Micro-structure and Lagrangian statistics of the scalar field with a mean gradient in isotropic turbulence

2003 ◽  
Vol 474 ◽  
pp. 193-225 ◽  
Author(s):  
G. BRETHOUWER ◽  
J. C. R. HUNT ◽  
F. T. M. NIEUWSTADT
Keyword(s):  
Scalar Field ◽  
Isotropic Turbulence ◽  
Numerical Study ◽  
Compressive Strain ◽  
Scalar Fields ◽  
Scalar Dissipation ◽  
Small Scale ◽  
Length Scale ◽  
Random Forcing ◽  
Kolmogorov Length Scale

This paper presents an analysis and numerical study of the relations between the small-scale velocity and scalar fields in fully developed isotropic turbulence with random forcing of the large scales and with an imposed constant mean scalar gradient. Simulations have been performed for a range of Reynolds numbers from Reλ = 22 to 130 and Schmidt numbers from Sc = 1/25 to 144.The simulations show that for all values of Sc [ges ] 0.1 steep scalar gradients are concentrated in intermittently distributed sheet-like structures with a thickness approximately equal to the Batchelor length scale η/Sc½ with η the Kolmogorov length scale. We observe that these sheets or cliffs are preferentially aligned perpendicular to the direction of the mean scalar gradient. Due to this preferential orientation of the cliffs the small-scale scalar field is anisotropic and this is an example of direct coupling between the large- and small-scale fluctuations in a turbulent field. The numerical simulations also show that the steep cliffs are formed by straining motions that compress the scalar field along the imposed mean scalar gradient in a very short time period, proportional to the Kolmogorov time scale. This is valid for the whole range of Sc. The generation of these concentration gradients is amplified by rotation of the scalar gradient in the direction of compressive strain. The combination of high strain rate and the alignment results in a large increase of the scalar gradient and therefore in a large scalar dissipation rate.These results of our numerical study are discussed in the context of experimental results (Warhaft 2000) and kinematic simulations (Holzer & Siggia 1994). The theoretical arguments developed here follow from earlier work of Batchelor & Townsend (1956), Betchov (1956) and Dresselhaus & Tabor (1991).

scholarly journals Correlation between small-scale velocity and scalar fluctuations in a turbulent channel flow

2009 ◽  
Vol 627 ◽  
pp. 1-32 ◽  
Author(s):  
HIROYUKI ABE ◽  
ROBERT ANTHONY ANTONIA ◽  
HIROSHI KAWAMURA
Keyword(s):  
Strain Rate ◽  
Channel Flow ◽  
Dissipation Rate ◽  
Compressive Strain ◽  
Outer Region ◽  
Turbulent Channel Flow ◽  
Scalar Fields ◽  
Scalar Dissipation ◽  
Small Scale ◽  
Scalar Dissipation Rate

Direct numerical simulations of a turbulent channel flow with passive scalar transport are used to examine the relationship between small-scale velocity and scalar fields. The Reynolds number based on the friction velocity and the channel half-width is equal to 180, 395 and 640, and the molecular Prandtl number is 0.71. The focus is on the interrelationship between the components of the vorticity vector and those of the scalar derivative vector. Near the wall, there is close similarity between different components of the two vectors due to the almost perfect correspondence between the momentum and thermal streaks. With increasing distance from the wall, the magnitudes of the correlations become smaller but remain non-negligible everywhere in the channel owing to the presence of internal shear and scalar layers in the inner region and the backs of the large-scale motions in the outer region. The topology of the scalar dissipation rate, which is important for small-scale scalar mixing, is shown to be associated with the organized structures. The most preferential orientation of the scalar dissipation rate is the direction of the mean strain rate near the wall and that of the fluctuating compressive strain rate in the outer region. The latter region has many characteristics in common with several turbulent flows; viz. the dominant structures are sheetlike in form and better correlated with the energy dissipation rate than the enstrophy.

Direct particle–fluid simulation of Kolmogorov-length-scale size particles in decaying isotropic turbulence

2017 ◽  
Vol 819 ◽  
pp. 188-227 ◽  
Author(s):  
Lennart Schneiders ◽  
Matthias Meinke ◽  
Wolfgang Schröder
Keyword(s):  
Kinetic Energy ◽  
Strain Rate ◽  
Viscous Dissipation ◽  
Dissipation Rate ◽  
Isotropic Turbulence ◽  
Bulk Flow ◽  
Small Scale ◽  
Length Scale ◽  
Kolmogorov Length Scale ◽  
Scale Size

The modulation of decaying isotropic turbulence by 45 000 spherical particles of Kolmogorov-length-scale size is studied using direct particle–fluid simulations, i.e. the flow field over each particle is fully resolved by direct numerical simulations of the conservation equations. A Cartesian cut-cell method is used by which the exchange of momentum and energy at the fluid–particle interfaces is strictly conserved. It is shown that the particles absorb energy from the large scales of the carrier flow while the small-scale turbulent motion is determined by the inertial particle dynamics. Whereas the viscous dissipation rate of the bulk flow is attenuated, the particles locally increase the level of dissipation due to the intense strain rate generated near the particle surfaces due to the crossing-trajectory effect. Analogously, the rotational motion of the particles decouples from the local fluid vorticity and strain-rate field at increasing particle inertia. The high level of dissipation is partially compensated by the transfer of momentum to the fluid via forces acting at the particle surfaces. The spectral analysis of the kinetic energy budget is supported by the average flow pattern about the particles showing a nearly universal strain-rate distribution. An analytical expression for the instantaneous rate of viscous dissipation induced by each particle is derived and subsequently verified numerically. Using this equation, the local balance of fluid kinetic energy around a particle of arbitrary shape can be precisely determined. It follows that two-way coupled point-particle models implicitly account for the particle-induced dissipation rate via the momentum-coupling terms; however, they disregard the actual length scales of the interaction. Finally, an analysis of the small-scale flow topology shows that the strength of vortex stretching in the bulk flow is mitigated due to the presence of the particles. This effect is associated with the energy conversion at small wavenumbers and the reduced level of dissipation at intermediate wavenumbers. Consequently, it damps the spectral flux of energy to the small scales.

Internal Stresses and Breakup of Porous Aggregates in Homogeneous Isotropic Turbulence

2014 ◽  
Author(s):  
Marco Vanni
Keyword(s):  
Numerical Simulation ◽  
Isotropic Turbulence ◽  
Internal Stresses ◽  
Length Scale ◽  
Homogeneous Isotropic Turbulence ◽  
Stokesian Dynamics ◽  
Disordered Structure ◽  
Kolmogorov Length Scale ◽  
Primary Particles ◽  
Internal Forces

The stresses acting on aggregates smaller than the Kolmogorov length scale in homogeneous isotropic turbulence were estimated by a two-scale numerical simulation. The fluid dynamics at the scales larger than the Kolmogorov length scale was calculated by a Direct Numerical Simulation of the turbulent flow, in which the aggregates were modeled as point particles. Then, we adopted Stokesian Dynamics to evaluate the phenomena governed by the smooth velocity field of the smallest scales. At this level the disordered structure of the aggregates was modeled in detail, in order to take into account the role that the primary particles have in generating and transferring the internal stress. From this result, it was possible to evaluate the internal forces acting at intermonomer contacts and determine the occurrence of breakup as a consequence of the failure of intermonomer bonds. The method was applied to disordered aggregates with isostatic and highly hyperstatic structures, respectively.

Effect of Schmidt number on small-scale passive scalar turbulence

2003 ◽  
Vol 56 (6) ◽  
pp. 615-632 ◽  
Author(s):  
RA Antonia ◽  
P Orlandi
Keyword(s):  
Turbulent Flows ◽  
Schmidt Number ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Scalar Dissipation ◽  
Small Scale ◽  
Homogeneous Isotropic Turbulence ◽  
Passive Scalars ◽  
First Case ◽  
Decaying Turbulence

Previous reviews of the behavior of passive scalars which are convected and mixed by turbulent flows have focused primarily on the case when the Prandtl number Pr, or more generally, the Schmidt number Sc is around 1. The present review considers the extra effects which arise when Sc differs from 1. It focuses mainly on information obtained from direct numerical simulations of homogeneous isotropic turbulence which either decays or is maintained in steady state. The first case is of interest since it has attracted significant theoretical attention and can be related to decaying turbulence downstream of a grid. Topics covered in the review include spectra and structure functions of the scalar, the topology and isotropy of the small-scale scalar field, as well as the correlation between the fluctuating rate of strain and the scalar dissipation rate. In each case, the emphasis is on the dependence with respect to Sc. There are as yet unexplained differences between results on forced and unforced simulations of homogeneous isotropic turbulence. There are 144 references cited in this review article.

scholarly journals A multifractal model for the momentum transfer process in wall-bounded flows

2017 ◽  
Vol 824 ◽  
Author(s):  
X. I. A. Yang ◽  
A. Lozano-Durán
Keyword(s):  
Turbulent Flows ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Small Scale ◽  
Length Scale ◽  
Cascade Process ◽  
Outer Length ◽  
Multifractal Formalism ◽  
Multiplicative Process ◽  
Stress Fluctuation

The cascading process of turbulent kinetic energy from large-scale fluid motions to small-scale and lesser-scale fluid motions in isotropic turbulence may be modelled as a hierarchical random multiplicative process according to the multifractal formalism. In this work, we show that the same formalism might also be used to model the cascading process of momentum in wall-bounded turbulent flows. However, instead of being a multiplicative process, the momentum cascade process is additive. The proposed multifractal model is used for describing the flow kinematics of the low-pass filtered streamwise wall-shear stress fluctuation $\unicode[STIX]{x1D70F}_{l}^{\prime }$, where $l$ is the filtering length scale. According to the multifractal formalism, $\langle {\unicode[STIX]{x1D70F}^{\prime }}^{2}\rangle \sim \log (Re_{\unicode[STIX]{x1D70F}})$ and $\langle \exp (p\unicode[STIX]{x1D70F}_{l}^{\prime })\rangle \sim (L/l)^{\unicode[STIX]{x1D701}_{p}}$ in the log-region, where $Re_{\unicode[STIX]{x1D70F}}$ is the friction Reynolds number, $p$ is a real number, $L$ is an outer length scale and $\unicode[STIX]{x1D701}_{p}$ is the anomalous exponent of the momentum cascade. These scalings are supported by the data from a direct numerical simulation of channel flow at $Re_{\unicode[STIX]{x1D70F}}=4200$.

scholarly journals A Lagrangian fluctuation–dissipation relation for scalar turbulence. Part I. Flows with no bounding walls

2017 ◽  
Vol 829 ◽  
pp. 153-189 ◽  
Author(s):  
Theodore D. Drivas ◽  
Gregory L. Eyink
Keyword(s):  
Isotropic Turbulence ◽  
Scalar Fields ◽  
Scalar Dissipation ◽  
Lagrangian Particle ◽  
Particle Trajectories ◽  
Navier Stokes Equation ◽  
Exact Relation ◽  
Molecular Diffusivity ◽  
Fluctuation Dissipation ◽  
Additional Support

An exact relation is derived between scalar dissipation due to molecular diffusivity and the randomness of stochastic Lagrangian trajectories for flows without bounding walls. This ‘Lagrangian fluctuation–dissipation relation’ equates the scalar dissipation for either passive or active scalars to the variance of scalar inputs associated with initial scalar values and internal scalar sources, as these are sampled backward in time by the stochastic Lagrangian trajectories. As an important application, we reconsider the phenomenon of ‘Lagrangian spontaneous stochasticity’ or persistent non-determinism of Lagrangian particle trajectories in the limit of vanishing viscosity and diffusivity. Previous work on the Kraichnan (Phys. Fluids, 1968, vol. 11, pp. 945–953) model of turbulent scalar advection has shown that anomalous scalar dissipation is associated in that model with Lagrangian spontaneous stochasticity. There has been controversy, however, regarding the validity of this mechanism for scalars advected by an actual turbulent flow. We here completely resolve this controversy by exploiting the fluctuation–dissipation relation. For either a passive or an active scalar advected by any divergence-free velocity field, including solutions of the incompressible Navier–Stokes equation, and away from walls, we prove that anomalous scalar dissipation requires Lagrangian spontaneous stochasticity. For passive scalars, we prove furthermore that spontaneous stochasticity yields anomalous dissipation for suitable initial scalar fields, so that the two phenomena are there completely equivalent. These points are illustrated by numerical results from a database of homogeneous isotropic turbulence, which provide both additional support to the results and physical insight into the representation of diffusive effects by stochastic Lagrangian particle trajectories.

scholarly journals The scaling of straining motions in homogeneous isotropic turbulence

2017 ◽  
Vol 829 ◽  
pp. 31-64 ◽  
Author(s):  
G. E. Elsinga ◽  
T. Ishihara ◽  
M. V. Goudar ◽  
C. B. da Silva ◽  
J. C. R. Hunt
Keyword(s):  
Reynolds Number ◽  
Shear Layer ◽  
Spatial Organization ◽  
Isotropic Turbulence ◽  
Local Strain ◽  
Scale Structure ◽  
Small Scale ◽  
Length Scale ◽  
Local Flow ◽  
Small Scale Structure

The scaling of turbulent motions is investigated by considering the flow in the eigenframe of the local strain-rate tensor. The flow patterns in this frame of reference are evaluated using existing direct numerical simulations of homogeneous isotropic turbulence over a Reynolds number range from $Re_{\unicode[STIX]{x1D706}}=34.6$ up to 1131, and also with reference to data for inhomogeneous, anisotropic wall turbulence. The average flow in the eigenframe reveals a shear layer structure containing tube-like vortices and a dissipation sheet, whose dimensions scale with the Kolmogorov length scale, $\unicode[STIX]{x1D702}$. The vorticity stretching motions scale with the Taylor length scale, $\unicode[STIX]{x1D706}_{T}$, while the flow outside the shear layer scales with the integral length scale, $L$. Furthermore, the spatial organization of the vortices and the dissipation sheet defines a characteristic small-scale structure. The overall size of this characteristic small-scale structure is $120\unicode[STIX]{x1D702}$ in all directions based on the coherence length of the vorticity. This is considerably larger than the typical size of individual vortices, and reflects the importance of spatial organization at the small scales. Comparing the overall size of the characteristic small-scale structure with the largest flow scales and the vorticity stretching motions on the scale of $4\unicode[STIX]{x1D706}_{T}$ shows that transitions in flow structure occur where $Re_{\unicode[STIX]{x1D706}}\approx 45$ and 250. Below these respective transitional Reynolds numbers, the small-scale motions and the vorticity stretching motions are progressively less well developed. Scale interactions are examined by decomposing the average shear layer into a local flow, which is induced by the shear layer vorticity, and a non-local flow, which represents the environment of the characteristic small-scale structure. The non-local strain is $4\unicode[STIX]{x1D706}_{T}$ in width and height, which is consistent with observations in high Reynolds number flow of a $4\unicode[STIX]{x1D706}_{T}$ wide instantaneous shear layer with many $\unicode[STIX]{x1D702}$-scale vortical structures inside (Ishihara et al., Flow Turbul. Combust., vol. 91, 2013, pp. 895–929). In the average shear layer, vorticity aligns with the intermediate principal strain at small scales, while it aligns with the most stretching principal strain at larger scales, consistent with instantaneous turbulence. The length scale at which the alignment changes depends on the Reynolds number. When conditioning the flow in the eigenframe on extreme dissipation, the velocity is strongly affected over large distances. Moreover, the associated peak velocity remains Reynolds number dependent when normalized by the Kolmogorov velocity scale. It signifies that extreme dissipation is not simply a small-scale property, but is associated with large scales at the same time.

scholarly journals Acceleration, pressure and related quantities in the proximity of the turbulent/non-turbulent interface

2009 ◽  
Vol 639 ◽  
pp. 153-165 ◽  
Author(s):  
MARKUS HOLZNER ◽  
B. LÜTHI ◽  
A. TSINOBER ◽  
W. KINZELBACH
Keyword(s):  
Fluid Velocity ◽  
Laboratory Frame ◽  
Small Scale ◽  
Length Scale ◽  
Viscous Effects ◽  
Turbulent Region ◽  
Ambient Flow ◽  
Kolmogorov Length Scale ◽  
Strain Interaction ◽  
Comparison Of The Results

This paper presents an analysis of flow properties in the proximity of the turbulent/non-turbulent interface (TNTI), with particular focus on the acceleration of fluid particles, pressure and related small scale quantities such as enstrophy, ω2 = ωiωi, and strain, s2 = sijsij. The emphasis is on the qualitative differences between turbulent, intermediate and non-turbulent flow regions, emanating from the solenoidal nature of the turbulent region, the irrotational character of the non-turbulent region and the mixed nature of the intermediate region in between. The results are obtained from a particle tracking experiment and direct numerical simulations (DNS) of a temporally developing flow without mean shear. The analysis reveals that turbulence influences its neighbouring ambient flow in three different ways depending on the distance to the TNTI: (i) pressure has the longest range of influence into the ambient region and in the far region non-local effects dominate. This is felt on the level of velocity as irrotational fluctuations, on the level of acceleration as local change of velocity due to pressure gradients, Du/Dt ≃ ∂u/∂t ≃ −∇ p/ρ, and, finally, on the level of strain due to pressure-Hessian/strain interaction, (D/Dt)(s2/2) ≃ (∂/∂t)(s2/2) ≃ −sijp,ij > 0; (ii) at intermediate distances convective terms (both for acceleration and strain) as well as strain production −sijsjkski > 0 start to set in. Comparison of the results at Taylor-based Reynolds numbers Reλ = 50 and Reλ = 110 suggests that the distances to the far or intermediate regions scale with the Taylor microscale λ or the Kolmogorov length scale η of the flow, rather than with an integral length scale; (iii) in the close proximity of the TNTI the velocity field loses its purely irrotational character as viscous effects lead to a sharp increase of enstrophy and enstrophy-related terms. Convective terms show a positive peak reflecting previous findings that in the laboratory frame of reference the interface moves locally with a velocity comparable to the fluid velocity fluctuations.

A general self-preservation analysis for decaying homogeneous isotropic turbulence

2015 ◽  
Vol 773 ◽  
pp. 345-365 ◽  
Author(s):  
L. Djenidi ◽  
R. A. Antonia
Keyword(s):  
Velocity Structure ◽  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Longitudinal Velocity ◽  
Length Scale ◽  
Grid Turbulence ◽  
Homogeneous Isotropic Turbulence ◽  
Kolmogorov Length Scale ◽  
Velocity Structure Function ◽  
Turbulence Data

A general framework of self-preservation (SP) is established, based on the transport equation of the second-order longitudinal velocity structure function in decaying homogeneous isotropic turbulence (HIT). The analysis introduces the skewness of the longitudinal velocity increment, $S(r,t)$ ($r$ and $t$ are space increment and time), as an SP controlling parameter. The present SP framework allows a critical appraisal of the specific assumptions that have been made in previous SP analyses. It is shown that SP is achieved when $S(r,t)$ varies in a self-similar manner, i.e. $S=c(t){\it\phi}(r/l)$ where $l$ is a scaling length, and $c(t)$ and ${\it\phi}(r/l)$ are dimensionless functions of time and $(r/l)$, respectively. When $c(t)$ is constant, $l$ can be identified with the Kolmogorov length scale ${\it\eta}$, even when the Reynolds number is relatively small. On the other hand, the Taylor microscale ${\it\lambda}$ is a relevant SP length scale only when certain conditions are met. The decay law for the turbulent kinetic energy ($k$) ensuing from the present SP is a generalization of the existing laws and can be expressed as $k\sim (t-t_{0})^{n}+B$, where $B$ is a constant representing the energy of the motions whose scales are excluded from the SP range of scales. When $B=0$, SP is achieved at all scales of motion and ${\it\lambda}$ becomes a relevant scaling length together with ${\it\eta}$. The analysis underlines the relation between the initial conditions and the power-law exponent $n$ and also provides a link between them. In particular, an expression relating $n$ to the initial values of the scaling length and velocity is developed. Finally, the present SP analysis is consistent with both experimental grid turbulence data and the eddy-damped quasi-normal Markovian numerical simulation of decaying HIT by Meldi & Sagaut (J. Turbul., vol. 14, 2013, pp. 24–53).

On the universality of local dissipation scales in turbulent channel flow

2015 ◽  
Vol 786 ◽  
pp. 234-252 ◽  
Author(s):  
S. C. C. Bailey ◽  
B. M. Witte
Keyword(s):  
Reynolds Number ◽  
Channel Flow ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Turbulent Channel Flow ◽  
Reynolds Numbers ◽  
Small Scale ◽  
Length Scale ◽  
Scaling Behaviour ◽  
Sixth Moment

Well-resolved measurements of the small-scale dissipation statistics within turbulent channel flow are reported for a range of Reynolds numbers from $Re_{{\it\tau}}\approx 500$ to 4000. In this flow, the local large-scale Reynolds number based on the longitudinal integral length scale is found to poorly describe the Reynolds number dependence of the small-scale statistics. When a length scale based on Townsend’s attached-eddy hypothesis is used to define the local large-scale Reynolds number, the Reynolds number scaling behaviour was found to be more consistent with that observed in homogeneous, isotropic turbulence. The Reynolds number scaling of the dissipation moments up to the sixth moment was examined and the results were found to be in good agreement with predicted scaling behaviour (Schumacher et al., Proc. Natl Acad. Sci. USA, vol. 111, 2014, pp. 10961–10965). The probability density functions of the local dissipation scales (Yakhot, Physica D, vol. 215 (2), 2006, pp. 166–174) were also determined and, when the revised local large-scale Reynolds number is used for normalization, provide support for the existence of a universal distribution which scales differently for inner and outer regions.

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