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]

In premixed combustion, an alternate diagram, known as the Borghi–Peters diagram, is also used to describe different regimes. This diagram is named after Roland Borghi (1985) and Norbert Peters (1986).^[5][6] The Borghi–Peters diagram uses ${\displaystyle l/\delta _{L}}$ as the x-axis and ${\displaystyle u'/S_{L}}$ as the y-axis, where ${\displaystyle \delta _{L}}$ and ${\displaystyle S_{L}}$ are the thickness and speed of the planar, laminar premixed flame. Since ${\displaystyle \delta _{L}Pr=\nu /S_{L}}$, where ${\displaystyle Pr}$ is the Prandtl number (set ${\displaystyle Pr=1}$), and ${\displaystyle t_{\mathrm {ch} }=\delta _{L}/S_{L}}$ in premixed flames, we have

${\displaystyle Re_{l}={\frac {u'}{S_{L}}}{\frac {l}{\delta _{L}}},\quad Da_{l}={\frac {l/\delta _{L}}{u'/S_{L}}},\quad \Rightarrow \quad {\frac {l}{\delta _{L}}}={\sqrt {Re_{l}Da_{l}}},\quad {\frac {u'}{S_{L}}}={\sqrt {\frac {Re_{l}}{Da_{l}}}}}$

The limitations of the Borghi–Peters diagram are that (1) it cannot be used for non-premixed combustion and (2) it is not suitable for practically relevant cases where both ${\displaystyle Re_{l}}$ and ${\displaystyle Da_{l}}$ are increased concurrently, such as increasing nozzle radius while maintaining constant nozzle exit velocity.^[7]