Hele-Shaw flow

About: Hele-Shaw flow is a research topic. Over the lifetime, 5451 publications have been published within this topic receiving 151320 citations.

Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem

Abstract: The classical rectangular lid-driven-cavity problem is considered in which the motion of an incompressible fluid is induced by a single lid moving tangentially to itself with constant velocity. In a system infinitely extended in the spanwise direction the flow is two-dimensional for small Reynolds numbers. By a linear stability analysis it is shown that this basic flow becomes unstable at higher Reynolds numbers to four different three-dimensional modes depending on the aspect ratio of the cavity’s cross section. For shallow cavities the most dangerous modes are a pair of three-dimensional short waves propagating spanwise in the direction perpendicular to the basic flow. The mode is localized on the strong basic-state eddy that is created at the downstream end of the moving lid when the Reynolds number is increased. In the limit of a vanishing layer depth the critical Reynolds number approaches a finite asymptotic value. When the depth of the cavity is comparable to its width, two different centrifugal-instability modes can appear depending on the exact value of the aspect ratio. One of these modes is stationary, the other one is oscillatory. For unit aspect ratio (square cavity), the critical mode is stationary and has a very short wavelength. Experiments for the square cavity with a large span confirm this instability. It is argued that this three-dimensional mode has not been observed in all previous experiments, because the instability is suppressed by side-wall effects in small-span cavities. For large aspect ratios, i.e., for deep cavities, the critical three-dimensional mode is stationary with a long wavelength. The critical Reynolds number approaches a finite asymptotic value in the limit of an infinitely deep cavity.

The wake of two side-by-side square cylinders

Abstract: Aerodynamic interference between two cylinders involves most of the generic flow features associated with multiple structures, thus providing an excellent model for gaining physical insight into the wake of multiple cylindrical structures. This work aims to provide an experimental systematic study of the flow behind two side-by-side square cylinders. The square cylinder is a representative model for bluff bodies with sharp corners, characterized by a fixed flow separation point, which are distinct from those of continuous curvature with oscillating separation points, typically represented by the circular cylinder. Experiments were performed at a Reynolds number Re of 4.7 × 10 4 and a cylinder centre-to-centre spacing ratio T/ d (d is the cylinder height) of 1.02–6.00. The flow was measured using different techniques, including hot wires, load cell, particle imaging velocimetry and laser-induced fluorescence flow visualization. Four distinct flow regimes and their corresponding T/ d ranges are identified for the first time on the basis of the flow structure and the Strouhal number. Physical aspects in each regime, such as interference between shear layers, gap flow deflection and changeover, multiple flow modes, entrainment, recirculation bubble, vortex interactions and formation lengths, are investigated in detail and are connected to the characteristics of the time-averaged and fluctuating fluid forces. The flow displays a marked difference in many facets from that behind two side-by-side circular cylinders, which is linked to their distinct flow separation natures. A crucial role played by the gap flow and its passage geometry in contributing to the observed difference is also unveiled.

Modeling pinchoff and reconnection in a Hele-Shaw cell. I. The models and their calibration

Abstract: This is the first paper in a two-part series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a Hele-Shaw cell with arbitrary density and viscosity contrast between the components. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [Lowengrub and Truskinovsky, Proc. R. Soc. London, Ser. A 454, 2617 (1998)]) to flow in a Hele-Shaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper [paper II, Phys. Fluids 14, 514 (2002)], we analyze the behavior of the models in the fully nonlinear regime. In the BHSCH system, the equilibrium concentration profile is obtained using the classical Maxwell construction [Rowlinson and Widom, Molecular Theory of Capillarity (Clarendon, Oxford, 1979)] and does not depend on the orientation of the gravitational field. We find that the equilibria in the HSCH model are somewhat surprising as the gravitational field actually affects the internal structure of an isolated interface by driving additional stratification of light and heavy fluids over that predicted in the Boussinesq case. A comparison of the linear growth rates indicates that the HSCH system is slightly more diffusive than the BHSCH system. In both, linear convergence to the sharp interface growth rates is observed in a parameter controlling the interface thickness. In addition, we identify the effect that each of the parameters, in the HSCH/BHSCH models, has on the linear growth rates. We then show how this analysis may be used to suggest a set of modified parameters which, when used in the HSCH/BHSCH systems, yield improved agreement with the sharp interface model at a finite interface thickness. Evidence of this improved agreement may be found in paper II.

Nonlinear Flow in Porous Media

Abstract: Many investigators are concerned about the validity of the Forchheimer equation which represents the relationship between the velocity of flow and pressure gradient in porous media. A theoretical development of this equation through analysis of the dimensionless form of the Navier-Stokes equation is presented. It shows that energy losses at high-flow velocities in porous medium are a result of convective acceleration effects not turbulent effects. In addition, two dimensionless terms representing the flow behavior are defined and evaluated. It is shown that a constant could be used to represent the geometric properties of the medium and that a characteristic length representative of the flow exist. Both of these quantities are easily evaluated through hydraulic measurements of gradients and flow velocities. Experimental data from many sources were used to evaluate the theoretical results.

Numerical simulations of laminar flow over a 3D backward‐facing step

Abstract: A numerical investigation of laminar flow over a three-dimensional backward-facing step is presented with comparisons with detailed experimental data, available in the literature, serving to validate the numerical results. The continuity constraint method, implemented via a finite element weak statement, was employed to solve the unsteady three-dimensional Navier–Stokes equations for incompressible laminar isothermal flow. Two-dimensional numerical simulations of this step geometry underestimate the experimentally determined extent of the primary separation region for Reynolds numbers Re greater than 400. It has been postulated that this disagreement between physical and computational experiments is due to the onset of three-dimensional flow near Re ≈ 400. This paper presents a full three-dimensional simulation of the step geometry for 100⩽ Re⩽ 800 and correctly predicts the primary reattachment lengths, thus confirming the influence of three-dimensionality. Previous numerical studies have discussed possible instability modes which could induce a sudden onset of three-dimensional flow at certain critical Reynolds numbers. The current study explores the influence of the sidewall on the development of three-dimensional flow for Re greater than 400. Of particular interest is the characterization of three-dimensional vortices in the primary separation region immediately downstream of the step. The complex interaction of a wall jet, located at the step plane near the sidewall, with the mainstream flow reveals a mechanism for the increasing penetration (with increasing Reynolds number) of three-dimensional flow structures into a region of essentially two-dimensional flow near the midplane of the channel. The character and extent of the sidewall-induced flow are investigated for 100⩽Re⩽ 800. © 1997 John Wiley & Sons, Ltd.