## Joukowski Active Figure

Active figures are executable files (software) that allow you to explore a topic.  This one is about potential flow over an airfoil.  This page is about how to use the active figure, describing the controls and what you can do with the software.

When you execute the program, your computer screen should look something like the following:

To stop the program, click on the X in the far upper right corner of the screen (just like any other window that you want to close).  Flow in the model enters from the left-hand side of the screen and leaves at the right hand side.

There is a group of controls towards the upper right side of the form that allows you to control the shape of the airfoil by changing the parameters that go into describing the shape.   Each of the controls is a "slider" that you can move with your mouse to alter the parameter.  It's a little difficult to describe how these will affect the shape of the airfoil. Just give each one a try.

Default shape of the airfoil with a b/l ratio of 0.15.  Because the "color selection" has been set to pressure, colors depict the distribution of pressure in the vicinity of the airfoil.

Here's a thinner airfoil generated by selecting a smaller value for b/l.

Here's a somewhat more curvaceous airfoil generated by changing the "camber" parameter.   This parameter is actually an angle and the value ranges between $$\pi/2$$ ( slider far left ) and $$\pi$$ (slider far right ).  In determining the shape, this parameter interacts with the b/l ratio.  The larger the value of b/l, the more the camber parameter affects the shape.

Here the previous airfoil has had the angle of attack changed.  Notice the significantly different pressure distribution from the previous figure.  This is how an airfoil generates lift!  There is a higher pressure on the underside of the structure than on the top side.

Switch back and forth between the Joukowski airfoil and a cylindrical geometry by clicking the appropriate radio button.  As will be discussed in the text, the solutions for the airfoil are nothing but a warping of the cylindrical geometry in a carefully prescribed way called the Joukowski transformation, an example of a conformal mapping.

UNchecking the "Auto Circulation" activates the slider/scrollbar that allows you to select a circulation of choice to see how the solution is affected.  When "Auto Circulation" is checked, circulation is set to 0.0 for the cylinder and a very specific value for the airfoil that has to do with airfoil geometry and angle of attack.  The flow field around the airfoil is very unrealistic when "Auto Circulation" is unchecked.  Circulation around an airfoil is the very thing about the flow (the one and only thing) that allows the airfoil to generate lift.

The upper left hand side of the screen includes a number of controls that affect the appearance of the image and also allow you to create colorful figures for your own use.

The "Resolution" slider affects the density of the grid used to display and compute various aspects of the solution.  As you can see, a denser grid provides more detail and makes a prettier picture.  However, there is a trade-off between increasing grid density and rapidity of the computations.  When the grid spacing decreases by a factor of 2 (increasing resolution),  computation time goes up by a factor of 4.  It's best to explore with the grid at a relatively low resolution.  This will allow you to change angle of attack, airfoil parameters, etc., and obtain results that are nearly instantaneous.  The maximal resolution makes for very slow solution updates indeed!
The resolution setting affects other aspects of the display also.  The contour lines (described below) are computed using the grid.  A sparse grid (low resolution)  makes the contour lines appear somewhat jagged instead of smooth.  Adding vectors to the image (see below) is also affected by the grid.  Increasing grid density (increased resolution) increases the number of vectors that are painted onto the image.

Move the "Scale" slider to change the apparent size of the image. This allows you to focus in on details of the solution happening close to the airfoil or back away to get the big picture.

If you like the way a particular solution looks, you can display an image in dense full-color by pressing the "Draw Image" button.  The solution has the same attributes as the one made with the grid, but requires considerably longer to create since the solution is computed at every pixel of the image.  A "Please Wait" message will appear while the image is being created.  Want to crash your computer and possibly have to reboot?  Go ahead and try to move sliders and change the image while the computer is rendering the image.  (Don't do this.)

Save an image file at any time by pressing the "Save File" button.  You can choose from various file formats including BMP, JPEG, TIF, GIF, and PNG.  What you see is what you get when the image is saved.

Press the "Animate" button to see an animation of flow past the airfoil or cylinder.  The animation is affected by the vector density slider (see below).  Increasing density on that slider increases the number of particles in the animation thereby increasing the detail and decreasing the solution rate.  A first-order integration is employed which is relatively inaccurate but intriguing to watch.  The animation will abort eventually, or press the red "Stop" button to and the animation. The "Stop" button is only available while the animation is running.  You cannot save the image while the animation is running.

Use the "Color Range" and "Color Offset" sliders to change the color of assignment to the physical values of the solution.  Understand -- these do not change the solution, only the way that it is displayed.  All three of the above images are of the same solution.  In the second image, or color range was decreased so that a smaller range of physical values is represented by the colors.  This tends to provide greater detail of the solution, but for a limited range of physical values. The "Color Offset" slider is active only when the checkbox next to the slider has been checked.  The slider allows you to change the color that corresponds to 0 of the physical solution. Changing this value of allows you to visualize solution detail better by concentrating color changes corresponding to a particular value of the physical solution.  In the third image, the color offset has been adjusted so that there is more latitude of colors in the low-pressure region at the top of the airfoil, thereby allowing better visualization of how the solution changes there.

Color represents pressure above.

Color represents velocity above.

Color represents the stream function above.

Color represents potential funtion above.

Use the "Color Selection" radio buttons to select which aspect of the physical solution is represented by color.  Obviously, you can only select one aspect at a time.  Pressure and velocity magnitude are inversely related so warm pressure colors correspond to cool velocity colors in the above.  The stream function and potential function will be described in the text for those with mathematical interest.  The "Draw Image" button has been used here to render these four images in full-color detail.

The next set of controls, on the left-hand side of the form, allows you to add contour lines, vectors, or a legend to the figure.

In the first figure of the above 3, pressure contour lines have been added and pressure is also represented by color.  The contour lines add detail to the color solution, giving a much more specific impression of how pressure varies in the solution and how rapidly it varies with distance.  Just like a topographic map, the solution varies rapidly with distance when contour lines are closer together.  The color representation of the solution is saturated (minimum value = blue) well away from the upper side of the airfoil and does not represent the dramatic variation of pressure here that is depicted by the contour lines.
In the second figure, pressure is still represented by color, but contours represent the stream function (particle paths). This allows for an added dimension to the solution, allowing you to visualize more than one aspect of the solution at a time.  Pressure affects the velocity and trajectory of fluid elements as they traverse the airfoil.
The third figure is the same as the second, having pressed the "Draw Image" button thereby giving a full-color representation.
As noted previously, the "Resolution" setting affects the appearance of the contour lines.  Greater resolution results in smoother lines.

Add vectors to the image by clicking 1 or more of the checkboxes in the "Vectors" group.  In the above, the "Composite" velocity vector box has been checked.  This is actually just the velocity of the fluid elements, but composite here refers to the fact that the velocity is the sum of 3 components corresponding to 3 parts of the potential function; uniform, dipole, and vortex.

Adjust the size of the vectors by moving the "Size" slider to the right of the vector group box.  In the above, vectors size has been decreased relative to the previous image.

Adjust the density of the vectors (number of vectors displayed) with the slider on the left side of the vector group box.  As noted above, the "Resolution" setting at the top of the active figure interacts with this setting, i.e. increasing the Resolution will increase the vector density, even if the Density setting itself has not been changed.
Vector density and size can be set to  best visualize whatever aspect of the solution you are interested in.  It certainly possible to set the vectors too dense or too large to enhance understanding of the solution.  Each vector represents the value located precisely at the tail end of the vector.

Select 1 or more of the vectors from the checkbox group.  In the above, only  the vortex component of the velocity vector is displayed corresponding to the circulation around the airfoil or cylinder.  Adding multiple vectors to the image starts to make things look pretty messy; for clarity it's usually best to display only one set of vectors at a time.

Here, only the dipole component of the composite velocity vector is displayed.

Click on the "Data" checkbox at the bottom left of the Windows form to add some technical data to the image to remind you what is being represented.

Click on the "Legend" checkbox to include a color scale legend to the right of the image so that the quantitative relationship between color and the  physical values is indicated.  Find out what these numbers mean in the text.