# Fluid Flow Around a Sphere: Theory Comparison

Compare theoretical and Caedium simulation results (generated by the Panel Flow add-on) of the pressure coefficient on a sphere.

### Goals

In this tutorial, you will learn how to:

- Read a file into Caedium
- Generate equations
- Export and import the data from an XY plot
- Compare the results of your simulation with a theoretical calculation of the same quantity

### Assumptions

- You have activated the Caedium Panel Flow add-on or Caedium Professional.
- You are familiar with Caedium essentials.
- You have completed the Fluid Flow Around a Sphere tutorial and saved your simulation file, or you have downloaded the flow-over-sphere.sym file.

## Read the Simulation File into Caedium

In the **Home** Toolbar click the **Open** button . Navigate to the location of the flow-over-sphere.sym file. Double-click on **flow-over-sphere.sym** to load it into Caedium.

## Plot Pressure Coefficient on the Sphere Versus theta

In the tutorial Fluid Flow Around a Sphere, you plotted pressure coefficient versus X-position along the curved edge of the sphere shown in the diagram below. In the first step of this tutorial, you will plot the pressure coefficient (as calculated in your Caedium simulation) versus theta, where theta is defined in the diagram below. (theta=0 is in the Y-direction.)

The steps to calculate values of theta are:

- Create scalar R (the radius of the sphere)
- Create x/R, which is equal to sin(theta)
- Create theta, which is equal to arcsin(sin(theta))

### Create Scalar R, the Radius of the Sphere

Select the **New** Toolbar and click the **Result** button .

In the **Create Result** dialog, select the **Constant** tab. For **Units**, select **Length**.

Click **Create** to create a scalar.

Select the **Results** Tool Palette. Select **Scalar Variables->Scalar** (the scalar variable you just created), click on it again to make its name editable, and change **Scalar** to **R**.

In the Properties Panel, set the **Value** of R to **5**. Press Enter on the keyboard to apply the changes to the Properties Panel.

### Create sin(theta)

Remember that sin(theta)=x/R.

In the **Create Result** dialog, select the **Binary** tab. Select **Divide** from the list.

In the **Results** Tool Palette, select **Vector Fields->XYZ** (position). Verify that **X** is selected as the **Scalar** in the Properties Panel. Drag and drop **Vector Fields->XYZ** onto the left-hand target in the **Create Result** dialog. Select **Scalar** to specify the X-position as the left-hand variable for the equation.

Drag **Scalar Variables->R** from the **Results** Tool Palette and drop it onto the right-hand target in the **Create Result** dialog.

Click **Create** to create the scalar field x/R, which is equal to sin(theta). In the **Results** Tool Palette, select **Scalar Fields->(XYZ:X / R)**, rename it **sin(theta)**, and press Enter on the keyboard.

### Create theta

Remember that theta=arcsin(sin(theta)).

In the **Create Result** dialog, select the **Unary** tab. Select **aSin** from the list. Drag **Scalar Variables->sin(theta)** from the **Results** Tool Palette and drop it onto the target in the **Create Result** dialog.

Click **Create** to create the scalar field theta. Click **Close** to close the dialog.

In the **Results** Tool Palette, select **Scalar Fields->aSin(sin(theta))**, rename it **theta**, and press Enter on the keyboard.

### Plot Pressure Coefficient Versus theta

Select the **Cp Plot** Window (if it is not already selected). Drag and drop **Scalar Fields->theta** onto the background of the Plot Window. Select **X Axis** to display values of theta along the X-axis of the XY plot.

This is the Caedium simulation result of pressure coefficient versus theta for the sphere.

## Calculate the Theoretical Values of Pressure Coefficient on the Sphere

In this step, you will use Caedium to perform a theoretical calculation of pressure coefficient on the sphere. The theory for this calculation is described in the example Potential Flow Around a Sphere.

The steps to calculate the theoretical values of pressure coefficient (Cp) are:

- Create scalars 1 and 9/4 (2.25)
- Create y/R, which is equal to cos(theta)
- Create cos
^{2}(theta) - Create the equation Cp = 1 - 9/4 x cos
^{2}(theta)

### Create the Scalars 1 and 9/4

Select the **New** Toolbar and click the **Result** button .

In the **Create Result** dialog, select the **Constant** tab. For **Units**, select **None**.

Click **Create** to create a scalar. Click **Create** again to create a second scalar.

In the **Results** Tool Palette, select **Scalar Variables->Scalar**. Click on it again to make its name editable, and change **Scalar** to **1**.

In the Properties Panel, set the **Value** of 1 to **1** and press Enter on the keyboard.

In the **Results** Tool Palette, rename **Scalar Variables->Scalar** to be **9/4**. In the Properties Panel, set its value to **2.25**.

### Create cos(theta)

Remember that cos(theta)=y/R.

In the **Create Result** dialog, select the **Binary** tab. Select **Divide** from the list.

In the **Results** Tool Palette, select **Vector Fields->XYZ**. In the Properties Panel, select **Y** as the **Scalar**.

Drag and drop **Vector Fields->XYZ** onto the left-hand target in the **Create Result** dialog. Select **Scalar** to specify the Y-position as the left-hand variable for the equation.

Drag **Scalar Variables->R** from the **Results** Tool Palette and drop it onto the right-hand target in the **Create Result** dialog.

Click **Create** to create the scalar field y/R. In the **Results** Tool Palette, select **Scalar Fields->(XYZ:Y / R)**, rename it **cos(theta)**, and press Enter.

You have now created all the elements necessary to calculate the theoretical values of pressure coefficient. Next you will assemble the parts of the equation Cp = 1 - 9/4 x cos^{2}(theta).

### Create cos^{2}(theta)

In the **Create Result** dialog, with the **Binary** tab already selected, select **Multiply** from the list.

In the **Results** Tool Palette, select **Scalar Fields->cos(theta)**, and drag and drop it onto the left-hand target in the **Create Result** dialog. Repeat the process to drop the **Scalar Fields->cos(theta)** tool onto the right-hand target. Click **Create** to create the scalar field cos^{2}(theta). In the **Results** Tool Palette, select **Scalar Fields->(cos(theta) x cos(theta))**, rename it **cos^2(theta)**, and press Enter.

### Create 9/4 x cos^{2}(theta)

In the **Results** Tool Palette, select **Scalar Variables->9/4**, and drag and drop it onto the left-hand target in the **Create Result** dialog. Drag and drop the **Scalar Fields->cos^2(theta)** tool onto the right-hand target.

Click **Create** to create the **(9/4 x cos^2(theta))** tool in the **Results** Tool Palette.

### Create Cp = 1 - 9/4 x cos^{2}(theta)

In the **Create Result** dialog, select **Subtract** from the list.

In the **Results** Tool Palette, select **Scalar Variables->1**, and drag and drop it onto the left-hand target in the **Create Result** dialog. Drag and drop the **Scalar Fields->(9/4 x cos^2(theta))** tool onto the right-hand target.

Click **Create** to create Cp. Click **Close** in the **Create Result** dialog.

In the **Results** Tool Palette, select **Scalar Fields->(1 - (9/4 x cos^2(theta)))**, rename it **Theory**, and press Enter.

## Plot Theoretical Values of Pressure Coefficient Versus theta

Select the View Window (**view**). Drag the **Scalar Fields->Theory** tool from the **Results** Tool Palette and drop it onto the edge of the sphere highlighted in the diagram below.

Double-click the edge in the **Select** dialog and select **XY Plot** to create an XY plot of pressure coefficient versus Y-position.

Note that the plot has XYZ:Y along the X-axis. By default, a variable is plotted against the scalar variable selected for XYZ (position). In a previous step, you selected Y as the scalar for position.

Drag and drop the **Scalar Fields->theta** tool onto the background of the Plot Window and select **X Axis** to plot theta along the X-axis.

Left-click the name of the edge in the **Theory Plot** Legend (edge_1 in this example). In the Properties Panel, set the **Name** to be **Theory**.

## Compare Theoretical and Caedium Simulation Pressure Coefficients

To compare the pressure coefficient values from your Caedium simulation with the theoretical values, you need to plot both data sets on the same XY plot. To do this, you will first export the theoretical results, reload them into Caedium as imported data, and then plot them on the same XY plot as the Caedium simulation results.

### Export the Theoretical Pressure Coefficient Data

Select the **File** Toolbar and click the **Export** button . Select the location to save the theoretical plot data, specify a **File name** (**sphere-cp-theory**) and select **Plot Series (*.csv)** as the file type. Click **Save** to export the theoretical pressure coefficient data.

### Import the Theoretical Pressure Coefficient Data

Select the **File** Toolbar and click the **Import** button . Select the location of the plot data you just saved. For the file type, select **Plot Series (*.csv)**. Double-click on **sphere-cp-theory.csv** to import it into Caedium.

The plot series will be visible in the **Results** Tool Palette, under **Imported**.

### Plot the Imported Data and the Caedium Simulation Data on the Same XY Plot

Select the **Cp Plot** Window. Drag and drop the **Imported->sphere-cp-theory.csv** tool onto the Plot Window and select **Done** to plot the theoretical data on the same XY plot as the simulation data.

Left-click the red symbol in the **Cp Plot** Legend. In the Properties Panel, set the **Line** type to **None**, and set the **Symbol** to **Circle Open**.

In the Plot Window, select the black symbol in the **Cp Plot** Legend. In the Properties Panel, rename it to be **Computation**, and set the **Symbol** to **None**.

Notice the good agreement between the computational and theoretical values of pressure coefficient.

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