Gas dynamics

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The gas dynamics is a field of fluid mechanics and fluid mechanics and deals with compressible (density varying) flows apart.

It includes both

• external currents, e.g. B. bodies on aircraft or reentry ( space shuttle , landing capsules of spaceships ), as well
• internal flows through nozzles and diffusers

Mathematical description of the one-dimensional, isentropic gas flow

Laws and Assumptions

If the assumptions of the streamline theory are fulfilled, a flow can be described as one-dimensional. Due to the relatively low density of gases, the effect of gravity can usually be neglected. It is also assumed for the mathematical description that no heat is added to or removed from the gas and that no friction losses occur. The entropy is therefore constant. And the isentropic relationship applies:

${\displaystyle p\sim \rho ^{\kappa }}$

The law of conservation of energy can be formulated as follows and states that the sum of kinetic energy and enthalpy is constant along the current filament .

${\displaystyle {\frac {u^{2}}{2}}+c_{\text{p}}T=c_{\text{p}}T_{t}}$ is constant

The continuity equation applies , which expresses that no mass is lost. The mass flow along the stream filament is constant.

${\displaystyle {\dot {m}}=\rho \cdot u\cdot A}$ is constant

The properties of the gas can be described by the equation of state of the ideal gas .

${\displaystyle p=\rho \cdot R_{s}\cdot T\!\,}$

This means that four equations are available to uniquely describe the four variables (speed u , pressure p , temperature T , density ρ ). With a mathematical transformation, the variable state variables of the flow can be expressed as dimensionless relationships. Pressure, temperature and density are related to the rest parameters (index t). The quiescent variables describe the state that occurs if the flow were decelerated to a standstill without loss. In the case of a flow that starts from a large pressure vessel, the vessel pressure, temperature and density are the rest parameters (assuming there is no loss).

The speed cannot be related to the state of rest (division by zero), but can be represented as Mach number Ma and Laval number M * . The speed of sound c is used for this purpose.

${\displaystyle c={\sqrt {\kappa R_{s}T}}}$

${\displaystyle M\!a={\frac {u}{\sqrt {\kappa R_{s}T}}}}$

${\displaystyle M^{*}={\frac {u}{\sqrt {{\frac {2\kappa }{\kappa +1}}R_{s}T_{t}}}}}$

Dimensionless relationships

The specified dimensionless values ​​are similarity indicators and can be converted into one another as follows.

	Square of the mach number	Square of the Laval number	Temperature ratio	Pressure ratio	Density ratio
	${\displaystyle M\!a^{2}\!\,}$	${\displaystyle M^{*2}\!\,}$	${\displaystyle {\frac {T}{T_{t}}}}$	${\displaystyle {\frac {p}{p_{t}}}}$	${\displaystyle {\frac {\rho }{\rho _{t}}}}$
${\displaystyle M\!a^{2}=\!\,}$	${\displaystyle M\!a^{2}\!\,}$	${\displaystyle {\frac {M^{*2}}{1-{\frac {\kappa -1}{2}}(M^{*2}-1)}}}$	${\displaystyle {\frac {2}{\kappa -1}}\left({\frac {T_{t}}{T}}-1\right)}$	${\displaystyle {\frac {2}{\kappa -1}}\left[\left({\frac {p_{t}}{p}}\right)^{\frac {\kappa -1}{\kappa }}-1\right]}$	${\displaystyle {\frac {2}{\kappa -1}}\left[\left({\frac {\rho _{t}}{\rho }}\right)^{\kappa -1}-1\right]}$
${\displaystyle M^{*2}=\!\,}$	${\displaystyle {\frac {M\!a^{2}}{1+{\frac {\kappa -1}{\kappa +1}}(M\!a^{2}-1)}}}$	${\displaystyle M^{*2}\!\,}$	${\displaystyle {\frac {\kappa +1}{\kappa -1}}\left(1-{\frac {T}{T_{t}}}\right)}$	${\displaystyle {\frac {\kappa +1}{\kappa -1}}\left[1-\left({\frac {p}{p_{t}}}\right)^{\frac {\kappa -1}{\kappa }}\right]}$	${\displaystyle {\frac {\kappa +1}{\kappa -1}}\left[1-\left({\frac {\rho }{\rho _{t}}}\right)^{\kappa -1}\right]}$
${\displaystyle {\frac {T}{T_{t}}}=}$	${\displaystyle \left(1+{\frac {\kappa -1}{2}}{M\!a}^{2}\right)^{-1}}$	${\displaystyle 1-{\frac {\kappa -1}{\kappa +1}}M^{*2}}$	${\displaystyle {\frac {T}{T_{t}}}}$	${\displaystyle \left({\frac {p}{p_{t}}}\right)^{\frac {\kappa -1}{\kappa }}}$	${\displaystyle \left({\frac {\rho }{\rho _{t}}}\right)^{\kappa -1}}$
${\displaystyle {\frac {p}{p_{t}}}=}$	${\displaystyle \left(1+{\frac {\kappa -1}{2}}{M\!a}^{2}\right)^{\frac {-\kappa }{\kappa -1}}}$	${\displaystyle \left(1-{\frac {\kappa -1}{\kappa +1}}M^{*2}\right)^{\frac {\kappa }{\kappa -1}}}$	${\displaystyle \left({\frac {T}{T_{t}}}\right)^{\frac {\kappa }{\kappa -1}}}$	${\displaystyle {\frac {p}{p_{t}}}}$	${\displaystyle \left({\frac {\rho }{\rho _{t}}}\right)^{\kappa }}$
${\displaystyle {\frac {\rho }{\rho _{t}}}=}$	${\displaystyle \left(1+{\frac {\kappa -1}{2}}{M\!a}^{2}\right)^{\frac {-1}{\kappa -1}}}$	${\displaystyle \left(1-{\frac {\kappa -1}{\kappa +1}}M^{*2}\right)^{\frac {1}{\kappa -1}}}$	${\displaystyle \left({\frac {T}{T_{t}}}\right)^{\frac {1}{\kappa -1}}}$	${\displaystyle \left({\frac {p}{p_{t}}}\right)^{\frac {1}{\kappa }}}$	${\displaystyle {\frac {\rho }{\rho _{t}}}}$

Formula symbols used

${\displaystyle T\!\,}$	Temperature (always as absolute temperature in Kelvin)
${\displaystyle T_{t}\!\,}$	Resting temperature
${\displaystyle p\!\,}$	Pressure (always as absolute pressure compared to vacuum)
${\displaystyle p_{t}\!\,}$	Resting pressure
${\displaystyle \rho \!\,}$	density
${\displaystyle c_{\text{p}}\!\,}$	specific heat capacity
${\displaystyle R_{s}\!\,}$	specific gas constant
${\displaystyle \kappa \!\,}$	Isentropic exponent
${\displaystyle u\!\,}$	Flow velocity
${\displaystyle c\!\,}$	Speed ​​of sound
${\displaystyle {\dot {m}}\!\,}$	Mass flow
${\displaystyle A\!\,}$	cross-sectional area flowed through
${\displaystyle M\!a\,}$	Mach number
${\displaystyle M^{*}\!\,}$	Laval number

See also

• Speed ​​of sound
• Supersonic speed
• Sound barrier
• Machscher cone
• Shock wave
• Laval nozzle

literature

• Werner Albring : Applied Fluid Mechanics , Akademie Verlag Berlin, 1990, ISBN 3-05-500206-7 .