Direct Numerical Simulation of Particle-Laden Swirling Flows on Turbulence Modulation

<table>The azimuthal fluctuation <svg height="17.9125" id="M183" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 16.387501 17.9125" width="16.387501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>, Reynolds stress <svg height="17.9125" id="M184" style="vertical-align:-0.11285pt" version="1.1" viewbox="0 0 29.6875 17.9125" width="29.6875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.017,-0,0,-.017,13.362,17.65)"><path d="M689 332q0 -81 -40 -161.5t-103.5 -131.5t-127.5 -51q-32 0 -58.5 16t-40.5 46q-70 -62 -139 -62q-45 0 -77.5 30t-32.5 86q0 8 7 52q11 77 45 187q6 19 6 34q0 6 -7 6q-24 0 -78 -64l-20 23q32 48 73 77t78 29q32 0 32 -43q0 -30 -12 -73q-27 -95 -38 -152
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</svg>, and the second invariant <svg height="15.25" id="M185" style="vertical-align:-3.13504pt" version="1.1" viewbox="0 0 13.8875 15.25" width="13.8875" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><path d="M414 650l-6 -28q-63 -4 -78.5 -17.5t-27.5 -71.5l-77 -416q-11 -58 0.5 -71.5t77.5 -17.5l-5 -28h-275l8 28q63 4 79.5 18.5t27.5 70.5l80 416q11 57 -2 71t-81 18l6 28h273z" id="x1D43C"></path></g>
<g transform="matrix(.012,-0,0,-.012,7.487,15.187)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
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</svg>.</table>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Direct Numerical Simulation of Particle-Laden Swirling Flows on Turbulence Modulation 