Coriolis force influence on the AKA effect

2021 ◽  
Author(s):  
Peter Rutkevich ◽  
Georgy Golitsyn ◽  
Anatoly Tur
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Nonlinear Equations ◽  
Coriolis Force ◽  
Large Scale ◽  
Fluid Velocity ◽  
Navier Stokes ◽  
Basic Solution ◽  
Navier Stokes Equation ◽  
Alpha Effect

<p>Large-scale instability in incompressible fluid driven by the so called Anisotropic Kinetic Alpha (AKA) effect satisfying the incompressible Navier-Stokes equation with Coriolis force is considered. The external force is periodic; this allows applying an unusual for turbulence calculations mathematical method developed by Frisch et al [1]. The method provides the orders for nonlinear equations and obtaining large scale equations from the corresponding secular relations that appear at different orders of expansions. This method allows obtaining not only corrections to the basic solutions of the linear problem but also provides the large-scale solution of the nonlinear equations with the amplitude exceeding that of the basic solution. The fluid velocity is obtained by numerical integration of the large-scale equations. The solution without the Coriolis force leads to constant velocities at the steady-state, which agrees with the full solution of the Navier-Stokes equation reported previously. The time-invariant solution contains three families of solutions, however, only one of these families contains stable solutions. The final values of the steady-state fluid velocity are determined by the initial conditions. After account of the Coriolis force the solutions become periodic in time and the family of solutions collapses to a unique solution. On the other hand, even with the Coriolis force the fluid motion remains two-dimensional in space and depends on a single spatial variable. The latter fact limits the scope of the AKA method to applications with pronounced 2D nature. In application to 3D models the method must be used with caution.</p><p>[1] U. Frisch, Z.S. She and P. L. Sulem, “Large-Scale Flow Driven by the Anisotropic Kinetic Alpha Effect,” Physica D, Vol. 28, No. 3, 1987, pp. 382-392.</p>

scholarly journals Solutions of Navier-Stokes Equation with Coriolis Force

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Sunggeun Lee ◽  
Shin-Kun Ryi ◽  
Hankwon Lim
Keyword(s):  
Diffusion Equation ◽  
Stokes Equation ◽  
Coriolis Force ◽  
Potential Flow ◽  
Convective Diffusion ◽  
Navier Stokes ◽  
Coriolis Effect ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Convective Diffusion Equation

We investigate the Navier-Stokes equation in the presence of Coriolis force in this article. First, the vortex equation with the Coriolis effect is discussed. It turns out that the vorticity can be generated due to a rotation coming from the Coriolis effect, Ω. In both steady state and two-dimensional flow, the vorticity vector ω gets shifted by the amount of -2Ω. Second, we consider the specific expression of the velocity vector of the Navier-Stokes equation in two dimensions. For the two-dimensional potential flow v→=∇→ϕ, the equation satisfied by ϕ is independent of Ω. The remaining Navier-Stokes equation reduces to the nonlinear partial differential equations with respect to the velocity and the corresponding exact solution is obtained. Finally, the steady convective diffusion equation is considered for the concentration c and can be solved with the help of Navier-Stokes equation for two-dimensional potential flow. The convective diffusion equation can be solved in three dimensions with a simple choice of c.

scholarly journals The Rise and Fall of α-Model Viscosity

1996 ◽  
Vol 158 ◽  
pp. 111-114
Author(s):  
F. V. Hessman ◽  
C. Obach
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Accretion Disks ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Averaged Models

AbstractFor nearly two decades, our only useful model for the viscosity in accretion disks has been the so-called ‘α-model’. However, it has become clear that the simplest models –in which α is constant – are inadequate to explain the range of behaviours seen in real disks. We show that the properties of steady-state, vertically-averaged models can be determined without any assumptions other than that the disks obey the classical Navier-Stokes equation. These solutions have derived values of α which vary with radius by many orders of magnitude even in small CV disks.

RELAXATION METHOD FOR NAVIER–STOKES EQUATION

2012 ◽  
Vol 23 (04) ◽  
pp. 1250021 ◽  
Author(s):  
P. M. C. DE OLIVEIRA
Keyword(s):  
Steady State ◽  
Stokes Equation ◽  
Wind Velocity ◽  
Linear Time ◽  
Navier Stokes ◽  
Cylinder Surface ◽  
Theoretical Problem ◽  
Navier Stokes Equation ◽  
Transient Time ◽  
Ball Speed

The motivation for this work was a simple experiment [P. M. C. de Oliveira, S. Moss de Oliveira, F. A. Pereira and J. C. Sartorelli, preprint (2010), arXiv:1005.4086], where a little polystyrene ball is released falling in air. The interesting observation is a speed breaking. After an initial nearly linear time-dependence, the ball speed reaches a maximum value. After this, the speed finally decreases until its final, limit value. The provided explanation is related to the so-called von Kármán street of vortices successively formed behind the falling ball. After completely formed, the whole street extends for some hundred diameters. However, before a certain transient time needed to reach this steady-state, the street is shorter and the drag force is relatively reduced. Thus, at the beginning of the fall, a small and light ball may reach a speed superior to the sustainable steady-state value. Besides the real experiment, the numerical simulation of a related theoretical problem is also performed. A cylinder (instead of a 3D ball, thus reducing the effective dimension to 2) is positioned at rest inside a wind tunnel initially switched off. Suddenly, at t = 0 it is switched on with a constant and uniform wind velocity [Formula: see text] far from the cylinder and perpendicular to it. This is the first boundary condition. The second is the cylinder surface, where the wind velocity is null. In between these two boundaries, the velocity field is determined by solving the Navier–Stokes equation, as a function of time. For that, the initial condition is taken as the known Stokes laminar limit V → 0, since initially the tunnel is switched off. The numerical method adopted in this task is the object of the current text.

History forces and the unsteady wake of a cylinder

1999 ◽  
Vol 393 ◽  
pp. 99-121 ◽  
Author(s):  
J. R. CHAPLIN
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Stokes Equation ◽  
Oscillatory Flow ◽  
Numerical Solutions ◽  
Asymptotic Properties ◽  
Quantitative Agreement ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Unsteady Wake

History forces on a stationary cylinder in arbitrary unsteady rectilinear flow are calculated by means of a model based on the asymptotic properties of the steady-state wake. The results capture many features found in numerical solutions of the Navier–Stokes equation for the same flows, though quantitative agreement deteriorates as the Reynolds number increases over the range 2 to 40. The cases studied are the impulsive start, stop, and reverse, and oscillatory flow.

scholarly journals Isotropic turbulence in compact space

2017 ◽  
Vol 822 ◽  
pp. 512-560
Author(s):  
Elias Gravanis ◽  
Evangelos Akylas
Keyword(s):  
Stokes Equation ◽  
Compact Space ◽  
Correlation Functions ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Scale Effects ◽  
Normal Modes ◽  
Formal Theory ◽  
Navier Stokes ◽  
Navier Stokes Equation

Isotropic turbulence is typically studied numerically through direct numerical simulations (DNS). The DNS flows are described by the Navier–Stokes equation in a ‘box’, defined through periodic boundary conditions. Ideal isotropic turbulence lives in infinite space. The DNS flows live in a compact space and they are not isotropic in their large scales. Hence, the investigation of important phenomena of isotropic turbulence, such as anomalous scaling, through DNS is affected by large-scale effects in the currently available Reynolds numbers. In this work, we put isotropic turbulence – or better, the associated formal theory – in a ‘box’, by imposing periodicity at the level of the correlation functions. This is an attempt to offer a framework where one may investigate isotropic theories/models through the data of DNS in a manner as consistent with them as possible. We work at the lowest level of the hierarchy, which involves the two-point correlation functions and the Karman–Howarth equation. Periodicity immediately gives us the discrete wavenumber space of the theory. The wavenumbers start from 1.835, 2.896, 3.923, and progressively approach integer values, in an interesting correspondence with the DNS wavenumber shells. Unlike the Navier–Stokes equation, infinitely smooth periodicity is obstructed in this theory, a fact expressed by a sequence of relations obeyed by the normal modes of the Karman–Howarth equation at the endpoints of a unit period interval. Similar relations are imparted to the two-point functions under the condition that the energy spectrum and energy transfer function are realizable. Hence, these relations are necessary conditions for realizability in this theory. Naturally constructed closure schemes for the Karman–Howarth equation do not conform to such relations, thereby destroying realizability. A closure can be made to conform to a finite number of them by adding corrective terms, in a procedure that possesses certain analogies with the renormalization of quantum field theory. Perhaps the most important one is that we can let the spectrum be unphysical (through sign-changing oscillations of decreasing amplitude) for the infinitely large wavenumbers, as long as we can controllably extend the regime where the spectrum remains physical, deep enough in the dissipation subrange so as to be realistically adequate. Indeed, we show that one or two such ‘regularity relations’ are needed at most for comparisons of the predictions of the theory with the current resolution level results of the DNS. For the implementation of our arguments, we use a simple closure scheme previously proposed by Oberlack and Peters. The applicability of our ideas to more complex closures is also discussed.

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