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1. Comparing local energy cascade rates in isotropic turbulence using structure-function and filtering formulations

   https://doi.org/10.1017/jfm.2023.1066  

   Yao, Hanxun ; Schnaubelt, Michael ; Szalay, Alexander S ; Zaki, Tamer A ; Meneveau, Charles ( February 2024 , Journal of Fluid Mechanics)

   Two common definitions of the spatially local rate of kinetic energy cascade at some scale$\ell$in turbulent flows are (i) the cubic velocity difference term appearing in the ‘scale-integrated local Kolmogorov–Hill’ equation (structure-function approach), and (ii) the subfilter-scale energy flux term in the transport equation for subgrid-scale kinetic energy (filtering approach). We perform a comparative study of both quantities based on direct numerical simulation data of isotropic turbulence at Taylor-scale Reynolds number 1250. While in the past observations of negative subfilter-scale energy flux (backscatter) have led to debates regarding interpretation and relevance of such observations, we argue that the interpretation of the local structure-function-based cascade rate definition is unambiguous since it arises from a divergence term in scale space. Conditional averaging is used to explore the relationship between the local cascade rate and the local filtered viscous dissipation rate as well as filtered velocity gradient tensor properties such as its invariants. We find statistically robust evidence of inverse cascade when both the large-scale rotation rate is strong and the large-scale strain rate is weak. Even stronger net inverse cascading is observed in the ‘vortex compression’$R>0$,$Q>0$quadrant, where$R$and$Q$are velocity gradient invariants. Qualitatively similar but quantitatively much weaker trends are observed for the conditionally averaged subfilter-scale energy flux. Flow visualizations show consistent trends, namely that spatially, the inverse cascade events appear to be located within large-scale vortices, specifically in subregions when$R$is large.

    

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2. Efficient simplicial replacement of semialgebraic sets

   https://doi.org/10.1017/fms.2023.36  

   Basu, Saugata ; Karisani, Negin ( January 2023 , Forum of Mathematics, Sigma)

   Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set$S \subset \mathbb {R}^k$by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex$\Delta $, whose geometric realization,$|\Delta |$, is semialgebraically homeomorphic toS. In this paper, we consider a weaker version of this question. We prove that for any$\ell \geq 0$, there exists an algorithm which takes as input a description of a semialgebraic subset$S \subset \mathbb {R}^k$given by a quantifier-free first-order formula$\phi $in the language of the reals and produces as output a simplicial complex$\Delta $, whose geometric realization,$|\Delta |$is$\ell $-equivalent toS. The complexity of our algorithm is bounded by$(sd)^{k^{O(\ell )}}$, wheresis the number of polynomials appearing in the formula$\phi $, andda bound on their degrees. For fixed$\ell $, this bound issingly exponentialink. In particular, since$\ell $-equivalence implies that thehomotopy groupsup to dimension$\ell $of$|\Delta |$are isomorphic to those ofS, we obtain a reduction (having singly exponential complexity) of the problem of computing the first$\ell $homotopy groups ofSto the combinatorial problem of computing the first$\ell $homotopy groups of a finite simplicial complex of size bounded by$(sd)^{k^{O(\ell )}}$.

    

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3. Bipartite-ness under smooth conditions

   https://doi.org/10.1017/S0963548323000019  

   Jiang, Tao ; Longbrake, Sean ; Ma, Jie ( February 2023 , Combinatorics, Probability and Computing)

   Given a family$\mathcal{F}$of bipartite graphs, theZarankiewicz number$z(m,n,\mathcal{F})$is the maximum number of edges in an$m$by$n$bipartite graph$G$that does not contain any member of$\mathcal{F}$as a subgraph (such$G$is called$\mathcal{F}$-free). For$1\leq \beta \lt \alpha \lt 2$, a family$\mathcal{F}$of bipartite graphs is$(\alpha,\beta )$-smoothif for some$\rho \gt 0$and every$m\leq n$,$z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta )$. Motivated by their work on a conjecture of Erdős and Simonovits on compactness and a classic result of Andrásfai, Erdős and Sós, Allen, Keevash, Sudakov and Verstraëte proved that for any$(\alpha,\beta )$-smooth family$\mathcal{F}$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\rho (\frac{2n}{5}+o(n))^{\alpha -1}$is bipartite. In this paper, we strengthen their result by showing that for every real$\delta \gt 0$, there exists$k_0$such that for all odd$k\geq k_0$and sufficiently large$n$, any$n$-vertex$\mathcal{F}\cup \{C_k\}$-free graph with minimum degree at least$\delta n^{\alpha -1}$is bipartite. Furthermore, our result holds under a more relaxed notion of smoothness, which include the families$\mathcal{F}$consisting of the single graph$K_{s,t}$when$t\gg s$. We also prove an analogous result for$C_{2\ell }$-free graphs for every$\ell \geq 2$, which complements a result of Keevash, Sudakov and Verstraëte.

    

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4. Surface layer response to heterogeneous tree canopy distributions: roughness regime regulates secondary flow polarity

   https://doi.org/10.1017/jfm.2022.583  

   Joshi, P. ; Anderson, W. ( September 2022 , Journal of Fluid Mechanics)

   Large-eddy simulation was used to model turbulent atmospheric surface layer (ASL) flow over canopies composed of streamwise-aligned rows of synthetic trees of height,$h$, and systematically arranged to quantify the response to variable streamwise spacing,$\delta _1$, and spanwise spacing,$\delta _2$, between adjacent trees. The response to spanwise and streamwise heterogeneity has, indeed, been the topic of a sustained research effort: the former resulting in formation of Reynolds-averaged counter-rotating secondary cells, the latter associated with the$k$- and$d$-type response. No study has addressed the confluence of both, and results herein show secondary flow polarity reversal across ‘critical’ values of$\delta _1$and$\delta _2$. For$\delta _2/\delta \lesssim 1$and$\gtrsim 2$, where$\delta$is the flow depth, the counter-rotating secondary cells are aligned such that upwelling and downwelling, respectively, occurs above the elements. The streamwise spacing$\delta _1$regulates this transition, with secondary cell reversal occurring first for the largest$k$-type cases, as elevated turbulence production within the canopy necessitates entrainment of fluid from aloft. The results are interpreted through the lens of a benchmark prognostic closure for effective aerodynamic roughness,$z_{0,{Eff.}} = \alpha \sigma _h$, where$\alpha$is a proportionality constant and$\sigma _h$is height root mean square. We report$\alpha \approx 10^{-1}$, the value reported over many decades for a broad range of rough surfaces, for$k$-type cases at small$\delta _2$, whereas the transition to$d$-type arrangements necessitates larger$\delta _2$. Though preliminary, results highlight the non-trivial response to variation of streamwise and spanwise spacing.

    

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5. Direct numerical simulations of turbulent pipe flow up to

   https://doi.org/10.1017/jfm.2022.1013  

   Yao, Jie ; Rezaeiravesh, Saleh ; Schlatter, Philipp ; Hussain, Fazle ( February 2023 , Journal of Fluid Mechanics)

   Well-resolved direct numerical simulations (DNS) have been performed of the flow in a smooth circular pipe of radius$R$and axial length$10{\rm \pi} R$at friction Reynolds numbers up to$Re_\tau =5200$using the pseudo-spectral code OPENPIPEFLOW. Various turbulence statistics are documented and compared with other DNS and experimental data in pipes as well as channels. Small but distinct differences between various datasets are identified. The friction factor$\lambda$overshoots by$2\,\%$and undershoots by$0.6\,\%$the Prandtl friction law at low and high$Re$ranges, respectively. In addition,$\lambda$in our results is slightly higher than in Pirozzoliet al.(J. Fluid Mech., vol. 926, 2021, A28), but matches well the experiments in Furuichiet al.(Phys. Fluids, vol. 27, issue 9, 2015, 095108). The log-law indicator function, which is nearly indistinguishable between pipe and channel up to$y^+=250$, has not yet developed a plateau farther away from the wall in the pipes even for the$Re_\tau =5200$cases. The wall shear stress fluctuations and the inner peak of the axial turbulence intensity – which grow monotonically with$Re_\tau$– are lower in the pipe than in the channel, but the difference decreases with increasing$Re_\tau$. While the wall value is slightly lower in the channel than in the pipe at the same$Re_\tau$, the inner peak of the pressure fluctuation shows negligible differences between them. The Reynolds number scaling of all these quantities agrees with both the logarithmic and defect-power laws if the coefficients are properly chosen. The one-dimensional spectrum of the axial velocity fluctuation exhibits a$k^{-1}$dependence at an intermediate distance from the wall – also seen in the channel. In summary, these high-fidelity data enable us to provide better insights into the flow physics in the pipes as well as the similarity/difference among different types of wall turbulence.

    

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