Two models and the generation mechanisms of the drag on an accelerating starting disk

Abstract

As a canonical problem, the impulsive starting of a circular disc contains the fundamental mechanisms of the force generation of the drag-based propulsions. In this paper, a circular disc is uniformly accelerated to a constant target velocity along a straight path, the instantaneous drag on and the flow fields around the disc are measured. A series of experiments were conducted by varying the two dimensionless numbers, i.e., the Reynolds number (Re) ranging from 40000 to 80000, and the acceleration number (A*)(double normalized uniform-acceleration distance) ranging from 0.5 to 2. Based on the quasi-steady and the impulse-based ideas, two analytical models are proposed for predicting and accounting for the drag force on the disc. Moreover, the two models distinguish the generation of the drag force into three phases. In the acceleration phase, the growth rate and initial peak of the drag on the disc strongly depend on A*, which make the drag-force histories exhibit a good scaling law for a given A*, and the whole drag is generally contributed by the increased growth-rate of the vortex ring circulation. In the transition phase, the drag decreases owing to the decrease of the circulation growth-rate of the vortex ring. In the vortex pinch-off phase, the circulation of the vortex ring nearly no longer grows and the size growth-rate of vortex ring gradually plays a dominant role in the drag generation. The present results suggest two implications for understanding the force generation in bio-propulsions.

Drag Force on a Starting Plate Scales with the Square Root of Acceleration

“…The conventional approach is to describe the forces on an accelerating, or decelerating, plate at high Reynolds number (Re ≫ 1) by considering the added mass force combined with a quasisteady drag force, where the added mass force is directly proportional to the acceleration [8][9][10]. The inviscid nature of added mass is a reason for doubting this approach [11][12][13] and recent experiments [7,14,15] demonstrate that this description is incomplete, and the dependence of the drag force on acceleration, velocity, and other parameters has not been described by a general scaling law.…”

“…Given the measured total force FðtÞ, the drag force F D on an accelerating plate with mass m submerged in a fluid with density ρ and kinematic viscosity ν, at high Reynolds number (Re ≫ 1), can be written as [8,15]…”

“…The value of m v depends on the object shape and direction of motion [10,20]. The ratio of the added mass force and quasisteady drag force defines the dimensionless acceleration a à ¼ al=V 2 a [14,15,21,22], which has previously been used in attempts to scale drag forces for accelerating objects. Furthermore, we present results as a function of the dimensionless time t à , defined as [23,24]…”

“…The vorticity that is advected and fed into the vortex at the plate edges increases the circulation Γ of the vortex. This can be directly related to the drag force, i.e., F D ðtÞ=l b ∝ ρ _ Γl a [15,33,34], given that the vortex remains…”

“…It is clear that the data do not collapse on a single curve, but rather organize on different curves that follow ða Ã Þ 0.5 for different target velocities V a . Evidently, scaling with C D [14,15] leaves a dependency on both a à and V a , which is resolved by using the scaling of Eq. ( 3) to obtain a uniform value for C a .…”