Lesson 4: Fluid Flow
Section 2 – Pressure Measurement in Ideal Fluid Flow
PRESSURE MEASUREMENT IN A FLOW
Let us now examine how total, static, and dynamic pressures in a flow are measured. In the case of the fluid flow about a simple hollow bent tube, called a pitot tube after its inventor, this is connected to a pressure measurement readout instrument.
The fluid dams up immediately at the tube entrance and comes to rest at the "stagnation point" while the rest of the fluid divides up to flow around the tube.
By Bernoulli's equation the static pressure at the stagnation point is the total pressure since the dynamic pressure reduces to zero when the flow stagnates. The pitot tube is, therefore, a total-pressure measuring device.
This is what is referred to as pressure measurement. In the case when the fluid flow about another hollow tube where the end facing the flow is closed and a number of holes have been drilled into the tube's side, this tube is called a static tube.
It may be connected to a pressure measuring readout instrument as before. Except at the stagnation point, the fluid is parallel to the tube everywhere. The static pressure of the fluid acts normal to the tube's surface.
A VENTURI TUBE
Now let us examine the complete set up of a Venturi tube and a set of manometers and static taps to measure static pressure. By the continuity equation the speed at station 2 V2 is greater than that at station 1 V1 as seen previously-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 if there is irrotational flow. Therefore, one can express the total pressure pt in terms of the static and dynamic pressures at stations 1 and 2 using equation (8), represented by, 1/2plV12 + p1 = 1/2p2V22 + p2 = pt (9) 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, therefore speed, increases, the static pressure must decrease to maintain a constant value of total pressure pt.
The block diagrams below the venturi tube show this interchange of dynamic and static pressures all along the venturi tube.
The conclusion to be drawn from this is that 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.
A critical part of ideal flow is flow through an airfoil. The following section expands the previous discussion of venturi flow to the ideal fluid flow past an airfoil. The following figure shows a "symmetric" (upper and lower surfaces the same) airfoil operating so that 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 ρ∞. Following the particle pathline (indicated by the dotted line and equal to a streamline in this steady flow) which follows the airfoil contour, the velocity decreases from the free- stream value as one approaches the airfoil nose (points 1 to 2).
At the airfoil nose, point 2, the flow comes to rest (stagnates). From Bernoulli's equation the static pressure at the nose, point 2, is equal to the total pressure.
Moving from the nose up along the front surface of the airfoil (points 2 to 3), the velocity increases and the static pressure decreases. By the continuity equation, as one reaches the thickest point on the airfoil, point 3, the velocity has acquired its highest value and the static pressure its lowest value.
Beyond this point as you moves along the rear surface of the airfoil, points 3 to 4, the velocity decreases and the static pressure increases until at the trailing edge, point 4, the flow comes to rest with the static pressure equal to the total pressure. Beyond the trailing edge the flow speed increases until the free-stream value is reached and the static pressure returns to free-stream static pressure.
Notice on the front surfaces of the airfoil (up to the station of maximum thickness), one has decreasing pressures (a negative pressure gradient) whereas on the rear surfaces one has increasing pressures (a positive pressure gradient).
The lift is defined as the force normal to the free-stream direction and the drag parallel to the free-stream direction. For a planar airfoil section operating in a perfect fluid, the drag is always zero no matter what the orientation of the airfoil is. This seems to defy physical intuition and is known as D'Alembert's paradox.
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 useful and the main function of the airfoil section.
Air is not a perfect fluid. It has viscosity. With slight adjustment, 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.
