A Mathematical Model Illustrating the Theory of Turbulence

Burgers, J. M.

doi:10.1016/s0065-2156(08)70100-5

Cited by 2,032 publications

(1,324 citation statements)

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“…These are the cases with the standard Smagorinsky subgrid-scale model, at a Reynolds number of approximately 2000, on two grids (viz. with [29]. The latter was based on the assumption of a uniform axial velocity distribution.…”

Section: Average Velocity Profilesmentioning

confidence: 99%

Simulations of confined turbulent vortex flow

Derksen

2005

Computers & Fluids

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Large-eddy simulations (LES) of the turbulent flow in a swirl tube with a tangential inlet have been performed. The geometry, and flow conditions were chosen according to an experimental study by [Escudier MP, Bornstein J, Zehnder N. Observations and LDA measurements of confined turbulent vortex flow. J Fluid Mech 1980;98:49-63]. Lattice-Boltzmann discretization was used to numerically solve the Navier-Stokes equations in the incompressible limit. Effects of spatial resolution and choices in subgridscale modeling were explicitly investigated with the experimental data set as the testing ground. Experimentally observed flow features, such as vortex breakdown and laminarization of the vortex core were well represented by the LES. The simulations confirmed the experimental observations that the average velocity profiles in the entire vortex tube are extremely sensitivity to the exit pipe diameter. For the narrowest exit pipe considered in the simulations, very high average velocity gradients are encountered. In this situation, the LES shows the most pronounced effects of spatial resolution and subgrid-scale modeling.

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Section: Average Velocity Profilesmentioning

confidence: 99%

Simulations of confined turbulent vortex flow

Derksen

2005

Computers & Fluids

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“…Stretched vortices of Burgers type, which are exact solutions of the Navier-Stokes equations, are often used as typical solutions to illustrate the tube-sheet paradigm of modern turbulence theory [1][2][3][4][5][6][7][8][9]. In reality they are pseudo-3D in nature as they are composed of 2D flows superimposed on a 3D vorticity free strain field, a fact exploited by Lundgren in his transformation [2].…”

Section: Introductionmentioning

confidence: 99%

“…Depending on whether a tube or shear layer symmetry has been chosen, corresponding to uni-axial or bi-axial strain [3], they have the properties that the vorticity ω ω ω firstly lies respectively either along the axis of the tube or in the plane of the layer and secondly that it aligns with an eigenvector of the strain matrix S. While numerical simulations [10,11] and experiments [12] have shown that, in a spatially averaged sense, alignment of the vorticity vector ω ω ω with the intermediate eigenvector of S is favoured (for a list of references see [13]), it is clear that local vorticity accumulation and alignment processes are more complicated than this [14,15]. For instance, the solution for the Burgers vortex has the drawback that it is stretched by a strain field that is decoupled from the flow around it and that its vorticity is unidirectional [1][2][3]. Complicated vortical structures caused by both vortex stretching and compression have a dynamic complexity that requires a more subtle theoretical explanation (see the recent review by Pullin and Saffman [16]).…”

Section: Introductionmentioning

confidence: 99%

Dynamically stretched vortices as solutions of the 3D Navier–Stokes equations

Gibbon

Fokas

Doering

1999

Physica D: Nonlinear Phenomena

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A well known limitation with stretched vortex solutions of the 3D Navier-Stokes (and Euler) equations, such as those of Burgers type, is that they possess uni-directional vorticity which is stretched by a strain field that is decoupled from them. It is shown here that these drawbacks can be partially circumvented by considering a class of velocity fields of the type u u u = (u 1 (x, y, t), u 2 (x, y, t), γ (x, y, t)z + W (x, y, t)) where u 1 , u 2 , γ and W are functions of x, y and t but not z. It turns out that the equations for the third component of vorticity ω 3 and W decouple. More specifically, solutions of Burgers type can be constructed by introducing a strain field into u u u such that u u u = (−(γ /2)x − (γ /2)y, γ z) + −ψ y , ψ x , W . The strain rate, γ (t), is solely a function of time and is related to the pressure via a Riccati equationγ + γ 2 + p zz (t) = 0. A constraint on p zz (t) is that it must be spatially uniform. The decoupling of ω 3 and W allows the equation for ω 3 to be mapped to the usual general 2D problem through the use of Lundgren's transformation, while that for W can be mapped to the equation of a 2D passive scalar. When ω 3 stretches then W compresses and vice versa. Various solutions for W are discussed and some 2π -periodic θ -dependent solutions for W are presented which take the form of a convergent power series in a similarity variable. Hence the vorticity ω ω ω = r −1 W θ , −W r , ω 3 has nonzero components in the azimuthal and radial as well as the axial directions. For the Euler problem, the equation for W can sustain a vortex sheet type of solution where jumps in W occur when θ passes through multiples of 2π .

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“…The Burgers-Rott vortex (Burgers, 1948) is similar in form to the Lamb-Oseen profile with two notable exceptions. First, rather than a time-dependent decay, the exponential function here is governed by the suction parameter, A. Secondly, the Burgers-Rott vortex possesses well-defined relations for the axial and radial velocities.…”

Section: Unidirectional Modelsmentioning

confidence: 76%