal-fluid flows.

The ideal-fluid solutions of hydrodynamicists agreed well with the observations of flows did

not involve solid surface, e.g. tides, however did not agree with observed behaviours in the

problems that concerned the hydraulicians, e.g, flow over immersed bodies. The following are

two typical examples.

(1) Flow over a thin plate

Figure 8-3 illustrates the differences between the ideal solutions by the hydrodynamicists and

the experimental observations by the hydraulicians for flow over a thin plate. For an ideal-

fluid flow (Figure 8-3a), the fluid is inviscid, there is no friction between the fluid and the

surface of the thin plate. Therefore, the fluid will maintain its free stream velocity when it

flows over the thin plate. However, for a real-fluid flow, the interaction between the viscous

fluid with the surface leads to a velocity gradient near the surface of the plate although the

fluid far away from the plate still maintains its free stream velocity.

(2) Flow over a circular cylinder

Figure 8-5 shows the patterns of ideal and real flow over a circular cylinder. In absence of

viscous effects, the wall shear stress is zero therefore the streamlines are symmetrical (see

Figure 8-6a). The fluid coasts from the front (point A), pasts the top (point C) and then

reaches to the rear (point F) of the cylinder.

However, the experimental observations by the hydraulicians are very different from the

above analysis by the hydrodynamicists, as shown in Figure 8-4b. A real-fluid flow cannot

coast along the cylinder surface down to point F. Instead, flow separation occurs from the

surface at a point between point C and F, leading to the formation of turbulent wake in the

downstream (see Figure 8-4b). The friction between the viscous fluid and the cylinder surface

leads to inevitable energy loss so that the flow does not have sufficient kinetic energy to travel

along the surface down to the rear of the cylinder.

From the above examples, it seems that the hydrodynamicists calculated what can NOT be

observed while the hydraulicians observed what can NOT be calculated [2]. By early 20th

century, the hydraulicians and hydrodynamicists had completely gone to live in their own

worlds. The hydraulicians continued to solve their problems by trial and error based on

experiments while the hydrodynamicists kept publishing academic papers based on

mathematics with little bearing on engineering problems.

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