We investigate the dissipation of linear, two-dimensional, interfacial waves in a setting comprising three fluids (an upper fluid of semi-infinite depth, a middle fluid-layer of finite thickness, and a lower fluid of semi-infinite depth) separated by two distinct interfaces, which we consider to be elastic. We derive analytic expressions for the dissipation rate of capillary-gravity waves in such a system, in both the barotropic and baroclinic modes of propagation. Using the dissipation rate model formulated herein, we conduct parametric studies of barotropic gravity waves in an air–oil–water system. We consider six different wavenumbers within the range of 0.0165 m⁻¹ (corresponding to ocean swell) to 44.5 m⁻¹ (corresponding to a typical laboratory gravity wave) and investigate the effects of three major mechanisms of loss of energy, which are the dissipation due to the (i) dynamics in the upper fluid (air), (ii) elastic interfaces, and (iii) viscous middle fluid (oil) layer of finite thickness. For waves with wavenumbers of 0.0165 m⁻¹ and 0.04 m⁻¹, the dominant mechanism for the energy loss is that due to the dynamics in air. For waves with wavenumbers of 1 m⁻¹ and 4 m⁻¹, the oil layer acts to increase the dissipation rates significantly but only when its thickness is beyond a threshold value. For waves with wavenumbers of 36.2 m⁻¹ and 44.5 m⁻¹, the elastic interfaces cause significant increases in the dissipation rates, when their elasticities change from a value of 0.01 N/m to 0.0225 N/m. The three-fluid model developed herein is applicable to capillary-gravity waves propagating in a generic fluid system with arbitrary values for the densities, viscosities, interfacial elasticities, and with an arbitrary value for the middle fluid-layer thickness within an upper limit. This model is useful in predicting the dissipation rates of waves on the ocean surface, which is (in general) covered with biofilms and oil layers of thicknesses ranging from a few μm to a few mm, and in predicting the dissipation rates of waves such as swell, for which the dynamics in the upper fluid (air) are important.

Two-dimensional linear waves at a contaminated interface separating two infinitely deep fluids of arbitrary densities and viscosities are investigated. The contamination is modeled as a massless monolayer, which may result from insoluble surfactants, and introduces interfacial elasticity. Thus, the interface supports two wave modes: transverse, capillary-gravity waves (CG-waves) and longitudinal, Marangoni waves (M-waves). A comprehensive dispersion relation is derived; it can be solved numerically to obtain the complex-valued frequency as a function of the irrotational wavenumber for the CG-waves and for the M-waves. The CG-waves are analyzed in this paper; the M-waves are analyzed in a separate work. The main result here is the derivation of an operational approximate formula for the temporal decay rate of the CG-waves. Its predictions are compared to the available measurements of (laboratory) gravity waves, ocean swell, and (laboratory) capillary waves. It is shown that the previously published decay rate models, which include either the effects due to an upper fluid or those due to interfacial contamination (but not both), are limiting cases of the present model. A parametric study of decay rate predictions shows that there are systems including ocean swell for which the influence of both the dynamics of the upper fluid and of the contamination are significant.

Results on flows in a draft tube of a constant-head, constant-specific speed, model Francis turbine are presented based on computational fluid dynamics (CFD) simulations and theoretical analysis. A three-dimensional, unsteady, Navier–Stokes solver with the detached-eddy simulation (DES) model and the realizable k–ϵ (RKE) model is used to analyze the vortex rope formed at different discharge coefficients. The dominant amplitude of the pressure fluctuations at a fixed point in the draft tube increases by 13 times, and the length of the rope increases by 3.4 times when the operating point of the turbine shifts from a discharge coefficient of 0.37 to 0.34. A perturbation analysis based on a steady, axisymmetric, inviscid, incompressible model for the mean flow is performed to obtain a Sturm–Liouville (SL) system, the solutions of which are oscillatory if the discharge coefficient is greater than 0.3635, and nonoscillatory otherwise.

Dissipation of standing gravity waves of frequencies within 1–2 Hz is investigated experimentally. The waves are generated in a rectangular tank filled with water, the surface of which is covered with an oil layer of mean thickness, d. Damping rates are measured as a function of d, and compared with results from established theoretical models—in particular, with those from a recently developed three-fluid dissipation model that considers waves in a system of semi-infinitely deep fluids that lie above and below an interfacial fluid layer of finite thickness. Based on a comparison of experimental data with predictions, the oil–water interfacial elasticity, E₂, is empirically determined to be a linear function of d. The theoretical predictions include contributions from the three-fluid dissipation model, which accounts for energy losses due to shear layers at the interfaces, friction in the fluid bulk, and compression–expansion oscillations of the elastic interfaces; and from a boundary-layer dissipation model, which accounts for energy losses due to boundary layers at the tank's solid surfaces. The linear function, $E_2(d)$ , is used to compute the three-fluid model damping rate. An effective viscosity of the oil–water system is used to compute the boundary-layer model damping rate. The theoretical predictions are, on average, within 5% of measurements for all the wave frequencies considered. The promise shown by the three-fluid model is highlighted, as are the assumptions involved in the analysis and comparisons.