A Similarity Approach to the Numerical Solution of Free Boundary Problems

Fazio, Riccardo

doi:10.1137/s0036144595285057

Cited by 34 publications

(26 citation statements)

References 53 publications

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“…It is an initial value method even if we solve a different model for each iteration when the governing differential equation is not invariant under every scaling group of point transformation. Its versatility has been shown by solving several problems of interest: free boundary problems [33,23,28,29], a hyperbolic moving boundary problem [25], the Homann and the Hiemenz flows governed by the Falkner-Skan equation in [26], one-dimensional parabolic moving boundary problems [30], two variants of the Blasius problem [32], namely: a boundary layer problem over moving surfaces, studied first by Klemp and Acrivos [38], and a boundary layer problem with slip boundary condition, that has found application to the study of gas and liquid flows at the micro-scale regime [18,40], parabolic problems on unbounded domains [34] and, recently, see the preprints: [22] parabolic moving boundary problems, and [21] an interesting problem in boundary layer theory: the so-called Sakiadis problem [45,46].…”

Section: Discussionmentioning

confidence: 99%

Blasius problem and Falkner–Skan model: Töpfer’s algorithm and its extension

Fazio

2013

Computers & Fluids

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a b s t r a c tIn this paper, we review the so-called Töpfer algorithm that allows us to find a non-iterative numerical solution of the Blasius problem, by solving a related initial value problem and applying a scaling transformation. Moreover, we remark that the applicability of this algorithm can be extended to any given problem, provided that the governing equation and the initial conditions are invariant under a scaling group of point transformations and that the asymptotic boundary condition is non-homogeneous. Then, we describe an iterative extension of Töpfer's algorithm that can be applied to a general class of problems. Finally, we solve the Falkner-Skan model, for values of the parameter where multiple solutions are admitted, and report original numerical results, in particular data related to the famous reverse flow solutions by Stewartson. The numerical data obtained by the extended algorithm are in good agreement with those obtained in previous studies.

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Section: Discussionmentioning

confidence: 99%

Blasius problem and Falkner–Skan model: Töpfer’s algorithm and its extension

Fazio

2013

Computers & Fluids

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“…Even though the problem is almost a century old, recent papers that employ the Blasius problem as an example include [2,1,5,6,11,15,16,21,18,17,23,25,26,27,28,29,30,32,33,34,36].…”

Section: Because All Fluid Flows Must Be Zero At a Solid Boundary Thmentioning

confidence: 99%

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Boyd¹

2008

SIAM Rev.

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Abstract. The Blasius flow is the idealized flow of a viscous fluid past an infinitesimally thick, semiinfinite flat plate. The Blasius function is the solution to 2fxxxWe use this famous problem to illustrate several themes. First, although the flow solves a nonlinear partial differential equation (PDE), Toepfer successfully computed highly accurate numerical solutions in 1912. His secret was to combine a Runge-Kutta method for integrating an ordinary differential equation (ODE) initial value problem with some symmetry principles and similarity reductions, which collapse the PDE system to the ODE shown above. This shows that PDE numerical studies were possible even in the precomputer age. The truth, both a hundred years ago and now, is that mathematical theorems and insights are an arithmurgist's best friend, and they can vastly reduce the computational burden. Second, we show that special tricks, applicable only to a given problem, can be as useful as the broad, general methods that are the fabric of most applied mathematics courses: the importance of "particularity." In spite of these triumphs, many properties of the Blasius function f (x) are unknown. We give a list of interesting projects for undergraduates and another list of challenging issues for the research mathematician.

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“…where c 1 and c 2 are two constants [9,19,22]. It is sometimes more advantageous to write down these relations in the form u = t c 1 ξ c 3Ṽ (ξ ) and h = t c 2 ξ c 4Z (ξ ), where c 3 and c 4 are two other constants,Ṽ (ξ ) = ξ −c 3V (ξ ) andZ (ξ ) = ξ −c 4Ẑ (ξ ), so that the governing equations (4) and (5) can be transformed into an autonomous differential equation [i.e., in Eq.…”

Section: Discussionmentioning

confidence: 99%

Existence and features of similarity solutions for non-Boussinesq gravity currents

Ancey

Cochard

Rentschler

et al. 2007

Physica D: Nonlinear Phenomena

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In this paper, the flow dynamics of gravity currents on a horizontal plane is investigated from a theoretical point of view by seeking similarity solutions. The current is generated by unleashing a varying volume of heavy fluid within an ambient fluid of much lower density. Unlike earlier investigators, we assume that the ambient fluid exerts no significant resisting action on the current, and therefore the flow depth is expected to drop to zero at the front in the absence of friction. In this context, the shallow-water equations are highly appropriate for computing the mean velocity and flow depth of the current. The boundary condition imposed at the front leads to technical mathematical difficulties. Indeed, unlike in the Boussinesq case, no regular solution to the shallow-water equations satisfies the downstream condition, but when the flow is supercritical at the channel inlet, it is possible to construct a piecewise solution by patching a regular solution to an exceptional solution, which represents the head behavior. To better understand this result and make sure that the result is physically relevant, we consider the Navier-Stokes equations within the high-Reynolds-number limit. Approximate similarity solutions can be worked out, which support our earlier analysis on the shallow-water equations. While the flow body is self-similar and weakly rotational, the head is not self-similar, but tends toward a self-similarity shape at long times. It is characterized by a strong vorticity, a straight free surface, and a nonuniform velocity profile, which becomes flatter and flatter with time.

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A Similarity Approach to the Numerical Solution of Free Boundary Problems

Cited by 34 publications

References 53 publications

Blasius problem and Falkner–Skan model: Töpfer’s algorithm and its extension

Blasius problem and Falkner–Skan model: Töpfer’s algorithm and its extension

The Blasius Function: Computations Before Computers, the Value of Tricks, Undergraduate Projects, and Open Research Problems

Existence and features of similarity solutions for non-Boussinesq gravity currents

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