Comparison of Presumed PDF Models of Turbulent Flames

<table>Dependencies of the normalized mean heat release rate on the Reynolds-averaged combustion progress variable, calculated for <svg height="13.6125" id="M450" style="vertical-align:-2.34499pt" version="1.1" viewbox="0 0 46.737499 13.6125" width="46.737499" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.6125)">
<g transform="translate(72,-61.11)">
<text transform="matrix(1,0,0,-1,-71.95,63.5)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑔</tspan>
<tspan style="font-size: 12.50px; " x="9.6148071" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="21.655195" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="27.906696" y="0">.</tspan>
<tspan style="font-size: 12.50px; " x="31.032446" y="0">9</tspan>
</text>
</g>
</g>
</svg> and various equivalence ratios, specified in legends, by invoking the beta-function PDF. Thin and bold curves are associated with <svg height="21.825001" id="M451" style="vertical-align:-2.21957pt" version="1.1" viewbox="0 0 64.300003 21.825001" width="64.300003" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,21.825)">
<g transform="translate(72,-54.54)">
<text transform="matrix(1,0,0,-1,-71.95,56.81)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑃</tspan>
<tspan style="font-size: 12.50px; " x="7.4267821" y="0">(</tspan>
<tspan style="font-size: 12.50px; " x="11.590281" y="0">𝑐</tspan>
<tspan style="font-size: 12.50px; " x="16.779026" y="0">,</tspan>
</text>
<g>
<path d="m -49.96,65.21 5.81,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<text transform="matrix(1,0,0,-1,-49.96,56.81)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑐</tspan>
<tspan style="font-size: 12.50px; " x="5.8013921" y="0">,</tspan>
</text>
<g>
<path d="m -38.94,70.71 14.16,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<text transform="matrix(1,0,0,-1,-38.94,56.81)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝑐</tspan>
</text>
<text transform="matrix(1,0,0,-1,-33.12,60.24)">
<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
<tspan style="font-size: 8.75px; " x="3.47" y="-1.67">2</tspan>
</text>
<text transform="matrix(1,0,0,-1,-24.78,56.81)">
<tspan style="font-size: 12.50px; " x="0" y="0">)</tspan>
</text>
</g>
</g>
</svg> and <svg height="21.9125" id="M452" style="vertical-align:-2.29482pt" version="1.1" viewbox="0 0 95.612503 21.9125" width="95.612503" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,21.9125)">
<g transform="translate(72,-54.47)">
<text transform="matrix(1,0,0,-1,-70.8,59.34)">
<tspan style="font-size: 12.50px; " x="0" y="0"></tspan>
<tspan style="font-size: 12.50px; " x="-1.15" y="2.53">𝑃</tspan>
<tspan style="font-size: 12.50px; " x="8.1397285" y="2.53">(</tspan>
<tspan style="font-size: 12.50px; " x="12.303228" y="2.53">𝑐</tspan>
<tspan style="font-size: 12.50px; " x="17.491974" y="2.53">,</tspan>
<tspan style="font-size: 12.50px; " x="28.494614" y="2.53">̃</tspan>
<tspan style="font-size: 12.50px; " x="22.718227" y="2.53">𝑐</tspan>
<tspan style="font-size: 12.50px; " x="28.519619" y="2.53">,</tspan>
</text>
<g>
<path d="m -37.06,70.71 27.58,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<text transform="matrix(1,0,0,-1,-37.06,56.81)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜌</tspan>
<tspan style="font-size: 12.50px; " x="6.2765059" y="0">𝑐</tspan>
</text>
<text transform="matrix(1,0,0,-1,-24.97,60.24)">
<tspan style="font-size: 8.75px; " x="10.62" y="-1.67">2</tspan>
<tspan style="font-size: 8.75px; " x="0" y="0"></tspan>
<tspan style="font-size: 8.75px; " x="2.9756801" y="0"></tspan>
</text>
<text transform="matrix(1,0,0,-1,-9.48,56.81)">
<tspan style="font-size: 12.50px; " x="0" y="0">/</tspan>
</text>
<g>
<path d="m -6,65.21 6.27,0" style="fill:none;stroke:#000000;stroke-width:0.82499999;stroke-linecap:butt;stroke-linejoin:miter;stroke-miterlimit:10;stroke-opacity:1;stroke-dasharray:none"></path>
</g>
<text transform="matrix(1,0,0,-1,-6,56.81)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝜌</tspan>
<tspan style="font-size: 12.50px; " x="6.2765059" y="0">)</tspan>
</text>
</g>
</g>
</svg>, respectively.</table>

Journal of Combustion

Comparison of Presumed PDF Models of Turbulent Flames

Figure 12