Similarity solution

About: Similarity solution is a research topic. Over the lifetime, 2074 publications have been published within this topic receiving 59790 citations.

Low-Aspect-Ratio Flat-Ship Theory

Abstract: The integral equation, first obtained by H. Maruo, which determines the pressure distribution generating flow past a slender ip of vanishing draft, is studied further. New results obtained include predictions of singular centerplane effects of gravity for pointed bodies, a similarity solution for ships with cusped parabolic waterplanes, and some preliminary numerical solutions of the integral equation in the general case. I. Introduction T HIS paper is an extension of a study by MaruoJ on ''ships of small draft," "flat ships/' or "planing surfaces," all of which are equivalent descriptions. The small-draft assumption allows linearization of the free-surface boundary condition, as in the comparable case of thin ships, or ships of small beam.2 However the present linearized problem is much harder to solve, since the generating singularity distribution (effectively a distribution of pressure points on the limiting waterplane) is not explicitly given in terms of hull shape, but requires solution of an integral equation. This problem is analogous to the lifting-surface problem of aerodynamics, whereas the thin-ship problem corresponds to the simpler thickness problem of aerodynamics. Although this paper contains a brief reconsideration of the general flat-ship problem, to emphasize some aspects not discussed by Maruo, the paper is devoted mainly to the lowaspect-ratio limit. Thus the wetted length of the ship is supposed much greater than its beam, the latter having already been assumed much greater than the draft by the flat-ship requirement. The ship is therefore not only flat, but also slender. An alternative derivation is given here of an integral equation equivalent to one obtained by Maruo, having as its unknown function a pressure distribution representing the ship. This integral equation is also obtainable from the highFroude-number slender-body theory of Ogilvie, 3 by assuming that the ship is not only slender, but also flat. Maruo's low-aspect-ratio flat-ship integral equation is formally valid only at moderately-high Froude numbers, specifically such that U2B/gL2 is a quantity of order unity, where U is ship speed, B its beam and L its length, and g is gravity. The equation reduces to that of low-aspect-ratio wing theory in aerodynamics as g->0. One approach to practical solution of any planing problem, whether or not the aspect ratio is low, is to expand in an asymptotic series for very high Froude number, commencing with the aerodynamic g = 0 limit as the leading term.4 Maruol obtains the first two terms in this series for the lift on a flat delta wing, and an alternative treatment of this class of expansion, for general hull shapes, is presented here. In particular, very strong effects of gravity near the center plane of pointed bodies are demonstrated. At all Froude numbers, the low-aspect-ratio flat-ship integral equation possesses a "similarity" solution, such that the pressure distribution has the same shape at all stations. This linearized but gravity-dependent result should not be confused with the well-known conical similarity solution for nonlinear planing or water entry in the absence of gravity (Gilbarg,5 p.360). In fact the present geometrical requirement is for a cusped parabolic waterplane shape but an arbitrary section shape, whereas the nonlinear zero-gravity solution requires a triangular plan form and section shape. The low-aspect-ratio flat-ship integral equation is amendable to direct computation, and we present here some preliminary examples of its numerical solution. Much more work needs to be done to derive efficient procedures, and the present computer program can only be considered as a crude first attempt. However, the results are of considerable interest, indicating rather dramatic gravity effects especially near the center plane, as predicted analytically, and confirming Maruo's1 estimate of the lift coefficient of a delta wing at sufficiently high Froude number.

Group method analysis of mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder

Abstract: The transformation group theoretic approach is applied to the system of equations governing the unsteady mixed convection boundary-layer flow of a micropolar fluid near a stagnation point on a horizontal cylinder. The application of a two-parameter group reduces the number of independent variables by two, and consequently the system of governing partial differential equations with boundary conditions reduces to a system of ordinary differential equations with appropriate boundary conditions. The possible forms of surface-temperature Tw, potential velocity U and sin Open image in new window with position and time are derived in steady and unsteady cases. New formulae of dimensionless temperature are presented using the group method analysis. Hiemenz and Falkner-Skan equations are obtained as special cases. The new similarity representations and similarity transformations in steady/unsteady states are obtained. The family of ordinary differential equations has been solved numerically using a fourth-order Runge-Kutta algorithm with the shooting technique. The effect of varying parameters governing the problem is studied.

Gravity-Driven Thin Film Flow of an Ellis Fluid

Abstract: The thin film lubrication approximation has been studied extensively for moving contact lines of Newtonian fluids. However, many industrial and biological applications of the thin film equation involve shear-thinning fluids, which often also exhibit a Newtonian plateau at low shear. This study presents new numerical simulations of the three-dimensional (i.e. two-dimensional spreading), constant-volume, gravity-driven, free surface flow of an Ellis fluid. The numerical solution was validated with a new similarity solution, compared to previous experiments, and then used in a parametric study. The parametric study centered around rheological data for an example biological application of thin film flow: topical drug delivery of anti-HIV microbicide formulations, e.g. hydroxyethylcellulose (HEC) polymer solutions. The parametric study evaluated how spreading length and front velocity saturation depend on Ellis parameters. A lower concentration polymer solution with smaller zero shear viscosity (η0), τ1/2, and λ values spread further. However, when comparing any two fluids with any possible combinations of Ellis parameters, the impact of changing one parameter on spreading length depends on the direction and magnitude of changes in the other two parameters. In addition, the isolated effect of the shear-thinning parameter, λ, on the front velocity saturation depended on τ1/2. This study highlighted the relative effects of the individual Ellis parameters, and showed that the shear rates in this flow were in both the shear-thinning and plateau regions of rheological behavior, emphasizing the importance of characterizing the full range of shear-rates in rheological measurements. The validated numerical model and parametric study provides a useful tool for future steps to optimize flow of a fluid with rheological behavior well-described by the Ellis constitutive model, in a range of industrial and biological applications.

The three-dimensional turbulent boundary layer near a plane of symmetry

Abstract: The asymptotic structure of the three-dimensional turbulent boundary layer near a plane of symmetry is considered in the limit of large Reynolds number A self-consistent two-layer structure is shown to exist wherein the streamwise velocity is brought to rest through an outer defect layer and an inner wall layer in a manner similar to that in two-dimensional boundary layers The cross-stream velocity distribution is more complex and two terms in the asymptotic expansion are required to yield a complete profile which is shown to exhibit a logarithmic region The flow in the inner wall layer is demonstrated to be collateral to leading order; pressure-gradient effects are formally of higher order but can cause the velocity profile to skew substantially near the wall at the large but finite Reynolds numbers encountered in practice The governing set of ordinary differential equations describing a self-similar flow is derived The calculated numerical solutions of these equations are matched asymptotically to an inner wall-layer solution and the results show trends that are consistent with experimental observations