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Dynamic Pressure in Compressible Flow

1930
  • Ernst Mach
  • Ludwig Prandtl
  • Theodore von Kármán
Aerospace engineers analyzing dynamic pressure data in a high-tech laboratory setting.

In compressible flows, particularly at high speeds, dynamic pressure is related to Mach number (\(M\)) and static pressure (\(p\)). For an ideal gas, the relationship is given by \(q = \frac{1}{2} \gamma p M^2\), where \(\gamma\) is the ratio of specific heats. This formulation is crucial for supersonic and hypersonic aerodynamics, where fluid density changes significantly.

When a fluid’s speed approaches a significant fraction of the speed of sound, the assumption of constant density (incompressibility) breaks down. Changes in pressure cause significant changes in density, and thermodynamic effects become important. This is the realm of compressible flow. The simple formula \(q = \frac{1}{2} \rho u^2\) is still used as a formal definition, but its relationship to pressure changes is more complex. The formula \(q = \frac{1}{2} \gamma p M^2\) provides a direct link between dynamic pressure and the key parameters of compressible flow: static pressure (\(p\)), the ratio of specific heats (\(\gamma\), which is a property of the gas, approximately 1.4 for air), and the Mach number (\(M = u/a\), where \(a\) is the local speed of sound).

This equation is derived from the definition of Mach number and the ideal gas equation of state. It is fundamental in high-speed aerodynamics. For instance, the pressure measured at a stagnation point (\(p_0\)) in supersonic flow is not given by the simple Bernoulli equation. Instead, it is related to the static pressure by the isentropic flow relations or, if a shock wave is present, by the Rankine-Hugoniot relations. In these calculations, the term \(\frac{1}{2} \gamma p M^2\) frequently appears, representing the kinetic energy component of the flow in a thermodynamically consistent way. This is crucial for accurately predicting the extreme pressures and temperatures experienced by supersonic aircraft, re-entry capsules, and meteorites entering the atmosphere. The concept is also sometimes referred to as “impact pressure” in this context, emphasizing the pressure rise due to the fluid’s momentum being brought to rest.

UNESCO Nomenclature: 3301
– Aeronautical engineering and technology

Type

Abstract System

Disruption

Substantial

Usage

Widespread Use

Precursors

  • Ideal gas law
  • Principles of thermodynamics
  • Bernoulli’s principle for incompressible flow
  • Concept of the speed of sound and Mach number
  • Euler equations for fluid dynamics

Applications

  • design of supersonic and hypersonic aircraft
  • re-entry vehicle thermal protection system design
  • rocket nozzle design and performance analysis
  • scramjet and ramjet engine development
  • high-speed wind tunnel testing
  • modeling of stellar winds and astrophysical jets

Patents:

NA

Potential Innovations Ideas

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Related to: compressible flow, supersonic, hypersonic, mach number, dynamic pressure, gas dynamics, specific heat ratio, shock wave, aerodynamics, impact pressure.

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Historical Context

(if date is unknown or not relevant, e.g. "fluid mechanics", a rounded estimation of its notable emergence is provided)

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