The modified cuinulant expansion for two-dimensional isotropic turbulence

1981 ◽  
Vol 110 ◽  
pp. 475-496 ◽  
Author(s):  
Tomomasa Tatsumi ◽  
Shinichiro Yanase
Keyword(s):  
Numerical Solutions ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Three Dimensional ◽  
Critical Time ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Inertial Subrange ◽  
Quasi Equilibrium ◽  
Two Dimensional

The two-dimensional isotropic turbulence in an incompressible fluid is investigated using the modified zero fourth-order cumulant approximation. The dynamical equation for the energy spectrum obtained under this approximation is solved numerically and the similarity laws governing the solution in the energy-containing and enstrophy-dissipation ranges are derived analytically. At large Reynolds numbers the numerical solutions yield the k−3 inertial subrange spectrum which was predicted by Kraichnan (1967), Leith (1968) and Batchelor (1969) assuming a finite enstrophy dissipation in the inviscid limit. The energy-containing range is found to satisfy an inviscid similarity while the enstrophy-dissipation range is governed by the quasi-equilibrium similarity with respect to the enstrophy dissipation as proposed by Batchelor (1969). There exists a critical time tc which separates the initial period (t < tc) and the similarity period (t > tc) in which the enstrophy dissipation vanishes and remains non-zero respectively in the inviscid limit. Unlike the case of three-dimensional turbulence, tc is not fixed but increases indefinitely as the viscosity tends to zero.

Similarity and self-preservation in isotropic turbulence

1951 ◽  
Vol 243 (867) ◽  
pp. 359-386 ◽  
Keyword(s):  
Energy Spectrum ◽  
Isotropic Turbulence ◽  
Initial Period ◽  
Reynolds Numbers ◽  
Wave Number ◽  
Inertial Subrange ◽  
Local Similarity ◽  
Mean Flow ◽  
Spectrum Function ◽  
Energy Spectrum Function

Measurements of the double and triple velocity correlation functions and of the energy spectrum function have been made in the uniform mean flow behind turbulence-producing grids of several shapes at mesh Reynolds numbers between 2000 and 100000. These results have been used to assess the validity of the various theories which postulate greater or less degrees of similarity or self-preservation between decaying fields of isotropic turbulence. It is shown that the conditions for the existence of the local similarity considered by Kolmogoroff and others are only fulfilled for extremely small eddies at ordinary Reynolds numbers, and that the inertial subrange in which the spectrum function varies as k -35 ( k is the wave-number) is non-existent under laboratory conditions. Within the range of local similarity, the spectrum function is best represented by an empirical function such as k -a log k , and it is concluded that all suggested forms for the inertial transfer term in the spectrum equation are in error. Similarity of the large scale structure of flows of differing Reynolds numbers at corresponding times of decay has been confirmed, and approximate measurements of the Loitsianski invariant in the initial period have been made. Its value, expressed non-dimensionally, decreases slowly with grid Reynolds number within the range of observation. Turbulence-producing grids of widely different shapes are found to produce flows identical in energy decay and in structure of the smaller eddies. The largest eddies depend markedly on the grid shape and are, in general, significantly anisotropic. Within the initial period of decay, the greater part of the energy spectrum function is self-preserving, and this part has a shape independent of the shape of the turbulence-producing grid. The part that is not self-preserving contains at least one-third of the total energy, and it is concluded that theories postulating quasi-equilibrium during decay must be considered with great caution.

Bifurcation Characteristics of Flows in Rectangular Sudden Expansion Channels

2005 ◽  
Author(s):  
Francine Battaglia ◽  
George Papadopoulos
Keyword(s):  
Reynolds Number ◽  
Laminar Flow ◽  
Critical Reynolds Number ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Expansion Ratio ◽  
Sudden Expansion ◽  
Two Dimensional ◽  
Effective Expansion ◽  
Three Dimensional Simulations

The effect of three-dimensionality on low Reynolds number flows past a symmetric sudden expansion in a channel was investigated. The geometric expansion ratio of in the current study was 2:1 and the aspect ratio was 6:1. Both experimental velocity measurements and two- and three-dimensional simulations for the flow along the centerplane of the rectangular duct are presented for Reynolds numbers in the range of 150 to 600. Comparison of the two-dimensional simulations with the experiments revealed that the simulations fail to capture completely the total expansion effect on the flow, which couples both geometric and hydrodynamic effects. To properly do so requires the definition of an effective expansion ratio, which is the ratio of the downstream and upstream hydraulic diameters and is therefore a function of both the expansion and aspect ratios. When the two-dimensional geometry was consistent with the effective expansion ratio, the new results agreed well with the three-dimensional simulations and the experiments. Furthermore, in the range of Reynolds numbers investigated, the laminar flow through the expansion underwent a symmetry-breaking bifurcation. The critical Reynolds number evaluated from the experiments and the simulations was compared to other values reported in the literature. Overall, side-wall proximity was found to enhance flow stability, helping to sustain laminar flow symmetry to higher Reynolds numbers in comparison to nominally two-dimensional double-expansion geometries. Lastly, and most importantly, when the logarithm of the critical Reynolds number from all these studies was plotted against the reciprocal of the effective expansion ratio, a linear trend emerged that uniquely captured the bifurcation dynamics of all symmetric double-sided planar expansions.

Cellular flow in a partially filled rotating drum: regular and chaotic advection

2017 ◽  
Vol 825 ◽  
pp. 631-650 ◽  
Author(s):  
Francesco Romanò ◽  
Arash Hajisharifi ◽  
Hendrik C. Kuhlmann
Keyword(s):  
Three Dimensional ◽  
Reynolds Numbers ◽  
Chaotic Advection ◽  
Two Dimensional ◽  
Rotating Drum ◽  
Three Dimensional Flow ◽  
Dimensional Flow ◽  
Heteroclinic Connection ◽  
Two Dimensional Flow ◽  
Chaotic Streamlines

The topology of the incompressible steady three-dimensional flow in a partially filled cylindrical rotating drum, infinitely extended along its axis, is investigated numerically for a ratio of pool depth to radius of 0.2. In the limit of vanishing Froude and capillary numbers, the liquid–gas interface remains flat and the two-dimensional flow becomes unstable to steady three-dimensional convection cells. The Lagrangian transport in the cellular flow is organised by periodic spiralling-in and spiralling-out saddle foci, and by saddle limit cycles. Chaotic advection is caused by a breakup of a degenerate heteroclinic connection between the two saddle foci when the flow becomes three-dimensional. On increasing the Reynolds number, chaotic streamlines invade the cells from the cell boundary and from the interior along the broken heteroclinic connection. This trend is made evident by computing the Kolmogorov–Arnold–Moser tori for five supercritical Reynolds numbers.

scholarly journals Nonlinear amplification in hydrodynamic turbulence

2021 ◽  
Vol 930 ◽  
Author(s):  
Kartik P. Iyer ◽  
Katepalli R. Sreenivasan ◽  
P.K. Yeung
Keyword(s):  
Reynolds Number ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Developed Turbulence ◽  
Advection Term ◽  
Nonlinear Amplification

Using direct numerical simulations performed on periodic cubes of various sizes, the largest being $8192^3$ , we examine the nonlinear advection term in the Navier–Stokes equations generating fully developed turbulence. We find significant dissipation even in flow regions where nonlinearity is locally absent. With increasing Reynolds number, the Navier–Stokes dynamics amplifies the nonlinearity in a global sense. This nonlinear amplification with increasing Reynolds number renders the vortex stretching mechanism more intermittent, with the global suppression of nonlinearity, reported previously, restricted to low Reynolds numbers. In regions where vortex stretching is absent, the angle and the ratio between the convective vorticity and solenoidal advection in three-dimensional isotropic turbulence are statistically similar to those in the two-dimensional case, despite the fundamental differences between them.

Experimental characterization of three-dimensional corner flows at low Reynolds numbers

2012 ◽  
Vol 707 ◽  
pp. 37-52 ◽  
Author(s):  
J. Sznitman ◽  
L. Guglielmini ◽  
D. Clifton ◽  
D. Scobee ◽  
H. A. Stone ◽  
...  
Keyword(s):  
Reynolds Number ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Secondary Flows ◽  
Channel Cross Section ◽  
Experimental Characterization ◽  
Low Reynolds Numbers ◽  
Low Reynolds

AbstractWe investigate experimentally the characteristics of the flow field that develops at low Reynolds numbers ($\mathit{Re}\ll 1$) around a sharp $9{0}^{\ensuremath{\circ} } $ corner bounded by channel walls. Two-dimensional planar velocity fields are obtained using particle image velocimetry (PIV) conducted in a towing tank filled with a silicone oil of high viscosity. We find that, in the vicinity of the corner, the steady-state flow patterns bear the signature of a three-dimensional secondary flow, characterized by counter-rotating pairs of streamwise vortical structures and identified by the presence of non-vanishing transverse velocities (${u}_{z} $). These results are compared to numerical solutions of the incompressible flow as well as to predictions obtained, for a similar geometry, from an asymptotic expansion solution (Guglielmini et al., J. Fluid Mech., vol. 668, 2011, pp. 33–57). Furthermore, we discuss the influence of both Reynolds number and aspect ratio of the channel cross-section on the resulting secondary flows. This work represents, to the best of our knowledge, the first experimental characterization of the three-dimensional flow features arising in a pressure-driven flow near a corner at low Reynolds number.

Three-dimensional instabilities in the boundary-layer flow over a long rectangular plate

2011 ◽  
Vol 681 ◽  
pp. 411-433 ◽  
Author(s):  
HEMANT K. CHAURASIA ◽  
MARK C. THOMPSON
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Three Dimensional ◽  
Reynolds Numbers ◽  
Vortical Flow ◽  
Forced Flow ◽  
Recirculation Region ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Dimensional Instability

A detailed numerical study of the separating and reattaching flow over a square leading-edge plate is presented, examining the instability modes governing transition from two- to three-dimensional flow. Under the influence of background noise, experiments show that the transition scenario typically is incompletely described by either global stability analysis or the transient growth of dominant optimal perturbation modes. Instead two-dimensional transition effectively can be triggered by the convective Kelvin–Helmholtz (KH) shear-layer instability; although it may be possible that this could be described alternatively in terms of higher-order optimal perturbation modes. At least in some experiments, observed transition occurs by either: (i) KH vortices shedding downstream directly and then almost immediately undergoing three-dimensional transition or (ii) at higher Reynolds numbers, larger vortical structures are shed that are also three-dimensionally unstable. These two paths lead to distinctly different three-dimensional arrangements of vortical flow structures. This paper focuses on the mechanisms underlying these three-dimensional transitions. Floquet analysis of weakly periodically forced flow, mimicking the observed two-dimensional quasi-periodic base flow, indicates that the two-dimensional vortex rollers shed from the recirculation region become globally three-dimensionally unstable at a Reynolds number of approximately 380. This transition Reynolds number and the predicted wavelength and flow symmetries match well with those of the experiments. The instability appears to be elliptical in nature with the perturbation field mainly restricted to the cores of the shed rollers and showing the spatial vorticity distribution expected for that instability type. Indeed an estimate of the theoretical predicted wavelength is also a good match to the prediction from Floquet analysis and theoretical estimates indicate the growth rate is positive. Fully three-dimensional simulations are also undertaken to explore the nonlinear development of the three-dimensional instability. These show the development of the characteristic upright hairpins observed in the experimental dye visualisations. The three-dimensional instability that manifests at lower Reynolds numbers is shown to be consistent with an elliptic instability of the KH shear-layer vortices in both symmetry and spanwise wavelength.

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

1992 ◽  
Vol 238 ◽  
pp. 1-30 ◽  
Author(s):  
George Em Karniadakis ◽  
George S. Triantafyllou
Keyword(s):  
Reynolds Number ◽  
Three Dimensional ◽  
Period Doubling ◽  
Reynolds Numbers ◽  
Transition To Turbulence ◽  
Two Dimensional ◽  
Near Wake ◽  
Vortex Filaments ◽  
Average Flow ◽  
Time Average

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast’ transition, from a laminar two-dimensional state at Reynolds number 200 to a turbulent state at Reynolds number 400. The process has been documented in several experimental investigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the Navier—Stokes equations at representative Reynolds numbers, up to 500. A high-order time-accurate, mixed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vortex street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vortex filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vortex filaments.Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance of intermittent phenomena. It is concluded that the wake undergoes transition to turbulence following the period-doubling route.

scholarly journals Planar flows past thin multi-blade configurations

1996 ◽  
Vol 324 ◽  
pp. 355-377 ◽  
Author(s):  
F. T. Smith ◽  
S. N. Timoshin
Keyword(s):  
Boundary Layer ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Laminar Flows ◽  
Two Dimensional ◽  
Boundary Layer Equations ◽  
Require Solution ◽  
Lift And Drag ◽  
Planar Flows ◽  
Steady Laminar

Two-dimensional steady laminar flows past multiple thin blades positioned in near or exact sequence are examined for large Reynolds numbers. Symmetric configurations require solution of the boundary-layer equations alone, in parabolic fashion, over the successive blades. Non-symmetric configurations in contrast yield a new global inner–outer interaction in which the boundary layers, the wakes and the potential flow outside have to be determined together, to satisfy pressure-continuity conditions along each successive gap or wake. A robust computational scheme is used to obtain numerical solutions in direct or design mode, followed by analysis. Among other extremes, many-blade analysis shows a double viscous structure downstream with two streamwise length scales operating there. Lift and drag are also considered. Another new global interaction is found further downstream. All the interactions involved seem peculiar to multi-blade flows.

A two-dimensional vortex condensate at high Reynolds number

2013 ◽  
Vol 715 ◽  
pp. 359-388 ◽  
Author(s):  
Basile Gallet ◽  
William R. Young
Keyword(s):  
Stokes Equation ◽  
Numerical Solutions ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Finite Limit ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Quasilinear Theory ◽  
Zero Integral ◽  
High Reynolds

AbstractWe investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

Flow separation on a spheroid at incidence

1979 ◽  
Vol 92 (4) ◽  
pp. 643-657 ◽  
Author(s):  
Taeyoung Han ◽  
V. C. Patel
Keyword(s):  
Reynolds Number ◽  
Wind Tunnel ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Inviscid Flow ◽  
Reynolds Numbers ◽  
Surface Flow ◽  
Laminar Separation ◽  
Low Reynolds Numbers ◽  
Turbulent Separation

Surface streamline patterns on a spheroid have been examined at several angles of attack. Most of the tests were performed at low Reynolds numbers in a hydraulic flume using coloured dye to make the surface flow visible. A limited number of experiments was also carried out in a wind tunnel, using wool tufts, to study the influence of Reynolds number and turbulent separation. The study has verified some of the important qualitative features of three-dimensional separation criteria proposed earlier by Maskell, Wang and others. The observed locations of laminar separation lines on a spheroid at various incidences have been compared with the numerical solutions of Wang and show qualitative agreement. The quantitative differences are attributed largely to the significant viscous-inviscid flow interaction which is present, especially at large incidences.

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