Unique laminar-flow stability limit based shallow-water theory

Abstract

Two approaches are generally taken in deriving the stability limit for the Froude number (𝐹𝑠) for laminar sheet flow: The first approach uses the Orr‐Sommerfeld equation, while the second uses the cross‐section‐averaged equations of continuity and motion. Because both approaches are based on shallow‐water theory, the values of 𝐹 𝑠 obtained from both approaches should be identical, yet in the literature they are not. This suggests that a defect exists in at least one of the two approaches. After examining the governing equations used in both approaches, one finds that the existing cross‐section‐averaged equation of motion is dependent on the frame of reference. To correct this defect, one can formulate a frame‐independent equation of motion relative to a coordinate system moving with constant velocity, then derive a new expression for 𝐹 𝑠 that is generally applicable to both laminar and turbulent flows in prismatic channels of arbitrary cross‐sectional geometry. For laminar sheet flow, the new expression for 𝐹 𝑠 obtained from the second approach yields 𝐹 𝑠 ≃0.527 , which agrees with that obtained from the first approach.