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Examples

Simple Examples / Examples using multiple applets / Advanced problems

**Example 1, WALZ: Laminar Integral Method**
__Problem:__

Consider 2D laminar flow of a fluid with a kinematic viscosity = 2.0x10^{-4}*m ^{2}/s*
at

We must provide input data for the kinematic viscosity as = 2.0x10

**Example 2, ILBLI Laminar Implicit Numerical Method**
__Problem:__

Consider 2D laminar flow of a fluid with a kinematic viscosity = 2.0x10^{-4}*m ^{2}/s*
at

This is the same flow problem solved with the Thwaites-Walz integral method using the code WALZ. Now, we can apply the implicit numerical method in code ILBLI to this problem for comparison in terms of accuracy of the predictions and computational effort required. Since the first part of the problem is flow over a flat plate Blasius solution can be used to obtain "initial" conditions at

**Example 3, MOSES Turbulent Integral Method**
__Problem:__

Consider 2D turbulent flow of a fluid with a kinematic viscosity =
1.0x10^{-5}*m ^{2}/s* at

Set "Starting x/L" = 4.0 and "Maximum x/L" = 7.0. We must provide input for the kinematic viscosity as = 1.0x10

**Example 4, ITBL** **Turbulent Numerical Method**
__Problem:__

Consider 2D turbulent flow of a fluid with a kinematic viscosity =
9.3x10^{-7}*m ^{2}/s *at

Set "Starting x/L" = 1.524, "Maximum x/L" = 1.829, "Kinematic viscosity" = 9.3x10

**Example 5, WALZHT Laminar Integral Method with Heat Transfer**
__Problem:__

Consider 2D laminar flow of a fluid with a kinematic viscosity 1.6x10^{-5}*m ^{2}/s*,

0.72 at

Enter the viscosity, density, specific heat, Prandtl number, free-stream velocity, reference length (1

0.0000e+000 1.0000e+000 2.0000e+001

2.5000e-001 1.0000e+000 2.0000e+001

5.0000e-001 1.0000e+000 2.0000e+001

7.5000e-001 1.0000e+000 2.0000e+001

1.0000e+000 1.0000e+000 2.0000e+001

1.2500e+000 9.8750e-001 2.0000e+001

1.5000e+000 9.7500e-001 2.0000e+001

1.7500e+000 9.6250e-001 2.0000e+001

2.0000e+000 9.5000e-001 2.0000e+001The code will fit a spline through the points used as input. Press the "Run" button and watch the skin friction and the integral quantities develop in the graphs. You may also select "Show -> Profile" and watch the assumed velocity profile develop at the same time. You may click on the graphs, or select "Change -> Plot options" to change the quantities plotted. Click here to see the window as it appears about halfway through the calculation, and at the end. Tabular values of the output can be accessed by selecting "Show -> Numerical Results" and copied into Excel for plotting and further analysis. Use "File -> Write parameter list" to save the input parameters of your calculation.

**Example 6 Turbulent Integral Method**
__Problem:__

Consider 2D turbulent flow of a fluid with a kinematic viscosity 1.0x10^{-5}*m ^{2}/s*,

5 at

Enter the viscosity, density, specific heat, Prandtl number, free-stream velocity, reference length (1

5.0000e+000 1.0000e+000 1.0000e+001

7.0000e+000 1.0000e+000 1.0000e+001Since the first part of the flow (to

**Example 7 Application of the Mixing Length Model to Jet Mixing**
__Problem:__

You are asked to analyze the flow of a two-dimensional jet with a nozzle half-height of 2.0 cm in a 10.0 m/s freestream flow with an initial jet to freestream velocity ratio, Uj/Ue = 3.0 in a fluid with v = 2.0x10^-5 m^2/s. Consider the region 0<= x <= 1.0 m. It is suggested that the mixing length model be employed.
__Solution:__

This problem can be solved using the JETMIX applet mentioned just above. Open the applet to calculate the flow. We must carefully enter the required input in the lower panel in the JETMIX window. Set: 1) Fluid Properties, viscosity = 2.0E-05 m^2/s; 2) Reference Properties, Freestream velocity = 10 m/s and reference length = 1.0m; 3) for Turbulence Model click on __Change__ and select Mixing Length; and 4) Calculation Parameters, Starting x/L = 0.0, Maximum x/L = 1.0, Uj = 3.0, Ue = 1.0, number of x steps = 101, y step size (m) = 0.0004 for fifty points across the inital jet half-height (once can and should vary these to study the effects of the step size), Number of y steps = 500, Mh = 51 to set the initial jet half-height as 0.02m. Click __Run__, and the solution appears as in Figure 9-29(A).

Click on __Show__ at the top and select __Numerical results__ and the window in Fig. 9-29(C) appears. One can copy and paste any or al of these and use them in *Microsoft Excel* or any other convenient plotting software.

One can pick off the predicted length of the *potential core* as the location where the centerline velocity begins to fall below the initial value as about x/L ≈ 0.36 or 0.36 m. The logarithmic plot of the centerline velocity shows the expected similarity behavior, i.e x^-n (a straight line on a logarithmic plot) for large x.

**Example 8. Comparing Methods**
__Problem:__

Repeat example 3 above using the integral and finite difference methods,
and compare results.
__Solution:__

Perform the calculation described in example 3 using MOSES. When the
calculation is complete, select "File -> Launch". Using the choice field
in the top left hand corner select "Launch ITBL". Use the choice field
a top center to select "From starting location" (dialog should look like
this).
Click "Launch" and the dialog will dissappear and an ITBL window will open.
Check the values and parameters already loaded into ITBL and you will find
that they are identical to those you used/specified with MOSES. Click "Run"
to perform the calculation with ITBL. Use "Show -> Numerical Results" in
both applets to copy and past results into Excel and compare. How does
the disagreements between the two methods compare to the differences from
using different turbulence models in ITBL?

**Example 9. Full Boundary Layer Undergoing Transition**
__Problem:__

Compute the drag on a 3m long 1m wide weather vane in a wind of 10*m/s*.
Due to the roughness of the weather-vane surface the flat-plate transition
Reynolds number Re_{x} is thought to be about 2.5x10^{5}.
__Solution:__

Using WALZ, enter the maximum *x/L* location as 3, reference length
*L*
as 1 *m*, and change the transition Reynolds number to 250,000. Using
the surface properties dialog "Change -> Surface properties", select "2D
body, sharp leading edge" and "Zero pressure gradient" (we are assuming
that the weather vane will move to zero angle of attack). Using standard
atmospheric conditions we may take the kinematic viscosity as 1.45x10^{-5}.
Press "Run" and observe the development of the
calculation. Note the that the calculation stops prematurely at a location
marked with a "T". This indicates that transition has been detected. Look
at the numerical results (Show -> Numerical Results) and you will see that,
since transition occured, only about 12 calculation steps were actually
performed. To improve the accuracy of this pre-transition calculation,
change the "Maximum *x*/L" to 0.5, just downstream of the transition
location. This effectively reduces the step size to 0.005*m. *Press
"Run" again and observe the calculation which
now has sufficient detail. To determine the exact transition location,
look at the numerical results (Show -> Numerical Results) and scroll to
the bottom of the table. Copy the position (3.2729e-001), and close the
numerical results dialog. Now, to start the turbulent boundary layer calculation
(which we will choose to do using MOSES), select "File -> Launch", and
then "Launch MOSES" (using the top left selector) and "From x/L = " (using
the top center selector) and paste the transition location you just copied
into the text area at the top right. Press "Launch" and MOSES will open
up, preset with all the necessary information, including a starting *x*
location and an initial boundary layer thickness corresponding to the transition
location determined by WALZ. Edit the maximum *x* location to 3.0,
and press "Run" to complete the boundary layer calculation using this turbulent
method and observe the development of its parameters
following transition. Finally open the numerical results windows in
both applets, and copy and paste the results into Excel (or another spreadsheet).
Integrate the distributions of skin friction coefficient to obtain the
total drag due to the laminar and turbulent portions of the boundary layer.
Don't forget to multiply by two to account for the two sides of the weather
vane.

**Example 10. Analysis of a NACA 0012 airfoil**
__Problem:__

Calculate the aerodynamic characteristics of a 2m chord NACA 0012 airfoil
flying at 2 degrees angle of attack at a speed of 50m/s into air at sea-level
conditions.
__Solution:__

Use the vortex panel method, and the 200 panel
description of a NACA 0012 airfoil provided along with it, to calculate
the inviscid solution for this foil at 2 degrees angle of attack. This
will immediately give you the lift and moment coefficients, but you need
to do a boundary layer calculation to get the drag and boundary layer characteristics.
Select "s, U (upper)" to output the edge length/velocity distribution for
the suction side of the airfoil. Start WALZ, and open the surface properties
dialog box. Select "2D body, rounded leading edge", and paste in the velocity
distribution from the vortex panel method (less column headers). Change
the viscosity to 1.45x10^{-5}, the reference length *L* to
2*m, * and the freestream velocity to 50 m/s. Set the maximum
x/L to 1.0249 (i.e. the trailing edge position, measured in chords
along the airfoil surface from stagnation) and choose the transition Reynolds
number, say 500,000. Set the number of *x* steps to 1000 - you want
a lot of detail around the leading edge. Press "Run" and the calculation
will develop as shown here until transition
is reached (at about 10% chord). Now transfer the transition location and
calculation to MOSES (or ITBL) using the same procedure applied in example
8. Perform the calculation and observe the boundary
layer development. Paste numerical results from WALZ and MOSES into
Excel, ready for analysis and integration. Now return to the vortex panel
method. Select "s, U (lower)" to output results for the pressure side of
the airfoil, and repeat the process.

**Example 11. **Repeat problem 9 at several larger angles of attack
until the airfoil stalls (see Notes).
Plot your results against actual measurements from Abbot and von Doenhoff's
"Theory of Wing Sections".

**Example 12.** Perform inviscid/boundary layer calculations on the
flow past a circular cylinder. Use potential flow theory or the vortex
panel method to obtain the inviscid velocity distribution. Compute
the separation location as a function of Reynolds number. At what Reynolds
number does the boundary layer undergo transition on the forward face of
the cylinder (for a flat plate transition Reynolds number of 500,000, say)?
What affect dows that have on the transition location? Compare your results
with published measurements?