When a fluid flows around a solid object, or an object moves through a stationary fluid, the interaction between the two is known as flow past immersed bodies. This is a fundamental concept in fluid mechanics, crucial for designing everything from high-speed trains to underwater pipelines.
The force exerted by the fluid on the body is generally resolved into two components: Drag (parallel to the flow) and Lift (perpendicular to the flow).
1. Drag Force
Drag is the resistance force acting in the direction of the relative external flow. It is composed of two primary types:
A. Pressure Drag (Form Drag)
This is caused by the difference in pressure between the front and rear of the body. When fluid flows past a body, a high-pressure zone is created at the stagnation point (front), while a low-pressure zone often develops at the rear due to flow separation.
- Dominance: High in “bluff” (non-streamlined) bodies like spheres or cylinders.
B. Skin Friction Drag (Viscous Drag)
This arises from the shear stresses at the surface of the body due to the fluid’s viscosity.
- Dominance: High in “streamlined” bodies like thin plates or airfoils.
The total drag force is calculated using the equation:
$$F_D = C_D \cdot \frac{1}{2} \rho A v^2$$
Where $C_D$ is the drag coefficient, $\rho$ is fluid density, $A$ is the projected frontal area, and $v$ is the velocity.
2. Lift Force ($F_L$)
Lift is the component of the total force acting at right angles to the direction of motion. It occurs when the body is shaped or oriented such that the fluid pressure is lower on the upper surface than on the lower surface (Bernoulliโs Principle).
The lift equation is:
Where CL is the lift coefficient.
3. The Boundary Layer Concept
As fluid flows over a body, the “no-slip condition” causes the fluid particles in contact with the surface to have zero velocity. This creates a thin region called the boundary layer where velocity gradients and shear stresses are concentrated.
- Laminar Boundary Layer: Smooth, predictable flow with lower friction but prone to early separation.
- Turbulent Boundary Layer: Erratic, swirling flow (eddies) with higher friction but better at “sticking” to the surface, which can actually reduce total drag on bluff bodies by delaying separation.
4. Flow Separation and Wake
Flow separation occurs when the fluid can no longer follow the contour of the body due to an adverse pressure gradient (pressure increasing in the direction of flow).
- The velocity in the boundary layer drops to zero and reverses.
- The flow “detaches” from the surface.
- A region of low pressure and recirculating flow, called a wake, forms behind the body.
- Result: A large wake significantly increases pressure drag.
5. Classification of Bodies
- Bluff Bodies: The length in the direction of flow is close to or less than the dimensions perpendicular to the flow (e.g., a sphere or a building). Drag is dominated by pressure differences.
- Streamlined Bodies: Shaped specifically to reduce the adverse pressure gradient and delay separation (e.g., an airplane wing). Drag is dominated by skin friction.
6. Stagnation Point
At the very front of the immersed body, the fluid velocity is reduced to zero. This point is called the stagnation point. According to the Bernoulli equation, the pressure here is at its maximum, known as the stagnation pressure ($P_s$):
Where $P_0$ is the static pressure of the undisturbed fluid.