💧Fluid Mechanics Unit 10 – Flow over Immersed Bodies

Key Concepts and Definitions

• Flow over immersed bodies involves the interaction between a fluid and an object submerged in it
• Fluid can be either a liquid (water) or a gas (air)
• Immersed body is any object partially or fully submerged in a fluid
• Streamlines are imaginary lines that represent the path of fluid particles in a flow
  • Streamlines are always parallel to the velocity vector at any point
• Stagnation point is a location on the immersed body where the fluid velocity is zero
• Pressure gradient is the change in pressure over a given distance in the direction of flow
• Viscosity is a measure of a fluid's resistance to deformation under shear stress
  • Higher viscosity fluids (honey) flow more slowly than lower viscosity fluids (water)

Fundamental Principles

• Continuity equation states that mass is conserved in a fluid flow
  • For incompressible fluids, this means that the volume flow rate is constant
• Bernoulli's principle relates velocity, pressure, and elevation in a fluid flow
  • As fluid velocity increases, pressure decreases, and vice versa
• No-slip condition states that fluid velocity at the surface of an immersed body is zero
• Pressure distribution around an immersed body is influenced by the body's shape and the flow conditions
  • High-pressure regions form at stagnation points and on the upstream side of the body
  • Low-pressure regions form on the downstream side and in areas of flow separation
• Fluid particles follow streamlines in steady, laminar flow
  • In turbulent flow, fluid particles exhibit random, chaotic motion

Types of Flow and Body Shapes

• Laminar flow is characterized by smooth, parallel streamlines and minimal mixing between fluid layers
  • Occurs at low Reynolds numbers (< 2300 for pipe flow)
• Turbulent flow is characterized by chaotic, swirling motion and significant mixing between fluid layers
  • Occurs at high Reynolds numbers (> 4000 for pipe flow)
• Bluff bodies have a wide, flat shape that promotes flow separation and wake formation (cylinder)
• Streamlined bodies have a smooth, tapered shape that minimizes flow separation (airfoil)
• Compressible flow involves significant changes in fluid density (high-speed gas flows)
• Incompressible flow assumes constant fluid density (low-speed liquid flows)
  • Most practical engineering applications involve incompressible flow

Drag and Lift Forces

• Drag is the force parallel to the flow direction that resists motion of the immersed body
  • Consists of pressure drag (due to pressure differences) and skin friction drag (due to viscous shear stress)
• Lift is the force perpendicular to the flow direction that acts to lift the immersed body
  • Generated by asymmetric pressure distribution around the body (airfoil)
• Drag coefficient ($C_D$) is a dimensionless number that quantifies the drag force on an immersed body
  • Depends on the body shape, flow conditions, and Reynolds number
• Lift coefficient ($C_L$) is a dimensionless number that quantifies the lift force on an immersed body
• Drag and lift forces can be calculated using the equations:
  • $F_D = \frac{1}{2} \rho v^2 A C_D$
  • $F_L = \frac{1}{2} \rho v^2 A C_L$
  • where $\rho$ is fluid density, $v$ is fluid velocity, and $A$ is the body's reference area

Boundary Layer Theory

• Boundary layer is the thin region near the surface of an immersed body where viscous effects are significant
  • Velocity in the boundary layer varies from zero at the surface to the freestream velocity at the edge
• Boundary layer thickness ($\delta$) is the distance from the surface to where the velocity reaches 99% of the freestream velocity
• Laminar boundary layer is characterized by smooth, parallel streamlines and minimal mixing
  • Occurs near the leading edge of the body and at low Reynolds numbers
• Turbulent boundary layer is characterized by chaotic, swirling motion and significant mixing
  • Occurs further downstream and at high Reynolds numbers
• Transition from laminar to turbulent boundary layer depends on the Reynolds number, surface roughness, and pressure gradient
• Flow separation occurs when the boundary layer detaches from the surface, forming a wake behind the body
  • Caused by adverse pressure gradients and geometry changes (sharp corners)

Flow Visualization Techniques

• Flow visualization is used to observe and analyze fluid motion around immersed bodies
• Streamlines can be visualized using smoke, dye, or oil streaks in wind tunnels or water channels
• Particle image velocimetry (PIV) uses laser light to illuminate tiny particles in the fluid and track their motion
  • Provides quantitative velocity data in a plane or volume
• Schlieren imaging uses light refraction to visualize density gradients in compressible flows
• Surface oil flow visualization reveals skin friction lines and separation patterns on the body surface
• Computational fluid dynamics (CFD) simulations provide detailed flow field data and visualizations
  • Requires solving the governing equations (Navier-Stokes) numerically on a discretized domain

Real-World Applications

• Aerodynamics of vehicles (cars, airplanes) involves optimizing body shapes to minimize drag and maximize lift
  • Streamlined designs and active flow control techniques (spoilers, vortex generators) are used
• Hydrodynamics of ships and submarines involves reducing drag to improve fuel efficiency and maneuverability
  • Bulbous bows and stern flaps are used to modify the pressure distribution and reduce wake
• Wind engineering of buildings and bridges involves understanding the effects of wind loads and vortex shedding
  • Aerodynamic shaping and damping devices (tuned mass dampers) are used to mitigate vibrations
• Biomedical flows in the cardiovascular system involve complex geometries and pulsatile flow
  • Stents and heart valves are designed to minimize flow disturbances and prevent thrombosis
• Environmental flows in rivers, oceans, and the atmosphere involve large-scale turbulent mixing and transport processes
  • Understanding these flows is crucial for predicting pollutant dispersion, sediment transport, and weather patterns

Problem-Solving Strategies

• Identify the type of flow (laminar vs. turbulent, compressible vs. incompressible) and the relevant governing equations
• Determine the appropriate boundary conditions (no-slip, symmetry, freestream) and initial conditions
• Simplify the problem using assumptions (steady-state, two-dimensional, inviscid) when appropriate
  • Justify the validity of these assumptions based on the flow conditions and geometry
• Use dimensional analysis to identify the relevant non-dimensional parameters (Reynolds number, Mach number)
  • These parameters help characterize the flow regime and enable comparison between different scenarios
• Apply conservation laws (mass, momentum, energy) and constitutive relations (Newton's law of viscosity) to derive the equations of motion
• Solve the equations analytically for simple geometries (flat plate, cylinder) or numerically using CFD for complex geometries
• Interpret the results in terms of the key flow features (pressure distribution, velocity profiles, drag and lift forces)
  • Validate the results using experimental data or benchmark solutions
• Perform sensitivity analysis to assess the impact of uncertainties in input parameters or modeling assumptions
  • Use this information to guide design decisions and optimize performance