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]

Borghi–Peters diagram

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]