Unified Friction Factor Model for Dune Beds from High to Low Submergence

Abstract

Abstract Accurate prediction of flow resistance in dune-bed streams is critical for river engineering and flood management. Traditional methods, which rely on empirically partitioning total shear stress into skin and form drag, often fail under low-submergence conditions where the flow is profoundly influenced by large roughness elements. This study presents a unified model that predicts the total Darcy-Weisbach friction factor without requiring such partitioning. The approach conceptualizes dunes as macro-roughness, adopting a gravel-bed resistance formula. The key finding is a physically based equivalent roughness height <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Subscript s" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> </mml:math> , derived from a momentum analysis of sudden-expansion head loss, which scales as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Subscript s Baseline tilde delta squared divided by lamda" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo>∼</mml:mo> <mml:msup> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">/</mml:mo> <mml:mi>λ</mml:mi> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="delta" display="inline" overflow="scroll"> <mml:mi>δ</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda" display="inline" overflow="scroll"> <mml:mi>λ</mml:mi> </mml:math> denote dune height and length, respectively. Calibration with fixed dune data yields <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k Subscript s Baseline equals 13.0 delta squared divided by lamda" display="inline" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>s</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>13.0</mml:mn> <mml:msup> <mml:mrow> <mml:mi>δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo stretchy="false">/</mml:mo> <mml:mi>λ</mml:mi> </mml:math> . The model demonstrates strong performance for fixed dunes and predicts independent mobile dune data acceptably without recalibration, validating its general applicability across a wide range of submergence.