💨Mathematical Fluid Dynamics Unit 7 – Compressible Flow in Fluid Dynamics

Key Concepts and Definitions

• Compressible flow involves fluids with density variations due to changes in pressure and temperature
• Mach number ($M$) represents the ratio of flow velocity to the local speed of sound and characterizes flow regimes (subsonic, transonic, supersonic, hypersonic)
• Stagnation properties (temperature, pressure, density) describe fluid properties when brought to rest isentropically
• Speed of sound ($a$) depends on fluid properties and determines the propagation of small disturbances in the fluid
  • Calculated using the formula $a = \sqrt{\gamma R T}$, where $\gamma$ is the specific heat ratio, $R$ is the gas constant, and $T$ is the absolute temperature
• Specific heat ratio ($\gamma$) is the ratio of specific heat at constant pressure ($c_p$) to specific heat at constant volume ($c_v$) and affects compressibility effects
• Compressibility effects become significant when $M > 0.3$, leading to density variations and non-linear behavior
• Ideal gas law ($pV = nRT$) relates pressure, volume, temperature, and gas properties for compressible fluids

Fundamental Equations

• Conservation of mass (continuity equation) ensures that mass is neither created nor destroyed in a system
  • For steady, one-dimensional flow: $\rho_1 A_1 v_1 = \rho_2 A_2 v_2$, where $\rho$ is density, $A$ is cross-sectional area, and $v$ is velocity
• Conservation of momentum (momentum equation) describes the balance of forces acting on a fluid element
  • Derived from Newton's second law: $\sum F = ma$
• Conservation of energy (energy equation) states that energy is conserved in a system, accounting for work done and heat transfer
  • For steady, adiabatic flow with no work: $h_1 + \frac{v_1^2}{2} = h_2 + \frac{v_2^2}{2}$, where $h$ is enthalpy
• Equation of state relates fluid properties (pressure, density, temperature) and depends on the fluid type (ideal gas, real gas)
  • For an ideal gas: $p = \rho R T$
• Isentropic flow relations describe the relationship between fluid properties in the absence of heat transfer and irreversibilities
  • Pressure-density relation: $\frac{p_2}{p_1} = \left(\frac{\rho_2}{\rho_1}\right)^\gamma$
  • Temperature-density relation: $\frac{T_2}{T_1} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma-1}$

Types of Compressible Flow

• Subsonic flow occurs when $M < 1$, characterized by smooth, continuous flow with small density variations
• Transonic flow occurs when $M \approx 1$, featuring a mix of subsonic and supersonic regions with shock waves and expansion waves
• Supersonic flow occurs when $M > 1$, characterized by thin shock waves, expansion waves, and significant density variations
  • Disturbances cannot propagate upstream in supersonic flow due to the flow velocity exceeding the speed of sound
• Hypersonic flow occurs at very high Mach numbers ($M > 5$), featuring strong shock waves, thin boundary layers, and chemical reactions
• Choked flow occurs when the flow reaches sonic conditions ($M = 1$) at a throat or constriction, limiting the mass flow rate
• Fanno flow describes adiabatic flow with friction in a constant-area duct, leading to a maximum entropy condition at the sonic point
• Rayleigh flow describes frictionless flow with heat transfer in a constant-area duct, resulting in a maximum entropy condition at the sonic point

Shock Waves and Expansion Waves

• Shock waves are thin regions of abrupt changes in fluid properties (pressure, density, temperature) that occur when a supersonic flow encounters an obstruction or a change in flow direction
  • Normal shock waves are perpendicular to the flow direction and cause a sudden decrease in velocity and increase in pressure, density, and temperature
  • Oblique shock waves are inclined to the flow direction and result from a change in flow direction (ramps, wedges, or corners)
• Expansion waves are regions of smooth, continuous changes in fluid properties that occur when a supersonic flow encounters a convex corner or a sudden expansion
  • Prandtl-Meyer expansion waves are isentropic and result in a decrease in pressure, density, and temperature, and an increase in velocity
• Shock wave relations describe the changes in fluid properties across a normal shock wave using the Rankine-Hugoniot equations
  • Pressure ratio: $\frac{p_2}{p_1} = \frac{2\gamma M_1^2 - (\gamma - 1)}{\gamma + 1}$
  • Density ratio: $\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2}{2 + (\gamma - 1)M_1^2}$
  • Temperature ratio: $\frac{T_2}{T_1} = \frac{[2\gamma M_1^2 - (\gamma - 1)][2 + (\gamma - 1)M_1^2]}{(\gamma + 1)^2 M_1^2}$
• Shock wave angle ($\beta$) depends on the upstream Mach number ($M_1$) and the deflection angle ($\theta$) for oblique shock waves
  • $\tan \theta = 2 \cot \beta \frac{M_1^2 \sin^2 \beta - 1}{M_1^2 (\gamma + \cos 2\beta) + 2}$

Isentropic Flow Analysis

• Isentropic flow assumes no heat transfer and no irreversibilities (friction, shocks) in the flow
• Isentropic flow relations connect fluid properties (pressure, density, temperature) at different points in the flow using the specific heat ratio ($\gamma$)
  • Pressure-density relation: $\frac{p_2}{p_1} = \left(\frac{\rho_2}{\rho_1}\right)^\gamma$
  • Temperature-density relation: $\frac{T_2}{T_1} = \left(\frac{\rho_2}{\rho_1}\right)^{\gamma-1}$
• Area-velocity relation (continuity equation) for isentropic flow: $\frac{A_2}{A_1} = \frac{v_1}{v_2} \left(\frac{\rho_1}{\rho_2}\right)$
• Critical properties (pressure, density, temperature) occur when the flow reaches sonic conditions ($M = 1$) and are denoted by an asterisk ($^*$)
  • Critical pressure ratio: $\frac{p^*}{p_0} = \left(\frac{2}{\gamma + 1}\right)^{\frac{\gamma}{\gamma - 1}}$
  • Critical density ratio: $\frac{\rho^*}{\rho_0} = \left(\frac{2}{\gamma + 1}\right)^{\frac{1}{\gamma - 1}}$
  • Critical temperature ratio: $\frac{T^*}{T_0} = \frac{2}{\gamma + 1}$
• Area-Mach number relation describes the variation of cross-sectional area with Mach number for isentropic flow
  • $\frac{A}{A^*} = \frac{1}{M} \left[\frac{2}{\gamma + 1} \left(1 + \frac{\gamma - 1}{2} M^2\right)\right]^{\frac{\gamma + 1}{2(\gamma - 1)}}$

Nozzles and Diffusers

• Nozzles are devices that accelerate a fluid from low velocity (high pressure) to high velocity (low pressure) by converting potential energy (pressure) into kinetic energy (velocity)
  • Converging nozzles accelerate subsonic flow and decelerate supersonic flow
  • Diverging nozzles accelerate supersonic flow and decelerate subsonic flow
  • Converging-diverging (CD) nozzles can accelerate a fluid from subsonic to supersonic velocities by achieving sonic conditions at the throat
• Diffusers are devices that decelerate a fluid from high velocity (low pressure) to low velocity (high pressure) by converting kinetic energy (velocity) into potential energy (pressure)
  • Subsonic diffusers have a gradually increasing cross-sectional area to prevent flow separation
  • Supersonic diffusers use a combination of shock waves and area changes to decelerate the flow efficiently
• Nozzle and diffuser performance is characterized by the pressure ratio, mass flow rate, and efficiency
  • Pressure ratio is the ratio of the outlet pressure to the inlet pressure
  • Mass flow rate depends on the inlet conditions and the throat area for choked flow
  • Efficiency measures the actual performance compared to the ideal (isentropic) case, accounting for losses due to friction, shocks, and flow separation

Practical Applications

• Jet engines (turbojets, turbofans) use compressible flow principles to generate thrust by accelerating a fluid through a nozzle
  • Inlet, compressor, combustor, turbine, and nozzle are the main components of a jet engine
  • Ramjets and scramjets are jet engines that rely on supersonic compression and combustion
• Rocket engines use converging-diverging nozzles to accelerate the exhaust gases to supersonic velocities and generate thrust
  • Solid-propellant rockets (boosters) and liquid-propellant rockets (main engines) are common types of rocket engines
• Supersonic wind tunnels use compressible flow principles to simulate high-speed flow conditions for testing aircraft, vehicles, and other objects
  • Converging-diverging nozzles, test sections, and diffusers are the main components of a supersonic wind tunnel
• Gas turbines (power generation, oil and gas) use compressible flow principles to compress and expand gases efficiently
  • Axial compressors, combustors, and turbines are the main components of a gas turbine
• High-speed vehicles (supersonic aircraft, hypersonic vehicles) encounter compressible flow effects and require careful design considerations
  • Swept wings, delta wings, and waverider configurations are used to mitigate shock wave effects and improve aerodynamic performance

Advanced Topics and Challenges

• Compressible boundary layers involve the interaction of viscous effects and compressibility, leading to phenomena such as shock-boundary layer interaction and flow separation
  • Laminar and turbulent compressible boundary layers have different characteristics and stability properties
  • Heat transfer and skin friction are affected by compressibility effects in boundary layers
• Real gas effects become significant at high pressures and temperatures, deviating from the ideal gas behavior
  • Van der Waals equation of state and other advanced equations of state are used to model real gas effects
  • Vibrational and chemical non-equilibrium effects can occur in high-temperature flows
• Numerical methods for compressible flow include finite difference, finite volume, and finite element methods
  • Shock-capturing schemes (Godunov, Roe, HLLC) are used to handle discontinuities and shock waves in the flow
  • Adaptive mesh refinement (AMR) and high-order methods improve the accuracy and efficiency of compressible flow simulations
• Multiphase compressible flow involves the interaction of different phases (gas, liquid, solid) and requires specialized modeling techniques
  • Eulerian-Lagrangian methods and two-fluid models are used to simulate multiphase compressible flows
  • Cavitation, condensation, and evaporation are examples of multiphase compressible flow phenomena
• Experimental techniques for compressible flow include schlieren imaging, shadowgraphy, and interferometry for visualizing density gradients and shock waves
  • Particle image velocimetry (PIV) and laser Doppler anemometry (LDA) are used to measure velocity fields in compressible flows
  • Pressure-sensitive paint (PSP) and temperature-sensitive paint (TSP) provide surface measurements in compressible flow experiments