Exact Solutions to the Three-Dimensional Navier–Stokes Equations Using the Extended Beltrami Method

Dyck, Nolan; Straatman, Anthony G.

doi:10.1115/1.4044927

Cited by 14 publications

(3 citation statements)

References 43 publications

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“…We present graphical results of such flows in each coordinate system. Our approach here shares some similarity with that in [23] and its extension in [24], but our emphasis is on developing new purely Beltrami flows.…”

Section: Work On Beltrami Flows Was Initiated In the Work Of Eugenio Beltrami Andmentioning

confidence: 99%

On the Axisymmetric Steady Incompressible Beltrami Flows

Bělík¹,

Su²,

Dokken³

et al. 2020

OJFD

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In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the rz-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of tornado-like flows that develop as swirl increases. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.

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Section: Work On Beltrami Flows Was Initiated In the Work Of Eugenio Beltrami Andmentioning

confidence: 99%

On the Axisymmetric Steady Incompressible Beltrami Flows

Bělík¹,

Su²,

Dokken³

et al. 2020

OJFD

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“…With a characteristic length scale being defined, it is thus expected that the spatial extension of vortex solution may also be rendered finite. The generalized Beltrami flows [10,38,39] and the extended Beltrami flows [40,41] will share the same features in common.…”

Section: Introductionmentioning

confidence: 99%

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

Takahashi

2023

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This paper is aimed at eliciting consistency conditions for the existence of unsteady incompressible axisymmetric swirling viscous Beltrami vortices and explicitly constructing solutions that obey the conditions as well as the Navier–Stokes equations. By Beltrami flow, it is meant that vorticity, i.e., the curl of velocity, is proportional to velocity at any local point in space and time. The consistency conditions are derived for the proportionality coefficient, the velocity field and external force. The coefficient, whose dimension is of [length−1], is either constant or nonconstant. In the former case, the well-known exact nondivergent three-dimensional unsteady vortex solutions are obtained by solving the evolution equations for the stream function directly. In the latter case, the consistency conditions are given by nonlinear equations of the stream function, one of which corresponds to the Bragg–Hawthorne equation for steady inviscid flow. Solutions of a novel type are found by numerically solving the nonlinear constraint equation at a fixed time. Time dependence is recovered by taking advantage of the linearity of the evolution equation of the stream function. The proportionality coefficient is found to depend on space and time. A phenomenon of partial restoration of the broken scaling invariance is observed at short distances.

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“…This idealisation admits a direct analytical study of the impact of energy input through its impact on various explicit flows. Indeed, explicit flows are frequently used as a tool for benchmarking analytical and numerical studies in this and other contexts, e.g., [3,5,6,11,12,16], and also for turbulence studies, e.g. [7,10,13].…”

Section: Introductionmentioning

confidence: 99%

Geophysical fluid models with simple energy backscatter: explicit flows and unbounded exponential growth

Prugger,

Rademacher,

Yang

2021

Preprint

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Motivated by numerical schemes for large scale geophysical flow, we consider the rotating shallow water and Boussinesq equations on the whole space with horizontal kinetic energy backscatter source terms built from constant negative viscosity and stabilising hyperviscosity. We study the impact of this energy input through various explicit flows, which are simultaneously solving the nonlinear equations and the linear equations that arise upon dropping the transport nonlinearity, i.e., the linearisation in the zero state. These include barotropic, parallel and Kolmogorov flows as well as monochromatic inertia gravity waves. With focus on stable stratification we find that the backscatter generates numerous solutions of this type that grow exponentially and unboundedly, also with vertical structure. This signifies the possibility of undesired energy concentration into specific modes due to the backscatter. Families of steady state flows of this type arise as well and superposition principles in the nonlinear equations provide explicit sufficient conditions for instability of some of these. For certain steady barotropic flows of this type we provide numerical evidence of eigenmodes whose growth rates are proportional to the amplitude factor. For all other arising steady solutions we prove this is not possible.

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Exact Solutions to the Three-Dimensional Navier–Stokes Equations Using the Extended Beltrami Method

Cited by 14 publications

References 43 publications

On the Axisymmetric Steady Incompressible Beltrami Flows

On the Axisymmetric Steady Incompressible Beltrami Flows

Three-Dimensional Unsteady Axisymmetric Viscous Beltrami Vortex Solutions to the Navier–Stokes Equations

Geophysical fluid models with simple energy backscatter: explicit flows and unbounded exponential growth

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