Figure 8

Schematic explanation for instability of a Krauklis wave traveling to the right. A solid particle (open circle) at the boundary moves along an ellipse (dotted line). White arrows represent the perturbed fluid and solid velocities associated with the Krauklis wave. The basic flow is represented by slender black arrows with its parabolic (or quartic) envelop. (a) The symmetric Krauklis wave inherently involves considerable friction at the boundary because of large amplitude of v _z , but it can be reduced if the basic flow is in the same direction as the wave propagation. (b) In the antisymmetric Krauklis wave, v _z is negligibly small but the solid particle motion at the boundary causes friction, which can be reduced if the basic flow is in the opposite direction to the wave propagation. This mechanism works only when the basic flow profile is nonparabolic.