Drag Forces on a 10 cm Disk and Sphere at Mach 2.0: Insights for Commercial Supersonic Flight

Introduction

The drag forces on various airfoils and shapes at supersonic speeds are critical for the design of commercial supersonic aircraft. This study delves into the drag forces on a 10 cm disk perpendicular to airflow and a 10 cm sphere, both at a Mach number of 2.0 at an altitude of 55,000 meters. The analysis is based on independent assumptions and adjusted for the absence of other structures or obstructions. This information is relevant for understanding the aerodynamics and performance requirements of commercial supersonic flights.

Understanding Drag Forces at Supersonic Speeds

Drag, the resistance that a moving object encounters in a fluid, such as air, is a critical factor in the design of aircraft, especially at supersonic speeds. The drag force can be calculated using the formula:

(D frac{1}{2} rho V^2 S_{ref} C_D)

Where:

(D) is the drag force (rho) is the air density (V) is the velocity of the object relative to the air (S_{ref}) is the reference area (usually the frontal area) (C_D) is the drag coefficient

Drag Coefficients (CD) for Different Shapes

The drag coefficient, a dimensionless number, is a measure of the drag force experienced by an object at a certain velocity and angle of attack. For objects that are blunt, such as 10 cm disks and spheres, the skin friction drag is often negligible compared to the pressure drag. At supersonic speeds, the drag coefficient (CD) for very blunt objects is almost constant, starting from about Mach 1.5 and upwards.

Drag on a 10 cm Disk

Considering a 10 cm disk, placed perpendicular to the airflow, we can estimate the drag force using an empirical drag coefficient (CD) of 1.5 for a blunt object. The reference area in this case is the frontal area of the disk. The steps to calculate the drag force are as follows:

Calculate the air density at 55,000 meters using the standard atmospheric model. Calculate the velocity in meters per second (since the Mach number is 2.0). Plug the values into the drag force formula: (D frac{1}{2} rho V^2 S_{ref} C_D)

Drag on a 10 cm Sphere

For a 10 cm sphere, the drag coefficient (CD) is typically around 1.0, also applicable for blunt shapes. The reference area for this shape is also the frontal area. Similar calculations can be done using the same formula:

Calculate the air density at 55,000 meters. Convert the Mach number to meters per second. Apply the drag force formula: (D frac{1}{2} rho V^2 S_{ref} C_D)

Adjusting for Altitude and Skin Friction Drag

The drag force calculations are often adjusted for altitude to account for variations in air density and velocity. For very blunt objects, skin friction drag is insignificant; however, at higher altitudes, the air is less dense, which affects the drag force. Assuming no other structures or obstructions, the calculations focus solely on the basic forces acting on the shapes.

Example Calculation for a 10 cm Disk

Assume the following values for the example:

Air density (rho) at 55,000 meters (using a standard atmospheric model) is approximately 32.6 kg/m3. Velocity (V) is 2,082 meters per second (since Mach 2.0 is approximately 2,082 meters per second). Reference area (frontal area) (S_{ref}) is 0.00785 m2. Drag coefficient (C_D) for a blunt object is 1.5.

Plugging these values into the drag force formula:

(D frac{1}{2} times 32.6 times (2082)^2 times 0.00785 times 1.5)

Calculate the drag force:

(D 2,332,470 N) (Newtons)

Example Calculation for a 10 cm Sphere

Using the same values as above:

Reference area (frontal area) (S_{ref}) is 0.00785 m2. Drag coefficient (C_D) for a blunt object is 1.0.

Plugging these into the drag force formula:

(D frac{1}{2} times 32.6 times (2082)^2 times 0.00785 times 1.0)

Calculate the drag force:

(D 1,554,980 N) (Newtons)

Conclusion

The drag forces on a 10 cm disk and a 10 cm sphere at Mach 2.0 at an altitude of 55,000 meters are significant considerations for the design of commercial supersonic aircraft. By understanding and accurately calculating these forces, engineers can optimize aircraft performance and design for efficient and safe flight. The drag coefficients and calculations provided here are based on empirical data and standard atmospheric assumptions, providing a foundational understanding for further aerodynamic studies and design optimization.