A round thin disk of radius R is oriented perpendicular to a...

A round thin disk of radius $R$ is oriented perpendicular to a fluid stream. The pressure distributions on the front and back surfaces are measured and presented in the form of pressure coefficients. The data are modeled with the following expressions for the front and back surfaces, respectively: $$\begin{array}{ll} \text { Front Surface } & C_{p}=1-\left(\frac{r}{R}\right)^{6} \\ \text { Rear Surface } & C_{p}=-0.42 \end{array}$$ Calculate the drag coefficient for the disk.

A round thin disk of radius $R$ is oriented perpendicular to a fluid stream. The pressure distributions on the front and back surfaces are measured and presented in the form of pressure coefficients. The data are modeled with the following expressions for the front and back surfaces, respectively:
$$\begin{array}{ll}
\text { Front Surface } & C_{p}=1-\left(\frac{r}{R}\right)^{6} \\
\text { Rear Surface } & C_{p}=-0.42
\end{array}$$
Calculate the drag coefficient for the disk.

Key Concepts

Pressure Coefficient

The pressure coefficient is a dimensionless number that relates the local pressure on a body immersed in a fluid flow to the dynamic pressure of the stream. It provides a normalized measure of how the pressure at a point on the surface deviates from the free?stream pressure, allowing comparison and integration over surfaces having variable pressure distributions.

Drag Coefficient

The drag coefficient is a nondimensional parameter that quantifies a body's resistance to motion through a fluid. It is defined by relating the net drag force acting on the object to the dynamic pressure and a reference area. This coefficient encapsulates the effects of shape, surface roughness, and flow conditions into a single number that characterizes drag performance.

Integration over Pressure Distributions

When a surface experiences a spatially varying pressure, the net force (and hence drag) is computed by integrating the pressure over the entire affected area. This process involves summing the contributions of all differential surface elements while accounting for their orientation relative to the flow, ultimately yielding an average coefficient such as the drag coefficient.

Polar Coordinate Integration

For bodies with radial symmetry, such as disks or circular plates, polar coordinate integration is an efficient method to evaluate surface integrals. By expressing the infinitesimal area elements in terms of a radius and angle, the integration reduces to combining a radial integral with an angular component, which often simplifies the calculation of overall force coefficients.

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