13.012 Hydrodynamics for Ocean Engineers Lab #2
13.012 Hydrodynamics for Ocean Engineers Prof. A.H. Techet Fall 2002
Laboratory #2: Hydrostatic and Dynamic Pressure Measurements
2.1 INTRODUCTION
Pressure in a fluid is defined as a normal force per unit area and has the dimensions of [Force/Length2] = [M/(TL)]. Fluid pressure exists whether the fluid is at rest or in motion. Hydrostatic pressure, the pressure resulting from a stationary fluid, and dynamic pressure, that arising from a moving fluid, are the subject of today's laboratory exercise.
HYDROSTATIC PRESSURE
In a static fluid, the forces acting on a small elemental fluid volume are a combination of the forces due to pressures exerted on the element from its surroundings and the force resulting from gravity acting on the element. As a result pressure variations only result in the vertical direction:
dp dp dp = −ρg = 0 = 0 dz dx dy
where ρ is the fluid density and g is the gravitational constant. A Right hand coordinate system is chosen such that z is positive in the direction opposite to gravity.
The above equation can be integrated to yield the hydrostatic equation:
∆p = ρg(∆z)
Examining this equation we can deduce that the pressure in a fluid will remain constant at all points at a given depth (unless the fluid density was to change - for example in the presence of higher salinity).
When measuring pressure in a fluid, the hydrostatic equation comes in very handy. Measurement devices such as a manometer are used to measure pressure relative to a reference pressure (most likely atmospheric pressure). Such measured pressures are
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referred to as gage pressures since they are relative to another pressure. The absolute pressure in the fluid can be found by adding the reference pressure to the gage pressure:
pabs = pref + pgage
BERNOULLI’S EQUATION
P + ½ ρ V2 + ρgz = constant
Where the dynamic pressure is 1/2ρV2 and the hydrostatic component is ρgz.
The pressure at the point where the flow directly impinges on a body is called the stagnation pressure. This is also referred to as the total pressure on the body. Using Bernoulli’s equation along a horizontal streamline (z∞ = zo) we can get a relation for the total pressure as a sum of the static pressure in the fluid and the dynamic pressure:
Po = P∞ + ½ ρV2
P_total = P_static + P_dynamic = constant along a streamline!
The coefficient of pressure on the cylinder can be written as
Po − P∞ C p = 1 2 2 ρV
CYLINDER DRAG FORCE:
And the force on the cylinder (DRAG) is attributed to the flow over the body and the pressure differential over the cylinder.
F= ∫∫ p nˆ ds
The coefficient of drag (non-dimensional drag) can be written in a similar fashion to the 2 pressure coefficient – as a ratio of the drag force, Fd, to the dynamic pressure, ½ ρV , times the cylinder diameter, D:
Fd Cd = 1 2 2 ρV D
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The velocity V is taken as the freestream velocity, and the Drag force is a force per unit length of the cylinder. This coefficient is termed the “sectional drag coefficient” since it related to only a small section of the cylinder length. For a uniform diameter cylinder this Cd is constant across the length!
2.2 BEFORE STARTING
1. Read the ENTIRE laboratory assignment handout BEFORE coming to lab. 2. Record the Ambient temperature and pressure 3. Listen to the lab instructor’s briefing.
2.3 KEEPING IN MIND THE LAB-WRITE UP
1. Complete all the exercises in Bold throughout this lab handout in addition to the questions at the end. 2. Take care to discuss your sources of error and estimate any error that occurred in calibration or data taking. 3. Briefly discuss each of the plots (1-2 sentences).
2.4 EXPERIMENTAL TASKS
The experimental procedure for this laboratory is discussed below. Please read this section carefully before commencing the exercises. If you have questions about the laboratory procedures please ask your instructor.
Pressure Transducers Calibration
The first step of this lab will be to calibrate the pressure sensors:
The pressure sensors are differential pressure transducers that measure the “difference” in pressure between the inlet and outlet. In order to calibrate the transducers each side is connected to a water column mounted on the lab wall to the left of the tunnel.
The gauges are connected to a data acquisition system that will display the pressure reading from the gauge in VOLTS.
TO CALIBRATE:
Apply a known pressure differential to the transducer
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o To calculate the applied pressure differential: measure the difference in height of the water columns and apply the hydrostatic equation to find the pressure. o Record this value (or at least ∆h) to use in your calibration curve Record the output VOLTAGE on the readout that corresponds with the applied pressure differential Repeat the above steps (4-5 times) by varying the water column height on one side by 0.5m increments. Plot the applied PRESSURE, in Pascals, VERSUS the VOLTAGE to obtain a calibration curve. Discuss the error in your calibration
This calibration curve will be used throughout the lab (for the write up) to convert the voltage readings from the transducers to actual pressure values.
While doing this remember to make notes as to potential sources of error in these measurements? Is it easy to read the height of the water column – is it a steady reading? Do the voltage readouts fluctuate? It is good to know your error in calibration as it will propagate throughout your calculations.
IMPORTANT NOTE: The tunnel gauge pressure transducer has only one port as the other is referenced to absolute zero.
Dynamic Pressure vs. Tunnel Speed
The nest step will be to measure the effects of flow speed on dynamic pressure.
The lab instructor will you reconnect the pressure gauges to proper location on the lab setup after the calibration step.
Two gauges will be used: 1. Measures the pressure in the test section relative to absolute zero (absolute pressure). 2. Is connected to a Pitot tube!! This gauge measures the difference between the total pressure at the inlet of the Pitot tube and the reference pressure at the static pressure port similar to the setup discussed in class. This gauge measures the dynamic pressure as:
2 P_dynamic = ½ ρV = Po-P∞ = P_total + P_static.
MEASURE THE PRESSURE AS A FUNCTION OF TUNNEL SPEED: Measure the pressure at 4-5 tunnel speeds by increasing the tunnel impeller RPM up to 200 rpm Max.
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Use the calibration chart given for speed vs. impeller RPM to calculate the flow speed in the test section Record the tunnel impeller speed and the ambient and pitot tube pressures. Plot the Pitot tube pressure vs. RPM, compare this with a direct calculation of dynamic pressure using the calibrated tunnel speed for each RPM.
Pressure Measurement around a Cylinder in a flow stream
In this final part of the lab the pressure around the cylinder will be measured using a static pressure port embedded in the cylinder. The cylinder can be rotated through 360 degrees in order to measure the pressure around the cylinder:
Pressure tap Rotation direction U ∞ α
The cylinder is mounted such that it can be rotated from outside the tunnel so that the pressure port can measure the cylinder static pressure at angles around the cylinder. A protractor is attached to the cylinder so that the rotated angle can be recorded.
MEASURE THE PRESSURE AROUND THE CYLINDER: Set the tunnel impeller RPM to 200 rpm. Set the cylinder angle to zero (zero angle will be with the port facing directly upstream). Vary the angle of the cylinder in 10 degree increments for a full 360 degree rotation. At each angle stop and record the pressure gauges. Be sure to record the tunnel speed as well as it can fluctuate!!! Plot the pressure recorded (in Pascals) versus angle around the cylinder
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2.5 POST LAB WRITE-UP
For this lab assume the density for water is 998 kg/m3 and acceleration due to gravity is 9.82 m/s2.
A. Pressure Transducers Calibration: a. Using the hydrostatic pressure equation compute the applied calibration pressure at each calibration point. b. Plot the output voltage for each pressure gauge as a function of this applied hydrostatic pressure. c. Fit a line to the dataset (using MATLAB or Excel, by hand, etc.) to obtain the sensitivity (slope) for each gauge – this will determine how many Pascals per volt are there for each gauge.
B. Dynamic Pressure vs. Tunnel Speed a. Plot the measured dynamic pressure vs. tunnel RPM. b. Use Bernoulli’s equation to calculate the dynamic pressure expected at each tunnel RPM based on the velocity/RPM calibration handed out in lab. c. Compare these two plots – is there a difference? If so, what could a source of error be? d. Using the cylinder diameter as the length scale (D = 0.0254m) what is the Reynold's number for this cylinder flow at 200 RPM impeller speed? (Kinematic viscosity for water is ν = 10-6 m2/s).
C. Pressure Measurement around a Cylinder in a flow stream a. Using the pressure calibration data compute the pressure measured on the cylinder at each rotated angle. b. Plot the cylinder pressure vs. angle. c. Where is the pressure maximum? Minimum? d. Compute the increment in vertical height Z for each angle increment :
dz = r d(sinα)
where r is the cylinder radius and alpha is the angle of rotation to the pressure tap. e. Integrate this data using MATLAB (or your preference) to get the pressure force on the cylinder in the streamwise direction (i.e. Drag force). Remember that force is the pressure times the area normal to the surface and the drag force is the component of the total force in the streamwise (x- axis) direction. f. What is the force/unit length on the cylinder in Newton/m? g. Compute the cylinder section drag coefficient using the computed force and measured free stream velocity.
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