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14:00   Boundary Free Turbulence 1 14:00 15 mins #196 Dynamics of the tetrad-based velocity gradient in turbulent flows Ping-Fan Yang, Alain Pumir, Haitao Xu Abstract: The statistical properties of the velocity gradient tensor in a turbulent flow are essential for the dynamics of turbulence. It is, however, extremely challenging to determine the true velocity gradient tensor in high-Reynolds-number flows. The perceived velocity gradient tensor, $\mathbf{M}$, based on 4 fluid particles separated at distances of order $R_0$, provides a useful way to investigate the inertial range dynamics of the flow. The tensor $\mathbf{M}$ is defined by $r^a_j M_{ji} = u^a_i$, where $a=1$, $\ldots$, $4$, refers to the four vertices, and $u_j^a$ and $r_j^a$ are the $j$-{th} velocity components and coordinates of the vertices with respect to the center of mass of the tetrad. Inspired by the equation for the true velocity gradient, an equation for the traceless part of $\mathbf{M}$, $\mathbf{M^*} = \mathbf{M} - \frac{1}{3}\tr(\mathbf{M})$, was proposed (see Eqs.~(3a) and (4) of \cite{bib:Chertkov1999}) to model the inertial range dymanics: \begin{equation} \frac{d \mathbf{M^*}}{dt} + \mathbf{M^{*2}} -\Pi \tr({\mathbf{M^{*2}}}) = \alpha[\mathbf{M^{*2}} -\Pi \tr({\mathbf{M^{*2}}})] - \zeta \mathbf{M^*} + \xi, \label{eq:tetradmodel} \end{equation} where $\Pi \equiv \mathbf{g}^{-1}/\tr(\mathbf{g}^{-1})$ and $\mathbf{g}$ being the moment of inertia tensor of the four particles: $g_{ij} \equiv \sum_{a=1}^4 x_i^a x_j^a$. On the r.h.s. of Eq.~(\ref{eq:tetradmodel}), the first term is the effect of pressure and the coefficient $\alpha$ represents the depression of nonlinearity \cite{bib:BORUE1998}, the $\zeta \mathbf{M^*}$ term could be considered as a damping term and the stochastic term $\xi$ represents the effects of the small scale jitter on the evolution of $\mathbf{M^*}$. Eq.~(\ref{eq:tetradmodel}) was proposed by analog with the equation for the true velocity gradient. Here we show that, starting from the definition of $\mathbf{M}$ in terms of the motion of the 4 tracer particles, the evolution of $\mathbf{M^*}$ can be derived from first principles as \begin{equation} \frac{d \mathbf{M^*}}{dt} + \mathbf{M^{*2}} -\frac{1}{3} \tr({\mathbf{M^{*2}}}) + \frac{2}{3} \tr({\mathbf{M}}) \mathbf{M^*} = \Pi \mathbf{H} - \frac{1}{3} \tr{(\Pi \mathbf{H})}, \label{eq:dMdt} \end{equation} where the tensor $H_{ij} \equiv \tr(\mathbf{g}^{-1}) \sum_{a=1}^4 x_i^a A_j^a$ depends on $\mathbf{A}$, the acceleration with respect to the center of mass. The coupling between $\tr(\mathbf{M})$ and $\mathbf{M^*}$ in Eq.~(\ref{eq:dMdt}) is absent in Eq.~(\ref{eq:tetradmodel}). The structure of the nonlinear terms, on the other hand, is qualitatively similar in the two equations. %One can see that on the one hand, the nonlinear term in the tetrad model ( Eq.~(\ref{eq:tetradmodel})) is similar to its counterpart in the exact evolution equation (Eq.~(\ref{eq:dMdt})), while on the other hand the tetrad model does not contain the effect produced by $tr({\mathbf{M}})$. The validity of Eq.~(\ref{eq:tetradmodel}) was tested directly by using Eq.~(\ref{eq:dMdt}). We focus here on the depression of nonlinearity, which we determine by evaluating numerically the evolution of the invariants of $\mathbf{M^*}$, $Q = - \frac{1}{2} \tr(\mathbf{M^*}^2)$ and $R = - \frac{1}{3} \tr( \mathbf{M^*}^3 )$, using Eq.~(\ref{eq:dMdt}), as originally done in~\cite{bib:BORUE1998}. We determined the joint PDFs between $R$ and $\tr(\mathbf{M^*H^*})$, and between $Q^2$ and $\tr(\mathbf{M^{*2}H^*})$ from the homogeneous isotropic flow at $R_\lambda = 430$ from the Johns Hopkins Turbulence Database~\cite{bib:YiLi2008}. We used regular tetrads over different scales $R_0$ in the range $25 \le R_0/\eta \le 400$. %The results, see Fig.~\ref{fig1}, suggest a linear relation %between $tr(\mathbf{M^*} \mathbf{H^*})$ and $R$ (left panel) and %between $tr(\mathbf{M^*}^2 \mathbf{H^*})$ and $Q^2$ (central panel). The %coefficients extracted from the fit are shown in panel c, and found to be %close to each other at moderate values of $R_0/\eta \le 250$, consistent %with the $\alpha$-term in Eq.~(\ref{eq:tetradmodel}). %The values of $\alpha$ appear to be approximately constant for %$25 \le R_0/\eta \le 250$: $\alpha \approx 0.35-0.40$, possibly %increasing when $R_0/eta$ towards the dissipative range. \begin{figure}[h] \setlength{\unitlength}{1cm} \begin{center} %\subfigure[]{ \includegraphics[width=5cm]{jointpdf_R_MH_all_97eta_2ndver.eps} %} %\subfigure[]{ \includegraphics[width=5cm]{jointpdf_Q_M2H_all_97eta_2ndver.eps} %} %\subfigure[]{ \includegraphics[width=5cm]{alpha_fit.eps} %} \caption{Joint PDFs between $R$ and $\tr(\mathbf{M^*H^*})$ (left) and $Q^2$ and $\tr(\mathbf{(M^*)^2H^*})$ (center) for regular tetrads, with size $R_0 = 97 \eta$. The black solid curves are the conditional averages of ordinates conditioned on abscissas, and the red dashed lines are the linear fitting. The coefficients $\alpha$ determined from these linear dependences (right) are very close to each other, consistent with Eq.~(\ref{eq:tetradmodel}).} \label{fig1} \end{center} \end{figure} The results, see Fig.~\ref{fig1}, suggest a linear relation between $\tr(\mathbf{M^*} \mathbf{H^*})$ and $R$ (left) and between $\tr(\mathbf{M^*}^2 \mathbf{H^*})$ and $Q^2$ (center). The coefficients extracted from the fit (right) are found to be close to each other at moderate values of $R_0/\eta \le 250$, consistent with the $\alpha$-term in Eq.~(\ref{eq:tetradmodel}), and approximately constant: $\alpha \approx 0.35-0.40$. %%%%%%%%%%% \begin{thebibliography}{1} \bibitem{bib:BORUE1998} V.~Borue and S.~A. Orszag. \newblock Local energy flux and subgrid-scale statistics in three-dimensional turbulence. \newblock {\em J. Fluid Mech.}, {\bf 366}:1--31, 1998. \bibitem{bib:Chertkov1999} M.~Chertkov, A.~Pumir, and B.~I. Shraiman. \newblock Lagrangian tetrad dynamics and the phenomenology of turbulence. \newblock {\em Phys. Fluids}, {\bf 11}:2394--2410, 1999. \bibitem{bib:YiLi2008} Li~Y., E.~Perlman, M.~Wan, Y.~Yang, C.~Meneveau, R.~Burns, S.~Chen, A.~Szalay, and G.~L. Eyink. \newblock A public turbulence database cluster and applications to study lagrangian evolution of velocity increments in turbulence. \newblock {\em J. Turbulence}, {\bf 9}(N31):1--29, 2008. \end{thebibliography} %\bibliographystyle{plainbv} %\bibliography{biblio} % Alternatively use this for the bibliography : % %\begin{thebibliography}{1} % %\bibitem{bib:momo1965} %A. Momo, B. Mimi, and C. Mama. Experimental study of blibli. \textit{Journal of Blibli} \textbf{15}: 43--62, 1965. % %\bibitem{bib:toto2002} %A. Toto, B. Titi, Tutu and C. Tutu. Effect of blibli on blublu. \textit{Journal of Blabla} \textbf{468}: 77--105, 2002. % %\end{thebibliography} \end{document} 14:15 15 mins #392 Kinematics and Dynamics of Scale-Local Stress/Strain Alignment in Turbulence Nicholas Ouellette, Joseph Ballouz Abstract: The directed flux of energy through the cascade is in many ways the defining feature of turbulence. This transfer, like any energy flux, can be naturally interpreted as the action of a stress against a rate of strain. As is well known, in the cascade, the required turbulent stress arises from the nonlinear term in the Navier-Stokes equations, and is an expression of the momentum exchange between different scales of motion. As we have recently argued, thinking about the cascade in this way allows one to cast it as a purely mechanical process, where some scales (large scales, in 3D turbulence) do work on other scales (small scales, in 3D turbulence) on average \cite{ballouz2018}. This interpretation of the cascade then highlights the fundamental importance of the geometric alignment between the turbulent stress tensor and the scale-local rate of strain tensor, since if they are misaligned with each other, no work can be done and no energy will be transferred down the cascade. As we have shown previously, these two tensors are (perhaps surprisingly) relatively poorly aligned with each other on average in both 2D and 3D turbulence \cite{ballouz2018,fang2016}, even though the cascade is such an essential feature of turbulence. One can rationalize this finding by appealing to the randomizing actions of turbulent motion. However, these tensors are not entirely randomly aligned outside of the dissipation range, suggesting that there are also effects that tend to pull their eigenframes into preferred relative orientations. Some of these effects are dynamic, in that the mechanics of the turbulence drives them; and some are purely kinematic, in that the behavior of these tensors is controlled to some degree by their mathematical form and the embedding dimension \cite{liao2015a}. Here, using experimental and DNS data for 2D and 3D turbulence as well as analysis of random solenoidal vector fields, we will begin to disentangle these disparate influences on the energy transfer in turbulence to isolate the essential mechanical processes that drive the cascade. 14:30 15 mins #521 Vortex stretching is not the main cause of the turbulent energy cascade Andrew Bragg, Maurizio Carbone Abstract: See attached pdf file for abstract. 14:45 15 mins #361 Inverse cascade in 3D Homogeneous Isotropic Turbulence Franck Plunian, Andrei Teimurazov, Rodion Stepanov, Mahendra Verma Abstract: The first evidence of an inverse cascade of energy in 3D Homogeneous Isotropic Turbulence is obtained by numerically solving the Navier-Stokes equations provided helicity is injected in the infrared range of scales, namely the scales larger than the energy forcing scale. The results are analyzed in terms of helical modes fluxes. 15:00 15 mins #522 A Lagrangian model for the velocity gradient tensor in turbulent flows based on strain-rate eigenframe variables Maurizio Carbone, Andrew Bragg, Michele Iovieno Abstract: We present a new model for the dynamics of the velocity gradient tensor along the fluid particle trajectory, in incompressible, statistically steady and isotropic turbulent flow. The equations for the velocity gradient are analyzed in the strain-rate eigenframe, in which the interplay between strain self-amplification and rotation of the fluid element can be sorted out. The form of the pressure Hessian is modelled by means of basic constraints deriving from the physics of the system and dimensional analysis. The equations should be invariant under permutation of the indexes of the eigendirections and under the arbitrary choice of the sign along the eigendirections. A set of six ordinary differential equations (ODE) and three algebraic equations is obtained. The resulting equations depend upon three dimensionless parameters and are similar to the equations derived through the assumption of Gaussian random field. However, our derivation shows that the model equations can be derived solely on the basis of symmetries and fundamental physical constraints, instead of assuming the form of the fluctuations of the random field. Moreover, the general structure of the system of ODEs allows for various closure methods for the deviatoric part of the pressure Hessian. The system of equations includes, for a suitable choice of the parameters, the closure based on Gaussian random field and the classic Restricted Euler (RE). While the classical RE exhibits almost surely a finite-time singularity, our model produces smooth dynamics for certain values of the dimensionless parameters and this can be interpreted as a "reduction of nonlinearity" due to the pressure Hessian, as described in previous works. In our derivation, the dimensionless coefficients are determined by means of the explicit expression of the pressure Hessian as a functional of the invariant Q. To deal with the non-locality of this expression, we employ the deformation tensor and the dependence of the velocity gradient invariants upon the initial conditions. The deformation of the fluid element has been employed also in the Recent Fluid Deformation (RFD) approximation. Even though the RFD produces overall results in good agreement with the numerical experiments at moderate Reynolds number, the predicted probability current in the Q-R plane due to the pressure Hessian vanishes along the axis Q=0, in contrast with the DNS data. This issue is addressed in our model, in which the hypothesis of proportionality between the pressure Hessian and the local value of the invariant Q is removed. In the inviscid case, where the model equations are formally time-reversible, the model nevertheless exhibits non-trivial dynamics in common with the viscous case. For example, the trajectories in the Q-R plane tend to be driven along the Vieillefosse tails: the strain self-amplifies until the regularizing effect due to the rotation of the fluid element pulls the system back towards moderate values of Q and R. This is a consequence of the reduction of non-linearity due to the pressure Hessian. Finally, the diffusive term is modelled in a similar fashion to the pressure Hessian, by reproducing the velocity field in the region of the sampled fluid particle as a function of the deformation tensor and the initial velocity gradient field. Diffusion breaks the time-reversibility of the system and the symmetry of the dynamics in the Q-R plane. A statistically steady state is maintained by means of an isotropic random forcing. The results from the stochastic model are compared with the DNS data. 15:15 15 mins #569 Effects of synthetic low-level jet on scaled-down wind farm performance Ali Doosttalab, Diego Siguenza, Josuenny O'Donnell, Venkatesh Pulletikurthi, Humberto Bocanegra Evans, Yaqing Jin, Leonardo P. Chamorro, Luciano Castillo Abstract: Nocturnal low-level jet (LLJ) is a relative maximum in wind velocity profile in the lower part of the atmosphere. These winds are associated with lower turbulent kinetic energy compared to the unstable atmospheric boundary layers. This regular periodic phenomenon may last hours and extends horizontally for several kilometers while its height ranges between 100 to 700 m [1]. LLJs are detected in many areas around the world and are often observed in the Great Plains of the United States [1], Germany [2] and South America [3] among others. The increasing tendency of wind turbine's heights raise concerns with wind speed shear associated with low-level jets for present and further wind farm designs. Low-level jets are important because they can increase the capacity factor of wind farms [4], however their interaction with wind farms remain unknown. This work aims to experimentally investigate the turbulence characteristics of LLJ-induced wake of scaled-down wind turbines. A synthetic low-level jet velocity profile was created using an inlet flow generator in the wind tunnel. Three turbine heights relative to the core of the LLJ were tested in this study, as shown in figure 1a. Velocity fields were obtained through particle image velocimetry (PIV) measurements in the wake of a single turbine and a 3 x 2 turbine array. Furthermore, power measurements were obtained from turbines in the array configurations. We observed that when the jet peak is at the turbine hub height, the wake recovery is weaker than in the other conditions. As it is shown in figures 1b & c, coinciding the jet peak with the turbine hub height will lead to a larger velocity deficit in the wake region of the wind turbine. This will result in significant decrease of power production in the array configuration. This substantial increase in the velocity deficit is due to lower turbulent mixing in the wake region. It is also shown that energy entrainment will significantly increase when the core of low-level jet is above or below the turbine hub height compared with unstable boundary layer and will decrease when the turbine hub height coincides with the low-level jet velocity peak. This increased energy entrainment will lead to higher efficiency of wind farms during LLJ events, provided that the jet core is not at the turbine hub height. 15:30 15 mins #316 HOW LARGE CAN VELOCITY GRADIENTS BE IN TURBULENT FLOWS ? Alain Pumir, Dhawal Buaria, Eberhard Bodenschatz, PK Yeung Abstract: Small-scale intermittency referring to the occurrence of extremely large fluctuations of velocity gradients, is a hallmark of turbulence. A natural question in this context is to understand how large can velocity gradients be in a flow at very large Reynolds number. Providing reliable empirical information on this problem, from experiments or numerical simulations, requires extremely high resolution, both spatially and temporally. We address the question with direct numerical simu- lations (DNS) of turbulent flows in a periodic domain using pseudo-spectral methods, going up to 81923 grid points and Taylor-scale Reynolds number (Rλ) of 650. The grid spacing was chosen to be ∆x ≈ η/2, where η is the Kolmogorov length scale, and the time step was chosen as explained in [2]. Fig. 1a shows the probability density functions (PDFs) of Ω, defined as Ω = ωiωi, where ω is vorticity. Once normalized by its variance, ⟨Ω⟩ = 1/τK2 , the distribution of Ω exhibits tails which extend further as the Reynolds number increases. A similar result is observed for the strain tensor S, parameterized by Σ = 2Sij Sij (such that ⟨Σ⟩ = ⟨Ω⟩), as shown in the inset of Fig. 1a. This broadening of the PDF tails of ΩτK2 (and ΣτK2 ) over the range of Rλ covered here, can be captured by rescaling them appropriately. Fig. 1b shows the PDFs of Ωτ 2 and Στ 2 , rescaled by Rδ , where ext ext λ τext = τK × R−β , with β = 0.775 ± 0.025 and δ ≈ 4. This shows that the extreme gradients in the flow are a factor λ ∼ Rλβ larger than their r.m.s. amplitude. To analyze this result, we recall the ostensibly simple relation: ∂u/∂x ∼ δur/r, where δu is the velocity increment over a scale r. Large gradients result from a large velocity increment over a small scale. Fig. 1c shows that the velocity increments can be as high as u′, the velocity r.m.s, over scales r ≤ η, much larger than the velocity increment, uK, predicted by Kolmogorov theory at this scale. Taking this observation into account, as well as the scaling of the extreme velocity gradients as a function of Rλ, leads to the expectation that the smallest scale in the flow, ηext, behaves as ηR−α, λ with α = β − 1/2. These predictions are also confirmed by a systematic analysis of the dependence of the PDFs of δur on the scale r. The dependence of the smallest scale, ηext ∼ ηR−α can be related to the observed dependence of strain, conditioned λ on vorticity. Our numerical results show that strain increases as a power law with vorticity, with an exponent γ smaller than 1. Phenomenological considerations suggest a relation between α and γ, which is consistent with the DNS results. In conclusion, our results [1] provide new insight on the extreme velocity gradients in turbulent flows over the range of Reynolds numbers 140 ≤ Rλ ≤ 650. This work implies stringent constraints on the spatial resolution necessary to investigate intermittency. How our results extend in the Rλ → ∞ limit remains an open question.