Glossary

3.4 Compressible and incompressible flows

4 min read•august 16, 2024

Compressible and incompressible flows are key concepts in fluid dynamics. They describe how fluids behave under different conditions, affecting and flow characteristics. Understanding these flow types is crucial for analyzing everything from everyday liquids to high-speed gases in aerospace applications.

The distinction between compressible and incompressible flows impacts equations, phenomena, and analysis methods in fluid dynamics. Compressible flows, common in high-speed gas dynamics, exhibit unique behaviors like . Incompressible flows, typical in low-speed situations, allow for simplified calculations in many engineering problems.

Compressible vs Incompressible Flows

Defining Characteristics and Applications

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  • Compressible flows involve significant changes in fluid density due to pressure variations
  • Incompressible flows maintain relatively constant density
  • Speed of sound in a fluid determines flow compressibility affects pressure disturbance propagation
  • assumptions valid for liquids and gases at low Mach numbers (M < 0.3)
  • effects significant in high-speed gas flows (aerodynamics and gas dynamics)
  • Equation of state for an ideal gas (p=ρRTp = \rho RT) essential in analyzing compressible flows
    • Relates pressure, density, and temperature

Phenomena and Behavior

  • Compressible flows exhibit unique phenomena
    • Shock waves form when flow exceeds speed of sound
    • Expansion fans occur in supersonic flow around corners
  • Incompressible flows lack these phenomena
  • Fluid behavior changes drastically as flow approaches speed of sound
  • Compressibility effects become pronounced at higher Mach numbers
  • Density variations lead to complex interactions between fluid motion and thermodynamic properties

Fluid Behavior Under Pressure and Density

Fluid Compressibility and Elasticity

  • Bulk modulus of elasticity (K) quantifies fluid's resistance to compression
    • Defined as ratio of pressure change to relative volume change
    • K=−VΔpΔVK = -V\frac{\Delta p}{\Delta V}
  • For gases, pressure-density relationship described by polytropic process equation
    • pρn=constant\frac{p}{\rho^n} = constant
    • n represents polytropic exponent
  • Important special cases of polytropic process
    • Isothermal processes (n = 1)
    • Adiabatic processes (n = γ, where γ is specific heat ratio)
  • Speed of sound in fluid (a) related to compressibility and density
    • For isentropic processes: a=(∂p∂ρ)sa = \sqrt{\left(\frac{\partial p}{\partial \rho}\right)_s}

Pressure Effects and Stagnation Properties

  • Pressure waves in compressible fluids lead to shock wave formation when local flow velocity exceeds speed of sound
  • Stagnation properties crucial in analyzing compressible flows
    • Represent fluid state if brought to rest isentropically
    • Include stagnation pressure, temperature, and density
  • Pressure changes in compressible flows affect temperature and density simultaneously
  • Rapid compression or expansion of gases leads to temperature changes (adiabatic heating/cooling)

Continuity, Momentum, and Energy Equations for Compressible Flows

Fundamental Equations

  • for compressible flows includes time rate of change of density
    • ∂ρ∂t+∇⋅(ρV)=0\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho\mathbf{V}) = 0
    • V\mathbf{V} represents velocity vector
  • Momentum equation incorporates variable density
    • ρ(∂V∂t+V⋅∇V)=−∇p+ρg+∇⋅τ\rho\left(\frac{\partial \mathbf{V}}{\partial t} + \mathbf{V} \cdot \nabla\mathbf{V}\right) = -\nabla p + \rho\mathbf{g} + \nabla \cdot \boldsymbol{\tau}
    • τ\boldsymbol{\tau} represents viscous stress tensor
  • Energy equation includes internal energy
    • ρ(∂e∂t+V⋅∇e)=−p∇⋅V+Φ+∇⋅(k∇T)\rho\left(\frac{\partial e}{\partial t} + \mathbf{V} \cdot \nabla e\right) = -p\nabla \cdot \mathbf{V} + \Phi + \nabla \cdot (k\nabla T)
    • e represents specific internal energy
    • Φ\Phi represents viscous dissipation
    • k represents thermal conductivity

Simplified Equations and Concepts

  • Steady, one-dimensional compressible flow equations simplify to quasi-one-dimensional flow equations
  • Total enthalpy (h0=h+V22h_0 = h + \frac{V^2}{2}) remains constant along streamline in adiabatic flows
  • Isentropic flow relations provide useful equations for ideal conditions
    • Relate pressure, density, and temperature to
    • Example: p0p=(1+γ−12M2)γγ−1\frac{p_0}{p} = \left(1 + \frac{\gamma-1}{2}M^2\right)^{\frac{\gamma}{\gamma-1}}
  • Conservation of mass, momentum, and energy principles applied to control volumes in compressible flows

Mach Number in Compressible Flow Regimes

Mach Number Definition and Significance

  • Mach number (M) defined as ratio of flow velocity to local speed of sound
    • M=VaM = \frac{V}{a}
  • Primary parameter for characterizing compressible flows
  • Determines nature of flow and governing equations
  • Critical in predicting formation of shock waves and other compressible flow phenomena

Flow Regimes and Characteristics

  • Subsonic flow (M < 1)
    • Smooth property variations
    • Elliptic governing equations allow upstream influence of disturbances
  • Transonic flow (0.8 < M < 1.2)
    • Mixed subsonic and supersonic regions
    • Often features normal shock waves
    • Requires special numerical treatment
  • Supersonic flow (M > 1)
    • Hyperbolic governing equations
    • Formation of oblique shock waves
    • Absence of upstream influence
  • Hypersonic flow (typically M > 5)
    • Significant high-temperature effects
    • Strong shock waves
    • Potential chemical reactions in fluid
  • Critical Mach number (M*)
    • Freestream Mach number at which sonic flow (M = 1) first appears locally on body
    • Marks onset of transonic effects
  • Area-Mach number relations in quasi-one-dimensional flow
    • Describe how changes in flow area affect Mach number
    • Different behaviors for subsonic and supersonic flows
    • Example: dAA=(M2−1)dVV\frac{dA}{A} = (M^2 - 1)\frac{dV}{V}

Key Terms to Review (17)

Compressible flow: Compressible flow refers to the behavior of fluid dynamics when the fluid's density changes significantly due to pressure or temperature variations. This is particularly important in high-speed flows, such as those encountered in aerodynamics, where the speed of the fluid approaches or exceeds the speed of sound. Understanding compressible flow is crucial for analyzing phenomena like shock waves and expansion fans that occur in these scenarios.
Continuity Equation: The continuity equation is a mathematical expression that describes the conservation of mass in a fluid flow system. It states that the mass flow rate must remain constant from one cross-section of a flow to another, reflecting that mass cannot be created or destroyed within a closed system. This principle is crucial in analyzing different flow regimes, including compressible and incompressible flows, as well as in understanding how mass behaves in magnetohydrodynamic systems involving plasma and magnetic fields.
Density: Density is defined as the mass of a substance per unit volume, typically expressed in units such as kilograms per cubic meter (kg/m³). It is a fundamental property that influences how fluids behave under various conditions, impacting their flow characteristics in compressible and incompressible states, the behavior of shock waves, and the resultant structures formed during such high-energy events.
Electromagnetic propulsion: Electromagnetic propulsion is a method of generating thrust by using electromagnetic fields to accelerate charged particles or conductive fluids. This technique relies on the interaction between magnetic fields and electric currents, allowing for movement without the need for traditional propellants. It can be applied in various systems, influencing both compressible and incompressible flow regimes, particularly in the context of plasma dynamics and magnetohydrodynamics.
Free-slip condition: The free-slip condition refers to a boundary condition in fluid dynamics where there is no shear stress acting on the surface of a fluid. This means that the fluid can slide freely along the boundary without any resistance, allowing for tangential motion while maintaining normal pressure. This condition is especially relevant when examining flows in magnetohydrodynamics, where the behavior of both fluids and magnetic fields must be considered.
Incompressible Flow: Incompressible flow refers to a fluid flow regime where the density of the fluid remains constant throughout the motion. This assumption simplifies the analysis and equations governing fluid dynamics, particularly in low-speed flows where changes in pressure do not significantly affect density. This concept plays a crucial role in understanding various fluid behaviors and is integral to the study of both compressible and incompressible flows, as well as numerical methods used to solve flow problems.
Irrotational Flow: Irrotational flow refers to a fluid motion where there is no rotation or angular momentum at any point in the flow field, meaning that the vorticity is zero throughout the entire flow. This concept is crucial in fluid dynamics as it simplifies the analysis of fluid motion, particularly when differentiating between compressible and incompressible flows. In irrotational flow, the fluid particles move in parallel layers without any swirling or eddies, allowing for a clearer understanding of how pressure and velocity behave within the fluid.
Lorentz force: The Lorentz force is the force experienced by a charged particle moving through an electromagnetic field, defined mathematically as the sum of electric and magnetic forces acting on it. This fundamental concept is crucial for understanding how charged particles interact with magnetic fields and how this interaction leads to various phenomena in magnetohydrodynamics, from instabilities to energy generation.
Mach Number: The Mach number is a dimensionless quantity that represents the ratio of the speed of an object to the speed of sound in the medium through which it is moving. This concept is crucial in understanding compressible flow, where variations in pressure and density are significant, as well as the behavior of shock waves in magnetohydrodynamics (MHD). It serves as an indicator for distinguishing between subsonic and supersonic flows, with implications for shock dynamics and astrophysical phenomena.
Magnetic Reynolds Number: The Magnetic Reynolds Number (M) is a dimensionless quantity that measures the relative importance of advection of magnetic fields to magnetic diffusion in a conducting fluid. It is defined as the ratio of the inertial forces to the magnetic diffusion forces, indicating whether magnetic fields are frozen into the fluid or can diffuse through it.
Navier-Stokes Equations: The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of viscous fluid substances. These equations express the conservation of momentum and mass for fluid flow, allowing us to understand how fluids behave under various conditions, including their response to forces like pressure and viscosity.
No-Slip Condition: The no-slip condition refers to the boundary condition in fluid dynamics where the fluid velocity at a solid boundary is equal to the velocity of that boundary itself. This condition is crucial in understanding how fluids interact with solid surfaces and plays a significant role in various fluid flow scenarios, including flows in magnetic fields, where the interaction between the fluid and magnetic forces must also be considered.
Plasma confinement: Plasma confinement refers to the methods and techniques used to contain plasma, a hot ionized gas composed of charged particles, in a controlled environment to facilitate processes such as nuclear fusion. Effective confinement is crucial for maintaining the stability and energy of the plasma, ensuring that it can achieve the necessary conditions for fusion reactions to occur without escaping into the surrounding environment.
Pressure: Pressure is defined as the force exerted per unit area on a surface, typically measured in pascals (Pa). It plays a critical role in fluid dynamics, influencing how fluids behave under various conditions, whether they are compressible or incompressible. The understanding of pressure is essential in analyzing fluid flow, conservation laws, and phenomena like shock waves, where abrupt changes can lead to jumps in pressure across discontinuities.
Shock waves: Shock waves are abrupt disturbances that move through a medium, creating a sharp change in pressure, temperature, and density. They occur when an object travels faster than the speed of sound in that medium, leading to a non-linear effect where the flow becomes compressible. These waves are crucial in understanding compressible flows, as they dictate the behavior of fluids when subjected to high speeds.
Steady flow: Steady flow refers to a fluid motion where the velocity of the fluid at any given point does not change over time. This concept is crucial when analyzing both compressible and incompressible flows, as it simplifies the equations governing fluid motion and allows for consistent pressure and velocity profiles throughout the flow field.
Velocity: Velocity is a vector quantity that describes the rate at which an object changes its position, encompassing both speed and direction. Understanding velocity is crucial for analyzing how fluids move under various conditions, such as when they are compressible or incompressible, and in relation to the forces acting on them. The concept of velocity also plays a vital role in the conservation laws that govern fluid behavior, as well as in phenomena like shock waves and dissipation mechanisms during fluid interactions.
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