Lesson 4: Fluid Flow
This will be a continuation of fluid properties, introducing Bernoulli’s theorem – the conservation of energy and simple equations related. The differences between rotational and irrotational flow will be discussed. We will get into Venturi tube flow, Ideal fluid flow, laminar and turbulent fluid flow. Discussion of the Reynolds Number and the dependence of flow will be presented. We will also introduce airfoils.
Section 1 - More Flow Types, Bernoulli's and Venturi
In this lesson, we will continue with flow types encountered in aerodynamics.
ROTATIONAL AND IRROTATIONAL FLOW
Rotational and irrotational flow - Fluid flow can be rotational or irrotational. If the elements of fluid at each point in the flow have no net angular (spin) velocity about the points, the fluid flow is said to be irrotational.
Imagine a small paddle wheel immersed in a moving fluid. If the wheel translates without rotating, the motion is irrotational. If the wheel rotates in a flow, the flow is rotational.
According to a theorem of Helmholtz, assuming zero viscosity, if a fluid flow is initially irrotational, it remains irrotational. Take the example of an observer fixed to the airfoil section. The flow far ahead of the airfoil section is uniform and of constant velocity. It is irrotational.
As the airflow passes about the airfoil section, it remains irrotational if zero viscosity is assumed. In real life, viscosity effects are limited to a small region near the surface of the airfoil and in its wake. Most of the flow may still be treated as irrotational.
A simplifying argument often used to aid in understanding basic ideas is that of a one-dimensional flow. In the case of a bundle of streamlines of a simple flow, each streamline can be thought of as a stream tube since fluid flows along it as if in a tube. In steady flow, the stream tube is permanent.
Taken together, the bundle of stream tubes comprise an even larger stream tube. Fluid flows through it as, for example, water flows through a pipe or channel. The velocity varies across the tube, in general, according to the individual streamline velocity variation. You can then easily imagine an "average" uniform value of velocity at the cross section to represent the actual varying value.
The velocity then is considered "one dimensional" since it varies only with the particular distance along the tube where observations are made. In addition to velocity, pressure, density, temperature, and other flow properties must also be uniform at each cross section for the flow to be one dimensional.
AERODYNAMIC FORCE PRINCIPLES
In order to understand how aerodynamic forces arise, two basic principles must be considered. They are the laws of conservation of mass and conservation of energy. These laws convey the facts that mass and energy can neither be created nor destroyed.
To illustrate, if the fluid is considered to be inviscid and incompressible (and essentially "perfect"), the flow is considered steady and one dimensional. Now let’s discuss Ideal Fluid Flow.
IDEAL FLUID FLOW
In Ideal Fluid Flow, the continuity equation is a statement of the conservation of mass in a system. Consider a pipe which is uniform in diameter at both ends, but has a constriction between the ends. This is called a venturi tube. In this case, it is assumed that the fluid, under the previously stated assumptions, is flowing in the direction indicated. Stations 1 and 2 have cross-sectional areas A1 and A2, respectively.
Let V1 and V2 be the average flow speeds at these cross sections (one-dimensional flow). We can also assume that there are no leaks in the pipe nor is fluid being pumped in through the sides.
The continuity equation states that the fluid mass passing station 1 per unit time must equal the fluid mass passing station 2 per unit time. In fact, this "mass flow rate" must be the same value at any cross section examined or there is an accumulation of mass- "mass creation"- and the steady flow assumption is violated. This can be represented by, (Mass rate)1 = (Mass rate)2 (1) where Mass rate = Density x Area x Velocity (2) This equation reduces to pl AlV1 = p2A2V2 (3) Since the fluid is assumed to be incompressible, p [Greek letter rho] is a constant and equation (3) reduces to AlV1 = A2V2 (4) This is the simple continuity equation for inviscid, incompressible, steady, onedimensional flow with no leaks. If the flow were viscous, the statement would still be valid as long as average values of V1 and V2 across the cross section are used.
By rearranging equation (4), you get V2 = (A1/A2)V1 (5) Since A1 is greater than A2, it can be concluded that V2 is greater than V1. This is a most important result. It states, under the assumptions made, that the flow speed increases where the area decreases and the flow speed decreases where the area increases.
By the continuity equation, the highest speed is reached at the station of smallest area. This is at the narrowest part of the constriction commonly called the throat of the venturi tube.
The fact that the product AV remains a constant along a tube of flow allows an interpretation of the streamline picture. In the area of the throat of the Venturi tube, the streamlines must crowd closer together than in the wide part.
Thus, the distance between streamlines decreases and the fluid speed increases. The conclusion is that, relatively speaking, widely spaced streamlines indicate regions of low-speed flow and closely spaced streamlines indicate regions of high-speed flow.
BERNOULLI'S THEOREM AND THE CONVERSATION OF ENERGY
Now onto Bernoulli's theorem and the conservation of energy. Assume a fluid flow which, as before, is inviscid, incompressible, steady, and one dimensional. The energy in the flow is composed of several energies. The kinetic energy arises because of the directed motion of the fluid; the pressure energy is due to the random motion within the fluid; and the potential energy is due to the position of the fluid above some reference level.
Bernoulli's theorem is an expression of the conservation of the total energy; that is, the sum total of these energies in a fluid flow remains a constant along a streamline. In other words, the sum of the kinetic energy, pressure energy and potential energy remains a constant.
If it is further assumed that the fluid flow is horizontal as in, for example, airflow approaching an aircraft in level flight, then the potential energy of the flow is a constant. Bernoulli's theorem reduces to Kinetic energy + Pressure energy = Constant
where the constant includes the constant value of potential energy. If you consider the energy per unit volume, you can obtain the dimensions of pressure and Bernoulli's theorem may be expressed in terms of pressure.
The kinetic energy per unit volume is called dynamic pressure q and is determined by q = 1/2pV2 where p and V are, respectively, the fluid flow density and speed at the point in question.
The pressure energy per unit volume (due to random motion within the fluid) is the static pressure of the fluid and is given the symbol p.
The constant energy per unit volume is called the total pressure pt.
Bernoulli's equation reduces to
Dynamic pressure + Static pressure = Total pressure (7)
or
1/2pV2 + p = pt (8) For rotational flow the total pressure pt is constant along a streamline but may vary from streamline to streamline. In an irrotational flow, the usual case considered for airflow approaching an aircraft, the total pressure is the same constant value everywhere.
Bernoulli's equation states that in a streamline fluid flow, the greater the speed of the flow, the less the static pressure; and the less the speed of the flow, the greater the static pressure. There exists a simple exchange between the dynamic and static pressures such that their total remains the same. As one increases, the other must decrease.
