Reynolds-number dependency in homogeneous, stationary two-dimensional turbulence

2010 ◽  
Vol 646 ◽  
pp. 517-526 ◽  
Author(s):  
ANNALISA BRACCO ◽  
JAMES C. MCWILLIAMS
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Large Scale ◽  
Velocity Structure ◽  
Stokes Equations ◽  
Turbulent Transport ◽  
Distribution Functions ◽  
Two Dimensional ◽  
Random Forcing ◽  
Statistical Measures

Turbulent solutions of the two-dimensional Navier–Stokes equations are a paradigm for the chaotic space–time patterns and equilibrium distributions of turbulent geophysical and astrophysical ‘thin’ flows on large horizontal scales. Here we investigate how homogeneous, stationary two-dimensional turbulence varies with the Reynolds number (Re) in stationary solutions with large-scale, random forcing and viscous diffusion, also including hypoviscous diffusion to limit the inverse energy cascade. This survey is made over the computationally feasible range in Re ≫ 1, approximately between 1.5 × 103 and 5.6 × 106. For increasing Re, we witness the emergence of vorticity fine structure within the filaments and vortex cores. The energy spectrum shape approaches the forward-enstrophy inertial-range form k−3 at large Re, and the velocity structure function is independent of Re. All other statistical measures investigated in this study exhibit power-law scaling with Re, including energy, enstrophy, dissipation rates and the vorticity structure function. The scaling exponents depend on the forcing properties through their influences on large-scale coherent structures, whose particular distributions are non-universal. A striking result is the Re independence of the intermittency measures of the flow, in contrast with the known behaviour for three-dimensional homogeneous turbulence of asymptotically increasing intermittency. This is a consequence of the control of the tails of the distribution functions by large-scale coherent vortices. Our analysis allows extrapolation towards the asymptotic limit of Re → ∞, fundamental to geophysical and astrophysical regimes and their large-scale simulation models where turbulent transport and dissipation must be parameterized.

Finite Reynolds number effect on the scaling range behaviour of turbulent longitudinal velocity structure functions

2017 ◽  
Vol 820 ◽  
pp. 341-369 ◽  
Author(s):  
S. L. Tang ◽  
R. A. Antonia ◽  
L. Djenidi ◽  
L. Danaila ◽  
Y. Zhou
Keyword(s):  
Reynolds Number ◽  
Structure Function ◽  
Large Scale ◽  
Velocity Structure ◽  
Longitudinal Velocity ◽  
Structure Functions ◽  
Third Order ◽  
Reynolds Number Effect ◽  
Rate Of Increase ◽  
Velocity Structure Function

The effect of large-scale forcing on the second- and third-order longitudinal velocity structure functions, evaluated at the Taylor microscale $r=\unicode[STIX]{x1D706}$, is assessed in various turbulent flows at small to moderate values of the Taylor microscale Reynolds number $R_{\unicode[STIX]{x1D706}}$. It is found that the contribution of the large-scale terms to the scale by scale energy budget differs from flow to flow. For a fixed $R_{\unicode[STIX]{x1D706}}$, this contribution is largest on the centreline of a fully developed channel flow but smallest for stationary forced periodic box turbulence. For decaying-type flows, the contribution lies between the previous two cases. Because of the difference in the large-scale term between flows, the third-order longitudinal velocity structure function at $r=\unicode[STIX]{x1D706}$ differs from flow to flow at small to moderate $R_{\unicode[STIX]{x1D706}}$. The effect on the second-order velocity structure functions appears to be negligible. More importantly, the effect of $R_{\unicode[STIX]{x1D706}}$ on the scaling range exponent of the longitudinal velocity structure function is assessed using measurements of the streamwise velocity fluctuation $u$, with $R_{\unicode[STIX]{x1D706}}$ in the range 500–1100, on the axis of a plane jet. It is found that the magnitude of the exponent increases as $R_{\unicode[STIX]{x1D706}}$ increases and the rate of increase depends on the order $n$. The trend of published structure function data on the axes of an axisymmetric jet and a two-dimensional wake confirms this dependence. For a fixed $R_{\unicode[STIX]{x1D706}}$, the exponent can vary from flow to flow and for a given flow, the larger $R_{\unicode[STIX]{x1D706}}$ is, the closer the exponent is to the value predicted by Kolmogorov (Dokl. Akad. Nauk SSSR, vol. 30, 1941a, pp. 299–303) (hereafter K41). The major conclusion is that the finite Reynolds number effect, which depends on the flow, needs to be properly accounted for before determining whether corrections to K41, arising from the intermittency of the energy dissipation rate, are needed. We further point out that it is imprudent, if not incorrect, to associate the finite Reynolds number effect with a consequence of the modified similarity hypothesis introduced by Kolmogorov (J. Fluid Mech., vol. 13, 1962, pp. 82–85) (K62); we contend that this association has misled the vast majority of post K62 investigations of the consequences of K62.

Turbulent diffusion in the geostrophic inverse cascade

2002 ◽  
Vol 469 ◽  
pp. 13-48 ◽  
Author(s):  
K. S. SMITH ◽  
G. BOCCALETTI ◽  
C. C. HENNING ◽  
I. MARINOV ◽  
C. Y. TAM ◽  
...  
Keyword(s):  
Drag Coefficient ◽  
Large Scale ◽  
Turbulent Transport ◽  
Passive Scalar ◽  
Turbulent Heat ◽  
Two Dimensional ◽  
Inverse Cascade ◽  
Random Forcing ◽  
Linear Drag ◽  
Geostrophic Turbulence

Motivated in part by the problem of large-scale lateral turbulent heat transport in the Earth's atmosphere and oceans, and in part by the problem of turbulent transport itself, we seek to better understand the transport of a passive tracer advected by various types of fully developed two-dimensional turbulence. The types of turbulence considered correspond to various relationships between the streamfunction and the advected field. Each type of turbulence considered possesses two quadratic invariants and each can develop an inverse cascade. These cascades can be modified or halted, for example, by friction, a background vorticity gradient or a mean temperature gradient. We focus on three physically realizable cases: classical two-dimensional turbulence, surface quasi-geostrophic turbulence, and shallow-water quasi-geostrophic turbulence at scales large compared to the radius of deformation. In each model we assume that tracer variance is maintained by a large-scale mean tracer gradient while turbulent energy is produced at small scales via random forcing, and dissipated by linear drag. We predict the spectral shapes, eddy scales and equilibrated energies resulting from the inverse cascades, and use the expected velocity and length scales to predict integrated tracer fluxes.When linear drag halts the cascade, the resulting diffusivities are decreasing functions of the drag coefficient, but with different dependences for each case. When β is significant, we find a clear distinction between the tracer mixing scale, which depends on β but is nearly independent of drag, and the energy-containing (or jet) scale, set by a combination of the drag coefficient and β. Our predictions are tested via high- resolution spectral simulations. We find in all cases that the passive scalar is diffused down-gradient with a diffusion coefficient that is well-predicted from estimates of mixing length and velocity scale obtained from turbulence phenomenology.

Two-dimensional global low-frequency oscillations in a separating boundary-layer flow

2008 ◽  
Vol 614 ◽  
pp. 315-327 ◽  
Author(s):  
UWE EHRENSTEIN ◽  
FRANÇOIS GALLAIRE
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
Perturbation Analysis ◽  
Boundary Layer Flow ◽  
Stokes Equations ◽  
Low Frequency ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Optimal Perturbation ◽  
Layer Flow

A separated boundary-layer flow at the rear of a bump is considered. Two-dimensional equilibrium stationary states of the Navier–Stokes equations are determined using a nonlinear continuation procedure varying the bump height as well as the Reynolds number. A global instability analysis of the steady states is performed by computing two-dimensional temporal modes. The onset of instability is shown to be characterized by a family of modes with localized structures around the reattachment point becoming almost simultaneously unstable. The optimal perturbation analysis, by projecting the initial disturbance on the set of temporal eigenmodes, reveals that the non-normal modes are able to describe localized initial perturbations associated with the large transient energy growth. At larger time a global low-frequency oscillation is found, accompanied by a periodic regeneration of the flow perturbation inside the bubble, as the consequence of non-normal cancellation of modes. The initial condition provided by the optimal perturbation analysis is applied to Navier–Stokes time integration and is shown to trigger the nonlinear ‘flapping’ typical of separation bubbles. It is possible to follow the stationary equilibrium state on increasing the Reynolds number far beyond instability, ruling out for the present flow case the hypothesis of some authors that topological flow changes are responsible for the ‘flapping’.

Axial flow in a two-dimensional microchannel induced by a travelling temperature wave imposed at the bottom wall

2018 ◽  
Vol 848 ◽  
pp. 1040-1072 ◽  
Author(s):  
Chenguang Zhang ◽  
Harris Wong ◽  
Krishnaswamy Nandakumar
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axial Flow ◽  
Temperature Wave ◽  
Bottom Wall ◽  
Moving Frame ◽  
Global Maximum ◽  
System Of Equations ◽  
Two Dimensional ◽  
Perturbation Solutions

Fluid flow in microchannels has wide industrial and scientific applications. Hence, it is important to explore different driving mechanisms. In this paper, we study the net transport or fluid pumping in a two-dimensional channel induced by a travelling temperature wave applied at the bottom wall. The Navier–Stokes equations with the Boussinesq approximation and the convection–diffusion heat equation are made dimensionless by the height of the channel and a velocity scale obtained by a dominant balance between buoyancy and viscous resistance in the momentum equation. The system of equations is transformed to an axial coordinate that moves with the travelling temperature wave, and we seek steady solutions in this moving frame. Four dimensionless numbers emerge from the governing equations and boundary conditions: the Reynolds number $Re$, a Reynolds number $Rc$ based on the wave speed, the Prandtl number $Pr$ and the dimensionless wavenumber $K$. The system of equations is solved by a finite-volume method and by a perturbation method in the limit $Re\rightarrow 0$. Surprisingly, the leading and first-order perturbation solutions agree well with the computed axial flow for $Re\leqslant 10^{3}$. Thus, the analytic perturbation solutions are used to study systematically the effects of $Re$, $Rc$, $Pr$ and $K$ on the dimensionless induced axial flow $Q$. We find that $Q$ varies linearly with $Re$, and $Q/Re$ versus any of the three remaining dimensionless groups always exhibits a maximum. The global maximum of $Q/Re$ in the parameter space is subsequently determined for the first time. This induced axial flow is driven solely by the Reynolds stress.

Fine Structure of Vortex Sheet Rollup by Viscous and Inviscid Simulation

1991 ◽  
Vol 113 (1) ◽  
pp. 31-36 ◽  
Author(s):  
G. Tryggvason ◽  
W. J. A. Dahm ◽  
K. Sbeih
Keyword(s):  
Reynolds Number ◽  
High Reynolds Number ◽  
Large Scale ◽  
Stokes Equations ◽  
Flow Region ◽  
Vortex Sheet ◽  
Navier Stokes ◽  
Vortex Sheets ◽  
Limiting Behavior ◽  
High Reynolds

Numerical simulations of the large amplitude stage of the Kelvin-Helmholtz instability of a relatively thin vorticity layer are discussed. At high Reynolds number, the effect of viscosity is commonly neglected and the thin layer is modeled as a vortex sheet separating one potential flow region from another. Since such vortex sheets are susceptible to a short wavelength instability, as well as singularity formation, it is necessary to provide an artificial “regularization” for long time calculations. We examine the effect of this regularization by comparing vortex sheet calculations with fully viscous finite difference calculations of the Navier-Stokes equations. In particular, we compare the limiting behavior of the viscous simulations for high Reynolds numbers and small initial layer thickness with the limiting solution for the roll-up of an inviscid vortex sheet. Results show that the inviscid regularization effectively reproduces many of the features associated with the thickness of viscous vorticity layers with increasing Reynolds number, though the simplified dynamics of the inviscid model allows it to accurately simulate only the large scale features of the vorticity field. Our results also show that the limiting solution of zero regularization for the inviscid model and high Reynolds number and zero initial thickness for the viscous simulations appear to be the same.

Direct numerical simulation of high-speed transition due to an isolated roughness element

2014 ◽  
Vol 748 ◽  
pp. 848-878 ◽  
Author(s):  
Pramod K. Subbareddy ◽  
Matthew D. Bartkowicz ◽  
Graham V. Candler
Keyword(s):  
Boundary Layer ◽  
Reynolds Number ◽  
High Speed ◽  
Large Scale ◽  
Stokes Equations ◽  
Roughness Element ◽  
Dynamic Mode Decomposition ◽  
Large Reynolds Number ◽  
Dynamic Mode ◽  
Low Dissipation

AbstractWe study the transition of a Mach 6 laminar boundary layer due to an isolated cylindrical roughness element using large-scale direct numerical simulations (DNS). Three flow conditions, corresponding to experiments conducted at the Purdue Mach 6 quiet wind tunnel are simulated. Solutions are obtained using a high-order, low-dissipation scheme for the convection terms in the Navier–Stokes equations. The lowest Reynolds number ($Re$) case is steady, whereas the two higher $Re$ cases break down to a quasi-turbulent state. Statistics from the highest $Re$ case show the presence of a wedge of fully developed turbulent flow towards the end of the domain. The simulations do not employ forcing of any kind, apart from the roughness element itself, and the results suggest a self-sustaining mechanism that causes the flow to transition at a sufficiently large Reynolds number. Statistics, including spectra, are compared with available experimental data. Visualizations of the flow explore the dominant and dynamically significant flow structures: the upstream shock system, the horseshoe vortices formed in the upstream separated boundary layer and the shear layer that separates from the top and sides of the cylindrical roughness element. Streamwise and spanwise planes of data were used to perform a dynamic mode decomposition (DMD) (Rowley et al., J. Fluid Mech., vol. 641, 2009, pp. 115–127; Schmid, J. Fluid Mech., vol. 656, 2010, pp. 5–28).

scholarly journals Direct numerical simulation of the oscillatory flow around a sphere resting on a rough bottom

2017 ◽  
Vol 822 ◽  
pp. 235-266 ◽  
Author(s):  
Marco Mazzuoli ◽  
Paolo Blondeaux ◽  
Julian Simeonov ◽  
Joseph Calantoni
Keyword(s):  
Reynolds Number ◽  
Large Scale ◽  
Oscillatory Flow ◽  
Stokes Equations ◽  
Coarse Sand ◽  
Spherical Particles ◽  
Navier Stokes ◽  
Sea Waves ◽  
Spherical Object ◽  
Navier Stokes Equations

The oscillatory flow around a spherical object lying on a rough bottom is investigated by means of direct numerical simulations of the continuity and Navier–Stokes equations. The rough bottom is simulated by a layer/multiple layers of spherical particles, the size of which is much smaller that the size of the object. The period and amplitude of the velocity oscillations of the free stream are chosen to mimic the flow at the bottom of sea waves and the size of the small spherical particles falls in the range of coarse sand/very fine gravel. Even though the computational costs allow only the simulation of moderate values of the Reynolds number characterizing the bottom boundary layer, the results show that the coherent vortex structures, shed by the spherical object, can break up and generate turbulence, if the Reynolds number of the object is sufficiently large. The knowledge of the velocity field allows the dynamics of the large-scale coherent vortices shed by the object to be determined and turbulence characteristics to be evaluated. Moreover, the forces and torques acting on both the large spherical object and the small particles, simulating sediment grains, can be determined and analysed, thus laying the groundwork for the investigation of sediment dynamics and scour developments.

The Flow of A Falling Ellipse: Numerical Method and Classification

2015 ◽  
Vol 31 (6) ◽  
pp. 771-782 ◽  
Author(s):  
R.-J. Wu ◽  
S.-Y. Lin
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Correction Method ◽  
Navier Stokes ◽  
Immersed Boundary ◽  
Two Dimensional ◽  
Inertia Moment ◽  
Navier Stokes Equations ◽  
Direct Forcing ◽  
Ib Method

AbstractA modified direct-forcing immersed-boundary (IB) pressure correction method is developed to simulate the flows of a falling ellipse. The pressure correct method is used to solve the solutions of the two dimensional Navier-Stokes equations and a direct-forcing IB method is used to handle the interaction between the flow and falling ellipse. For a fixed aspect ratio of an ellipse, the types of the behavior of the falling ellipse can be classified as three pure motions: Steady falling, fluttering, tumbling, and two transition motions: Chaos, transition between steady falling and fluttering. Based on two dimensionless parameters, Reynolds number and the dimensionless moment of inertia, a Reynolds number-inertia moment phase diagram is established. The behaviors and characters of five falling regimes are described in detailed.

GRANULAR TURBULENCE IN TWO DIMENSIONS: MICROSCALE REYNOLDS NUMBER AND FINAL CONDENSED STATES

2012 ◽  
Vol 23 (04) ◽  
pp. 1250032 ◽  
Author(s):  
MASAHARU ISOBE
Keyword(s):  
Reynolds Number ◽  
Energy Spectrum ◽  
Large Scale ◽  
Strong Relationship ◽  
Two Dimensions ◽  
Condensed State ◽  
Einstein Condensation ◽  
Two Dimensional ◽  
Developed Turbulence ◽  
Granular Gas

Granular gases from the viewpoint of "two-dimensional turbulence" are investigated. In the quasi-elastic and thermodynamic limit, we obtained clear evidence for an enstrophy (square of vorticity) cascade and -3 exponent in the Kraichnan–Leith–Bachelor energy spectrum by performing large-scale (N ~ 16.8 million number of disks) event-driven molecular dynamics simulations. In these calculations, the enstrophy dissipation rate showed a strong relationship with the evolution of the exponent in the energy spectrum. The growth of the Reynolds number based on the microscale confirmed that the enstrophy cascade regime was that of fully developed turbulence. Moreover, a condensed state resembling Bose–Einstein condensation in decaying two-dimensional Navier–Stokes turbulence also appeared as the final attractor of the evolving granular gas in the long time limit.

scholarly journals Turbulence in the molecular interstellar medium

2006 ◽  
Vol 2 (S237) ◽  
pp. 9-16
Author(s):  
Mark H. Heyer ◽  
Chris Brunt
Keyword(s):  
Star Formation ◽  
Interstellar Medium ◽  
Structure Function ◽  
Large Scale ◽  
Velocity Structure ◽  
Scaling Law ◽  
Local Environment ◽  
Cloud Data ◽  
Star Formation Activity ◽  
Velocity Structure Function

AbstractThe observational record of turbulence within the molecular gas phase of the interstellar medium is summarized. We briefly review the analysis methods used to recover the velocity structure function from spectroscopic imaging and the application of these tools on sets of cloud data. These studies identify a near-invariant velocity structure function that is independent of the local environment and star formation activity. Such universality accounts for the cloud-to-cloud scaling law between the global line-width and size of molecular clouds found by Larson (1981) and constrains the degree to which supersonic turbulence can regulate star formation. In addition, the evidence for large scale driving sources necessary to sustain supersonic flows is summarized.

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