2.2 Fluid dynamics and compressible flow

Fluid dynamics and compressible flow are crucial in aerospace propulsion. They govern how gases and liquids behave in engines, nozzles, and other components. Understanding these concepts is key to designing efficient propulsion systems and predicting their performance.

This topic covers fluid properties, conservation laws, and flow equations. It also explores the differences between incompressible and compressible flows, as well as important phenomena like shock waves. These principles are essential for analyzing and optimizing propulsion technologies.

Fluid dynamics in motion

Characteristics of fluids

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• Fluids are substances that continuously deform under an applied shear stress, including liquids and gases
• Density is the mass per unit volume of a fluid, which can vary with changes in pressure and temperature
• Viscosity is a measure of a fluid's resistance to deformation under shear stress, influencing the development of velocity gradients and boundary layers
• Examples of fluids include water, air, and oil

Behavior of fluids in motion

• Velocity is the speed and direction of fluid motion, typically varying throughout a flow field
• Pressure is the force per unit area exerted by a fluid, which acts perpendicular to any surface in contact with the fluid
• Turbulence is characterized by chaotic, irregular fluid motion with rapid mixing and increased dissipation of energy
• Examples of fluid behavior in motion include the flow of water in a pipe and the movement of air over an airplane wing

Fluid flow analysis in propulsion

Conservation principles

• The continuity equation expresses the conservation of mass in a fluid flow, relating changes in density, velocity, and cross-sectional area
• The momentum equation, derived from Newton's second law, relates the forces acting on a fluid to the rate of change of its momentum
• The energy equation, based on the first law of thermodynamics, accounts for changes in kinetic, potential, and internal energy along a streamline
• Examples of conservation principles in propulsion include the flow through a rocket nozzle and the operation of a gas turbine engine

Fluid flow equations

• Bernoulli's equation is a simplified form of the energy equation for steady, inviscid, incompressible flow along a streamline
• The Navier-Stokes equations are a set of partial differential equations that describe the motion of viscous fluids, incorporating the effects of pressure, viscosity, and external forces
• These equations are used to analyze fluid flow in various propulsion components such as fuel injectors, combustion chambers, and turbomachinery
• Examples of fluid flow equations in action include the design of wind tunnel experiments and computational fluid dynamics (CFD) simulations

Incompressible vs Compressible flow

Incompressible flow

• Incompressible flow assumes constant fluid density, which is a reasonable approximation for liquids and low-speed gas flows
• In incompressible flow, the velocity field is divergence-free, meaning that the continuity equation simplifies to $\nabla \cdot \vec{v} = 0$
• Incompressible flow analysis is often used for low-speed applications such as hydraulic systems and low-speed aerodynamics
• Examples of incompressible flow include the flow of water through a pipe and the flow around a slow-moving vehicle

Compressible flow

• Compressible flow accounts for changes in fluid density due to variations in pressure and temperature, which become significant at high speeds or in the presence of shock waves
• The Mach number, defined as the ratio of the flow velocity to the local speed of sound, is a key parameter in characterizing compressible flow regimes
• Subsonic flow (Mach < 1) exhibits relatively minor density changes, while supersonic flow (Mach > 1) is characterized by the formation of shock waves and significant density variations
• Compressibility effects influence the design of propulsion systems, such as the shape of nozzles, diffusers, and turbomachinery components
• Examples of compressible flow include the flow through a supersonic wind tunnel and the exhaust of a rocket engine

Compressible flow phenomena in propulsion

Shock waves and expansion waves

• Shock waves are thin regions of abrupt changes in fluid properties (pressure, density, and velocity) that occur when a supersonic flow encounters an obstacle or a sudden change in flow direction
• Normal shock waves are perpendicular to the flow direction and cause a sudden decrease in velocity, increase in pressure, and increase in entropy
• Oblique shock waves form at an angle to the flow direction and can be attached to sharp-edged objects or generated by flow turning
• Expansion waves occur when a supersonic flow encounters a convex corner or a sudden expansion in flow area, causing a decrease in pressure and an increase in velocity
• Examples of shock waves and expansion waves in propulsion include supersonic inlet design and rocket engine exhaust plumes

Compressible flow effects on propulsion performance

• Shock-boundary layer interactions can lead to flow separation, increased drag, and reduced propulsion efficiency
• Shock-induced flow phenomena, such as shock-shock interactions and shock-vortex interactions, can affect the stability and performance of propulsion systems
• Compressible flow effects must be considered in the design of supersonic inlets, nozzles, and other propulsion components to optimize performance and avoid adverse flow phenomena
• Examples of compressible flow effects on propulsion performance include the design of scramjet engines and the analysis of supersonic aircraft aerodynamics