SP-367 Introduction to the Aerodynamics of Flight

- Appendix B -

DIMENSIONS AND UNITS

 There is a fundamental difference between dimensions and units. A dimension represents the definition of an inherent physical property which remains independent of the particular scheme used to denote its measure. For example, the quantity of matter present in a lump of metal has the dimension of mass and the physical size of the edge of a book has the dimension of length.

A unit represents the particular, arbitrary scheme used to denote the magnitude of a physical property. Thus, the mass of matter in the lump of metal may be expressed in kilograms or slugs and the length of the book expressed in meters or feet depending on the system of units selected. Usually the quantity to be measured influences the choice of units to be employed, that is, meters or feet to measure the length of the book rather than kilometers or miles.

Basic Dimensions

There are four basic dimensions of general interest to aerodynamicists. These are called the basic or primary dimensions and are length, mass, time, and temperature. They may be abbreviated by using, respectively, L, M, T, and ø [Greek letter theta].

Derived Dimensions

The dimensions of all other quantities may be found to be combinations of quantities expressible in terms of the basic or primary dimensions. These are known as derived or secondary dimensions. For example, area may be represented as a length times a length or L2. A list of the more common quantities encountered in aerodynamics and their dimensions are included in table II.

Angular Measurement

The measure of the central angle of a circle is defined as the ratio of the subtended arc of the circle divided by the radius, that is, a ratio of two lengths. Thus, this measure is dimensionless but is assigned a special name of radians. Additionally one may express the angle in degrees by noting that an angle of 1 radian equals about 57.3°. The fact that both radian measure and degree measure are dimensionless means that the numerical value of an angle does not change from one system of units to another.

 Systems of Units

There are two basic engineering systems of units in use in aerodynamics. They are the International System of Units (SI) and the British Engineering System of Units (B.E.S.). In 1964 the United States National Bureau of Standards officially adopted the International System of Units to be used in all of its publications. The National Aeronautics and Space Administration has adopted a similar policy and this is the system of units used in this report. Table II lists the SI and B.E.S. units for both the basic dimensions and some of the more common aerodynamic quantities.

Vectors and Scalars

Vectors are quantities that have both a magnitude and a direction. Examples of physical quantities that are vectors are force, velocity, and acceleration. Thus, when one states that a car is moving north at 100 kilometers per hour, with respect to a coordinate system attached to the Earth, one is specifying the vector quantity velocity with a magnitude (100 kilometers per hour) and a direction (north).

Scalars are quantities that have a magnitude only. Examples of physical quantities that are scalars are mass, distance, speed, and density. Thus, when one states only the fact that a car is moving at 100 kilometers per hour one has specified a scalar, speed, since only a magnitude (100 kilometers per hour) is given (that is, no direction is specified).

To represent a vector on a diagram, an arrow is drawn. The length of the arrow is proportional to the magnitude of the vector and the direction of the arrow corresponds to the direction of the vector. Figure 161 shows the side view of a wing called the airfoil cross section (or simply airfoil section). Two aerodynamic forces are known to act on the section: lift and drag. They are vectors and may be drawn to act through a special point called the center of pressure discussed in the text. In the first step a scale is chosen and the force magnitudes are scaled. The second step is to place the vectors at the center-of-pressure point in the directions specified from the physical definition that lift always acts perpendicularly to the incoming velocity of the air and drag always acts parallel to and away from the incoming velocity of the air.

Vectors may be added together (composition) to form one vector (the resultant) or one vector may be broken down (resolution) into several components. In figure 161 the lift and drag have been composed into the resultant shown. The resultant can be resolved back into the lift and drag components.

 TABLE II.- SYSTEMS OF UNITS

Quantity

Basic dimensions

Units

SI

B.E.S

Length

L

meter

foot

Mass

M

kilogram

slug

Time

T

second

second

Temperature

ø [Greek letter theta]

°C (relative)

°F (relative

K (absolute)

°R (absolute)

Quantity

Derived dimensions

Units

SI

B.E.S

Area

L2

meters2

feet2

Volume

L3

meters3

feet3

Velocity

LT-1

meters/second

feet/second

Acceleration

LT-2

meters/second2

feet/second2

Force

MLT-2

newton

pound

Pressure

ML-1T-2

newtons/meter2

pounds/foot2

Density

ML-3

kilogram/meter 3

slugs/foot3

Kinematic viscosity

L2T-1

meters2/second

feet2/second

Momentum

MLT-1

newton- second

pound-second

Energy

ML2T-2

joule

foot-pound

Power

ML2T-3

watt

foot - pound/second

Angle

-

Angular velocity

T-1

Angular acceleration

T-2

Moment of inertia

ML2

kilogram - meter2

slug-ft2

[
190] Figure 161.- Vector representation.

Motion

Motion is the movement or change in position of a body. Motion is always with respect to a particular observer. Consider the flight of an aircraft through the air. One may adopt two points of view. First an observer fixed in the air sees the aircraft approach at velocity (See fig. 162(a).) On the other hand an observer fixed on the aircraft sees the air (or observer fixed in air) approach him at velocity from the opposite direction. (See fig. 162(b).) The two observers read the same magnitude of velocity (that is, speed) but indicate opposite directions. In many cases, for example, in the use of a wind tunnel, the second point of view is adopted where the aircraft or airfoil is fixed in the tunnel and air is forced to flow past it. (See fig. 162(c).)

[
191] (a) Observer fixed in the air.
(b) Observer fixed on aircraft.
(c) Wind-tunnel operation- wing fixed in place and air placed in motion over wing with velocity .
Figure 162.- Relative motion.   