Example 1, WALZ:  Laminar Integral Method
Problem:
Consider 2D laminar flow of a fluid with a kinematic viscosity = 2.0x10^-4m²/s at U_inf = 10.0 m/s over a surface that is a flat plate from the leading edge to x = 1.0 m. At that station, a ramp begins that produces an inviscid velocity distribution U_e(x) = 10.5 - x/2, m/s. This is an adverse pressure gradient, since U_e is decreasing so that p increases. Calculate the boundary layer development over this surface up to x = 2.0 m. Does the flow separate?
Solution:
We must provide input data for the kinematic viscosity as = 2.0x10^-4m²/s and the freestream velocity as U_inf = 10.0 m/s. Select "Number of x steps" = 41, and "Maximum x/L" = 2, and "Reference Length L" = 1.0 m, to give a step size of 0.05 m. Using the surface characteristics dialog ("Change -> Surface properties", change the body shape to "2D body, sharp leading edge", and type in point pairs to define the bilinear inviscid velocity distribution required, e.g. 0   1.0, 0.5  1.0, 1.0  1.0, 1.5  0.975, 2.0 0.95.  The code will fit a spline through the points used as input. Then, 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 implied velocity profile develop at the same time. 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 2, ILBLI Laminar Implicit Numerical Method
Problem:
Consider 2D laminar flow of a fluid with a kinematic viscosity = 2.0x10^-4m²/s at U_inf = 10.0 m/s over a surface that is a flat plate from the leading edge to x = 1.0 m. At that station, a ramp begins that produces an inviscid velocity distribution U_e(x) = 10.5 - x/2, m/s. This is an adverse pressure gradient, since U_e is decreasing so that p increases. Calculate the boundary layer development over this surface up to x = 2.0 m. Does the flow separate?
Solution:
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 x = 1.0. Thus, the numerical calculation will begin at x = 1.0 and go to x = 2.0. Set "Starting x/L" = 1.0 and "Maximum x/L" = 2.0. We must again provide input data for the kinematic viscosity as = 2.0x10^-4m²/s and the freestream velocity as U_inf = 10.0 m/s. Select  "Reference Length L" = 1.0m, The Blasius solution at x = 1.0 gives the initial boundary layer thickness of delta = 0.0224. Choose the "Number of y steps" to be 100 and the step size to be 0.00112, giving about 20 steps across the initial boundary layer thickness. Since the implicit method is unconditionally stable, no stability criterion need be followed, and we can select the "Number of x steps" based only on accuracy considerations. Select  41 to give a step size of 0.025, which is about the size of the initial boundary layer thickness. Initial profiles for u and v are required. For simplicity, we use a Polhausen profile for u and set v = 0 as adequate approximations to the exact Blasius solution ("Change -> Initial Profile"). Since this case has a linear edge velocity variation velocities at only two points need be specified. Using the surface characteristics dialog, type in point pairs 1.0   1.0, 2.0  0.95. 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 computed u and v velocity profiles develop at the same time. 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 3, MOSES Turbulent Integral Method
Problem:
Consider 2D turbulent flow of a fluid with a kinematic viscosity = 1.0x10^-5m²/s at U_inf = 10.0 m/s over a surface that is a flat plate from x = 0.0 to 5.0 m. Then, a ramp begins that produces an inviscid velocity distribution U_e(x) = 15 - x, m/s.  Calculate the boundary layer to x = 7.0 m and determine if the flow separates.
Solution:
Set "Starting x/L" = 4.0 and "Maximum x/L" = 7.0. We must provide input for the kinematic viscosity as = 1.0x10^-5m²/s and the freestream velocity as U_inf = 10.0 m/s. Select "Number of x steps" = 21, and "Reference Length L" = 1.0 m, to give a step size of 0.10 m. Since the part of the calculation upstream of x=4.0 is over a flat plate, the simple integral solution (see section 7-7 of Schetz, 1993, "Boundary Layer Analysis") can be used to infer the initial boundary layer thickness at x=4.0 of 0.072m.  Using the surface characteristics dialog ("Change -> Surface properties") type in two point pairs to define the bilinear inviscid velocity distribution, e.g. 4.0  1.0, 4.25  1.0, 4.5  1.0, 4.75  1.0, 5.0 1.0, 5.5  0.95, 6.0  0.9, 6.5  0.85, 7.0  0.8.  The code will fit a spline through the points used as input. Then, 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 implied velocity profile develop at the same time. 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 4, ITBL Turbulent Numerical Method
Problem:
Consider 2D turbulent flow of a fluid with a kinematic viscosity = 9.3x10^-7m²/s at U_inf = 3.049 m/s (10 ft/s) over a flat plate from x = 1.524 to 1.829 m (5 to 6ft.) Use all three available turbulence models and compare the results.
Solution:
Set "Starting x/L" = 1.524, "Maximum x/L" = 1.829, "Kinematic viscosity" = 9.3x10^-7m²/s , "Freestream velocity"  = 10.0 m/s, and "Reference Length L" = 1.0 m. The simple integral solution (see Sec. 7-7 in Schetz, 1993, "Boundary Layer Analysis") can be used to obtain "initial" conditions at x = 1.524m, giving delta = 0.0261 m. Choose "Number of y steps"= 1000, and a "y step size"=  4.3x10^-5m - this will result in about 600 points across the initial boundary layer thickness, and set "Number of x steps"= 101 to give a step size about one tenth of the initial boundary layer thickness. For simplicity, we use a Coles profile for u and set v = 0 at the upstream boundary ("Change -> Initial Profile"). Using the surface characteristics dialog ("Change -> Surface properties") we select "Zero pressure gradient". Lastly, the turbulence model must be selected ("Change -> Turbulence Model"). The default value is a mixing length model. 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 computed velocity profiles develop at the same time. 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. Finally, repeat the calculation with the other two turbulence models. Use "File -> Write parameter list" to save the input parameters of your calculation.

Example 5, WALZHT Laminar Integral Method with Heat Transfer
Problem:
Consider 2D laminar flow of a fluid with a kinematic viscosity 1.6x10^-5m²/s, C_p = 1005 J/kg/K, density = 1.2 kg/m³ and Prandtl number Pr =
0.72 at U_inf = 2.0 m/s over a surface that is a flat plate from the leading edge to x = 1.0 m. At that station, a ramp begins that produces an inviscid velocity distribution U_e(x) = 2.1 - x/10, m/s. The wall to freestream temperature difference, T_w - T_e = 20^oC. Calculate the boundary layer development over this surface up to x = 2.0 m. Does the flow separate? Note how the dimensionless wall shear, C_f, and dimensionless heat transfer, Nu, vary in the constant pressure and varying pressure regions along the surface.
Solution:
Enter the viscosity, density, specific heat, Prandtl number, free-stream velocity, reference length (1m) and maximum x/L (2.0). Using the surface characteristics dialog ("Change -> Surface properties"), select "2D body, sharp leading edge" and type in groups of three numbers to define the inviscid velocity and temperature difference distributions, e.g.

 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+001

Example 6 Turbulent Integral Method
Problem:
Consider 2D turbulent flow of a fluid with a kinematic viscosity 1.0x10^-5m²/s, C_p = 4187 J/kg/K, density = 1.2 kg/m³ and Prandtl number Pr =
5 at U_inf = 10.0 m/s over a surface that is a flat plate from x=0.0 to 7.0 m assuming a simple inviscid velocity distribution U_e(x) = 10 m/s = constant. Calculate the boundary layer properties betwee x=5.0 and 7.0m where the temperature difference is 10K. Determine the heat transfer?
Solution:
Enter the viscosity, density, specific heat, Prandtl number, free-stream velocity, reference length (1m), starting x/L (5.0) maximum x/L (7.0). Using the surface characteristics dialog ("Change -> Surface properties"), specify the surface velocity and temperature distributions. Since both are constant it is only necessary to specify 2 points;

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

 7.0000e+000  1.0000e+000  1.0000e+001

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 10m/s. Due to the roughness of the weather-vane surface the flat-plate transition Reynolds number Re_x is thought to be about 2.5x10⁵.
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 0.005m. 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 2m,  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?