Sensitivity of horizontal flows to forcing geometry

2001 ◽  
Vol 432 ◽  
pp. 419-441
Author(s):  
ISAO KANDA ◽  
P. F. LINDEN
Keyword(s):  
Reynolds Number ◽  
Analytical Solutions ◽  
Stokes Flow ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Vortex Pattern ◽  
Inverse Energy Cascade ◽  
Horizontal Flows ◽  
Source Sink ◽  
Special Case

We investigate the horizontal flow produced by source–sink forcing in a stably stratified fluid. The forcing jets are kept laminar and are placed along the boundary of a square domain. We find that the resultant flow patterns are extremely sensitive to the forcing geometry. The single dominant vortex pattern, interpreted as the result of inverse energy cascade of two-dimensional turbulence in our previous work (Boubnov, Dalziel & Linden 1994), turns out to be a special case. We show that some of the steady patterns resemble the eigenmodes of the Helmholtz equation as the inviscid vorticity equation. Although there are significant discrepancies in the streamfunction vs. vorticity relations between the observed flows and the analytical solutions, we identify the differences as a result of viscous diffusion of vorticity from the source flows. We also study the transition from forced to decaying flow. The flow assumes the properties of Stokes flow at quite large Reynolds number, indicating transformation into patterns with small advective acceleration.

Infrared Reynolds number dependency of the two-dimensional inverse energy cascade

2011 ◽  
Vol 667 ◽  
pp. 463-473 ◽  
Author(s):  
ANDREAS VALLGREN
Keyword(s):  
Reynolds Number ◽  
Cascade Range ◽  
Energy Cascade ◽  
Energy Spectra ◽  
Small Scale ◽  
Two Dimensional ◽  
Inverse Energy Cascade ◽  
Linear Friction ◽  
Non Local ◽  
Reynolds Number Dependency

High-resolution simulations of forced two-dimensional turbulence reveal that the inverse cascade range is sensitive to an infrared Reynolds number, Reα = kf/kα, where kf is the forcing wavenumber and kα is a frictional wavenumber based on linear friction. In the limit of high Reα, the classic k−5/3 scaling is lost and we obtain steeper energy spectra. The sensitivity is traced to the formation of vortices in the inverse energy cascade range. Thus, it is hypothesized that the dual limit Reα → ∞ and Reν = kd/kf → ∞, where kd is the small-scale dissipation wavenumber, will lead to a steeper energy spectrum than k−5/3 in the inverse energy cascade range. It is also found that the inverse energy cascade is maintained by non-local triad interactions.

Computing internal viscous flow problems for the circle by integral methods

1977 ◽  
Vol 79 (3) ◽  
pp. 609-624 ◽  
Author(s):  
R. D. Mills
Keyword(s):  
Reynolds Number ◽  
Circular Cylinder ◽  
Integral Representation ◽  
Analytical Solutions ◽  
Reynolds Numbers ◽  
Cylinder Wall ◽  
Two Dimensional ◽  
Discontinuous Surface ◽  
Flow Problems ◽  
Integral Methods

Steady two-dimensional viscous motion within a circular cylinder generated by (a) the rotation of part of the cylinder wall and (b) fluid entering and leaving through slots in the wall is considered. Studied in particular are moving-surface problems with continuous and discontinuous surface speeds, simple inflow–outflow problems and the impinging-jet problem within a circle. The analytical solutions at zero Reynolds number are given for the last two types of problem. Numerical results are obtained at various Reynolds numbers from the integral representation of the solution. At zero Reynolds number this approach involves a quadrature around the circumference of the cylinder. At other Reynolds numbers it involves an iterative–integral technique based on the use of the ‘clamped-plate’ biharmonic Green's function.

Non-axisymmetric vortices in two-dimensional flows

2000 ◽  
Vol 406 ◽  
pp. 175-198 ◽  
Author(s):  
STÉPHANE LE DIZÈS
Keyword(s):  
Reynolds Number ◽  
External Field ◽  
Strain Field ◽  
Asymptotic Methods ◽  
Angular Frequency ◽  
Critical Layer ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Axisymmetric Structures ◽  
Two Dimensional Flows

Slightly non-axisymmetric vortices are analysed by asymptotic methods in the context of incompressible large-Reynolds-number two-dimensional flows. The structure of the non-axisymmetric correction generated by an external rotating multipolar strain field to a vortex with a Gaussian vorticity profile is first studied. It is shown that when the angular frequency w of the external field is in the range of the angular velocity of the vortex, the non-axisymmetric correction exhibits a critical-point singularity which requires the introduction of viscosity or nonlinearity to be smoothed. The nature of the critical layer, which depends on the parameter h = 1/(Re ε3/2), where ε is the amplitude of the non-axisymmetric correction and Re the Reynolds number based on the circulation of the vortex, is found to govern the entire structure of the correction. Numerous properties are analysed as w and h vary for a multipolar strain field of order n = 2, 3, 4 and 5. In the second part of the paper, the problem of the existence of a non-axisymmetric correction which can survive without external field due to the presence of a nonlinear critical layer is addressed. For a family of vorticity profiles ranging from Gaussian to top-hat, such a correction is shown to exist for particular values of the angular frequency. The resulting non-axisymmetric vortices are analysed in detail and compared to recent computations by Rossi, Lingevitch & Bernoff (1997) and Dritschel (1998) of non-axisymmetric vortices. The results are also discussed in the context of electron columns where similar non-axisymmetric structures were observed (Driscoll & Fine 1990).

Application of the Cosserat Spectrum Theory to Stokes Flow

1999 ◽  
Vol 66 (3) ◽  
pp. 811-814
Author(s):  
W. Liu ◽  
A. Plotkin
Keyword(s):  
Reynolds Number ◽  
Velocity Field ◽  
Laplace Equation ◽  
Analytical Solutions ◽  
Stokes Flow ◽  
Low Reynolds Number ◽  
Flow Problems ◽  
Cosserat Spectrum ◽  
Linear Shear ◽  
Flow Past A Sphere

This paper presents an application of the Cosserat spectrum theory in elasticity to the solution of low Reynolds number (Stokes flow) problems. The velocity field is divided into two components: a solution to the vector Laplace equation and a solution associated with the discrete Cosserat eigenvectors. Analytical solutions are presented for the Stokes flow past a sphere with uniform, extensional, and linear shear freestream profiles.

scholarly journals Peristatic solutions for finite one- and two-dimensional systems

2016 ◽  
Vol 22 (8) ◽  
pp. 1639-1653 ◽  
Author(s):  
Vinesh V Nishawala ◽  
Martin Ostoja-Starzewski
Keyword(s):  
Numerical Method ◽  
Continuum Mechanics ◽  
Analytical Solutions ◽  
Differential Form ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Nonlocal Continuum ◽  
One Dimensional ◽  
Static Version ◽  
Special Case

Peridynamics is a nonlocal continuum mechanics theory where its governing equation has an integro-differential form. This paper specifically uses bond-based peridynamics. Typically, peridynamic problems are solved via numerical means, and analytical solutions are not as common. This paper analytically evaluates peristatics, the static version of peridynamics, for a finite one-dimensional rod as well as a special case for two dimensions. A numerical method is also implemented to confirm the analytical results.

Chaotic Mixing of Highly Viscous Liquids With Rectangular or Elliptical Rotors

2005 ◽  
Author(s):  
M. Erol Ulucakli
Keyword(s):  
Reynolds Number ◽  
Fixed Point ◽  
Fixed Points ◽  
Stokes Flow ◽  
Two Dimensional ◽  
Stable And Unstable Manifolds ◽  
Rectangular Cell ◽  
Viscous Liquids ◽  
Unstable Manifolds ◽  
Time Periodic

The objective of this research is to experimentally investigate various mixing regions in a two-dimensional Stokes flow driven by a rectangular or elliptical rotor. Flow occurs in a rectangular cell filled with a very viscous fluid. The Reynolds number based on rotor size is in the order of 0.5. The flow is time-periodic and can be analyzed, both theoretically and experimentally, by considering the Poincare map that maps the position of a fluid particle to its position one period later. The mixing regions of the flow are determined, theoretically, by the fixed points of this map, either hyperbolic or degenerate, and their stable and unstable manifolds. Experimentally, the mixing regions are visualized by releasing a blob of a passive dye at one of these fixed points: as the flow evolves, the blob stretches to form a streak line that lies on the unstable manifold of the fixed point.

Laminar Flow in a Uniformly Porous Channel

1964 ◽  
Vol 15 (3) ◽  
pp. 299-310 ◽  
Author(s):  
Thein Wah
Keyword(s):  
Differential Equation ◽  
Reynolds Number ◽  
Laminar Flow ◽  
Numerical Solutions ◽  
Large Reynolds Number ◽  
Two Dimensional ◽  
Series Solutions ◽  
Porous Walls ◽  
Linear Differential ◽  
Good Agreement

SummaryThe flow in a two-dimensional channel with porous walls through which fluid is uniformly injected or extracted is considered. Following Berman, a solution is obtained giving a fourth-order non-linear differential equation which depends on a suction Reynolds number R. Numerical solutions of this equation have been obtained. Series solutions of this equation for small and large Reynolds number are given and are shown to give good agreement with the numerical solutions.

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