A Survey on a Class of Exact Solutions of the Navier–Stokes Equations and a Model for Turbulence

Ohkitani, Koji

doi:10.2977/prims/1145475447

Cited by 7 publications

(6 citation statements)

References 36 publications

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“…In accordance with Lundgren, 4 the axial vorticity contributes to give Ϫ5/3 spectrum. The same approach was applied to the contribution of the azimuthal vorticity in Ohkitani 15 and it was shown that this vorticity gives Ϫ7/3 spectrum.…”

Section: Introductionmentioning

confidence: 86%

Multimode stretched spiral vortex and nonequilibrium energy spectrum in homogeneous shear flow turbulence

Horiuti

Ozawa

2011

Physics of Fluids

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The stretched spiral vortex ͓T. S. Lundgren, "Strained spiral vortex model for turbulent structures," Phys. Fluids 25, 2193 ͑1982͔͒ is identified in turbulence in homogeneous shear flow and the spectral properties of this flow are studied using direct-numerical simulation data. The effects of mean shear on the genesis, growth, and annihilation processes of the spiral vortex are elucidated, and the role of the spiral vortex in the generation of turbulence is shown. As in homogeneous isotropic turbulence ͓K. Horiuti and T. Fujisawa, "The multi mode stretched spiral vortex in homogeneous isotropic turbulence," J. Fluid Mech. 595, 341 ͑2008͔͒, multimodes of the spiral vortex are extracted. Two symmetric modes of configurations with regard to the vorticity alignment along the vortex tube in the core region and dual vortex sheets spiraling around the tube are often educed. One of the two symmetric modes is created by a conventional rolling-up of a single spanwise shear layer. Another one is created by the convergence of the recirculating flow or streamwise roll ͓F. Waleffe, "Homotopy of exact coherent structures in plane shear flows," Phys. Fluids 15, 1517 ͑2003͔͒ caused by the upward and downward motions associated with the streaks. The vortex tube is formed by axial straining and lowering of pressure in the recirculating region. The spanwise shear layers are entrained by the tube and they form spiral turns. The latter symmetric mode tends to be transformed into the former mode with lapse of time due to the action of the pressure Hessian term. The power law in the inertial subrange energy spectrum is studied. The base steady spectrum fits the equilibrium Kolmogorov Ϫ5/3 spectrum, to which a nonequilibrium component induced by the fluctuation of the dissipation rate is added. This component is extracted using the conditional sampling on , and it is shown that it fits the Ϫ7/3 power in accordance with the statistical theory. The correlation between these spectra and the appearance and diminution of the streaks and the two modes of the spiral vortex is discussed. The temporal variations of the spectrum are divided into two regimes, Phases 1 and 2. Large energy contained in the low-wavenumber range in Phase 1 is cascaded to the small scales in Phase 2. This energy transfer is accomplished by the reversal in the sign of Ϫ7/3 power component.

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Section: Introductionmentioning

confidence: 86%

Multimode stretched spiral vortex and nonequilibrium energy spectrum in homogeneous shear flow turbulence

Horiuti

Ozawa

2011

Physics of Fluids

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“…Note that the solution for constant pressure gradient is similar to that for zero pressure gradient (17) because there will be a constant term in (5b), (6) and (7), and it can be expressed as the same polynomial in (8). It is interesting to note that more general solutions to (13) can be found by substituting additional terms which then resemble the following:…”

Section: Solutions For Pmentioning

confidence: 92%

“…They may help explain the issue of global smoothness in time [5]. Moreover, the solutions may be examined as models for turbulence [6].…”

Section: Introductionmentioning

confidence: 99%

On a special class of analytical solutions to the three-dimensional incompressible Navier–Stokes equations

Nugroho

Ali

Karim

2009

Applied Mathematics Letters

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a b s t r a c tThe three-dimensional incompressible Navier-Stokes equations with the continuity equation are solved analytically in this work. The spatial and temporal coordinates are transformed into a single coordinate ξ . The solution is proposed to be in the form V = ∇Φ + ∇ × Φ where Φ is a potential function that is defined as Φ = P(x, ξ )R(ξ ). The potential function is firstly substituted into the continuity equation to produce the solution for R and the resultant expression is used sequentially in the Navier-Stokes equations to reduce the problem to the class of nonlinear ordinary differential equations in P terms. Here, more general solutions are also obtained based on the particular solutions of P. Explicit analytical solutions are found to be mathematically similar for the cases of zero and constant pressure gradient. Two examples are given to illustrate the applicability of the method. It is also concluded that the selection of variables for the potential function can be interchanged from the beginning, resulting in similar explicit solutions.

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“…Lundgren 17 has obtained a largetime asymptotic solution to the Navier-Stokes equations for the spiral layers of the axial vorticity subjected to an axisymmetric irrotational strain and shown that his spiral vortices exhibit the energy spectrum with a range of a Kolmogorov k −5/3 falloff ͑see also the works of Lundgren 24 and Gilbert 25 ͒. The spirals of the cross-axial vorticity in the axisymmetric straining flow have a k −7/3 spectrum at large wavenumbers, 26,27 and the energy dissipation for the spirals of the axial and the cross-axial vorticities can be of comparable order. 26 Recently, in their direct numerical simulation of isotropic turbulence, Kida and Miura 28 have found the existence of double spiral vortex layers with the dominant azimuthal compoment of vorticity, that is, the latter class of spirals.…”

Section: Introductionmentioning

confidence: 96%

Energy dissipation in spiral vortex layers wrapped around a straight vortex tube

Kawahara

2005

Physics of Fluids

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Energy dissipation in spiral layers of high azimuthal vorticity around a straight vortex tube is investigated analytically. Asymptotic expressions of local and total viscous dissipation are obtained for the spiral vortex layers. When a vortex tube, which aligns with a uniform shear flow of a shear rate S, starts with a vortex filament of circulation ⌫ at an initial instant t = 0, it wraps and stretches background vorticity lines around itself to form double spiral vortex layers of intense dissipation. The contribution of the spiral layers to total dissipation per unit axial length is evaluated to be 1.29 2 S 2 ͑⌫ /2͒ 4/3 t at large vortex Reynolds numbers ⌫ / ӷ 1, being the kinematic viscosity of fluid. There exists the critical time after which the contribution of the spirals to the total dissipation dominates that of the tube. If the tube is tilted at a small angle ␣ in the direction of the uniform shear vorticity, the spirals around the tube are cross axially sheared into different shapes depending on the sign of ␣, which leads to local reduction ͑or enhancement͒ of the energy dissipation in the spirals at ␣ Ͼ 0 ͑or Ͻ0͒. The primary effect of the cross-axial shear on the total dissipation is shown to be − 1 4 ␣S 3 ⌫ ln͑⌫ /2͒t 2 at St͉␣͉ Ӷ 1 for ⌫ / ӷ 1. The contribution to turbulent energy dissipation from spiral structures around a tubular vortex at a large-Reynolds-number limit is also discussed based upon recently reported direct numerical simulations and the present analysis.

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A Survey on a Class of Exact Solutions of the Navier–Stokes Equations and a Model for Turbulence

Cited by 7 publications

References 36 publications

Multimode stretched spiral vortex and nonequilibrium energy spectrum in homogeneous shear flow turbulence

Multimode stretched spiral vortex and nonequilibrium energy spectrum in homogeneous shear flow turbulence

On a special class of analytical solutions to the three-dimensional incompressible Navier–Stokes equations

Energy dissipation in spiral vortex layers wrapped around a straight vortex tube

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