10.2 Drag and Lift Forces

Drag and Lift Forces

Drag and lift are the two fundamental forces that a fluid exerts on an immersed body. Drag acts parallel to the flow direction and resists motion, while lift acts perpendicular to it. Both arise from the distribution of pressure and shear stress over the body's surface, and quantifying them through non-dimensional coefficients is central to designing anything that moves through a fluid.

Drag vs lift forces

Drag force acts parallel to the direction of fluid flow. It resists the motion of a body through the fluid. Think of the force you feel pushing back on your hand when you stick it out a car window. Drag results from the combined effects of pressure and shear stress acting on the body's surface.

Lift force acts perpendicular to the flow direction. It's generated whenever there's an asymmetric pressure distribution around the body, or in certain cases by viscous effects. Airplane wings are the classic example, but lift shows up in many geometries. A spinning soccer ball curving through the air is also experiencing a lift force.

Pressure and shear stress contributions

Drag and lift each have two physical sources: pressure forces (acting normal to the surface) and shear stress (acting tangent to the surface). Their relative importance depends on the body shape and the flow regime.

Contributions to drag:

• Form drag (pressure drag) comes from the pressure difference between the upstream (high-pressure) and downstream (low-pressure) sides of a body. For blunt objects like spheres or cylinders, form drag dominates because the flow separates early and creates a large low-pressure wake behind the body. Form drag also dominates at high Reynolds numbers.
• Skin friction drag comes from viscous shear stress acting directly on the body's surface. For streamlined shapes like airfoils, where flow stays attached over most of the surface, skin friction is the primary drag source. It prevails at low Reynolds numbers and on slender bodies.

Contributions to lift:

• Pressure-driven lift is the dominant mechanism for most lift-generating bodies. An asymmetric pressure distribution develops with higher pressure on the bottom surface and lower pressure on the top surface, producing a net upward force. This is how airplane wings generate lift.
• Shear-driven lift occurs in specific situations like the Magnus effect. When a body spins (a cylinder or sphere, for example), the boundary layer develops asymmetrically: the side spinning with the flow has a thinner boundary layer and delayed separation, while the opposite side separates earlier. This asymmetry creates a net lateral force. That's why a spinning baseball curves.

Drag and lift coefficient calculations

Engineers use non-dimensional coefficients to compare drag and lift across different bodies, fluids, and speeds. These coefficients normalize the actual force by the dynamic pressure and a reference area.

The drag coefficient is defined as:

$C_D = \frac{F_D}{\frac{1}{2} \rho U^2 A}$

The lift coefficient follows the same form:

$C_L = \frac{F_L}{\frac{1}{2} \rho U^2 A}$

where:

• $F_D$ and $F_L$ are the drag and lift forces
• $\rho$ is the fluid density
• $U$ is the freestream velocity
• $A$ is the reference area

The choice of reference area matters and varies by convention. For blunt bodies (spheres, cylinders), $A$ is typically the frontal area (the cross-section the flow "sees"). For wings and airfoils, $A$ is the planform area (the area viewed from above). Always check which convention is being used before comparing $C_D$ or $C_L$ values.

Both coefficients depend on several factors:

1. Body geometry (shape, size, and orientation relative to the flow)
2. Flow conditions (Reynolds number, Mach number, angle of attack)
3. Surface properties (roughness can trip the boundary layer and change both drag and lift)

Reynolds number and drag relationship

The Reynolds number characterizes the ratio of inertial forces to viscous forces in a flow:

$Re = \frac{\rho U L}{\mu}$

where $L$ is a characteristic length (diameter for a sphere, chord length for an airfoil) and $\mu$ is the fluid's dynamic viscosity. The Reynolds number is the single most important parameter governing how $C_D$ behaves for a given geometry.

How $C_D$ varies with $Re$:

• At low $Re$ (laminar flow), viscous forces dominate and $C_D$ is inversely proportional to $Re$: $C_D \propto \frac{1}{Re}$. For a sphere in Stokes flow ($Re < 1$), the exact result is $C_D = \frac{24}{Re}$.
• At high $Re$ (turbulent flow), $C_D$ becomes relatively independent of $Re$ and levels off to a roughly constant value that depends mainly on body shape.
• The transition from laminar to turbulent flow occurs at a critical $Re$ that depends on body geometry and surface roughness.

Why this matters through boundary layer behavior:

• A laminar boundary layer (low $Re$) produces lower skin friction drag on the surface. However, laminar layers separate from the surface more easily, which can create a large wake and high form drag on blunt bodies.
• A turbulent boundary layer (high $Re$) has higher skin friction, but it carries more momentum near the wall and resists separation much better. This delays separation, shrinks the wake, and can actually reduce total drag on blunt bodies. Golf ball dimples exploit exactly this effect: the dimples trip the boundary layer to turbulent, delaying separation and cutting the ball's drag roughly in half compared to a smooth sphere.

This tradeoff between skin friction and form drag is one of the most important ideas in external flow. For streamlined bodies, you want laminar flow as long as possible to minimize skin friction. For blunt bodies, a turbulent boundary layer can be beneficial because reducing form drag matters more.