Williams diagram

In combustion, Williams diagram refers to a classification diagram of different turbulent combustion regimes in a plane, having turbulent Reynolds number ${\displaystyle Re_{l}}$ as the x-axis and turbulent Damköhler number ${\displaystyle Da_{l}}$ as the y-axis.^[1] The diagram is named after Forman A. Williams (1985).^[2] The definition of the two non-dimensionaless numbers are^[3]

${\displaystyle Re_{l}={\frac {u'l}{\nu }},\quad Da_{l}={\frac {l/u'}{t_{\mathrm {ch} }}}}$

where ${\displaystyle u'}$ is the rms turbulent velocity flucturation, ${\displaystyle l}$ is the integral length scale, ${\displaystyle \nu }$ is the kinematic viscosity and ${\displaystyle t_{\mathrm {ch} }}$ is the chemical time scale. The Reynolds number ${\displaystyle Re_{\lambda }}$ based on the Taylor microscale ${\displaystyle \lambda =l/{\sqrt {Re_{l}}}}$ becomes ${\displaystyle Re_{\lambda }={\sqrt {Re_{l}}}}$. The Damköhler number based on the Kolmogorov time scale ${\displaystyle t_{\eta }={\sqrt {\nu l/u^{\prime 3}}}}$ is given by ${\displaystyle Da_{\eta }=Da_{l}/{\sqrt {Re_{l}}}}$. The Karlovitz number ${\displaystyle Ka=t_{\mathrm {ch} }/t_{\eta }}$ is defined by ${\displaystyle Ka={\sqrt {Re_{l}/Da_{l}}}}$.

The Williams diagram is universal in the sense that it is applicable to both premixed and non-premixed combustion. In supersonic combustion and detonations, the diagram becomes three-dimensional due to the addition of the Mach number ${\displaystyle Ma=u'/c}$ as the z-axis, where ${\displaystyle c}$ is the sound speed.^[4]