On the Constitutive Relation for the Reynolds Stresses and the Prandtl-Kolmogorov Hypothesis of Effective Viscosity in Axisymmetric Strained Turbulence

1999 ◽  
Vol 122 (1) ◽  
pp. 48-50 ◽  
Author(s):  
J. Jovanovic´ ◽  
I. Otic´
Keyword(s):  
Strain Rate ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Constitutive Relation ◽  
Effective Viscosity ◽  
Constitutive Relations ◽  
Reynolds Stresses ◽  
Reynolds Stress Tensor ◽  
The Mean ◽  
Mean Strain

The constitutive relation for the Reynolds stress tensor is considered for turbulence developing in axisymmetric strain fields. It is confirmed that the Reynolds stress tensor is aligned linearly with the mean strain rate. In contrast to the Prandtl-Kolmogorov, hypothesis, the effective viscosity is found to grow in proportion to the anisotropy of turbulence and the length scale based on the magnitude of the mean strain rate. Using invariant theory the effective viscosity is determined for the limiting states of turbulence. Additional analysis of the constitutive relations is supplemented for the dissipation and pressure-strain correlations. It is shown that analytical derivations are in excellent agreement with the data obtained from direct numerical simulations. [S0098-2202(00)02801-7]

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 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.

Effects of Axial Casing Grooves on the Structure of Turbulence in the Tip Region of an Axial Turbomachine Rotor

2020 ◽  
Author(s):  
Huang Chen ◽  
Yuanchao Li ◽  
Subhra Shankha Koley ◽  
Joseph Katz
Keyword(s):  
Strain Rate ◽  
Reynolds Stress ◽  
Suction Side ◽  
Reynolds Stresses ◽  
Flow Rates ◽  
Tip Leakage ◽  
Anisotropy Tensor ◽  
Stress Models ◽  
Turbomachine Rotor ◽  
Mean Strain

Abstract Challenges in predicting the turbulence in the tip region of turbomachines include anisotropy, inhomogeneity, and non-equilibrium conditions, resulting in poor correlations between the Reynold stresses and the corresponding mean strain rate components. The geometric complexity introduced by casing grooves exacerbates this problem. Taking advantage of a large database collected in the refractive index-matched liquid facility at JHU, this paper examines the evolution of turbulence in the tip region of an axial turbomachine with and without axial casing grooves, and for two flow rates. The semi-circular axial grooves are skewed by 45° in the positive circumferential direction, similar to that described in Müller et al. [1]. Comparison to results obtained for an untreated endwall includes differences in the distributions of turbulent kinetic energy (TKE), Reynolds stresses, anisotropy tensor, and dominant terms in the TKE production rate. The evolution of TKE at high flow rates for blade sections located downstream of the grooves is also investigated. Common features include: with or without casing grooves, the TKE is high near the tip leakage vortex (TLV) center, and in the shear layer connecting it to the blade suction side tip corner. The turbulence is highly anisotropic and inhomogeneous, with the anisotropy tensor demonstrating shifts from one dimensional (1D) to 2D and to 3D structures over small distances. Furthermore, the correlation between the mean strain rate and Reynolds stress tensor components is poor. With the grooves, the flow structure, hence the distribution of Reynolds stresses, becomes much more complex. Turbulence is also high in the corner vortex that develops at the entrance to the grooves and in the flow jetting out of the grooves into the passage. Consistent with trends of production rates of normal Reynolds stress components, the grooves increase the axial and reduce the radial velocity fluctuations compared to the untreated endwall. These findings introduce new insight that might assist the future development of Reynolds stress models suitable for tip flows.

On the application of successive plane strains to grid-generated turbulence

1979 ◽  
Vol 93 (3) ◽  
pp. 501-513 ◽  
Author(s):  
J. N. Gence ◽  
J. Mathieu
Keyword(s):  
Shear Flow ◽  
Detailed Analysis ◽  
Stress Tensor ◽  
Plane Strain ◽  
Reynolds Stress ◽  
Homogeneous Turbulence ◽  
Reynolds Stress Tensor ◽  
Principal Axes ◽  
The Mean

A grid-generated turbulence is subjected to a pure plane strain and the principal axes of the Reynolds stress tensor become those of the strain. This ‘oriented’ homogeneous turbulence is then submitted to a new strain the principal axes of which have a different orientation. We show that in such a situation it is possible to observe a transfer of energy from the fluctuating motion to the mean one. Such transfer is necessarily associated with a forced decay of the anisotropy of the motion. A detailed analysis of the reorientation of the principal axes of the Reynolds stress tensor in the frame of those of the second strain gives an explanation of the evolution of the principal axes of the Reynolds stress tensor in a shear flow.

An Evaluation of Mean Reynolds Stress Turbulence Models: The Triple Velocity Correlation

1978 ◽  
Vol 100 (1) ◽  
pp. 47-54 ◽  
Author(s):  
D. E. Cormack ◽  
L. G. Leal ◽  
J. H. Seinfeld
Keyword(s):  
Experimental Data ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Turbulence Models ◽  
Velocity Correlation ◽  
Reynolds Stress Tensor ◽  
Closure Models ◽  
Proposed Model ◽  
The Mean ◽  
The One

A four-parameter model is presented for the triple velocity correlation which appears in the mean Reynolds stress tensor equations of turbulence. A comparison with the closure models of Hanjalic and Launder [1], Daly and Harlow [2], Wyngaard, Cote´, and Rao [3], and Shir [4], in the context of existing experimental data, indicates that the proposed model is superior. It is also demonstrated that, of the one-parameter models, the Hanjalic-Launder model best approximates the data. Due to the complexity of the four-parameter representation, it is recommended that the Hanjalic-Launder model be used for computational purposes.

Determination of the Reynolds-Stress Tensor With a Single Slanted Hot-Wire in Periodically Unsteady Turbomachinery Flow

1979 ◽  
Author(s):  
P. Kool
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Sampling Rate ◽  
Axial Compressor ◽  
Sampling Technique ◽  
Reynolds Stresses ◽  
Hot Wire ◽  
Mean Flow ◽  
Tip Leakage Flow ◽  
Reynolds Stress Tensor

The pitchwise distribution of the components of the Reynolds stress tensor downstream of a rotating blade row can be obtained with a single slanted hot-wire and the technique of periodic sampling and averaging. The requirements imposed on the hot-wire and the sensitivity coefficients to the six stresses are given. Information concerning the electronic set-up and the problems associated with measuring the Reynolds stresses are discussed. It is shown that the sampling technique does not modify the mean square value of the sampled signal if the correlation time of the turbulence is smaller than half the sampling rate, or when the frequency bandwidth of the turbulence is higher than twice the sampling frequency. Typical results, including the 3D-mean flow data obtained downstream of a low-speed axial compressor rotor at different radii are presented. The results show typical modifications of the stresses in the rotor wake region and the regions influenced by secondary flow or tip-leakage flow.

Budget of Equations for Reynolds Stresses in a Turbulent Patch Arising from Internal Wave Breaking

2012 ◽  
Vol 7 (4) ◽  
pp. 87-95
Author(s):  
Sergey Yakovenko
Keyword(s):  
Internal Wave ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Wave Breaking ◽  
Spatial Behavior ◽  
Statistical Moments ◽  
Mixing Efficiency ◽  
Reynolds Stresses ◽  
Local Isotropy ◽  
Reynolds Stress Tensor

Results of the direct numerical simulations are used to obtain statistical moments in a quasi-steady turbulent patch arising in a stably stratified flow above an obstacle after internal wave breaking. Temporal evolution and spatial behavior of the Reynolds-stress tensor components and budgets of the Reynolds-stress transport equations have been studied. Such an analysis is helpful to explore the turbulent patch and its energetic characteristics by means of statistical moments, to examine closure hypotheses in turbulence models, and to evaluate geophysically important quantites. In particular, the calculated global value of the mixing efficiency is about 0.2 as in oceanic applications. Moreover, simple algebraic relations are proposed between the turbulent kinetic energy and its dissipation rate. In the Reynolds stress tensor equation, the pressure-strain correlation term provides the redistribution of the normal stress components according to the linear returnto-isotropy approximation, whereas the dissipation tensor roughly follows the local-isotropy assumption

scholarly journals Measurements and Characterization of Turbulence in the Tip Region of an Axial Compressor Rotor

2017 ◽  
Vol 139 (12) ◽  
Author(s):  
Yuanchao Li ◽  
Huang Chen ◽  
Joseph Katz
Keyword(s):  
Strain Rate ◽  
Shear Layer ◽  
Turbulent Flows ◽  
Reynolds Stress ◽  
Suction Side ◽  
Stress And Strain ◽  
Spatial And Temporal Variability ◽  
Mean Flow ◽  
The Mean ◽  
Stress And Strain Rate

Modeling of turbulent flows in axial turbomachines is challenging due to the high spatial and temporal variability in the distribution of the strain rate components, especially in the tip region of rotor blades. High-resolution stereo-particle image velocimetry (SPIV) measurements performed in a refractive index-matched facility in a series of closely spaced planes provide a comprehensive database for determining all the terms in the Reynolds stress and strain rate tensors. Results are also used for calculating the turbulent kinetic energy (TKE) production rate and transport terms by mean flow and turbulence. They elucidate some but not all of the observed phenomena, such as the high anisotropy, high turbulence levels in the vicinity of the tip leakage vortex (TLV) center, and in the shear layer connecting it to the blade suction side (SS) tip corner. The applicability of popular Reynolds stress models based on eddy viscosity is also evaluated by calculating it from the ratio between stress and strain rate components. Results vary substantially, depending on which components are involved, ranging from very large positive to negative values. In some areas, e.g., in the tip gap and around the TLV, the local stresses and strain rates do not appear to be correlated at all. In terms of effect on the mean flow, for most of the tip region, the mean advection terms are much higher than the Reynolds stress spatial gradients, i.e., the flow dynamics is dominated by pressure-driven transport. However, they are of similar magnitude in the shear layer, where modeling would be particularly challenging.

Education Committee Best Paper of 1995 Award: Methods of Classical Mechanics Applied to Turbulence Stresses in a Tip Leakage Vortex

1996 ◽  
Vol 118 (4) ◽  
pp. 622-629 ◽  
Author(s):  
J. G. Moore ◽  
S. A. Schorn ◽  
J. Moore
Keyword(s):  
Stress Tensor ◽  
Reynolds Stress ◽  
Classical Mechanics ◽  
Three Dimensional ◽  
Surface Model ◽  
Reynolds Stresses ◽  
Tip Leakage Vortex ◽  
Tip Leakage ◽  
Stress Tensors ◽  
Principal Directions

Moore et al. measured the six Reynolds stresses in a tip leakage vortex in a linear turbine cascade. Stress tensor analysis, as used in classical mechanics, has been applied to the measured turbulence stress tensors. Principal directions and principal normal stresses are found. A solid surface model, or three-dimensional glyph, for the Reynolds stress tensor is proposed and used to view the stresses throughout the tip leakage vortex. Modeled Reynolds stresses using the Boussinesq approximation are obtained from the measured mean velocity strain rate tensor. The comparison of the principal directions and the three-dimensional graphic representations of the strain and Reynolds stress tensors aids in the understanding of the turbulence and what is required to model it.

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