Figure (a) shows a pipe (a venturi tube) that is uniform in diameter at both ends, but has a constriction between the ends. The cross-section A1 is greater than the cross-section A2 and it can be concluded that V2 is greater than V1, since the flow speed increases where the area decreases and the flow speed decreases where the area increases, as shown in figure (b). Figure (c) shows the streamline pattern in the venturi tube. In the area of the throat, the streamlines crowd closer together than in the wide part, indicating regions of relatively high-speed flow.
For rotational flow the total pressure pt is constant along a streamline but may vary from streamline to streamline as shown in figure (a). In an irrotational flow, the usual case considered for airflow approaching an aircraft, the total pressure is the same constant value everywhere as shown in figure (b).
Figure (a) Pitot tube. Figure (b) Static tube. Figure (c) Pitot-static tube.
Ideal Fluid Flow and Pressure Distribution
The term static pressure, p, is used to describe the pressure exerted by a static, or still, air mass. Static pressure + dynamic pressure = constant.
One uses a Venturi tube, a pipe that is uniform in diameter at both ends but has a constriction between the ends, along with the continuity and the Bernoulli equations to demonstrate pressure distribution.
Continuity equation: AlV1 = A2V2
Bernoulli's equation: Static pressure + Dynamic pressure = Total pressure
p + 1/2ρV2 = pt
The static pressure of the undisturbed free-stream fluid flow entering the tube may be used as a reference value. Any variation of static pressure in the tube then is a greater or lesser value than the free-stream static pressure. Holes can be drilled into the walls of the Venturi tube to measure the static pressure. These holes are commonly called "static taps" and are connected to a "U-tube manometer"a tube having a U-shape with a liquid such as colored alcohol within it. When the static pressure measured at the static tap equals the free-stream static pressure, the fluid levels in the tube are at some equal reference level. But static pressures above or below the free-stream pressure are indicated by a decrease or increase in the level of fluid in the tube.
Using a complete setup of a Venturi tube and a set of manometers and static taps to measure static pressure, by the continuity equation, the speed at the throat of the tube, designated V2, is greater than the speed at tube's widest point, V1. The speed at the throat also is the highest speed achieved in the Venturi tube. By Bernoulli's equation, the total pressure pt is constant everywhere in the flow (assuming irrotational flow). Therefore, one can express the total pressure pt in terms of the static and dynamic pressures at the widest point and at the throat using equation (9):
1/2plV12 + p1 = 1/2p2V22 + p2 = pt
Since V2 is greater than V1 and p2 = p1 (fluid is incompressible), it follows that p2 is less than p1, for as the dynamic pressure, hence speed, increases, the static pressure must decrease to maintain a constant value of total pressure pt. Thus, the static pressure decreases in the region of high-speed flow and increases in the region of low-speed flow. This is also demonstrated by the liquid levels of the manometers where as one reaches the throat, the liquid level has risen above the reference level and indicates lower than free-stream static pressure. At the throat, this is the minimum static pressure since the flow speed is the highest.
In a “symmetric” (upper and lower surfaces the same) airfoil, a line drawn through the nose and tail of the airfoil is parallel to the free stream direction. The free-stream velocity is denoted by V∞ and the free-stream static pressure by p∞. The particle pathline follows the airfoil contour, and the velocity decreases from the free-stream value as one approaches the airfoil nose. At the airfoil nose, the flow comes to rest (stagnates). From Bernoulli's equation, the static pressure at the nose is equal to the total pressure (there is no dynamic pressure at this point). Moving from the nose up along the front surface of the airfoil, the velocity increases and the static pressure decreases. By the continuity equation, as one reaches the thickest point on the airfoil, the velocity has acquired its highest value and the static pressure its lowest value (the dynamic pressure is at its greatest value).
Beyond this point, as one moves along the rear surface of the airfoil, the velocity decreases and the static pressure increases until, at the trailing edge, the flow comes to rest with the static pressure equal to the total pressure. Beyond the trailing edge, the airflow speed increases until the free-stream value is reached and the static pressure returns to free-stream static pressure.
Note particularly that on the front surfaces of an airfoil (up to the point of maximum thickness), pressure decreases whereas on the rear surfaces, pressure increases.
Lift is defined as the force at right angles to (normal to) the free-stream direction and drag is the force parallel to the free-stream direction. For a planar airfoil section operating in a perfect fluid (one that is inviscid and incompressible), the drag is always zero no matter what the orientation of the airfoil is. This seemingly defies physical intuition and is known as D'Alembert's paradox (named after Jean leRond d'Alembert, a great mathematician and physicist of the 18th century). It is the result of assuming a fluid of zero viscosity. The components of the static-pressure forces parallel to the free-stream direction on the front surface of the airfoil always exactly balance the components of the pressure forces on the rear surface of the airfoil. The lift is determined by the static-pressure difference between the upper and lower surfaces and is zero for this particular case since the pressure distribution is symmetrical. If, however, the airfoil is tilted at an angle to the free stream, the pressure distribution symmetry between the upper and lower surfaces no longer exists and a lift force results. This is very desirable and the main function of the airfoil section.
Air is not a perfect fluid. It possesses viscosity. With slight modification, the continuity and Bernoulli principles still apply in the real world. The airflow over an airfoil will appear to be slightly different with an accompanying reduction in lift and the existence of drag in several forms.
Adapted from Talay, Theodore A. Introduction to the Aerodynamics of Flight. SP-367, Scientific and Technical Information Office, National Aeronautics and Space Administration, Washington, D.C. 1975. Available at http://history.nasa.gov/SP-367/cover367.htm
For Further Reading:
Anderson, Jr., John D. A History of Aerodynamics. Cambridge, England: Cambridge University Press, 1997.
Hewitt, Paul G. Conceptual Physics. Sixth Edition. Glenview, Ill.: Scott, Foresman and Company, 1989.
Smith, Hubert. The Illustrated Guide to Aerodynamics, 2nd edition. Blue Ridge Summit, Pa.: TAB Books, 1992.
Wegener, Peter P. What Makes Airplanes Fly? New York: Springer-Verlag, 1991.