Laval number

Physical key figure
Surname	Laval number
Formula symbol	${\displaystyle M^{*}}$ ${\displaystyle {\mathit {La}}}$
dimension	dimensionless
definition	${\displaystyle M^{*}={\frac {u}{c^{*}}}}$
${\displaystyle u}$ Flow velocity ${\displaystyle c^{*}}$ critical speed of sound
Named after	Gustav de Laval
scope of application	compressible flows

The Laval-number  (after Gustav de Laval ) or Mach number is a similarity measure of gas dynamics . It results from the ratio of the local flow velocity to the critical speed of sound : ${\displaystyle {\mathit {La}}}$  ${\displaystyle M^{*}}$  ${\displaystyle u}$  ${\displaystyle c^{*}}$

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

It is

• The critical speed of sound is  the speed of sound that would result if the flow were to accelerate (or decelerate) to the speed of sound without loss of heat or friction ${\displaystyle c^{*}}$
• ${\displaystyle \kappa }$the isentropic exponent
• ${\displaystyle R_{\mathrm {s} }}$the specific gas constant
• ${\displaystyle T^{*}}$ the critical temperature (explanation see below)
• ${\displaystyle T_{\mathrm {t} }}$the resting temperature .

The speed of sound is temperature dependent and the temperature decreases in an accelerated gas flow. The speed of sound is therefore subject to changes along the flow path. The condition in which a gas flow flows at the speed of sound is called the critical point . This point is reached in the narrowest cross-section of a Laval nozzle (with a sufficiently large pressure gradient) . The critical temperature prevails there , which is in a fixed ratio to the resting temperature.

Individual evidence

1. ^ E. Truckenbrodt: Fluid Mechanics: Basics and Technical Applications . Springer-Verlag, 2013, ISBN 3-662-41599-2 , p. 270 ( limited preview in Google Book search).