💨Fluid Dynamics Unit 10 – Hydrodynamics

Key Concepts and Definitions

• Hydrodynamics studies the motion of fluids and forces acting on immersed bodies
• Fluids include liquids and gases that continuously deform under applied shear stress
• Viscosity measures a fluid's resistance to deformation and is a key property in hydrodynamics
  • Dynamic viscosity ($\mu$) relates shear stress to velocity gradient
  • Kinematic viscosity ($\nu$) is the ratio of dynamic viscosity to density ($\rho$)
• Reynolds number ($Re$) characterizes the ratio of inertial forces to viscous forces in a fluid
  • Defined as $Re = \frac{\rho VL}{\mu}$, where $V$ is velocity and $L$ is characteristic length
  • Low $Re$ indicates laminar flow, while high $Re$ suggests turbulent flow
• Compressibility describes a fluid's density change in response to pressure changes
  • Incompressible fluids (liquids) maintain constant density, while compressible fluids (gases) experience density variations
• Streamlines are curves tangent to velocity vectors at each point in a fluid flow field

Fundamental Equations

• Conservation of mass (continuity equation) ensures that mass is neither created nor destroyed in a fluid system
  • For incompressible flow: $\nabla \cdot \vec{V} = 0$, where $\vec{V}$ is the velocity vector
• Conservation of momentum (Navier-Stokes equations) describes the motion of viscous fluids
  • For incompressible flow: $\rho \frac{D\vec{V}}{Dt} = -\nabla p + \mu \nabla^2 \vec{V} + \rho \vec{g}$
  • $\frac{D}{Dt}$ is the material derivative, $p$ is pressure, and $\vec{g}$ is the gravitational acceleration vector
• Bernoulli's equation relates pressure, velocity, and elevation along a streamline for inviscid, steady, incompressible flow
  • $p + \frac{1}{2}\rho V^2 + \rho gz = \text{constant}$, where $z$ is elevation
• Potential flow theory simplifies the analysis of inviscid, irrotational flows by introducing a velocity potential function ($\phi$)
  • Velocity components are given by $u = \frac{\partial \phi}{\partial x}$, $v = \frac{\partial \phi}{\partial y}$, and $w = \frac{\partial \phi}{\partial z}$
• Vorticity ($\vec{\omega}$) measures the local rotation of fluid elements and is defined as the curl of the velocity vector
  • $\vec{\omega} = \nabla \times \vec{V}$
  • Irrotational flows have zero vorticity everywhere

Types of Fluid Flow

• Laminar flow occurs at low Reynolds numbers and is characterized by smooth, parallel streamlines
  • Fluid layers slide past each other without mixing, resulting in minimal cross-stream momentum transfer
• Turbulent flow occurs at high Reynolds numbers and features chaotic, irregular motion with rapid mixing
  • Eddies, vortices, and fluctuations enhance momentum, heat, and mass transfer
• Transitional flow exists between laminar and turbulent regimes, exhibiting intermittent turbulent bursts
• Steady flow maintains constant velocity, pressure, and other properties at each point over time
  • Streamlines coincide with pathlines in steady flow
• Unsteady flow experiences temporal variations in flow properties
  • Streamlines and pathlines differ in unsteady flow due to the time-dependent velocity field
• Uniform flow has constant velocity magnitude and direction across any cross-section perpendicular to the flow
• Non-uniform flow exhibits spatial variations in velocity profile, such as developing pipe flow

Boundary Layer Theory

• Boundary layers form near solid surfaces due to the no-slip condition, which states that fluid velocity matches the surface velocity at the interface
• Velocity gradient within the boundary layer gives rise to shear stress and viscous effects
• Boundary layer thickness ($\delta$) is defined as the distance from the surface where velocity reaches 99% of the freestream value
  • $\delta$ increases with downstream distance ($x$) as $\delta \propto \sqrt{\frac{\nu x}{U_\infty}}$ for laminar flow over a flat plate
• Displacement thickness ($\delta^*$) quantifies the distance by which streamlines are displaced due to the reduced mass flow in the boundary layer
• Momentum thickness ($\theta$) represents the loss of momentum flux in the boundary layer compared to inviscid flow
• Shape factor ($H$) is the ratio of displacement thickness to momentum thickness and indicates the nature of the boundary layer
  • $H \approx 2.6$ for laminar boundary layers, while $H \approx 1.3-1.4$ for turbulent boundary layers
• Boundary layer separation occurs when adverse pressure gradients cause flow reversal near the surface
  • Separated flows lead to increased drag, loss of lift, and vortex shedding

Pressure and Velocity Fields

• Pressure field in a fluid is governed by the momentum equations and boundary conditions
  • Pressure acts normal to fluid elements and varies spatially in the presence of velocity gradients or external forces
• Hydrostatic pressure increases linearly with depth in a stationary fluid due to the weight of the overlying fluid column
  • $p = p_0 + \rho gz$, where $p_0$ is the reference pressure at $z=0$
• Dynamic pressure arises from the fluid's motion and is proportional to the square of velocity
  • $p_d = \frac{1}{2}\rho V^2$
• Velocity field describes the speed and direction of fluid motion at each point in space
  • Obtained by solving the continuity and momentum equations subject to boundary conditions
• Streamfunction ($\psi$) is a scalar function that defines the velocity field in 2D incompressible flow
  • Velocity components are given by $u = \frac{\partial \psi}{\partial y}$ and $v = -\frac{\partial \psi}{\partial x}$
  • Streamlines are lines of constant $\psi$
• Potential flow solutions, such as uniform flow, source/sink flow, and doublet flow, can be superposed to model complex velocity fields

Applications in Engineering

• Aerodynamics studies the motion of air and its interaction with solid bodies, such as aircraft wings and vehicles
  • Lift generation, drag reduction, and stability are key considerations in aerodynamic design
• Hydrodynamics plays a crucial role in the design of ships, submarines, and offshore structures
  • Hull form optimization, propulsion systems, and wave-structure interactions are important aspects
• Pipe flow analysis involves determining pressure drop, flow rate, and velocity profile in closed conduits
  • Friction factor correlations (Moody diagram) and minor loss coefficients are used in pipe system design
• Open channel flow deals with the motion of liquids with a free surface, such as rivers and canals
  • Froude number ($Fr$) characterizes the ratio of inertial to gravitational forces and determines the flow regime (subcritical, critical, or supercritical)
• Turbomachinery, including pumps, turbines, and compressors, relies on hydrodynamic principles to convert energy between fluid and mechanical forms
  • Blade design, flow passages, and performance curves are essential aspects of turbomachinery analysis
• Environmental fluid mechanics applies hydrodynamics to study atmospheric and oceanic flows, pollutant dispersion, and sediment transport

Experimental Methods

• Flow visualization techniques allow qualitative observation of fluid motion and flow patterns
  • Dye injection, smoke tracers, and particle image velocimetry (PIV) are common methods
• Pressure measurements can be performed using manometers, pressure transducers, or pressure-sensitive paint (PSP)
  • Pitot tubes measure local flow velocity by converting dynamic pressure to static pressure
• Velocity measurements employ various techniques depending on the flow conditions and desired spatial and temporal resolution
  • Hot-wire anemometry measures velocity based on heat transfer from a thin wire exposed to the flow
  • Laser Doppler velocimetry (LDV) and acoustic Doppler velocimetry (ADV) use the Doppler effect to measure velocity non-intrusively
• Force and moment measurements on immersed bodies can be conducted using load cells, strain gauges, or pressure integration
  • Wind tunnel testing of scaled models is common in aerodynamics to determine lift, drag, and moment coefficients
• Particle tracking methods, such as radioactive particle tracking (RPT) and magnetic particle tracking (MPT), provide Lagrangian velocity data
• Computational fluid dynamics (CFD) complements experimental methods by numerically solving the governing equations on discretized domains
  • Validation and verification with experimental data are crucial for ensuring the accuracy and reliability of CFD results

Advanced Topics and Current Research

• Turbulence modeling remains a challenging aspect of hydrodynamics due to the complex, multi-scale nature of turbulent flows
  • Reynolds-averaged Navier-Stokes (RANS) models, large eddy simulation (LES), and direct numerical simulation (DNS) are approaches with varying levels of resolution and computational cost
• Multiphase flows involve the simultaneous presence of multiple fluid phases or components, such as gas-liquid or solid-liquid mixtures
  • Interface tracking methods (volume of fluid, level set) and particle-based methods (discrete element method, smoothed particle hydrodynamics) are used to model multiphase flows
• Biological fluid mechanics explores the role of fluids in living systems, including blood flow, respiratory flow, and swimming and flying in nature
  • Non-Newtonian fluid behavior, compliant walls, and fluid-structure interaction are important considerations
• Micro- and nanoscale flows exhibit unique phenomena due to the increased importance of surface forces and molecular effects
  • Slip boundary conditions, Knudsen number effects, and electrokinetic flows are relevant in microfluidic devices and lab-on-a-chip applications
• Fluid-structure interaction (FSI) involves the coupled dynamics of fluids and deformable solids, such as flutter of aircraft wings or flow-induced vibration of structures
  • Partitioned or monolithic numerical approaches are used to solve the coupled fluid and structural equations
• Machine learning and data-driven methods are being increasingly applied to hydrodynamics problems for flow control, optimization, and reduced-order modeling
  • Neural networks, Gaussian processes, and sparse regression techniques are popular tools in this emerging field