Cover Page for Precalculations – Individual Portion
Lift on a Wing
Prepared by Professor J. M. Cimbala, Penn State University
Latest revision: 11 January 2012
Name:
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Date:
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Section number:
ME 325._____
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Precalculations
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Precalculations
A.
Dimensional Analysis
The lift force, FL, on a wing or fin is a function of average its average chord length c, the freestream velocity V, the
fluid’s density , the fluid’s coefficient of viscosity , the speed of sound a, and the wing’s angle of attack . This parametric
relation is written as
FL  FL  c,V ,  , , a,  
(2)
1.
(1)
In the table below, write the primary dimensions of each variable in Equation (1) in the {m}, {L}, {t}, and {T}
(mass, length, time, and temperature) primary dimension system. E.g., pressure: {P} = {F/L2} = {m1L-1t-2}.
Variable
Dimension
Variable
FL

c
a
V

Dimension

Since there are seven variables and three primary dimensions, we expect four dimensionless parameters, or s,
according to the Buckingham Pi technique.
(8)
2.
1 =
Choosing c, V, and  as repeating variables, work through the method of repeating variables, showing all your
work in the space below. Summarize your results (list your s) in the spaces provided.
2 =
13 =
4 =
You may recall that these s can be multiplied by any constant or raised to any exponent (including negative
exponents) without changing their nondimensionality. Your s should be reducible to the following relationship:
CL  CL  Re, Ma,  
(2)
where CL is the lift coefficient, CL = 2FL /(V2A); A is the airfoil’s planform area, A = sc; Rec is the Reynolds number based
on chord length, Re = Vc/; and Ma is the Mach number, Ma = V/a. Note that since area A has the same dimensions {L2} as
does c2, we have also substituted A for c2 in the lift coefficient (to agree with standard convention).
(2)
3.
Flows can be considered incompressible when the Mach number is less than about 0.3. Our wind tunnel has a
maximum speed of about 60 m/s. The speed of sound in room temperature air is approximately 340 m/s. Are
compressibility effects important in this lab? Justify your answer.
For the remainder of this lab, Mach number will be neglected. Equation (2) then reduces to
CL  CL  Re, 
(2)
4.
B.
Scaling From Model to Prototype
(2)
1.
(3)
Write out a relationship, similar to Eq. (3), but for drag coefficient. Note that it is not necessary to repeat the
dimensional analysis, since lift force FL and drag force FD have the same dimensions (force) and are functions of
the same variables. Be sure to name and define any new dimensionless parameters you generate.
The prototype human powered submarine operates in sea water. For standard sea level conditions, at 20 o C, look
up the density and viscosity of sea water. Write down your reference for these data.
sea water = ________________________ kg/m3.
sea water = ________________________ Ns/m2.
(2)
2.
Reference:
Our model of the fin is one-half scale, and will be tested in air at nearly standard conditions. For air at 20 C, look
up the density and viscosity.
air = ________________________ kg/m3.
air = ________________________ Ns/m2.
(3)
3.
Reference:
At approximately what wind tunnel speed should the model be tested, in order to accurately simulate the operation
of the prototype at 3.0 knots (in sea water)? Show all your work in the space below:
(3)
4.
The wind tunnel can reach a maximum speed of around 60 m/s. Is this speed adequate to achieve dynamic
similarity between the model and the prototype? Explain.
(3)
5.
Suppose the wind tunnel is operated at a velocity of 52.8 m/s. To approximately what Reynolds number on the
model fin does this correspond? Show all your work in the space below:
(3)
6.
Suppose that at an angle of attack of 20o, the lift force on the model fin is measured to be 22.4 N at a wind tunnel
velocity of 35.8 m/s. Estimate the lift force on the actual prototype fin at an angle of attack of 20 o in sea water.
Show all your work in the space below:
Experimental Objectives
a.
b.
c.
Calibrate the electronic pressure transducer.
Calibrate the wind tunnel lift balance.
Measure lift and calculate lift coefficient on a model fin as a function of angle of attack and Reynolds number.