Fundamental Physical Quantities

1. Pressure.

Consider a surface immersed in a fluid. Note that molecules of fluid are constantly in motion. Pressure is the normal force per unit area exerted on the surface due to the time rate of change of momentum of the gas molecules impacting on that surface.

We normally define pressure at a point. If dA is the elemental area of that point, dF is the force on one side of dA due to pressure, then pressure at that point in the fluid is defined as

 p = lim (dF/dA)                       dA -> 0

So, pressure is a point property, it can vary from one point to another point.

2. Density

It is defined as the mass per unit volume. It is also a point property. If dV = elemental volume about point B, dm = elemental mass of the fluid inside dV, then density at point B is

 ρ = lim (dm/dV)          dV-> 0

 ρ is the limiting form of mass per unit volume.

3. Temperature

The temperature T of a gas is directly proportional to the average kinetic energy of the molecules of the fluid. If KE is the mean molecular kinetic energy, then temperature is given by

KE = 3/2 kT                             where k is Boltzmann constant.

T is also a point property. In a high temperature gas, the molecules are randomly moving about at high speeds, as compared to the same in a gas at lower temperature.

4. Flow velocity

In case of solid objects moving at 30 m/s all parts of the solid are simultaneously translating at the same 30 m/s velocity. In contrast, the fluid in motion, one part of the fluid may be traveling at a different velocity from the other part. Now in case of flow of air over an airfoil, look at infinitesimally small element of fluid mass, called fluid element, moving with time. Both the speed and direction of this fluid element can vary as it moves from point to point in the gas. Now fix your eyes on a specific fixed point in space (pt. B) in the gas. This we define as follows:

The velocity of a flowing gas at any point B in space is the velocity of an infinitesimally small fluid element as it sweeps through B. This flow velocity at any point has magnitude and direction; hence it is a vector quantity.  In contrast to this, p, ρ and T are scalar variables.

Consider two adjacent streamlines a and b. The streamlines are infinitesimally small distance dy apart. We can imagine that streamline a is rubbing against streamline b and due to friction, exert a force of magnitude dF_f on streamline b acting tangentially towards right. Further, imagine this force acting on dA where dA is perpendicular to y-axis and tangent to streamline b.

The total shear stress   τ at point 1 is:                 τ  = lim (dF_f /dA)           dA -> 0

The shear stress τ is the limiting form of the magnitude of the frictional force per unit area (perpendicular to y-axis). Shear stress acts tangentially along the streamline. The value of shear stress at a point on a streamline is proportional to the spatial rate of change of velocity normal to the streamline at that point.

τ  µ dV /dy

Hence, we have, τ  = m(dV /dy)  where m is the viscosity coefficient and dV /dy  is the velocity gradient.

Units:   p          N/m²  or  lb/ft²

                ρ          kg/m³   Or  slug/ft³

                                               

Lifting and non-lifting surfaces

Lifting surfaces are those which produce lift when placed in airflow. In a conventional aircraft the following surfaces produce lift:

(a)  Wings along with their flaps and ailerons.

(b)  Canard surfaces.

(c)  Horizontal stabilizer along with the elevators installed aft of them

Non-lifting surfaces are those which either do not produce any lift or the lift produced is insignificant. Examples of such surfaces are:

(a)  Fuselage

(b)  Vertical stabilizer along with the rudder mounted on it.

(c)  Ventral fin

(d)  Engine nacelle and engine mount.

(e)  Undercarriage along with the undercarriage doors

(f)   Fuselage doors (for pilot’s cabin and passenger / cargo )

Origin of Lift
A lifting surface is nominally an extrusion of a streamlined cross-section: the cross-section has a rounded leading edge, sharp trailing edge, and a smooth surface. The theory of lifting surfaces centers on the Kutta condition, which requires that fluid particle streamlines do not wrap around the trailing edge of the surface, but instead rejoin with streamlines from the other side of the wing at the trailing edge. The fluid must travel faster over one side of the surface than the other. The reduced Bernoulli pressure so induced can be thought of as the lift-producing mechanism. More formally, lift arises from circulation, which is the integral of velocity around the cross-section, and a lifting surface requires circulation in order to meet the Kutta condition.

L = ρ U Г where      Г =v.ds