A multi-pyrazolyl cyclotriphosphazene containing polymerizable group N(3)P(3)(3,5-Me(2)Pz)(5)(O-C(6)H(4)-p-C(6)H(4)-p-CH=CH(2)) (2) has been prepared from the corresponding chloro derivative N(3)P(3)Cl(5)(O-C(6)H(4)-p-C(6)H(4)-p-CH=CH(2)) (1). The X-ray structures of 1 and 2 have been determined. Compound 2 undergoes ready metalation with CuCl(2) to afford N(3)P(3)(3,5-Me(2)Pz)(5)(O-C(6)H(4)-p-C(6)H(4)-p-CH=CH(2)).CuCl(2) (3). Model compound N(3)P(3)(3,5-Me(2)Pz)(5)(O-C(6)H(4)-p-CHO).CuCl(2) (6) has been prepared and characterized by spectroscopy and X-ray crystallography. In this compound, the coordination around copper is distorted trigonal bipyramidal, and the cyclotriphosphazene coordinates in a non-gem N(3) mode. Compound 2 has been copolymerized with divinylbenzene to afford cross-linked multisite coordinating polymer CPPL which is readily metalated with CuCl(2) to afford copper-containing polymer CPPL-Cu. The coordination environment around copper in CPPL-Cu has been evaluated by obtaining its EPR, optical, and IR spectra and comparing them with those of model compounds 3 and 6. The utility of CPPL-Cu as a heterogeneous catalyst has been demonstrated in the phosphate ester hydrolysis involving three model phosphate esters: p-nitrophenyl phosphate (pNPP), bis(p-nitrophenyl) phosphate (bNPP), and 2-(hydroxypropyl)-p-nitrophenyl phosphate (hNPP). In all of these reactions, a significant rate enhancement of ester hydrolysis is observed. Detailed kinetic analyses to evaluate Michaelis-Menten parameters have also been carried out along with experiments to elucidate the effect of pH, solvent, and temperature on the rate of hydrolysis. Recycling experiments on the hydrolysis of pNPP with CPPL-Cu shows that it can be recycled several times over without affecting the rates.

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A new accuracy-preserving parallel algorithm employing compact schemes is presented for direct numerical simulation of the Navier-Stokes equations. Here the connotation of accuracy preservation is having the same level of accuracy obtained by the proposed parallel compact scheme, as the sequential code with the same compact scheme. Additional loss of accuracy in parallel compact schemes arises due to necessary boundary closures at sub-domain boundaries. An attempt to circumvent this has been done in the past by the use of Schwarz domain decomposition and compact filters in “A new compact scheme for parallel computing using domain decomposition,” J. Comput. Phys. 220, 2 (2007), 654--677, where a large number of overlap points was necessary to reduce error. A parallel compact scheme with staggered grids has been used to report direct numerical simulation of transition and turbulence by the Schwarz domain decomposition method. In the present research, we propose a new parallel algorithm with two benefits. First, the number of overlap points is reduced to a single common boundary point between any two neighboring sub-domains, thereby saving the number of points used, with resultant speed-up. Second, with a proper design, errors arising due to sub-domain boundary closure schemes are reduced to a user designed error tolerance, bringing the new parallel scheme on par with sequential computing. Error reduction is achieved by using global spectral analysis, introduced in “Analysis of central and upwind compact schemes,” J. Comput. Phys. 192, 2, (2003) 677--694, which analyzes any discrete computing method in the full domain integrally. The design of the parallel compact scheme is explained, followed by a demonstration of the accuracy of the method by solving benchmark flows: (1) periodic two-dimensional Taylor-Green vortex problem; (2) flow inside two-dimensional square lid-driven cavity (LDC) at high Reynolds number; and (3) flow inside a non-periodic three-dimensional cubic LDC with the staggered grid arrangement.

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Comprehensive understanding of the routes of instability and transition for many flows is not complete yet. For a zero pressure gradient (ZPG) boundary layer, linear spatial theory predicted Tollmien-Schlichting (TS) waves, which have been experimentally verified by vortically exciting the flow by a monochromatic source. This is the well-known frequency response of dynamical system theory. Natural transition in real flows occurs due to polychromatic excitation, and to simulate such transition, the ZPG boundary layer has been excited via an impulse response in some of our recent direct numerical simulations. Such impulse responses cause transition even when TS waves are not excited. In the present exercise, we show the theoretical basis of natural transition by spatiotemporal stability analysis, as used in the work of Sengupta et al. [“Spatiotemporal growing wave fronts in spatially stable boundary layers,” Phys. Rev. Lett. 96(22), 224504 (2006)], by invoking finite start-up of the frequency response to wall excitation. There appear to be different instability mechanisms active for the frequency and the impulse responses to localized wall excitation. Here, we show that in both the frequency and impulse responses, the spatiotemporal wave-front (STWF) is the common element. Additionally, we also consider cases, where following different start-ups, the wall excitation remains constant, which also show the presence of the STWF. The presented results for the ZPG boundary layer show that the TS wave is not necessary for transition to turbulence and help us to re-evaluate our understanding of the transition mechanism for this canonical flow.

Turbulence has remained an unsolved problem in physics, despite the availability of some numerical results. The onset and growth of disturbances leading to two-and three-dimensional turbulence have been theoretically and computationally shown for wall excitation with the response having modal and nonmodal components in the spectrum. The nonmodal component is seen to be dominant. Here, we conclusively show the nonmodal growth for a transition to turbulence, with the same equilibrium flow excited from the free stream with results from linearized and nonlinear two-dimensional Navier-Stokes equations. We establish that transition to turbulence definitely requires nonlinearity, even if the onset process is similar to a linear mechanism. Thus, the transition to turbulence in wall-bounded flows is due to a nonmodal nonlinear mechanism for any disturbance.

Transition to turbulence for flow over a semi-infinite flat plate can occur due to disturbances convecting in the free stream, and various routes have been studied in the literature. Specifically, this has been discussed in great detail as a global receptivity problem [Sengupta et al., “Nonmodal nonlinear route of transition to two-dimensional turbulence,” Phys. Rev. Res. 2, 012033(R) (2020)]. Here, the same is studied for the deterministic excitation of a vortex translating with a fixed speed at a fixed height outside the boundary layer. The study aims to capture the fundamental differences in the mechanisms followed by linear and nonlinear global formulations, the existence of the nonmodal spatio-temporal wave front (STWF), onset of nonlinearity, and nonlinear saturation of the STWF by dispersion and phase effects. Other issues related to the initial location of the vortex and its strength are addressed in the present work. The main finding of the investigation is that the onset of the STWF is captured by both formulations, but it occurs at different locations and times. The nonlinear global receptivity study shows earlier onset and saturation of the STWF, while the corresponding linear study shows a delayed onset and no saturation. Overall, it is essential to operate within a nonlinear framework to capture the later stages of transition, phase shift due to nonlinear dispersion and saturation, which depend strictly on nonlinearity.