Aeronautical Engineer’s Data Book - part 4 pot

81 Basic fluid mechanics The equations for 1D flow are derived by considering flow along a straight stream tube see Figure 5.2.. 82 Table 5.3 2D flow: fundamental equations Laplace’s eq

in constant motion; a liquid adopts the shape of a vessel containing it whilst a gas expands to fill any container in which it is placed Some basic fluid relationships are given in Table 5.1

Table 5.1 Basic fluid relationships

Density ( ) Mass per unit volume

Units kg/m 3 (lb/in 3 )

Specific gravity (s) Ratio of density to that of

water, i.e s = / water

Specific volume (v) Reciprocal of density, i.e s =

1/  Units m 3 /kg (in 3 /lb) Dynamic viscosity () A force per unit area or shear

stress of a fluid Units Ns/m 2 (lbf.s/ft 2 )

Kinematic viscosity (  ) A ratio of dynamic viscosity to

density, i.e  = µ/ Units m2 /s (ft 2 /sec)

5.1.2 Perfect gas

A perfect (or ‘ideal’) gas is one which follows

Boyle’s/Charles’ law pv = RT where:

p = pressure of the gas

v = specific volume

T = absolute temperature

R = the universal gas constant

Although no actual gases follow this law totally, the behaviour of most gases at temperatures

77 Basic fluid mechanics

well above their liquefication temperature will

approximate to it and so they can be considered

where n is known as the polytropic exponent

Figure 5.1 shows the four main changes of state relevant to aeronautics: isothermal, adiabatic: polytropic and isobaric

where a = the velocity of propagation of a

pressure wave in the fluid

5.1.5 Fluid statics

Fluid statics is the study of fluids which are at rest (i.e not flowing) relative to the vessel containing

it Pressure has four important characteristics:

• Pressure applied to a fluid in a closed vessel (such as a hydraulic ram) is transmitted to all parts of the closed vessel at the same value (Pascal’s law)

• The magnitude of pressure force acting at any point in a static fluid is the same, irrespective of direction

• Pressure force always acts perpendicular to the boundary containing it

• The pressure ‘inside’ a liquid increases in proportion to its depth

Other important static pressure equations are:

• Absolute pressure = gauge pressure + atmospheric pressure

• Pressure (p) at depth (h) in a liquid is given

79 Basic fluid mechanics

The stream tube for conservation of mass

dp ds

The stream tube and element for the momentum equation

W The forces on the element

Fig 5.2 Stream tube/fluid elements: 1-D flow

(3D) depending on the way that the flow is constrained

5.2.1 1D Flow

1-D flow has a single direction co-ordinate x and

a velocity in that direction of u Flow in a pipe

or tube is generally considered one dimensional

80

Table 5.2 Fluid principles

created or destroyed

algebraic sum of the forces acting in that direction (Newton’s second law of motion)

and are in balance in a steadily operating system

81 Basic fluid mechanics

The equations for 1D flow are derived by considering flow along a straight stream tube (see Figure 5.2) Table 5.2 shows the principles, and their resulting equations

5.2.2 2D Flow

2D flow (as in the space between two parallel flat plates) is that in which all velocities are

parallel to a given plane Either rectangular (x,y)

or polar (r, ) co-ordinates may be used to describe the characteristics of 2D flow Table 5.3 and Figure 5.3 show the fundamental equations Rectangular co-ordinates

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Table 5.3 2D flow: fundamental equations

Laplace’s equation

2

∂x The principle of force = mass (Newton’s law of motion) applies to fluids acceleration

and fluid particles

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β qcos β

q

Fig 5.6 Velocity potential basis

� � �

2

85 Basic fluid mechanics

5.2.3 The Navier-Stokes equations

The Navier-Stokes equations are written as:

x

If q>O this is a source of strength |q|

If q<O this is a sink of strength |q|

Flow due to a combination

of source and sink

Fig 5.7 Sources, sinks and combination

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5.2.4 Sources and sinks

A source is an arrangement where a volume of fluid (+q) flows out evenly from an origin

toward the periphery of an (imaginary) circle

around it If q is negative, such a point is termed

a sink (see Figure 5.7) If a source and sink of

equal strength have their extremities infinitesi­mally close to each other, whilst increasing the

strength, this is termed a doublet

5.3 Flow regimes

5.3.1 General descriptions

Flow regimes can be generally described as follows (see Figure 5.8):

Steady Flow parameters at any point do

they may differ between points) Unsteady Flow parameters at any point vary

Laminar Flow which is generally considered flow smooth, i.e not broken up by eddies Turbulent Non-smooth flow in which any

causing eddies and turbulence Transition The condition lying between flow laminar and turbulent flow regimes

5.3.2 Reynolds number

Reynolds number is a dimensionless quantity which determines the nature of flow of fluid over a surface

Low Reynolds numbers (below about 2000)result in laminar flow High Reynolds numbers(above about 2300) result in turbulent flow.

Basic fluid mechanics 87

‘Wake’ eddies move slower than the rest

of the fluid

Steady flow

Unsteady flow

Boundary layer

Velocity distributions in laminar and turbulent flows

The flow is not steady

relative to any axes

Wake Area of laminar flow

Area of turbulent flow Boundary layer of

thickness ( δ)

Turbulent flow

Laminar flow v

umax

u

The flow is steady, relative

to the axes of the body

Fig 5.8 Flow regimes

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Values of Re for 2000 < Re < 2300 are gener­

ally considered to result in transition flow Exact flow regimes are difficult to predict in this region

5.4 Boundary layers

5.4.1 Definitions

The boundary layer is the region near a surface

or wall where the movement of the fluid flow

is governed by frictional resistance

The main flow is the region outside the

boundary layer which is not influenced by frictional resistance and can be assumed to be

‘ideal’ fluid flow

Boundary layer thickness: it is convention

to assume that the edge of the boundary layer lies at a point in the flow which has a velocity equal to 99% of the local mainstream velocity

5.4.2 Some boundary layer equations

Figure 5.9 shows boundary layer velocity profiles for dimensional and non-dimensional cases The non-dimensional case is used to allow comparison between boundary layer profiles of different thickness

Dimensional case Non-dimensional case

where:

µ = velocity parallel to the surface

y = perpendicular distance from the surface

= boundary layer thickness

U1 = mainstream velocity

u = velocity parameters u/U1(non-dimensional)

y = distance parameter y/ (non-dimensional)

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Table 5.4 Isentropic flows

5.7 Normal shock waves

5.7.1 1D flow

A shock wave is a pressure front which travels

at speed through a gas Shock waves cause anincrease in pressure, temperature, density andentropy and a decrease in normal velocity.Equations of state and equations of conser-vation applied to a unit area of shock wave give(see Figure 5.10):

Shock wave travels into area of stationary gas

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Pressure and density relationships across the

shock are given by the Rankine-Hugoniot

In axisymmetric flow the variables are indepen­

dent of  so the continuity equation can be

�





93 Basic fluid mechanics

5.7.2 The pitot tube equation

An important criterion is the Rayleigh super­sonic pitot tube equation (see Figure 5.11)

into a fluid flow Figure 5.12 shows the layout

of spherical co-ordinates used to analyse these types of flow

Relationships between the velocity compo­nents and potential are given by:

Figures 5.13(a) and (b) show drag types and

‘rule of thumb’ coefficient values

Fig 5.13(a) Relationship between pressure and fraction

drag: ‘rule of thumb’

95 Basic fluid mechanics

Cylinder (right angles to flow)

4

π – d 2

4

dl

π – d 2

4

π – d 2

4

Bluff bodies

Hollow semi-sphere opposite stream 1.42 Hollow semi-sphere facing stream 0.38 Hollow semi-cylinder opposite stream 1.20 Hollow semi-cylinder facing stream 2.30

Open wheel, rotating, h/D = 0.28 0.58

Streamlined bodies

Laminar flat plate (Re = 10 6 ) 0.001

0.006 0.025 0.025 0.05 0.05 0.16 0.005 0.09 n.a

Turbulent flat plate (

Airfoil section, minimum

Airfoil section, at stall

2-element airfoil

4-element airfoil

Subsonic aircraft wing, minimum

Subsonic aircraft wing, at stall

Subsonic aircraft wing, minimum

Subsonic aircraft wing, at stall

Aircraft wing (supersonic)

Subsonic transport aircraft

Supersonic fighter,

Fig 5.13(b) Drag coefficients for standard shapes

Section 6

Basic aerodynamics

6.1 General airfoil theory

When an airfoil is located in an airstream, the flow divides at the leading edge, the stagna­tion point The camber of the airfoil section means that the air passing over the top surface has further to travel to reach the trail­ing edge than that travelling along the lower surface In accordance with Bernoulli’s equation the higher velocity along the upper airfoil surface results in a lower pressure, producing a lift force The net result of the velocity differences produces an effect equiv­alent to that of a parallel air stream and a rotational velocity (‘vortex’) see Figures 6.1 and 6.2

For the case of a theoretical finite airfoil section, the pressure on the upper and lower surface tries to equalize by flowing round the tips This rotation persists downstream of the wing resulting in a long U-shaped vortex (see Figure 6.1) The generation of these vortices needs the input of a continuous supply of energy; the net result being to increase the drag

of the wing, i.e by the addition of so-called

induced drag

6.2 Airfoil coefficients

Lift, drag and moment (L, D, M) acting on an

aircraft wing are expressed by the equations:

U Lift (L) per unit width = C L l 2

2

97 Basic aerodynamics

An effective rotational velocity (vortex) superimposed on the parallel airstream

+ + + + +

– – – –

– (a)

Pressures equalize by flows

Tip Midspan

Tip

Core of vortex (c)

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U Drag (D) per unit width = C D l2

2

Moment (M) about LE or

U 1/4 chord = C M l2

2 per unit width

C L , C D and C M are the lift, drag and moment coefficients, respectively Figure 6.3 shows typical values plotted against the angle of attack, or incidence, () The value of CD is

small so a value of 10 C D is often used for the

characteristic curve C L rises towards stall point and then falls off dramatically, as the wing

enters the stalled condition C D rises gradually, increasing dramatically after the stall point Other general relationships are:

• As a rule of thumb, a Reynolds number of

Re  106 is considered a general flight condition

Reynolds numbers between 105 and 107

• C D decreases rapidly up to Reynolds numbers of about 106, beyond which the rate of change reduces

• Thickness and camber both affect the

maximum C L that can be achieved As a

general rule, C L increases with thickness and then reduces again as the airfoil

becomes even thicker C L generally increases as camber increases The

minimum C D achievable increases fairly steadily with section thickness

6.3 Pressure distributions

The pressure distribution across an airfoil section varies with the angle of attack () Figure 6.4 shows the effect as  increases, and

the notation used The pressure coefficient C p

reduces towards the trailing edge

99 Basic aerodynamics

Characteristics for an asymmetrical ‘infinite-span 2D airfoil’

Characteristic curves of a practical wing

–8˚ –4˚ 0˚ 4˚ 8˚ 12˚ 16˚ 20˚

α

Fig 6.3 Airfoil coefficients

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Arrow length represents the magnitude of pressure coefficient Cp

moment coefficient (C M) is constant, i.e does

not vary with lift coefficient (C L) Its theoreti­cal positions are indicated in Table 6.1

Table 6.1 Position of aerodynamic centre

Condition Theoretical positon of the AC

101 Basic aerodynamics

Using common approximations, the following equations can be derived:

where C Ma = pitching moment coefficient at

distance a back from LE

xAC = position of AC back from LE

c = chord length

6.5 Centre of pressure

The centre of pressure (CP) is defined as the point in the section about which there is no pitching moment, i.e the aerodynamic forces

on the entire section can be represented by lift and drag forces acting at this point The CP does not have to lie within the airfoil profile and can change location, depending on the

magnitude of the lift coefficient C L The CP is

conventionally shown at distance kCP back from the section leading edge (see Figure 6.5) Using

Lift and drag only cut at the CP

and fluid particles

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5.2 .4 Sources and sinks

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Pressure and density relationships across the

shock are given by the Rankine-Hugoniot

In