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The influence of inertia and cavity topography is examined for the steady-state flow inside thin cavities of arbitrary shape. The problem is closely related to the die casting process. The flow is determined by solving the thin-film equations, using a coupled spectral/finite-difference method. The flows inside flat, contracting, and modulated cavities of rectangular cross section are examined. The interplay between inertia and cavity modulation is particularly emphasized. Regions of shear and elongation are identified; their size and location are strongly dependent on inertia and cavity thickness. The flow inside a grooved cavity (of nonrectangular cross section) is also investigated. In this case, the problem reduces to Poisson’s equation for the streamwise velocity component, and an exact solution is obtained, which is compared against the thin-film solution. The range of validity of the thin-film approximation is thus identified, and is influenced by the cross-section aspect ratio, the modulation (groove) amplitude and wavelength.

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PurposeThis paper presents a new numerical model that, unlike most existing ones, can solve the whole liquid sloshing, nonlinear, moving boundary problem with free surface undergoing small to very large deformations without imposing any linearization assumptions.Design/methodology/approachThe time‐dependent, unknown, irregular physical domain is mapped onto a rectangular computational domain. The explicit form of the mapping function is unknown and is determined as part of the solution. Temporal discretization is based on one‐step implicit method. Second‐order, finite‐difference approximations are used for spatial discretizations.FindingsThe performance of the algorithm has been verified through convergence tests. Comparison between numerical and experimental results has indicated that the algorithm can accurately predict the sloshing motion of the liquid undergoing large interfacial deformations.Originality/valueThe ability to model liquid sloshing motion under conditions leading to large interfacial deformations utilizing the model presented in this paper improves our ability to understand the problem of sloshing motion in tuned liquid dampers (TLDs), which would eventually help in constructing more effective TLDs.

SUMMARYA hybrid spectral=ÿnite-di erence scheme is proposed to determine the inertial ow inside narrow channels. The ow ÿeld is represented spectrally in the depthwise direction, which together with the Galerkin projection lead to a system of equations that are solved using a variable step ÿnite di erence discretization. The method is particularly e ective for non-linear ow, and its validity is here demonstrated for a ow with inertia. The problem is closely related to high-speed lubrication ow. The validity of the spectral representation is assessed by examining the convergence of the method, and comparing with the fully two-dimensional ÿnite-volume solution (FLUENT), and the widely used depth-averaging method from shallow-water theory. It is found that a low number of modes are usually su cient to secure convergence and accuracy. Good agreement is obtained between the low-order description and the ÿnite-volume solution at low to moderate modiÿed Reynolds number. The depth-averaging solution is unable to predict accurately (qualitatively and quantitatively) the high-inertia ow. The in uence of inertia is examined on the ow.