scholarly journals Multiscale analysis of the topological invariants in the logarithmic region of turbulent channels at a friction Reynolds number of 932

2016 ◽  
Vol 803 ◽  
pp. 356-394 ◽  
Author(s):  
A. Lozano-Durán ◽  
M. Holzner ◽  
J. Jiménez
Keyword(s):  
Velocity Gradient ◽  
Turbulent Channel Flow ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Filter Width ◽  
Strain Components ◽  
Order Of Magnitude ◽  
Self Similar ◽  
The Mean ◽  
Orbital Periods

The invariants of the velocity gradient tensor,$R$and$Q$, and their enstrophy and strain components are studied in the logarithmic layer of an incompressible turbulent channel flow. The velocities are filtered in the three spatial directions and the results are analysed at different scales. We show that the$R$–$Q$plane does not capture the changes undergone by the flow as the filter width increases, and that the enstrophy/enstrophy-production and strain/strain-production planes represent better choices. We also show that the conditional mean trajectories may differ significantly from the instantaneous behaviour of the flow since they are the result of an averaging process where the mean is 3–5 times smaller than the corresponding standard deviation. The orbital periods in the$R$–$Q$plane are shown to be independent of the intensity of the events, and of the same order of magnitude as those in the enstrophy/enstrophy-production and strain/strain-production planes. Our final goal is to test whether the dynamics of the flow is self-similar in the inertial range, and the answer turns out to be that it is not. The mean shear is found to be responsible for the absence of self-similarity and progressively controls the dynamics of the eddies observed as the filter width increases. However, a self-similar behaviour emerges when the calculations are repeated for the fluctuating velocity gradient tensor. Finally, the turbulent cascade in terms of vortex stretching is considered by computing the alignment of the vorticity at a given scale with the strain at a different one. These results generally support a non-negligible role of the phenomenological energy-cascade model formulated in terms of vortex stretching.

scholarly journals Evolution of the velocity gradient tensor invariant dynamics in a turbulent boundary layer

2017 ◽  
Vol 815 ◽  
pp. 223-242 ◽  
Author(s):  
P. Bechlars ◽  
R. D. Sandberg
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Velocity Gradient ◽  
Evolution Equations ◽  
Turbulence Models ◽  
Turbulent Structures ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
The Mean ◽  
The Individual

In order to improve the physical understanding of the development of turbulent structures, the compressible evolution equations for the first three invariants $P$, $Q$ and $R$ of the velocity gradient tensor have been derived. The mean evolution of characteristic turbulent structure types in the $QR$-space were studied and compared at different wall-normal locations of a compressible turbulent boundary layer. The evolution of these structure types is fundamental to the physics that needs to be captured by turbulence models. Significant variations of the mean evolution are found across the boundary layer. The key features of the changes of the mean trajectories in the invariant phase space are highlighted and the consequences of the changes are discussed. Further, the individual elements of the overall evolution are studied separately to identify the causes that lead to the evolution varying with the distance to the wall. Significant impact of the wall-normal location on the coupling between the pressure-Hessian tensor and the velocity gradient tensor was found. The highlighted features are crucial for the development of more universal future turbulence models.

scholarly journals Measurements of the coupling between the tumbling of rods and the velocity gradient tensor in turbulence

2015 ◽  
Vol 766 ◽  
pp. 202-225 ◽  
Author(s):  
Rui Ni ◽  
Stefan Kramel ◽  
Nicholas T. Ouellette ◽  
Greg A. Voth
Keyword(s):  
Strain Rate ◽  
Velocity Gradient ◽  
Strong Dependence ◽  
Joint Probability ◽  
Weighted Average ◽  
Seeding Density ◽  
Full Description ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
The Mean

AbstractWe present simultaneous experimental measurements of the dynamics of anisotropic particles transported by a turbulent flow and the velocity gradient tensor of the flow surrounding them. We track both rod-shaped particles and small spherical flow tracers using stereoscopic particle tracking. By using scanned illumination, we are able to obtain a high enough seeding density of tracers to measure the full velocity gradient tensor near the rod. The alignment of rods with the vorticity and the eigenvectors of the strain rate from experimental results agree well with numerical findings. A full description of the tumbling of rods in turbulence requires specifying a seven-dimensional joint probability density function (jPDF) of five scalars characterizing the velocity gradient tensor and two scalars describing the relative orientation of the rod. If these seven parameters are known, then Jeffery’s equation specifies the rod tumbling rate and any statistic of rod rotations can be obtained as a weighted average over the jPDF. To look for a lower-dimensional projection to simplify the problem, we explore conditional averages of the mean-squared tumbling rate. The conditional dependence of the mean-squared tumbling rate on the magnitude of both the vorticity and the strain rate is strong, as expected, and similar. There is also a strong dependence on the orientation between the rod and the vorticity, since a rod aligned with the vorticity vector tumbles due to strain but not vorticity. When conditioned on the alignment of the rod with the eigenvectors of the strain rate, the largest tumbling rate is obtained when the rod is oriented at a certain angle to the eigenvector that corresponds to the smallest eigenvalue, because this particular orientation maximizes the contribution from both the vorticity and strain.

Effects of pressure gradient on the evolution of velocity-gradient tensor invariant dynamics on a controlled-diffusion aerofoil at

2019 ◽  
Vol 868 ◽  
pp. 584-610 ◽  
Author(s):  
H. Wu ◽  
S. Moreau ◽  
R. D. Sandberg
Keyword(s):  
Pressure Gradient ◽  
Velocity Gradient ◽  
Stream Velocity ◽  
Pressure Gradients ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Controlled Diffusion ◽  
Mean Pressure ◽  
The Mean ◽  
The Impact

A weakly compressible flow direct numerical simulation of a controlled-diffusion aerofoil at $8^{\circ }$ geometrical angle of attack, a chord-based Reynolds number of $Re_{c}=150\,000$ and a Mach number of $M=0.25$ based on the free-stream velocity relevant to many industrial applications was conducted to improve the understanding of the impact of the pressure gradient on the development of turbulent structures. The evolution equations for the two invariants $Q$ and $R$ of the velocity-gradient tensor have been studied at various locations along the aerofoil chord on its suction side. The shape of the mean evolution of the velocity-gradient tensor invariants were found to vary strongly when the flow encounters favourable, zero and adverse pressure gradients and as well for different wall-normal locations. The coupling between the pressure-Hessian tensor and the velocity-gradient tensor was found to be the major factor that causes these changes and is greatly influenced by the mean pressure-gradient condition and the wall-normal distance. Striking differences exist from the mean trajectories of this coupling at least in the log layer and outer layer subject to different mean pressure gradients. The nonlinearity and viscous diffusion effects keep their respective invariant characters regardless of the pressure-gradient effects and wall-normal locations. The wall and the mean adverse pressure gradient were both found to suppress the vortical stretching features of the flow. These features are of great importance for the development of future turbulence models on wall-bounded flows, especially on surfaces with significant curvature such as cambered aerofoils and blades for which significant mean pressure gradients exist.

A study of the evolution and characteristics of the invariants of the velocity-gradient tensor in isotropic turbulence

1999 ◽  
Vol 381 ◽  
pp. 141-174 ◽  
Author(s):  
ANDREW OOI ◽  
JESUS MARTIN ◽  
JULIO SORIA ◽  
M. S. CHONG
Keyword(s):  
Velocity Gradient ◽  
Phase Plane ◽  
Isotropic Turbulence ◽  
Invariant Space ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor ◽  
Conditional Mean ◽  
Kinematic Vorticity ◽  
The Mean ◽  
Local Value

Since the availability of data from direct numerical simulation (DNS) of turbulence, researchers have utilized the joint PDFs of invariants of the velocity gradient tensor to study the geometry of small-scale motions of turbulence. However, the joint PDFs only give an instantaneous static representation of the properties of fluid particles and dynamical Lagrangian information cannot be extracted. In this paper, the Lagrangian evolution of the invariants of the velocity gradient tensor is studied using conditional mean trajectories (CMT). These CMT are derived using the concept of the conditional mean time rate of change of invariants calculated from a numerical simulation of isotropic turbulence. The study of the CMT in the invariant space (RA, QA) of the velocity-gradient tensor, invariant space (RS, QS) of the rate-of-strain tensor, and invariant space (RW, QW) of the rate-of-rotation tensor show that the mean evolution in the (Σ, QW) phase plane, where Σ is the vortex stretching, is cyclic with a characteristic period similar to that found by Martin et al. (1998) in the cyclic mean evolution of the CMT in the (RA, QA) phase plane. Conditional mean trajectories in the (Σ, QW) phase plane suggest that the initial reduction of QW in regions of high QW is due to viscous diffusion and that vorticity contraction only plays a secondary role subsequent to this initial decay. It is also found that in regions of the flow with small values of QW, the local values of QW do not begin to increase, even in the presence of self-stretching, until a certain self-stretching rate threshold is reached, i.e. when Σ≈0.25 〈QW〉1/2. This study also shows that in regions where the kinematic vorticity number (as defined by Truesdell 1954) is low, the local value of dissipation tends to increase in the mean as observed from a Lagrangian frame of reference. However, in regions where the kinematic vorticity number is high, the local value of enstrophy tends to decrease. From the CMT in the (−QS, RS phase plane, it is also deduced that for large values of dissipation, there is a tendency for fluid particles to evolve towards having a positive local value of the intermediate principal rate of strain.

Invariants of velocity gradient tensor in supersonic turbulent pipe, nozzle, and diffuser flows

2018 ◽  
Vol 30 (1) ◽  
pp. 015104 ◽  
Author(s):  
Komal Kumari ◽  
Susila Mahapatra ◽  
Somnath Ghosh ◽  
Joseph Mathew
Keyword(s):  
Velocity Gradient ◽  
Gradient Tensor ◽  
Velocity Gradient Tensor
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