Drag and Lift Coefficients for Immersed Bodies

Hello! Welcome to your next lesson on fluid dynamics.

In our last session, we used the linear momentum equation to calculate the total force a moving fluid exerts on objects like pipe bends and jet engines. We saw that this force is a direct result of the fluid changing its momentum.

Today, we will focus on what happens when a body is completely immersed in a moving fluid, like an airplane wing in the air or a submarine in the water. We'll decompose the total fluid force into two critical components that are fundamental to aerospace engineering: lift and drag. Your learning outcome for this lesson is to analyze flow around immersed bodies using drag and lift coefficients.

By the end of this lesson, you will understand where lift and drag come from and how to use dimensionless coefficients to calculate their magnitude—a core skill for any aspiring aerospace engineer.

1. From Total Force to Lift and Drag

When a fluid flows over a body, it exerts forces on the body's surface. As we saw in the last lesson, these forces arise from two sources:

1. Pressure distribution acting perpendicular to the surface.
2. Wall shear stress (friction) acting parallel to the surface.

Integrating these pressure and shear stress distributions over the entire surface gives the resultant force, ${\vec{F}}_R$, on the body. For engineering purposes, especially in aerospace, it's far more useful to resolve this resultant force into two components relative to the direction of the oncoming fluid flow (the free-stream velocity, $V$):

• Drag ($F_D$): The component of the force parallel to the direction of the upstream flow. It represents the fluid's resistance to the body's motion.
• Lift ($F_L$): The component of the force perpendicular to the direction of the upstream flow.

2. The Origins of Drag

The drag force itself is also composed of two parts, corresponding to its physical origins:

1. Pressure Drag (or Form Drag): Caused by the pressure difference between the front and rear of the body. This is dominant for blunt, non-streamlined bodies where the flow separates from the surface, creating a low-pressure wake region behind the body.
2. Friction Drag (or Skin Friction Drag): Caused by the shear stress of the fluid "rubbing" against the body's surface. This is the dominant form of drag for highly streamlined bodies like an airfoil at a small angle of attack.

The following video provides an excellent visual explanation of these concepts.

3. The Drag and Lift Equations

Calculating lift and drag by integrating pressure and shear stress is extremely difficult. Instead, we use a practical engineering approach based on dimensionless numbers, which you've already seen with the Reynolds number and friction factor. Here, we introduce the drag coefficient ($C_D$) and the lift coefficient ($C_L$).

The drag and lift forces are calculated using these simple and powerful equations:

Drag Equation:
$F_D=C_D\cdot \frac{1}{2}\rho V^{2}A$

Lift Equation:
$F_L=C_L\cdot \frac{1}{2}\rho V^{2}A$

Let's break down the terms:

• $C_D$ and $C_L$: The drag and lift coefficients. These are dimensionless numbers that depend on the body's shape, its orientation (angle of attack), and the flow regime (i.e., the Reynolds Number). They are typically determined experimentally in a wind tunnel or through computational fluid dynamics (CFD).
• $\rho$: The density of the fluid.
• $V$: The free-stream velocity of the fluid.
• $\frac{1}{2}\rho V^2$: This group of terms is called the dynamic pressure, representing the kinetic energy per unit volume of the fluid.
• $A$: A characteristic reference area. The choice of reference area is critical and must be specified. For bluff bodies like spheres or cars, it is typically the frontal area (the projected area you see from the front). For streamlined bodies like wings, it is the planform area (the area you see from above).

The formal definitions are provided in the following reading.

4. Factors Affecting Lift and Drag Coefficients

The values of $C_D$ and $C_L$ are not constant. They are primarily influenced by the body's shape, its angle of attack, and the Reynolds number of the flow.

The Effect of Reynolds Number and Shape on Drag

For a given shape, the drag coefficient changes with the Reynolds number ($Re=\frac{\rho VL}{\mu }$). This is because the Reynolds number dictates the nature of the boundary layer (laminar or turbulent), which in turn affects flow separation.

Drag coefficient $C_D$ as a function of Reynolds number for a smooth sphere and cylinder. Note the sudden drop, known as the "drag crisis," where the boundary layer becomes turbulent, delaying flow separation and drastically reducing pressure drag. This chart is similar to Figure 9.35 in the Purdue notes.

As you saw in the video, a turbulent boundary layer has more energy and can resist flow separation longer than a laminar one. This is why golf balls have dimples—to trigger turbulence, delay separation, and lower the pressure drag.

The document below provides tables of typical drag coefficients for various common shapes, which are invaluable for quick engineering estimates.

The Effect of Angle of Attack on Lift

For lift-generating bodies like airfoils, the most important factor is the angle of attack ($\alpha$).

As you can see from the graph:

• The lift coefficient ($C_L$, blue line) increases almost linearly with the angle of attack up to a certain point.
• At a critical angle of attack (around 14-16° here), the flow separates abruptly from the top surface of the wing. This causes $C_L$ to drop sharply, a dangerous condition known as stall.
• The drag coefficient ($C_D$, red line) is low at small angles but increases dramatically as the wing approaches stall.

The video below explains how lift is generated and how devices like flaps are used to increase lift during takeoff and landing.

5. Worked Example: Force on a Car Spoiler

Let's apply these concepts to a simple aerospace-related problem: calculating the downforce from a car's rear spoiler, which is essentially an inverted airfoil.

Conclusion

In this lesson, we've explored the crucial concepts of lift and drag, which are central to analyzing the motion of vehicles through fluids. You now have the tools to quantify these forces, a foundational skill for your aerospace engineering goals.

Key Takeaways:

• Lift and drag are the components of the total aerodynamic force acting perpendicular and parallel to the fluid flow, respectively.
• Drag is comprised of pressure drag (due to shape and flow separation) and friction drag (due to surface shear stress).
• Lift and drag are calculated using the lift and drag equations, which rely on the dimensionless coefficients $C_L$ and $C_D$.
• These coefficients are not constant; they depend on the body's shape, its orientation (angle of attack), and the Reynolds number of the flow.
• Stall is a critical flight condition where an increase in the angle of attack leads to a sudden loss of lift.

Preview of the Next Module:
We have now concluded our module on fluid flow applications. We've seen how to analyze energy in pipe systems and how to calculate forces on immersed bodies.

Now, we will pivot to the first module in Dynamics. In our next lesson, we will step back from forces and focus on Kinematics—the pure geometry of motion. We'll learn to apply kinematic equations to solve problems of particle motion, developing the mathematical language to describe velocity and acceleration. This will build the foundation for later connecting forces (like lift and drag) to the motion they produce, via Newton's Second Law.