Fluid Dynamics

The movement of fluids is generally a complex subject. Most of the significant problems in fluid dynamics have been solved only recently, due to the advent of computers. But if the fluid velocities are not too high and "turbulence" is not too significant, a few simple principles (and the corresponding equations) can provide insights into the behavior of plumbing systems and into the response of buildings subjected to wind.

The Bernoulli Principle

Daniel Bernoulli was a well-known mathematician and all around scientist. He made many contributions to the field of fluid dynamics. The Bernoulli principle applies to fluid flow and relates the pressure in fluid with its velocity. (You may have already been introduced to this relationship if you took SCI 100.)

Fluid Velocity

First, a few reminders of the basic definitions in fluid dymanics. A fluid is either a gas (like air) or liquid (like water) that does not hold its own shape. The pressure in a fluid is the force per unit area (P = F/A). Pressure in a fluid increases with depth. The density is the mass per unit volume (r = mass/volume). For incompressible fluid such as liquids, the density remains essentially constant with depth. However, for gases the density typically increases with depth, as it does in our atmosphere, for example.

The fluid velocity is the velocity of the molecules that make up the fluid. That is not necessarily the same everywhere. The fluid velocity of the air around us is what we call the windspeed, which is certainly not the same everywhere. We use streamlines to help visualize this. A streamline is the path taken by any small element or parcel of the fluid. Imagine small leaves floating in a river and being carried along with water. The leaves trace out streamlines. Here's what you should remember about streamlines:

1. The velocity of a fluid is always tangent to the streamlines.
2. Streamlines can never cross one another.
3. The velocity is higher where streamlines are close together and lower where they are far apart.

The first statement comes from the definition of a streamline. The second statement means that one element of fluid cannot pass through another. The third statement can be understood if we examine the river example again. Assume the river narrows but the depth stays the same. We know the river must speed up there. Why? Because the same amount of water must flow in the narrow region as in the wider regions. Since there is less water in a given length along the stream in the narrow region, it must flow proportionately faster. Now think about leaves floating along the streamlines. They will bunch together as they approach the narrow region.

Flow Rate

We can express the connection between area and velocity in an equation, called the equation of continuity. Let v represent the velocity and A represent the cross sectional area (perpendicular to the flow) of the river any point as shown in the diagram. The equation of continuity for an incompressible fluid is:

This is a common sense equation. If A₁ is twice A₂, then v₁ is half v₂. The water will have to flow twice as fast where the area is half as much in order for the same amount of water to flow. (Note: The equation of continuity for compressible fluids is only slightly more complicated, but we won't need it for this course.)

The product Av is not just an abstract quantity. It represents the volumetric flow rate of the fluid. It has units of volume per time and represents the volume of fluid that passes by each second. In plumbing, it represents the rate at which water flows from a faucet or flows down a drain waste vent (DWV) pipe.

Velocity and Pressure

Consider a garden hose in which water flows. The walls of the hose encloses a tube of flow or a bundle of streamlines. No fluid enters or leaves a tube of flow. We can follow a tube of flow even if it is not confined by walls. For example, deep ocean currents act like rivers within the ocean and represent tubes of flow. This idea allows us to keep track of just where the fluid is going.

Now look at the tube of flow which has a constricted region as shown. Fluid is moving to the right. We know that the fluid must speed up as it moves from region #1 into the constricted region (region #2) and slow down again as it moves into region #3. The fluid has mass. If we are going to accelerate it to the right, we must have a net force to the right. This occurs because the pressure in region #1 is greater than in region #2. Conversely, the fluid must be decelerated with a force back to left as it enter region #3. So the pressure in region #3 must be higher than that of region #2. The general trend is clear.

The pressure is higher where the velocity is lower, and the pressure is lower where the velocity is higher.

Using conservation of energy, Bernoulli derived an expression for the relationship between velocity and pressure in a tube of flow. It is

The net force that acts on a parcel of the fluid or objects in the fluid is often the result of pressure differences. If we re-arrange the equation above, we have

It is worth noting at this point that the equation above is a special case of Bernoulii's Equation in which we have not taken into account the effect that depth has on pressure. But if the tube of flow is nearly horizontal, this equation is a good approximation. We will incorporate depth into the equation later.

UNITS

The SI system of units for all the quantities we've been discussing is as follows:

velocity (v)	meters / second = m/s
density (r)	kilograms / cubic meter = kg/m3
pressure (P)	Newtons / square meters = N/m2 = Pascal (Pa)

Since the British system is still commonly used, here's a few conversions to make calculations a bit easier.

• 1 mph = .447 m/s    (miles per hour to meters per second)
• 1 psi = 6,890 Pa    (pounds per square inch to Pascals)
• 1 lb/ft³ is equivalent to 16 kg/m³    (weight density to mass density)

Bernoulli Lift

This is the essential physics that explains how airplanes fly. The wing is shaped (the shape is called an airfoil) such that the air flows faster over the top of the wing than over the bottom. So the pressure is higher on the bottom than the top. The pressure difference creates a net lift (called the Bernoulli lift) that counters the airplanes weight. It is amazing, but a 500,000 lb jet can fly because it can generate Bernoulli lift nearly equal to its weight. (About 10% of the lift comes other sources.) But that same lift can have catastrophic effects on houses during hurricanes.

Note that the pressure difference is proportional to the difference in v². That means that pressure differences are more dramatic at higher speeds. Damage from hurricanes increases dramatically with sustained windspeed increases due in large part to this behavior.

Problem: Estimate the net force generated on a house window during hurricane Marilyn. According to the National Weather Service, the sustained winds were just over 100 mph. (Nevermind that those of us who were here thought they were much higher.) This is a speed of about 45 m/s.

Solution: Let's assume you have the house closed up. The speed inside is close to zero. Taking r of air to be about 1.3 kg/m³, the pressure difference between inside and outside the house would be

DP = ¹/₂ 1.3 (45² - 0²) = 1300 Pa = .2 psi.

At first, .2 psi may not seem that large. But consider the surface area of 3 ft by 3 ft glass window. The net force acting to push the window outward would be

F_net = DP (A) = .2 psi x (36 in x 36 in) = 250 lb!

That's correct, 250 lb! Most people who have sat near a window during a storm have witnessed the unnerving flex in the glass. What about the roof? For a typical 10 m by 10 m roof, you should be to show that the net lift force would be 1.3 x 10⁵ N, which is almost 30,000 lb or 14 tons! Most often, a roof lost during a hurricane is not battered off by direct wind blast, but "sucked" off the roof by the Bernoulli lift.

Why Have a Tall Ceiling?

If you have a properly vented high ceiling, the Bernoulli principle will work to keep fresh air moving into your home. Recall from SCI 100 that hot air naturally rises. As air heats up, it expands, becomes less dense and hence more buoyant. If the roof is properly ventilated, the hot air is drawn out (the process is also called drafting) and cooler air is drawn into the house through door and window openings. The taller the roof, the greater the draft. It is the same principle that ensures that the dangerous hot ash and smoke from a fireplace travels up the chimney. For a nice description of how a chimney works, check out the Chimney.com site. Maybe you'll want a fireplace in your home for those chilly winter nights! The Bernouilli effect can considerably enhance the draft. When even a gentle breeze blows across the roof, a low pressure area is created. This helps to "pull" the hot air out of the house.

Bernoulli lift shows up in many interesting areas, but the physics is the same. Check out these other links.

• Newton's Apple provides a few more examples applying Bernoulli's Principle.
• The Rochester Museum and Science Center has a few Bernoulli experiments for you to try.
• Here's a nice tutorial on the airfoil that has detailed flow diagrams of how air moves past an airplane wing.

Bernoulli's Equation

Bernoulli's Equation as given in the previous section is a special case and does not include the effect of depth on pressure. But the modification to the equation is simple. The "full" Bernoulli's Equation is:

The only change is that a "rgy" term has been added to each side of the equation. The symbol "y" has a slightly different meaning from "h" that we used before. It is the position, rather than depth. When using h for depth, we take h as positive measured downward. For the position y, upward is taken as positive. The absolute values for y₁ and y₂ are not as important as their relative values. If position y₁ is above y₂, then y₁ > y₂. The example below will show you how this works. (As an exercise, convince yourself that if the velocity is zero everywhere, the equation above reduces to the static pressure equation from the Fluid Statics section, if we realize that h = y₁ - y₂.)

There are names for the three different terms found on either side of Bernoulli's equation. They are names commonly used in engineering:

P	Pressure (absolute pressure, not a gauge pressure)
1/2 rv2	Dynamic Head (or dynamic pressure head)
rgy	Static Head (or static pressure head)

Hydraulic engineers certainly must be concerned with all three of these terms when designing hydroelectric dams, city water supply systems, or plumbing for large buildings. Although these considerations are less important for single story homes connected to a city supply system, most homes built in the islands must provide both the reservoir (the cistern) and the pressure (the electric pump.) It is wise to have a working knowledge of Bernoulli's Equation. Let's do a few calculations.

Problem: You have a cistern that, when full, has a water level 2.0 meters above the bottom. You have installed a 1" pipe with a spigot (faucet) near the bottom of the cistern so that you drain the cistern and water plants. If the cistern is full, with what speed will water flow out of the spigot and at what flow rate? What would happen if you attached a hose to the spigot and tried to water plants above the spigot level?

Solution: We need to chose two points in the "tube of flow" of water from the cistern to the spigot. Since water is moving, we can't just apply the static pressure equation. The two places where we know the pressure is 1) at the top of the cistern and 2) just outside the spigot. Both are at atmospheric pressure because they are open the atmosphere at those points. What are the velocities at these two points? At the top of the cistern we know it is nearly zero since water can't drain that fast from a 1" pipe. The velocity just outside the spigot is what we want to find. Let's take the height at the top to be zero (the surface is often taken to be zero). The bottom must be -2 meters. That is, the bottom value ( y₂) must be 2 meters less than the top value (y₁). Using P_atm = 10⁵ N/m² and r = 10³ kg/m³, we have ...

Solving the equation for the velocity, we get v₂ = square root of (2 x 9.8 x 2) = 6.3 m/s. Note that we could just as easily chosen the bottom position (y₂) to be zero. In that case, the top position would have been y₁ = +2m. The result would be the same.

Now consider the flow rate given by Av. Since v has units of m/s, the cross sectional area A must be converted. The diameter is d = 1" x 2.54 in/cm x .01 m/cm = .0254 m. The area is A = p(.0254/2)² = 5.1 x 10^-4 m². And the flow rate would be 5.1 x 10^-4 m² x 6.3 m/s = 3.2 x 10^-3 m³/s. At first, this might seem small, but let's convert this flow rate into something more familiar. A liter is 1/1000 th of a cubic meter, so the flow rate is 3.2 liters/second. That's close to a gallon per second! This number might be a bit high, since we have not accounted for friction and turbulence, but it gives a good estimate.

What about watering plants at higher ground? Can you connect a hose to the spigot and get 3.2 liters/s anywhere? It depends upon the vertical postion of the end of the hose. The new position where the pressure is known will be just outside the nozzle of the hose. The end of hose is treated exactly as we did the spigot. If the end of the hose is held .5 m above the spigot, then y₂ = -1.5 m and the velocity will be less. What happens if the end of the hose is carried above the top of the water level in the cistern? Well, Bernoulli's equations will give you an answer. The velocity will be the square root of a negative number. In mathematics, that is called an imaginary number. It tells us that it never happened!. Water cannot flow uphill! (Actually, it can if it also flows downhill a greater distance. You'll get a chance to investigate the peculiar behavior of the siphon in lab.)

Viscosity

Earlier we stated that liquids do not maintain a shape because they cannot sustain shear forces and the molecules slide easily by one another. That does not mean there are no attractive forces between molecules, but simply that those forces are insufficient to maintain a shape. The molecules' mutual attraction to one another is called cohesion and is the cause of viscosity. Viscosity in common language refers to the "thickness" of the liquid. Salad oil is more viscous than water, and honey is more viscous than salad oil. (Viscosity is also function of temperature. In fact, as most liquids cool, they become more viscous and eventually become solid. But this behavior is not simple and we won't focus on it in this course.)

In fluid dynamics, the viscosity is denoted by the greek letter eta, h. It is defined by the fluid's resistance to flow in a way that is reminiscent of how we defined Young's Modulus for solids in the first module. Here's the idea. A thin layer of the liquid of thickness t is trapped between two flat plates of area A. The bottom plate is fixed and the top plate is pulled to the right with a velocity v. A certain force F will be required to maintain this velocity. The more viscous the fluid, the greater the force needed. The definition is

h = Shear Stress / Strain rate = F / A /V / t

Viscosity has the rather odd SI unit of Pa^.seconds. But the most commonly used unit is the Poise = .1 Pa^.s. There are a few things to note about the behavior of the fluid. First, there is a thin layer of the fluid (perhaps only a molecule thick) that is "stuck" to each plate. It is dragged along with the plate by an attractive force between the fluid and plate called adhesion. In the diagram, the velocity of the fluid is zero at the bottom plate and v at the top. The fluid velocity increases linearly from the zero at the bottom to v at the top. Think of the fluid as many layers sliding across one another, exerting cohesive forces on each other. This internal friction has a marked effect on how fluid flows through pipes.

The velocity profile for water flowing through a pipe is similar to that of the two plates, except the velocity is zero at the walls of the pipe and the maximum value v is at the center of the pipe. So how does the viscosity effect the flow rate of a fluid through a pipe? A Frenchman name Poiseuille derived an expression for the flow rate through a horizontal pipe of radius R and length L with a pressure difference of DP from one end to the other. The derivation is challenging even for our physics majors, so we'll just state the result and discuss what it tells us about installing plumbing.

We want to understand the implications of this equation. The viscosity for a given fluid is a fixed value. What are the variables that can we control? They are the radius of the pipe, the pressure difference, and the length of the pipe. What does Poiseuille's equation say about each of those?

• The flow rate depends upon R⁴! This result may seem suprising, but remember that the Equation of Continuity told us that a pipe with a 1" diameter will have four times the flow rate of that of a 1/2 " pipe for the same flow velocity. In addition, the drag decreases as the pipe size inceases. Poiseuille's Eq. says that for a given pressure difference and given length, the flowrate through a 1" pipe is 16 times that for a 1/2 " pipe!

• The pressure difference is what drives the fluid through the pipe. It is determine by your water pump and the elevation difference. Clearly a stronger pump can provide a greater pressure gradient. Note that our previous calculation for checking whether a 20 psi pump could push water up 6 meters to the second floor, did not consider what flow rate could be maintained with such a pressure. We revisit this problem below.

• The effect of the length of the pipe is easy to understand. A pipe with twice the length will produce twice as much drag and cut the flow rate in half. So we don't want to make our pipes any longer than we have to. And if there is a particularly long run (for example, out to the workshop located a hundred feet from the pump), then perhaps we should invest in a larger pipe!

There are two ways to look at P's Eq. The pressure difference DP maintains the flow rate OR for a given flow rate, there will be a pressure drop DP. Plumbers generally look at it in the latter way. They estimate the flow rate that will be needed and from that calculate the pressure loss along a given length of pipe to determine whether or not they will have use a larger pipe.

Problem: Recall the problem in the previous section in which we had a pump capable of producing a 20 psi (1.4 x 10⁵ Pa) gauge pressure. Using the static pressure equation we found that it could support water to a height of 14 m, well above the 6 m from the cistern water level to the second floor faucet. But what flow rate could it produce at this faucet?

Solution: The first thing to note is that Poiseulle's equation is for a horizontal pipe. The pressure difference in the equation is due to viscosity and does not include the pressure difference due to height. To support water in the pipe to a height of 6 m requires a pressure of P = rgh = 5.9 x 10⁴ Pa, called the static pressure head. This much pressure is needed just to hold the water up to this height. So the pressure available to move the water is DP = 1.4 x 10⁵ - 5.9 x 10⁴ = 8.1 x 10⁴ Pa. Next we need to know the pipe length and radius. The industry standard for branch supply water pipes is 1/2 " (also called Schedule 40), so this is a radius of .64 cm. The pipe will have to be at least 6 m long. Finally, we need the viscosity of water. Viscosity values can be found in most physics texts or engineering handbooks. For water, h = .010 Poise = .0010 Pa^.s. If we plug all this into Poiseuille's equation, we get

Flow Rate = (3.14/8) x (.0064⁴ / .001) x (8.1 x 10⁴ / 6) = 8.9 x 10^-3 m³/s

This is nearly 9 liters/s! So it seems we have more than enough pressure. The 20 psi pressure is certainly sufficient, but there is one more important consideration.

Turbulence

All of the equations we have introduced for fluid flow apply only for laminar flow. Laminar flow is flow in which the fluid "layers" slide smoothly over one another. In laminar flow there are no vortices (whorls) or cavitation. Vortices are where the fluid rotates back upon itself and cavitation is where bubbles and voids are created. Such turbulence often occurs at barriers or wherever there are abrupt changes is the direction of the fluid flow. Turbulence introduces frictional energy losses into the system that are very difficult, if not impossible to predict. Even the seemingly simplist of problems typically must be solve with computers. As the velocity of a fluid increases, so does the onset of turbulence. (The Reynolds number provides an estimate of the transition between laminar and turbulent flow. It is a number calculated from flow rate, pipe size, fluid viscosity and density.) The diagram below shows how the shape of a junction between two different sized pipes can effect the onset of turbulence.

In laminar flow, the streamlines converge (or diverge) smoothly. In turbulenct flow, the streamlines become erratic and the increase in friction can become very significant. Despite its complex nature, there are a few simple things you can do to reduce turbulence in your plumbing.

• Have as few unions, joints and bends as possible. Wherever there is an abrupt change in direction, turbulence can result. Curved joints are better than abrupt angle joints.

• Join pipes of different diameters with connectors designed for the job. These connectors typically provide a smooth gradual transistion from one diameter to the next.

• The National Building Code has set out installation guidelines. These guidelines will tell you how many branches (typically 1/2 ") can be supplied by a main (typically 3/4 " or 1") line. It provides table of the pressure loss per foot for different size pipes. Most of the values in the Code are the result of experiments. We will perform a few of our own experiments in the laboratory.

Is your mind feeling a bit viscous at this point? Maybe you can loosen it up with a few QUESTIONS about fluid dynamics.