On the topology of the eigenframe of the subgrid-scale stress tensor

2016 ◽  
Vol 798 ◽  
pp. 598-627 ◽  
Author(s):  
Zixuan Yang ◽  
Bing-Chen Wang
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Isotropic Turbulence ◽  
Rotation Angle ◽  
A Priori ◽  
Strain Rate Tensor ◽  
Subgrid Scale ◽  
Geometrical Properties ◽  
Stress Models ◽  
Geometric Relationship

In this paper, the geometrical properties of the subgrid-scale (SGS) stress tensor are investigated through its eigenvalues and eigenvectors. The concepts of Euler rotation angle and axis are utilized to investigate the relative rotation of the eigenframe of the SGS stress tensor with respect to that of the resolved strain rate tensor. Both Euler rotation angle and axis are natural invariants of the rotation matrix, which uniquely describe the topological relation between the eigenframes of these two tensors. Different from the reference frame fixed to a rigid body, the eigenframe of a tensor consists of three orthonormal eigenvectors, which by their nature are subjected to directional aliasing. In order to describe the geometric relationship between the SGS stress and resolved strain rate tensors, an effective method is proposed to uniquely determine the topology of the eigenframes. The proposed method has been used for testing three SGS stress models in the context of homogeneous isotropic turbulence at three Reynolds numbers, using both a priori and a posteriori approaches.

Subgrid-scale structure and fluxes of turbulence underneath a surface wave

2019 ◽  
Vol 878 ◽  
pp. 768-795
Author(s):  
Kuanyu Chen ◽  
Minping Wan ◽  
Lian-Ping Wang ◽  
Shiyi Chen
Keyword(s):  
Strain Rate ◽  
Eddy Viscosity ◽  
Isotropic Turbulence ◽  
Wave Theory ◽  
Viscosity Model ◽  
Strain Rate Tensor ◽  
Linear Wave ◽  
Subgrid Scale ◽  
Eddy Viscosity Model ◽  
Wave Induced

In this study, the behaviours of subgrid-scale (SGS) turbulence are investigated with direct numerical simulations when an isotropic turbulence is brought to interact with imposed rapid waves. A partition of the velocity field is used to decompose the SGS stress into three parts, namely, the turbulent part $\unicode[STIX]{x1D749}^{T}$, the wave-induced part $\unicode[STIX]{x1D749}^{W}$ and the cross-interaction part $\unicode[STIX]{x1D749}^{C}$. Under strong wave straining, $\unicode[STIX]{x1D749}^{T}$ is found to follow the Kolmogorov scaling $\unicode[STIX]{x1D6E5}_{c}^{2/3}$, where $\unicode[STIX]{x1D6E5}_{c}$ is the filter width. Based on the linear Airy wave theory, $\unicode[STIX]{x1D749}^{W}$ and the filtered strain-rate tensor due to the wave motion, $\tilde{\unicode[STIX]{x1D64E}}^{W}$, are found to have different phases, posing a difficulty in applying the usual eddy-viscosity model. On the other hand, $\unicode[STIX]{x1D749}^{T}$ and the filtered strain-rate tensor due to the turbulent motion, $\tilde{\unicode[STIX]{x1D64E}}^{T}$, are only weakly wave-phase-dependent and could be well related by an eddy-viscosity model. The linear wave theory is also used to describe the vertical distributions of SGS statistics driven by the wave-induced motion. The predictions are in good agreement with the direct numerical simulation results. The budget equation for the turbulent SGS kinetic energy shows that the transport terms related to turbulence are important near the free surface and they compensate the imbalance between the energy flux and the SGS energy dissipation.

On the properties of similarity subgrid-scale models as deduced from measurements in a turbulent jet

1994 ◽  
Vol 275 ◽  
pp. 83-119 ◽  
Author(s):  
Shewen Liu ◽  
Charles Meneveau ◽  
Joseph Katz
Keyword(s):  
Stress Tensor ◽  
Eddy Viscosity ◽  
Weighting Function ◽  
A Priori ◽  
Scale Model ◽  
Solid Boundary ◽  
Strain Rate Tensor ◽  
Subgrid Scale ◽  
The Real ◽  
Subgrid Scale Model

The properties of turbulence subgrid-scale stresses are studied using experimental data in the far field of a round jet, at a Reynolds number of Rλ ≈ 310. Measurements are performed using two-dimensional particle displacement velocimetry. Three elements of the subgrid-scale stress tensor are calculated using planar filtering of the data. Using a priori testing, eddy-viscosity closures are shown to display very little correlation with the real stresses, in accord with earlier findings based on direct numerical simulations at lower Reynolds numbers. Detailed analysis of subgrid energy fluxes and of the velocity field decomposed into logarithmic bands leads to a new similarity subgrid-scale model. It is based on the ‘resolved stress’ tensor Lij, which is obtained by filtering products of resolved velocities at a scale equal to twice the grid scale. The correlation coefficient of this model with the real stress is shown to be substantially higher than that of the eddy-viscosity closures. It is shown that mixed models display similar levels of correlation. During the a priori test, care is taken to only employ resolved data in a fashion that is consistent with the information that would be available during large-eddy simulation. The influence of the filter shape on the correlation is documented in detail, and the model is compared to the original similarity model of Bardina et al. (1980). A relationship between Lij and a nonlinear subgrid-scale model is established. In order to control the amount of kinetic energy backscatter, which could potentially lead to numerical instability, an ad hoc weighting function that depends on the alignment between Lij and the strain-rate tensor, is introduced. A ‘dynamic’ version of the model is shown, based on the data, to allow a self-consistent determination of the coefficient. In addition, all tensor elements of the model are shown to display the correct scaling with normal distance near a solid boundary.

LES Investigation of Boussinesq Constitutive Relation Validity in a Corner Separation Flow

2018 ◽  
Author(s):  
Jean-François Monier ◽  
Nicolas Poujol ◽  
Mathieu Laurent ◽  
Feng Gao ◽  
Jérôme Boudet ◽  
...  
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Constitutive Relation ◽  
Strain Rate Tensor ◽  
Separation Flow ◽  
Compressor Cascade ◽  
Corner Separation ◽  
Reynolds Stress Tensor ◽  
Mean Strain

The present study aims at analysing the Boussinesq constitutive relation validity in a corner separation flow of a compressor cascade. The Boussinesq constitutive relation is commonly used in Reynolds-averaged Navier-Stokes (RANS) simulations for turbomachinery design. It assumes an alignment between the Reynolds stress tensor and the zero-trace mean strain-rate tensor. An indicator that measures the alignment between these tensors is used to test the validity of this assumption in a high fidelity large-eddy simulation. Eddy-viscosities are also computed using the LES database and compared. A large-eddy simulation (LES) of a LMFA-NACA65 compressor cascade, in which a corner separation is present, is considered as reference. With LES, both the Reynolds stress tensor and the mean strain-rate tensor are known, which allows the construction of the indicator and the eddy-viscosities. Two constitutive relations are evaluated. The first one is the Boussinesq constitutive relation, while the second one is the quadratic constitutive relation (QCR), expected to render more anisotropy, thus to present a better alignment between the tensors. The Boussinesq constitutive relation is rarely valid, but the QCR tends to improve the alignment. The improvement is mainly present at the inlet, upstream of the corner separation. At the outlet, the correction is milder. The eddy-viscosity built with the LES results are of the same order of magnitude as those built as the ratio of the turbulent kinetic energy k and the turbulence specific dissipation rate ω. They also show that the main impact of the QCR is to rotate the mean strain-rate tensor in order to realign it with the Reynolds stress tensor, without dilating it.

scholarly journals Comparative Study of the Grid-Scale and Subgrid-Scale Velocity Fields - A Priori Test

1970 ◽  
Vol 30 ◽  
pp. 19-31
Author(s):  
M Ashraf Uddin ◽  
M Matiar Rahman ◽  
M Saiful Islam Mallik
Keyword(s):  
Isotropic Turbulence ◽  
A Priori ◽  
Velocity Fields ◽  
Subgrid Scale ◽  
Homogeneous Isotropic Turbulence ◽  
Eddy Simulation ◽  
Large Eddy ◽  
Spatial Spectra ◽  
Sharp Cutoff ◽  
Better Than

Generation of grid-scale (GS) and subgrid-scale (SGS) velocity fields is performed by direct filtering of DNS (Direct Numerical Simulation) data at a low Reynolds number in homogeneous isotropic turbulence in order to assess the spectral accuracy as well as the performance of filter functions for LES (Large Eddy Simulation). The filtering is performed using three classical filter functions: Gaussian, Tophat and Sharp cutoff filters and in all three cases the results are compared with three different filter widths for LES. Comparing the distributions of GS and SGS velocities, and the decay of turbulence with those from DNS fields through out the whole calculation we have found that among the three filter functions, the performance of Sharp cutoff filter is better than that of the other two filter functions in terms of both spatial spectra and the distribution of velocities. Furthermore, it is shown that the accuracy of the filtering approach does not depend only on the filter functions but also on the filter widths for LES. GANIT J. Bangladesh Math. Soc. (ISSN 1606-3694) 30 (2010) 19-31   DOI: http://dx.doi.org/10.3329/ganit.v30i0.8499

scholarly journals Tempered fractional LES modeling

2021 ◽  
Vol 932 ◽  
Author(s):  
Mehdi Samiee ◽  
Ali Akhavan-Safaei ◽  
Mohsen Zayernouri
Keyword(s):  
Isotropic Turbulence ◽  
Stokes Equations ◽  
A Priori ◽  
Correlation Coefficients ◽  
Scale Model ◽  
Navier Stokes ◽  
Model Parameters ◽  
Subgrid Scale ◽  
Modelling Framework ◽  
Non Local

The presence of non-local interactions and intermittent signals in the homogeneous isotropic turbulence grant multi-point statistical functions a key role in formulating a new generation of large-eddy simulation (LES) models of higher fidelity. We establish a tempered fractional-order modelling framework for developing non-local LES subgrid-scale models, starting from the kinetic transport. We employ a tempered Lévy-stable distribution to represent the source of turbulent effects at the kinetic level, and we rigorously show that the corresponding turbulence closure term emerges as the tempered fractional Laplacian, $(\varDelta +\lambda )^{\alpha } (\cdot )$ , for $\alpha \in (0,1)$ , $\alpha \neq \frac {1}{2}$ and $\lambda >0$ in the filtered Navier–Stokes equations. Moreover, we prove the frame invariant properties of the proposed model, complying with the subgrid-scale stresses. To characterize the optimum values of model parameters and infer the enhanced efficiency of the tempered fractional subgrid-scale model, we develop a robust algorithm, involving two-point structure functions and conventional correlation coefficients. In an a priori statistical study, we evaluate the capabilities of the developed model in fulfilling the closed essential requirements, obtained for a weaker sense of the ideal LES model (Meneveau, Phys. Fluids, vol. 6, issue 2, 1994, pp. 815–833). Finally, the model undergoes the a posteriori analysis to ensure the numerical stability and pragmatic efficiency of the model.

scholarly journals ANALYSIS OF INFLAMMABLE STATE OF PURCHASE AT THE WALKING

2019 ◽  
pp. 80-85
Author(s):  
Irina Bubnovska
Keyword(s):  
Stress State ◽  
Strain Rate ◽  
Stress Tensor ◽  
Plane Strain ◽  
Equilibrium Equation ◽  
State Parameter ◽  
Strain Rate Tensor ◽  
Workpiece Material

On the basis of the equilibrium equation, the plasticity conditions in the zones of plane strain, the equation of the connection of the components of the stress tensor and the strain rate tensor, an expression was obtained for determining the stress state parameter, which makes it possible to estimate the deformity of the workpiece material during rolling.

scholarly journals Simultaneous Invariants of Strain and Rotation Rate Tensors and Their Admitted Region

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Igor Vigdorovich ◽  
Holger Foysi
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Turbulent Flows ◽  
Reynolds Stress ◽  
Building Block ◽  
Rotation Rate ◽  
Constitutive Relations ◽  
Three Dimensional ◽  
Strain Rate Tensor ◽  
Reynolds Stress Tensor

The purpose of this paper is to establish the admitted region for five simultaneous, functionally independent invariants of the strain rate tensorSand rotation rate tensorΩand calculate some simultaneous invariants of these tensors which are encountered in the theory of constitutive relations for turbulent flows. Such a problem, as far as we know, has not yet been considered, though it is obviously an integral part of any problem in which scalar functions of the tensorsSandΩare studied. The theory provided inside this paper is the building block for a derivation of new algebraic constitutive relations for three-dimensional turbulent flows in the form of expansions of the Reynolds-stress tensor in a tensorial basis formed by the tensorsSandΩ, in which the scalar coefficients depend on simultaneous invariants of these tensors.

Statistical geometry of subgrid-scale stresses determined from holographic particle image velocimetry measurements

2002 ◽  
Vol 457 ◽  
pp. 35-78 ◽  
Author(s):  
BO TAO ◽  
JOSEPH KATZ ◽  
CHARLES MENEVEAU
Keyword(s):  
Strain Rate ◽  
Nonlinear Model ◽  
Mixed Model ◽  
Particle Image ◽  
The Other ◽  
Strain Rate Tensor ◽  
Conditional Sampling ◽  
Subgrid Scale ◽  
Image Velocimetry ◽  
Holographic Particle Image Velocimetry

Three-dimensional velocity distributions of a turbulent flow in the core region of a square duct at ReH = 1.2 × 105 are measured using holographic particle image velocimetry (HPIV). Spatial filtering of the 136 × 130 × 128 velocity vector maps enables calculation of subgrid-scale (SGS) stresses and parameters based on the filtered velocity gradients, such as the filtered strain-rate tensor and vorticity vector. Probability density functions (p.d.f.) of scalar parameters characterizing eigenvalue structures confirm that the most probable strain-rate topology is axisymmetric extension, and show that the most probable SGS stress state is axisymmetric contraction. Conditional sampling shows that high positive SGS dissipation occurs preferentially in regions with these preferred strain-rate and stress topologies. High negative SGS dissipation (backscatter) occurs preferentially in regions of axisymmetric contracting SGS stress topology, but is not associated with any particular strain-rate topology. The nonlinear model produces the same trends but tends to overpredict the likelihood of the preferred stress state.Joint p.d.f.s of relative angles are used to investigate the alignments of the SGS stress eigenvectors relative to the vorticity and eigenvectors associated with eddy viscosity and similarity/nonlinear models. The results show that the most extensive SGS stress eigenvector is preferentially aligned at 32° to the most contracting strain-rate eigenvector. This alignment trend persists, with some variations in angle and peak probability, during conditional samplings based on the SGS dissipation rate, vorticity and strain-rate magnitudes. The relative alignment of the other two stress and strain-rate eigenvectors has a bimodal behaviour with the most contracting and intermediate stress eigenvectors ‘switching places’: from being aligned at 32° to the most extensive strain-rate eigenvector to being parallel to the intermediate strain-rate eigenvector. Conditional sampling shows that one of the alignment configurations occurs preferentially in regions of high vorticity magnitude, whereas the other one dominates in regions where the filtered strain-rate tensor has axisymmetric contracting topology. Analysis of DNS data for isotropic turbulence at lower Re shows similar trends.Conversely, the measured stress eigenvectors are preferentially aligned with those of the nonlinear model. This alignment persists in various regions of the flow (high vorticity, specific flow topologies, etc). Furthermore, the alignment between the strain-rate and nonlinear model tensors also exhibits a bimodal behaviour, but the alignment angle of both configurations is 42°. Implications of alignment trends on SGS dissipation are explored and conditions for high backscatter are identified based on the orientation of the stress eigenvectors. Several dynamical and kinematical arguments are presented that may explain some of the observed preferred alignments among tensors. These arguments motivate further analysis of the mixed model, which shows good alignment properties owing to the dominance of the Leonard stress on the alignments. Nevertheless, the data also show that the mixed model produces some unrealistic features in probability distributions of SGS dissipation, and unphysical eigenvector alignments in selected subregions of the flow.

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