Problem Set #1



Homework 1

Due: Wed Sept 9 (due to Labor Day holiday)

10% bonus if turned in on Friday 9/4/09

Note: There are 4 problems. No more will be added to this assignment

1. [5 points]

A flat plate of zero thickness sits in a uniform free stream flow at an angle of attack of 15deg. The free stream pressure is 0.5 , the free stream density is 0.2 and the free stream velocity is 20. The chord length of the plate is 3. The pressure and shear stress distributions on the upper and lower surface of the plate are given by

Upper surface:

p = 0.35 - 0.03x

τ = .01/(1+x)

Lower surface

p = .65 + 0.2x

τ = .02/(1+x)

where x is the distance along the plate from the leading edge, measured in units of chord (so x=1 is the trailing edge.).

The units are all consistent.

a) Calculate the free stream dynamic pressure

b) Calculate and plot the distribution of pressure and shear stress coefficients along the upper and lower surfaces

c) Calculate the Forces normal and tangential to the plate Sketch a free body diagram showing these forces with the correct direction.

d) Calculate the net Lift and Drag forces acting on the plate

e) Calculate the lift and drag coefficients

Note: the forces are all actually forces per unit width (since we are considering a 2D airfoil.)

2. [5 points]

All variables we will consider in this course (333) will have units consisting of combinations of Mass (M), Length (L) and Time (T).

For example, velocity has units L/T, or, for example, m/s in MKS.

Express the units of each of the following in this form (ie as a combination of M, L and T.). Also state it in, specifically, MKS units: (Results given for velocity)

Variable name MLT units MKS units

Velocity L/T m/s

Density

Pressure

Pressure/Density

Density*Velocity^2

Shear stress

3. (5 pts)

Consider a problem where the Power output from a wind turbine depends on the wind velocity V, the diameter D of the turbine, the air density ρ, and the turbine rotational speed ω :

P = function (ω , V, D, ρ)

(recall power has the units of force times velocity, or work/time)

Use dimensional analysis to determine how V,D,and ρ can be combined to define nondimensional power coefficient C_P and nondimensional rotational speed coefficient C_ω, so

C_P = function (C_ω)

(where the specific wind speed, diameter and air density would not effect the curve…. This is the way engineers report wind turbine data).

4. [10 points] For this problem you will use a Matlab program to help you learn about the pressure and skin friction distribution on an airfoil.

You can download the jonesairfoil program from the webpage

and run it from your computer. Note: You do not have to be familiar with MATLAB to run this.

When you start the program there are two windows that come up. (Note that the “MATLAB command window” is below these two windows). The left window shows the geometry of an airfoil and a plot of the pressure coefficient distribution, Cp, as functions of the fraction of chord, x/c. Note that the axis for Cp is reversed so that positive values are plotted downward (a standard method followed in aerodynamics). Several sliders control geometric parameters (xc, yc, xt, yt, delta) and also the angle of attack (alpha).

The left window also shows the value of the lift coefficient, Cl, which is computed from the pressure and skin friction distributions. (We will learn later in the semester how to compute Cp(x) and Cf(x). For now take them as given.) The right window shows a plot of the local skin friction coefficient, Cf(x), for the upper and lower surfaces. Also shown is the thickness of the viscous region (theta) on the upper and lower surfaces.

The curves can be distinguished because theta continually increases while Cf(x) has a more complicated behavior and decreases toward the trailing edge of the airfoil. The value of the drag coefficient, Cd, and the ratio of lift to drag, L/D, are given. Two other parameters xTransition and xSeparation are also shown. There is one slider bar, which controls the Reynolds number, Re = ρU[pic]c/μ (where ρ is the fluid density, U[pic] is the freestream velocity, c is the chord length, and μ is the fluid viscosity).

a) Use the default geometry settings and an angle of attack of 3 degrees, and answer or do the following.

Print the plots in the two figures.

Is the airfoil symmetric top-to-bottom?

Label which Cp curve is for the upper surface and which is for the lower surface.

The value of the boundary layer thickness, theta, given by the program is the nondimensional value (theta = θ/c), where θ is the dimensional boundary layer thickness and c is the chord length. If the airfoil has a chord length c=3m, what is the dimensional thickness of the viscous region at the end of the airfoil?

The skin friction curve increases rapidly near the leading edge and then decreases. Further back on the airfoil Cf again rapidly increases and then decreases. At the point where Cf rapidly increases the second time, the flow transitions from laminar (smooth) flow to turbulent (chaotic) flow. This nondimensional location is printed as xTransition in figure 2. In terms of x/c where is this transition point located for the upper and lower surfaces? (Note the program also gives xSeparation. At this location the thin viscous boundary layer separates from the surface. If the flow is separated over a large portion of the airfoil then the airfoil is stalled.)

b) Use the alpha slider to increase the angle of attack to 6 degrees. (Note it is difficult to see the value of alpha in the window and, if too many digits are printed, the leading digit will move out of view. To overcome this you can click in the alpha window and delete some of the digits. When you are done changing alpha hit “Return” to re-compute with this new value.)

Describe what happens to the pressure distribution. Note especially the location of minimum pressure and the difference in the pressure on the upper and lower surfaces.

What happens to Cl as alpha is increased?

Edit the Reynolds number, Re, by clicking in the box with the value and changing the number. Hit “Return” to re-compute the solution with the new Reynolds number. Use Re = 6e+05, 6e+06 and 6e+07 and answer the following questions.

What happens to the transition location on the upper surface as the Reynolds number increases?

What happens to the boundary layer thickness (theta) on the upper surface as the Reynolds number increases?

What happens to the drag coefficient as the Reynolds number increases?

What happens to the lift coefficient as the Reynolds number increases (see Cl in figure 1)?

Assume the airfoil has a chord length c=2 m and is flying at sea level under standard atmospheric conditions.

If the Reynolds number is Re = 6e+06, what is the flight speed U[pic]?

What is the flight speed if the Reynolds number is increased to Re = 6e+07, keeping the chord length and atmospheric conditions fixed?

What is the dimensional drag (per unit span) for these two cases?

Does a lower drag coefficient necessarily mean a lower dimensional drag? Explain your answer.

Consider the definition of the Reynolds number and answer the following questions:

• What happens to the Reynolds number if the airfoil is made larger?

• If the free-stream velocity is increased?

• If the altitude is changed from sea level to 12,000 m, while the Mach number and airfoil size are held fixed?

To exit the joneplot program click on the Done button just above the alpha slider. Then type and exit in the MATLAB command window to exit MATLAB.

(Note: There was a typo in pb 4. It read 6deg in both a) and b). Corrected 8AM, Wed)

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download