2.1 Principles of fluid dynamics in underwater environments

Fluid dynamics in underwater environments is crucial for understanding how submerged vehicles move and interact with water. Density, viscosity, and pressure play key roles in determining fluid behavior, affecting everything from buoyancy to drag forces on underwater robots.

Laminar and turbulent flow regimes significantly impact vehicle performance underwater. The Reynolds number helps predict flow characteristics, while boundary layers, flow separation, and vortex shedding influence drag and stability. These principles are essential for designing efficient underwater vehicles.

Fluid properties in underwater environments

Characteristics of fluids

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• Fluids are substances that continuously deform under an applied shear stress, which includes liquids (water) and gases (air) underwater
• The two primary properties that govern fluid behavior are density and viscosity
  • Density is mass per unit volume
    • Seawater has a typical density around 1025 kg/m^3 which varies with temperature, salinity and pressure
  • Dynamic viscosity is a fluid's resistance to flow and shear deformation
    • Kinematic viscosity is dynamic viscosity divided by density
    • Viscosity of seawater is affected by temperature and salinity

Hydrostatic pressure and buoyancy

• Fluids exert a force perpendicular to any surface they are in contact with, known as hydrostatic pressure
  • Static pressure increases linearly with depth in a fluid
• Archimedes' principle states that the buoyant force on a submerged object is equal to the weight of the fluid displaced by the object
  • An object's buoyancy determines if it will float, sink or remain neutrally buoyant
• Underwater vehicles experience drag forces that resist their motion through the fluid
  • Drag depends on the vehicle's shape, surface roughness, velocity and fluid properties

Pressure and force on submerged objects

Calculating hydrostatic pressure and force

• Hydrostatic pressure at a given depth is calculated using the equation $P = ρgh$, where $P$ is pressure, $ρ$ is fluid density, $g$ is acceleration due to gravity, and $h$ is depth below the surface
• Pascal's law states that pressure applied to an enclosed fluid is transmitted undiminished to every part of the fluid and enclosing walls
  • This principle governs the operation of hydraulic systems
• The total hydrostatic force acting on a submerged surface is found by integrating the pressure over the area
  • For a flat plate, $F = ρghA$ where $A$ is the surface area

Hydrostatic forces on submerged surfaces

• Hydrostatic forces always act perpendicular to the surface
  • On a curved or angled surface, the force can be resolved into horizontal and vertical components
• The center of pressure is the point where the resultant hydrostatic force acts on a submerged surface
  • It is located below the centroid, at a distance that depends on the shape of the surface
• Hydrostatic forces and moments on submerged vehicles affect their stability and control
  • They contribute to restoring moments that keep the vehicle upright
  • They also influence pitching moments that affect trim angles

Viscosity and density effects on flow

Navier-Stokes equations and Reynolds number

• Fluid flow around a moving underwater vehicle is governed by the Navier-Stokes equations, which relate velocity, pressure, density and viscosity
  • These form a set of coupled differential equations
• The Reynolds number (Re) is a dimensionless quantity that characterizes the ratio of inertial forces to viscous forces
  • It is calculated as $Re = ρVL/μ$, where $ρ$ is density, $V$ is velocity, $L$ is a characteristic length, and $μ$ is dynamic viscosity
• In general, higher Reynolds numbers (Re > 10^5) indicate turbulent flow dominated by inertial forces, while lower Reynolds numbers (Re < 10^3) indicate laminar flow where viscous forces dominate
  • Many underwater vehicles operate in the turbulent regime

Boundary layers, separation and vortex shedding

• The boundary layer is a thin region near a surface where viscous effects are significant
  • Flow velocity transitions from zero at the wall to the freestream velocity
  • Its thickness depends on the Reynolds number
• Flow separation occurs when the boundary layer detaches from a surface, often due to an adverse pressure gradient
  • This creates regions of recirculating flow and increases pressure drag on vehicles
• Vortex shedding is an unsteady flow phenomenon where alternating vortices are shed downstream of a bluff body (cylinder)
  • The shedding frequency depends on the Strouhal number, $St = fL/V$ where $f$ is the shedding frequency

Laminar vs turbulent flow underwater

Characteristics of laminar and turbulent flow

• Laminar flow is characterized by smooth, parallel streamlines and minimal mixing between fluid layers
  • Velocity varies only perpendicular to the flow direction
  • It occurs at low Reynolds numbers
• Turbulent flow exhibits chaotic, fluctuating velocity fields with significant mixing
  • Velocity varies in all directions and eddies of many length scales are present
  • It occurs at high Reynolds numbers
• The transition from laminar to turbulent flow depends on the critical Reynolds number, which varies with geometry
  • For flow over a flat plate, $Re_{crit} ≈ 5 × 10^5$

Effects and occurrence of flow regimes

• Turbulent flow enhances momentum and heat transfer compared to laminar flow
  • Fluctuating velocities lead to increased shear stress and convective mixing
• Laminar flow produces less skin friction drag than turbulent flow, but is less stable and more prone to separation
  • Turbulent boundary layers remain attached to surfaces longer
• In underwater environments, surface roughness of vehicles and structures can trip the boundary layer from laminar to turbulent
  • Marine growth and biofouling increase roughness
• Vehicles operating at low speeds (gliders, AUVs) may have laminar flow over portions of their hull
  • Faster vehicles (submarines, torpedoes) are usually fully turbulent
  • Propellers and control surfaces often experience local turbulent flow