This study derives and investigates a scaling law for the internal wave-making resistance on an obstacle moving along a two-layer fluid, expressed as D/M5/3 = fD(Γ). Here, D is a resistance coefficient, Γ = (Fr − 1)/M2/3, Fr is the two-layer Froude number, and M is the dimensionless draft relative to the upper fluid depth. By introducing two small asymptotic parameters under the assumptions of weak nonlinearity and shallow water, the law is derived from an asymptotic theory extended from the single-layer scenario. It quantitatively describes the dependence of resistance on the obstacle's speed and geometry. Under this scaling, the resistance data for different obstacle sizes collapse onto a unified curve. The validity of the scaling law is assessed using a strongly nonlinear numerical model. The results demonstrate that the law holds well for −1.0 Γ 1.0, with optimal accuracy for the draft of M ≤ 0.5. As the depth ratio between the two layers increases, maintaining the law's validity requires a smaller ratio of the two asymptotic parameters. Furthermore, the scaling law is found to be unsuitable for obstacles with a large width-to-length ratio or a sharp body shape.
Sun et al. (Thu,) studied this question.