Drag coefficient of a rough grain submerged in low-Reynolds-number flow

Controversy exists regarding whether grain morphology reduces or enhances the drag of a single grain in creeping flows. Further complication occurs when orientation dependence of aspherical grains comes into play. To bridge this gap, this study investigates numerically the drag on fractally rough grains depicted by spherical harmonics. Rough shapes could induce drag reduction, indicated by a lower mean value of drag coefficients upper C Subscript upper D <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:math> $C_{\!D}$ at various rotation angles, compared with that of the corresponding smooth sphere. Moreover, the derived power law between upper C Subscript upper D <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> </mml:math> $C_{\!D}$ and the projected area upper A Subscript p <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:math> $A_{\!p}$ perpendicular to the flow direction, expressed as upper C Subscript upper D Baseline proportional to upper A Subscript p Baseline Superscript negative 0.8 <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content"> <mml:msub> <mml:mi>C</mml:mi> <mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:mi>D</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∝</mml:mo> <mml:msup> <mml:mrow> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mspace width="negativethinmathspace"/> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.8</mml:mn> </mml:mrow> </mml:msup> </mml:math> $C_{\!D}\propto {A_{\!p}}^{-0.8}$ for spheroidal and triaxial-ellipsoidal grains, remains valid for irregular shapes. Such a rotational dependence helps to explain the paradox where drag enhancement is consistently encountered in settling grain experiments prevalent in geophysics. The macroscopic observations are elucidated by microanalysis on the fluid–grain contact pressure differences relative to the volume-equivalent sphere, revealing that the net drag reduction is mainly rooted in the frictional drag. By gaining a deeper understanding of the drag force on rough grains, this research provides valuable insights into particle–fluid interactions in creeping flows, and holds promising implications for unresolved simulations of fluid–particle systems.