Creeping Flow Analyses of Free Surface Cavity Flows

1996 ◽  
Vol 8 (6) ◽  
pp. 415-433
Author(s):  
P. H. Gaskell ◽  
J. L. Summers ◽  
H. M. Thompson ◽  
M. D. Savage
Keyword(s):  
Free Surface ◽  
Creeping Flow ◽  
Cavity Flows ◽  
Surface Cavity

Creeping flow analyses of free surface cavity flows

1996 ◽  
Vol 8 (6) ◽  
pp. 415-433 ◽  
Author(s):  
P. H. Gaskell ◽  
J. L. Summers ◽  
H. M. Thompson ◽  
M. D. Savage
Keyword(s):  
Free Surface ◽  
Creeping Flow ◽  
Cavity Flows ◽  
Surface Cavity

Shear flow over a translationally symmetric cylindrical bubble pinned on a slot in a plane wall

1994 ◽  
Vol 275 ◽  
pp. 351-378 ◽  
Author(s):  
James Q. Feng ◽  
Osman A. Basaran
Keyword(s):  
Reynolds Number ◽  
Finite Element ◽  
Free Surface ◽  
Shear Flow ◽  
Continuation Method ◽  
Turning Points ◽  
Creeping Flow ◽  
Simple Shear Flow ◽  
Bubble Deformation ◽  
Structure Of Flow

Steady states of a translationally-symmetric cylindrical bubble protruding from a slot in a solid wall into a liquid undergoing a simple shear flow are investigated. Deformations of and the flow past the bubble are determined by solving the nonlinear free-boundary problem comprised of the two-dimensional Navier–Stokes system by the Galerkin/finite element method. Under conditions of creeping flow, the results of finite element computations are shown to agree well with asymptotic results. When the Reynolds number Re is finite, flow separates from the free surface and a recirculating eddy forms behind the bubble. The length of the separated eddy measured in the flow direction increases with Re, whereas its width is confined to within the region that lies between the supporting solid surface and the separation point at the free surface. By tracking solution branches in parameter space with an arc-length continuation method, curves of bubble deformation versus Reynolds number are found to exhibit turning points when Re reaches a critical value Rec. Therefore, along a family of bubble shapes, solutions do not exist when Re > Rec. The locations of turning points and the structure of flow fields are found to be governed virtually by a single parameter, We = Ca Re, where We and Ca are Weber and capillary numbers. Two markedly different modes of bubble deformation are identified at finite Re. One is dominant when Re is small and is tantamount to a plain skewing or tilting of the bubble in the downstream direction; the other becomes more pronounced when Re is large and corresponds to a pure upward stretching of the bubble tip.

Gravity-driven creeping flow of two adjacent layers through a channel and down a plane wall

1998 ◽  
Vol 371 ◽  
pp. 345-376 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Free Surface ◽  
Numerical Simulations ◽  
Channel Flow ◽  
Film Flow ◽  
Finite Amplitude ◽  
Creeping Flow ◽  
Plane Wall ◽  
Surface Velocity ◽  
Inclined Plane ◽  
Gravity Driven

We study the stability of the interface between (a) two adjacent viscous layers flowing due to gravity through an inclined or vertical channel that is confined between two parallel plane walls, and (b) two superimposed liquid films flowing down an inclined or vertical plane wall, in the limit of Stokes flow. In the case of channel flow, linear stability analysis predicts that, when the fluids are stably stratified, the flow is neutrally stable when the surface tension vanishes and the channel is vertical, and stable otherwise. This behaviour contrasts with that of the gravity-driven flow of two superimposed films flowing down an inclined plane, where an instability has been identified when the viscosity of the fluid next to the plane is less than that of the top fluid, even in the absence of fluid inertia. We investigate the nonlinear stages of the motion subject to finite-amplitude two-dimensional perturbations by numerical simulations based on boundary-integral methods. In both cases of channel and film flow, the mathematical formulation results in integral equations for the unknown interface and free-surface velocity. The properties of the integral equation for multi-film flow are investigated with reference to the feasibility of computing a solution by the method of successive substitutions, and a deflation strategy that allows an iterative procedure is developed. In the case of channel flow, the numerical simulations show that disturbances of sufficiently large amplitude may cause permanent deformation in which the interface folds or develops elongated fingers. The ratio of the viscosities and densities of the two fluids plays an important role in determining the morphology of the emerging interfacial patterns. Comparing the numerical results with the predictions of a model based on the lubrication approximation shows that the simplified approach can only describe a limited range of motions. In the case of film flow down an inclined plane, we develop a method for extracting the properties of the normal modes, including the ratio of the amplitudes of the free-surface and interfacial waves and their relative phase lag, from the results of a numerical simulation for small deformations. The numerical procedure employs an adaptation of Prony's method for fitting a signal described by a time series to a sum of complex exponentials; in the present case, the signal is identified with the cosine or sine Fourier coefficients of the interface and free-surface waves. Numerical simulations of the nonlinear motion confirm that the deformability of the free surface is necessary for the growth of small-amplitude perturbations, and show that the morphology of the interfacial patterns developing subject to finite-amplitude perturbations is qualitatively similar to that for channel flow.

Some Remarks on the Progress of Cavity Flow Studies

1975 ◽  
Vol 97 (4) ◽  
pp. 439-449 ◽  
Author(s):  
C. C. Hsu
Keyword(s):  
Experimental Data ◽  
Free Surface ◽  
Cavity Flow ◽  
Cavity Flows ◽  
Practical Applications ◽  
Flow Problems ◽  
Mathematical Difficulties ◽  
Approximate Expressions ◽  
Finite Disturbance ◽  
Theoretical Developments

Theoretical developments on cavity flow studies are briefly reviewed. Physical and mathematical difficulties involved in cavity flow problems are discussed. Particular attention, with regard to practical applications, is given to the development of linearized theories. Based on the existing analyses, efforts to develop simple approximate expressions for the force coefficients of supercavitating hydrofoils (including the effects of free surface, cascade, aspect ratio, and finite disturbance) are made. Numerical results calculated from these expressions are compared with existing experimental data. Special problems involving unsteady cavity flows, such as pulsation of the finite cavities, are also discussed.

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