On a potential-velocity formulation of Navier-Stokes equations

Marner, Florian; Gaskell, Philip; Scholle, Markus

doi:10.1134/s1029959914040110

Cited by 21 publications

(43 citation statements)

References 8 publications

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“…In the following, proof is given that the equations derived in 7,11 for 2D flow uniquely result from equation (22) as a special case of general 3D flow. For steady flow with ∂ t Ψ n = 0 equation (16) is fulfilled via the identity ϕ n = ηΨ n , while the auxiliary vector field defined by (24) reads:…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Though not immediately obvious, the above rudimentary form of the first integral corresponds to that of the 2D first integral 7,11 ; a more conveniently recognisable form is arrived at via the following reformulation.…”

Section: B First Integral Of the Field Equationsmentioning

confidence: 99%

“…This formulation embodies the classical complex-valued Goursat representation for steady Stokes flow, allowing the streamfunction to be expressed in terms of two analytic functions 7 . While essentially a rediscovery of the result of Legendre 8 , along similar lines to the work of Coleman 9 and Ranger 10 , a hallmark of Scholle, Haas, and Gaskell 6 's particular derivation is it provides a clear hint apropos generalisation to unsteady, three-dimensional (3D) viscous flow: the attainment of which has hitherto remained out of reach; providing the impetus for the present work.…”

Section: Introductionmentioning

confidence: 99%

“…Boundary conditions, physical and auxiliary, in the framework of the above are provided, with the condition essential to the investigation of 3D free-surface flow problems derived in the form of a first integral of the usual dynamic boundary condition 6,7 . As a whole the approach followed together with the established first integral, represents an important step forward; demonstrated via the solution of three classical, yet diverse, fluid flow problems of differing complexity, two of which are approached analytically, the other numerically.…”

Section: Introductionmentioning

confidence: 99%

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Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

Scholle

Gaskell

Marner

2018

Journal of Mathematical Physics

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The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. An exact first integral of the full, unsteady, incompressible Navier-Stokes equations is achieved in its most general form via the introduction of a tensor potential and parallels drawn with Maxwell's theory. Subsequent to this gauge freedoms are explored, showing that when used astutely they lead to a favourable reduction in the complexity of the associated equation set and number of unknowns, following which the inviscid limit case is discussed. Finally, it is shown how a change in gauge criteria enables a variational principle for steady viscous flow to be constructed having a self-adjoint form. Use of the new formulation is demonstrated, for different gauge variants of the first integral as the starting point, through the solution of a hierarchy of classical three-dimensional flow problems; two of which are tractable analytically, the third being solved numerically. In all cases the results obtained are found to be in excellent accord with corresponding solutions available in the open literature. Concurrently, the prescription of appropriate commonly occurring physical and necessary auxiliary boundary conditions, incorporating for completeness the derivation of a first integral of the dynamic boundary condition at a free surface, is established, together with how the general approach can be advantageously reformulated for application in solving unsteady flow problems with periodic boundaries.

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Section: Acknowledgmentsmentioning

confidence: 99%

Section: B First Integral Of the Field Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

Scholle

Gaskell

Marner

2018

Journal of Mathematical Physics

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“…Scholle et al [2] used the Finite Element Method (FEM) and complex variable method for modeling the film over corrugated surfaces. Marner et al [3] developed a potential-velocity formulation of the Navier-Stokes (N-S) equations by using the least square FEM approach. More recently, Marner et al [4] proposed a generalized complex-valued first integral of the N-S equations for the unsteady Couette flows in a corrugated channel system.…”

Section: Introductionmentioning

confidence: 99%

Study on SPH Viscosity Term Formulations

Zheng

Shao

2018

Applied Sciences

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Featured Application: The proposed study contributes to the mesh-free numerical modeling field through the examinations of several viscosity term formulations. By simulating benchmark viscosity-dominated flows in engineering practice, the optimal viscosity scheme has been found to work with the Smoothed Particle Hydrodynamics (SPH) method.Abstract: For viscosity-dominated flows, the viscous effect plays a much more important role. Since the viscosity term in SPH-governing (Smoothed Particle Hydrodynamics) equations involves the discretization of a second-order derivative, its treatment could be much more challenging than that of a first-order derivative, such as the pressure gradient. The present paper summarizes a series of improved methods for modeling the second-order viscosity force term. By using a benchmark patch test, the numerical accuracy and efficiency of different approaches are evaluated under both uniform and non-uniform particle configurations. Then these viscosity force models are used to compute a documented lid-driven cavity flow and its interaction with a cylinder, from which the most recommended viscosity term formulation has been identified.

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On a potential‐velocity formulation of incompressible Navier‐Stokes equations

Marner

Scholle

2014

Proc Appl Math and Mech

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Computational fluid dynamics has emerged as an essential investigative tool in nearly every field of technology. Despite a well-developed mathematical theory and the existence of readily available commercial software codes, computing solutions to the governing equations of fluid motion remains challenging, especially due to the non-linearity involved. Additionally, in the case of free surface film flows the dynamic boundary condition at the free surface complicates the mathematical treatment notably.Recently, by introduction of an auxiliary potential field, a first integral of the two-dimensional incompressible Navier-Stokes equations has been constructed leading to a set of equations, the differential order of which is lower than that of the original equations [1]. A useful application to free surface simulation was found in [2]. Moreover the new formulation is naturally extendible to three dimensions via tensor calculus, involving a non-unique symmetric tensor potential. The corresponding degrees of freedom can be used in order to achieve a numerically convenient representation. Finally an efficient staggered-grid finite difference scheme is applied to a Stokes flow problem in a 3D lid-driven cavity to demonstrate the capabilities of the new approach.

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On a potential-velocity formulation of Navier-Stokes equations

Cited by 21 publications

References 8 publications

Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

Exact integration of the unsteady incompressible Navier-Stokes equations, gauge criteria, and applications

Study on SPH Viscosity Term Formulations

On a potential‐velocity formulation of incompressible Navier‐Stokes equations

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