9.5 Compressibility effects

Compressibility effects play a crucial role in fluid dynamics, especially for high-speed flows. As fluid velocity increases, density changes become significant, leading to phenomena like shock waves and choked flow in aerospace applications.

The Mach number, the ratio of flow velocity to local sound speed, is key in characterizing compressible flows. It determines flow regimes from subsonic to hypersonic, with compressibility effects becoming more pronounced at higher Mach numbers.

Compressibility in fluid dynamics

• Compressibility refers to the ability of a fluid to change its density in response to changes in pressure
• Understanding compressibility effects is crucial for analyzing high-speed flows encountered in aerospace applications, such as aircraft and rocket propulsion systems
• Compressibility introduces additional complexity to fluid dynamics equations and can lead to phenomena like shock waves and choked flow

Mach number

Definition of Mach number

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• Mach number ($M$) is a dimensionless quantity that represents the ratio of the flow velocity to the local speed of sound
• Mathematically expressed as $M = \frac{v}{a}$, where $v$ is the flow velocity and $a$ is the local speed of sound
• Mach number is a key parameter in characterizing compressible flows and determining the flow regime

Mach number regimes

• Subsonic flow: $M < 1$, flow velocity is less than the local speed of sound
• Transonic flow: $M \approx 1$, flow velocity is near the local speed of sound and can exhibit both subsonic and supersonic regions
• Supersonic flow: $M > 1$, flow velocity is greater than the local speed of sound and is characterized by the presence of shock waves
• Hypersonic flow: $M \gg 1$ (typically $M > 5$), flow velocity is much greater than the local speed of sound and exhibits strong compressibility effects and high-temperature effects

Mach number vs compressibility

• As Mach number increases, compressibility effects become more significant
• Compressibility effects are negligible for low Mach numbers ($M < 0.3$) and the flow can be treated as incompressible
• For higher Mach numbers, compressibility effects must be considered, and the flow is treated as compressible
• Compressibility effects lead to changes in density, pressure, and temperature, which affect the flow properties and behavior

Speed of sound

Definition of speed of sound

• Speed of sound ($a$) is the speed at which small pressure disturbances propagate through a fluid medium
• It represents the maximum speed at which information can travel in a fluid and is a function of the fluid properties
• For an ideal gas, the speed of sound is given by $a = \sqrt{\gamma R T}$, where $\gamma$ is the specific heat ratio, $R$ is the specific gas constant, and $T$ is the absolute temperature

Factors affecting speed of sound

• Speed of sound depends on the fluid properties, primarily the compressibility and density
• In gases, speed of sound is affected by temperature, with higher temperatures resulting in higher speed of sound
• Composition of the gas also influences the speed of sound, as different gases have different specific heat ratios and gas constants
• In liquids, speed of sound is affected by the bulk modulus and density of the liquid

Speed of sound in gases vs liquids

• Speed of sound is generally higher in liquids compared to gases due to the higher density and lower compressibility of liquids
• In gases, speed of sound is typically in the range of a few hundred meters per second (air at room temperature: ~343 m/s)
• In liquids, speed of sound is typically in the range of a thousand meters per second (water at room temperature: ~1,480 m/s)
• The higher speed of sound in liquids has implications for the design of hydraulic systems and underwater acoustics

Compressible vs incompressible flow

Characteristics of compressible flow

• Compressible flow is characterized by significant changes in density, pressure, and temperature as the flow velocity changes
• Density variations are a key feature of compressible flow, and the flow properties are strongly coupled with the density changes
• Compressible flow can exhibit phenomena like shock waves, expansion waves, and choked flow
• Examples of compressible flow include high-speed flow around aircraft, flow in rocket nozzles, and flow in gas pipelines

Characteristics of incompressible flow

• Incompressible flow assumes that the density of the fluid remains constant throughout the flow field
• Pressure changes in incompressible flow do not cause significant density changes, and the flow properties are decoupled from density variations
• Incompressible flow is generally easier to analyze and solve compared to compressible flow, as the governing equations are simplified
• Examples of incompressible flow include low-speed flow of liquids, such as water in pipes and channels, and low-speed airflow around vehicles

Transition from incompressible to compressible flow

• The transition from incompressible to compressible flow occurs as the Mach number increases and compressibility effects become significant
• For Mach numbers less than 0.3, the flow can be treated as incompressible with negligible compressibility effects
• As the Mach number approaches 1, the flow enters the transonic regime, where both subsonic and supersonic regions can coexist
• For Mach numbers greater than 1, the flow is fully compressible, and compressibility effects dominate the flow behavior
• The transition from incompressible to compressible flow is gradual, and the extent of compressibility effects depends on the specific flow conditions and geometry

Density changes in compressible flow

Density variations with Mach number

• In compressible flow, density varies significantly with changes in Mach number
• As Mach number increases, the density of the fluid decreases due to the increased compressibility effects
• The relationship between density and Mach number is described by the isentropic flow relations, which assume adiabatic and reversible flow
• For isentropic flow, the density ratio ($\frac{\rho}{\rho_0}$) is given by $\left(1 + \frac{\gamma - 1}{2}M^2\right)^{-\frac{1}{\gamma - 1}}$, where $\rho_0$ is the stagnation density

Density ratio across shock waves

• Shock waves are thin regions of abrupt changes in flow properties, including density, pressure, and temperature
• Across a normal shock wave, the density increases abruptly, leading to a density ratio greater than 1
• The density ratio across a normal shock wave is given by $\frac{\rho_2}{\rho_1} = \frac{(\gamma + 1)M_1^2}{(\gamma - 1)M_1^2 + 2}$, where $\rho_1$ and $\rho_2$ are the densities upstream and downstream of the shock, respectively, and $M_1$ is the upstream Mach number
• The density increase across a shock wave is accompanied by a decrease in flow velocity and an increase in pressure and temperature

Density effects on flow properties

• Density variations in compressible flow have significant effects on other flow properties, such as pressure, temperature, and velocity
• Changes in density affect the mass flow rate through a given cross-section, as the mass flow rate is the product of density, velocity, and area
• Density variations also influence the dynamic pressure ($\frac{1}{2}\rho v^2$), which is a measure of the kinetic energy of the flow
• The relationship between density, pressure, and temperature is described by the equation of state, such as the ideal gas law ($p = \rho R T$) for perfect gases

Pressure changes in compressible flow

Pressure variations with Mach number

• Pressure in compressible flow varies with changes in Mach number, similar to density variations
• As Mach number increases, the pressure decreases due to the increased compressibility effects
• The relationship between pressure and Mach number for isentropic flow is given by the isentropic pressure ratio ($\frac{p}{p_0}$), which is expressed as $\left(1 + \frac{\gamma - 1}{2}M^2\right)^{-\frac{\gamma}{\gamma - 1}}$, where $p_0$ is the stagnation pressure

Pressure ratio across shock waves

• Across a normal shock wave, the pressure increases abruptly, leading to a pressure ratio greater than 1
• The pressure ratio across a normal shock wave is given by $\frac{p_2}{p_1} = \frac{2\gamma M_1^2 - (\gamma - 1)}{\gamma + 1}$, where $p_1$ and $p_2$ are the pressures upstream and downstream of the shock, respectively, and $M_1$ is the upstream Mach number
• The pressure increase across a shock wave is accompanied by a decrease in flow velocity and an increase in density and temperature

Pressure effects on flow properties

• Pressure variations in compressible flow affect other flow properties, such as density, temperature, and velocity
• Changes in pressure are related to changes in density through the equation of state, such as the ideal gas law ($p = \rho R T$) for perfect gases
• Pressure gradients in compressible flow can lead to the acceleration or deceleration of the flow, depending on the direction of the gradient
• The relationship between pressure, density, and velocity is described by the momentum equation, which takes into account the effects of pressure forces on the flow

Temperature changes in compressible flow

Temperature variations with Mach number

• Temperature in compressible flow varies with changes in Mach number, similar to density and pressure variations
• As Mach number increases, the temperature increases due to the conversion of kinetic energy into thermal energy through compressibility effects
• The relationship between temperature and Mach number for isentropic flow is given by the isentropic temperature ratio ($\frac{T}{T_0}$), which is expressed as $\left(1 + \frac{\gamma - 1}{2}M^2\right)^{-1}$, where $T_0$ is the stagnation temperature

Temperature ratio across shock waves

• Across a normal shock wave, the temperature increases abruptly, leading to a temperature ratio greater than 1
• The temperature ratio across a normal shock wave is given by $\frac{T_2}{T_1} = \frac{[2 + (\gamma - 1)M_1^2][2\gamma M_1^2 - (\gamma - 1)]}{(\gamma + 1)^2 M_1^2}$, where $T_1$ and $T_2$ are the temperatures upstream and downstream of the shock, respectively, and $M_1$ is the upstream Mach number
• The temperature increase across a shock wave is accompanied by a decrease in flow velocity and an increase in density and pressure

Temperature effects on flow properties

• Temperature variations in compressible flow affect other flow properties, such as density, pressure, and velocity
• Changes in temperature are related to changes in density and pressure through the equation of state, such as the ideal gas law ($p = \rho R T$) for perfect gases
• Temperature variations can lead to changes in the speed of sound, as the speed of sound is a function of temperature ($a = \sqrt{\gamma R T}$)
• High temperatures in compressible flow can also lead to real gas effects, such as dissociation and ionization, which can further influence the flow properties

Compressible flow equations

Continuity equation for compressible flow

• The continuity equation for compressible flow represents the conservation of mass in a flow field
• In differential form, the continuity equation is given by $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0$, where $\rho$ is the density, $t$ is time, and $\mathbf{v}$ is the velocity vector
• For steady, one-dimensional flow, the continuity equation simplifies to $\rho_1 v_1 A_1 = \rho_2 v_2 A_2$, where subscripts 1 and 2 denote two different locations in the flow, and $A$ is the cross-sectional area

Momentum equation for compressible flow

• The momentum equation for compressible flow represents the conservation of momentum in a flow field
• In differential form, the momentum equation is given by $\rho \frac{D\mathbf{v}}{Dt} = -\nabla p + \nabla \cdot \mathbf{\tau} + \rho \mathbf{f}$, where $\frac{D}{Dt}$ is the material derivative, $p$ is the pressure, $\mathbf{\tau}$ is the viscous stress tensor, and $\mathbf{f}$ represents body forces
• For inviscid, steady, one-dimensional flow, the momentum equation simplifies to $\rho v \frac{dv}{dx} = -\frac{dp}{dx}$, relating changes in velocity to changes in pressure

Energy equation for compressible flow

• The energy equation for compressible flow represents the conservation of energy in a flow field
• In differential form, the energy equation is given by $\rho \frac{D}{Dt}\left(e + \frac{v^2}{2}\right) = -\nabla \cdot (p\mathbf{v}) + \nabla \cdot (k\nabla T) + \Phi$, where $e$ is the internal energy, $k$ is the thermal conductivity, and $\Phi$ represents the dissipation function
• For steady, adiabatic, inviscid flow, the energy equation simplifies to the total enthalpy ($h_0 = h + \frac{v^2}{2}$) being constant along a streamline, where $h$ is the static enthalpy
• The compressible flow equations, along with the equation of state and appropriate boundary conditions, form a complete set of equations for describing compressible fluid motion

Compressibility effects on aerodynamics

Compressibility effects on lift and drag

• Compressibility effects can significantly influence the lift and drag characteristics of aerodynamic bodies, such as airfoils and wings
• As Mach number increases, the lift coefficient typically increases up to a certain point, known as the critical Mach number, beyond which the lift coefficient starts to decrease
• The drag coefficient also increases with increasing Mach number, particularly in the transonic regime, where wave drag becomes significant due to the formation of shock waves
• Compressibility effects can lead to the formation of shock waves on the surface of aerodynamic bodies, which can cause flow separation and a sudden increase in drag (known as the drag divergence Mach number)

Compressibility effects on flow patterns

• Compressibility effects can alter the flow patterns around aerodynamic bodies compared to incompressible flow
• At high Mach numbers, shock waves can form on the surface of aerodynamic bodies, leading to abrupt changes in flow properties and potentially causing flow separation
• Compressibility effects can also lead to the formation of expansion waves, which are regions of gradual changes in flow properties
• The interaction between shock waves and boundary layers can lead to complex flow phenomena, such as shock-induced separation and shock-boundary layer interaction

Compressibility effects on aircraft design

• Compressibility effects play a crucial role in the design of high-speed aircraft, such as supersonic and hypersonic vehicles
• Aircraft designers must consider the effects of compressibility on the aerodynamic performance, structural integrity, and stability of the vehicle
• Swept wings and thin airfoil sections are often used in high-speed aircraft to delay the onset of compressibility effects and reduce wave drag
• Area ruling, which involves shaping the aircraft's cross-sectional area distribution to minimize wave drag, is another design technique used to mitigate compressibility effects
• Supersonic and hypersonic aircraft may also employ specialized propulsion systems, such as ramjets and scramjets, which are designed to operate efficiently in high-speed, compressible flow conditions

Compressibility effects in propulsion systems

Compressibility effects in jet engines

• Compressibility effects are significant in jet engines, particularly in the compressor and turbine stages, where the flow velocities are high
• In the compressor stage, compressibility effects can limit the pressure ratio that can be achieved in each stage, requiring multiple stages to reach the desired overall pressure ratio
• In the turbine stage, compressibility effects can lead to the formation of shock waves and flow separation, reducing the efficiency of the turbine and potentially causing structural damage
• Jet engine designers must carefully consider the effects of compressibility when selecting the appropriate compressor and turbine designs, as well as the overall engine cycle and operating conditions

Compressibility effects in rocket nozzles

• Compressibility effects are crucial in the design and operation of rocket nozzles, which are used to accelerate the high-temperature, high-pressure exhaust gases to supersonic velocities
• The flow in a rocket nozzle is highly compressible, and the nozzle geometry is designed to achieve efficient expansion of the exhaust gases
• The converging-diverging (De Laval) nozzle is commonly used in rocket propulsion systems to accelerate the flow from subsonic to supersonic velocities
• Compressibility effects in rocket nozzles can lead to the formation of shock waves, which can reduce the nozzle efficiency and cause flow separation
• Rocket nozzle designers must optimize the nozzle geometry, including the throat area and exit area ratio, to maximize the thrust and specific impulse while minimizing the effects of compressibility

Compressibility effects on engine performance

• Compressibility effects can have a significant impact on the performance of propulsion systems, such as jet engines and rocket