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In this work, we consider Anderson acceleration for numerical solutions of nonlinear time dependent partial differential equations discretized by space–time spectral methods, where classical fixed-point methods converge slowly or even diverge. Specifically, we apply Anderson acceleration with finite window size w to speed up fixed-point methods in solving nonlinear reaction diffusion, nonlinear Schrödinger and Navier Stokes equations. We focus on studying the influence of the window size w on the number of iterations to numerical convergence. Numerical results show the high efficiency of Anderson acceleration in solving a variety of nonlinear time dependent problems discretized by space–time spectral methods, and a small value of w is enough to achieve good performance. [ABSTRACT FROM AUTHOR]

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We consider both stationary and time-dependent solutions of the 3-D Navier–Stokes equations (NSE) on a multi-connected bounded domain Ω ⊂ R 3 with inhomogeneous boundary values on ∂ Ω = Γ ; here Γ is a union of disjoint surfaces Γ 0 , Γ 1 , ... , Γ l . Our starting point is Leray's classic problem, which is to find a weak solution u ∈ H 1 (Ω) of the stationary problem assuming that on the boundary u = β ∈ H 1 / 2 (Γ). The general flux condition ∑ j = 0 l ∫ Γ j β · n d S = 0 must be satisfied due to compatibility considerations. Early results on this problem including the initial results in (J. Math. Pures Appl.12 (1933) 1–82) assumed the more restrictive flux condition ∫ Γ j β · n d S = 0 for each j = 1 , ... , l. More recent results, of which those in (An Introduction to the Mathematical Theory of the Navier–Stokes Equations, vol. II1994 Springer–Verlag) and (In Lectures on the Analysis of Nonlinear Partial Differential Equations2013 237–290 Int. Press) are particularly representative, assume only the general flux condition in exchange for size restrictions on the data. In this paper we also assume only the general flux condition throughout, and for virtually the same size restrictions on the data as in (In Lectures on the Analysis of Nonlinear Partial Differential Equations2013 237–290 Int. Press) we obtain the existence of a weak solution that matches that found in (In Lectures on the Analysis of Nonlinear Partial Differential Equations2013 237–290 Int. Press) when the assumptions imposed here and those assumed in (In Lectures on the Analysis of Nonlinear Partial Differential Equations2013 237–290 Int. Press) are both met; additionally we demonstrate that this solution is unique. For slightly stronger size restrictions we obtain the existence and uniqueness of solutions of both Leray's problem and global mild solutions of the corresponding time-dependent problem, while showing that both the stationary and time-dependent solutions we construct are a bit stronger than weak solutions. The settings in which we establish our results allow us to culminate our discussion by showing that our time-dependent solutions converge to each other exponentially in time, so that in particular our stationary solutions are asymptotically stable. We also discuss additional features which allow for data of increased size on certain domains, including those which are thin in a generalized sense. [ABSTRACT FROM AUTHOR]

Hypersonic boundary-layer stability has significant importance in vehicle design and successful operation. This paper investigates the stabilization effects of local wall cooling on the hypersonic boundary layers over a 5 deg half-angle blunt cone with a nose radius of 0.0254 mm. We employed a high-order-accurate flow solver to calculate the steady flow for a freestream Mach number of 6.0 and a unit Reynolds number of 25.59×106/m. In simulations, we considered partial wall cooling, entire wall cooling, and adiabatic wall scenarios. Furthermore, we examined partial cooling parameters such as strip location, length, and temperature profiles. We calculated the growth rates, phase speed, and N-factor diagrams using a linear stability analysis. The results showed that complete wall cooling destabilizes the boundary layer. However, the cooling strip upstream of the synchronization point stabilized the boundary layer by damping the disturbances. The longer cooling strip further stabilized the boundary layer. The cooling strip placed downstream of the synchronization point destabilized the boundary layer. [ABSTRACT FROM AUTHOR]

The image registration is always a strongly ill-posed problem, a stable numerical approach is then desired to better approximate the deformation vectors. This paper introduces an efficient numerical implementation of the Navier Stokes equation in the fluid image registration context. Although fluid registration approaches have succeeded in handling large image deformations, the numerical results are sometimes inconsistent and unexpected. This is related, in fact, to the used numerical scheme which does not take into consideration the different properties of the continuous operators. To take into account these properties, we use a robust numerical scheme based on finite volume with pressure correction. This scheme, which is called by the Semi-Implicit Method for Pressure-Linked Equation-Consistent (SIMPLEC), is known for its stability and consistency in fluid dynamics context. The experimental results demonstrate that the proposed method is more efficient and stable, visually and quantitatively, compared to some classical registration methods.

The frequency properties and the typical flow features within the interaction zone in conical shock-wave/turbulent boundary-layer interactions (CSBLIs) are analyzed by performing proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) on a time-resolved direct numerical simulation database. The streamwise velocity and pressure fluctuations are processed according to their energy and frequency, respectively. As is the case of planar shock-wave/boundary-layer interactions, fluctuations in some bands of frequencies are more energetic downstream of the flow separation zone. The energetic low-frequency modes of the streamwise velocity fluctuations are reminiscent of Görtler-like vortices, whereas those of pressure fluctuations are connected with the "breathing" motion of the separation bubble. Higher-frequency modes, on the other hand, are related to wrinkling of the impinging conical shock. Although they bear resemblance with those in planar shock/turbulent boundary-layer interactions, it was found that the flow structures herein are less coherent owing to spanwise inhomogeneity of the impinging shock strength. The POD and DMD results of CSBLI support the notion that Görtler-like vortices might be an intrinsic mechanism of the interaction zone, which is probably responsible for the low-frequency unsteadiness of CSBLI. The present results may serve as a basic guidance for the design of fluidic control strategies of shock-induced flow separation. [ABSTRACT FROM AUTHOR]

Recent studies about the Lamb vector have led to the development of the vortex-force theory: a formulation able to predict the aerodynamic force in compressible flows and decompose it into lift, lift-induced drag, and profile drag. Here, a revised formulation of the vortex-force theory developed at ONERA in collaboration with the University of Naples is presented and applied to steady transonic flows. In the mathematical developments, special care is given to the presence of shock wave discontinuities within the flowfield. The equivalence between the new definition, the Kutta-Joukowski lift theorem, Maskell's lift-induced drag formula ("Progress Towards a Method for the Measurement of the Components of the Drag of a Wing of Finite Span," Procurement Executive, Ministry of Defence, Royal Aircraft Establishment TR 72232, Farnborough, England, U.K., 1972) and Betz's profile drag formula ("A Method for the Direct Determination of Wing-Section Drag," NACA TM 337, 1925) are emphasized. The revised formulation also presents several practical advantages: the decomposition is naturally invariant to the reference point chosen for the calculation of moment transformations, and the computation can be performed in a fine part of the grid, close to the body skin. The formulation is finally tested on the NASA Common Research Model wing-fuselage configuration in cruise flight conditions. [ABSTRACT FROM AUTHOR]

Computational investigations of a tangent ogive-cylinder geometry with varying forebodies at zero degrees angle of attack are presented. The model geometry and conditions are selected to match experiments conducted in the Air Force Research Laboratory (AFRL) Mach 6 Ludwieg Tube. Specifically, five forebodies of interest consisting of sharp and blunt ogives of 4 and 2 caliber and a hemispherical shape are studied at a freestream unit Reynolds number of 7.01×106 m-1. Boundary-layer-edge properties and wall-normal profiles of velocity and temperature are compared across streamwise locations past the ogive-cylinder junction. Modal stability analysis is used to characterize the most amplified frequencies corresponding to Mack's first and second modes and those are found to agree with experimental results for the sharp forebodies. The entropy layer induced by blunt forebodies envelopes the boundary layer and stabilizes modal disturbances. Nonmodal analysis revealed a broadband set of disturbances present for the blunter forebodies, in agreement with experimental observations. Flow perturbation contours of most amplified planar and oblique disturbances are shown to qualitatively match wind tunnel schlieren images, with a switch from rope-like to elongated structures, i.e., from high-frequency Mack's second modes to low-frequency Mack's first modes, as the forebody angle for the sharp tip is increased, and from boundary-layer to entropy-layer disturbances as the bluntness is increased. [ABSTRACT FROM AUTHOR]

Many industrial applications, such as gas turbines or aeroengines, are subject to thermoacoustic instability, especially in the lean combustion regime. In this paper, the linearized Navier-Stokes equations (LNSEs) framework is applied to predict the acoustic liner damping effects with real geometry and different bias flow velocities. The results show that the LNSEs can capture the tilt angle effect of the orifice or grazing and bias flow coupling phenomenon. The forced large-eddy simulation results of premixed methane flame present the same trend in gain and phase of the flame transfer function with the G equation. Then, the flame response model is coupled into the Helmholtz and LNSE solvers. The mean flow from Reynolds-averaged Navier-Stokes calculation is used in the LNSEs, and no assumption of liner geometry is made. The linear stability analysis reveals that the velocity fluctuation in the combustion is also damped when a stable acoustic mode is achieved with the acoustic liner. The flame dynamic, the hydrodynamic, and the acoustic phenomena can all be considered when using the LNSEs. This indicates that the LNSEs may allow us to consider a more complex mechanism that can affect the thermoacoustic inability in the combustor system and help us to design better dampers to control. [ABSTRACT FROM AUTHOR]

Adjoint methods for aerodynamic design optimization have been a considerable area of research in computational fluid dynamics. Several approaches have been developed to fully couple the flow, adjoint, and design equations for steady flows. However, only limited attempts have been made to apply those to unsteady flows using a monolithic approach. This study aims to fully couple the primal, adjoint, and design equations based on a harmonic balance technique allowing the unsteady flow optimization to be performed in a single monolithic computation. The harmonic balance is the appropriate method for such problems as it reformulates unsteady time-periodic problems as "mathematically steady," where one does not need to store the entire time history of the primal solution. The novel unsteady technique will be applied to a time-periodic lid-driven cavity. Here, the Reynolds number is taken to be the design variable so as to match a target vorticity distribution at the first subtime level of the time-spectral (harmonic balance) solution. Finally, mean drag minimization of a rotationally oscillating cylinder in crossflow is performed. In all cases, it is shown that the proposed method is quite efficient, requiring less than a couple of hundred total iterations to identify the "optimum" conditions. [ABSTRACT FROM AUTHOR]

A widely used approach to mathematically describe the atmosphere is to consider it as a geophysical fluid in a shallow domain and thus to model it using classical fluid dynamical equations combined with the explicit inclusion of an ϵ ϵ parameter representing the small aspect ratio of the physical domain. In our previous paper [ 14 ] we proved a weak convergence theorem for the polluted atmosphere described by the Navier-Stokes equations extended by an advection-diffusion equation. We obtained a justification of the generalised hydrostatic limit model including the pollution effect described for the case of classical, east-north-upwards oriented local Cartesian coordinates. Here we give a two-fold improvement of this statement. Firstly, we consider a meteorologically more meaningful coordinate system, incorporate the analytical consequences of this coordinate change into the governing equations, and verify that the weak convergence still holds for this altered system. Secondly, still considering this new, so-called downwind-matching coordinate system, we prove an analogous strong convergence result, which we make complete by providing a closely related existence theorem as well. [ABSTRACT FROM AUTHOR]

We present an implicit large-eddy simulation of transonic buffet over the OAT15A supercritical airfoil at Mach number 0.73, angle of attack 3.5 deg, and Reynolds number 3×106. The simulation is performed using a matrix-free discontinuous Galerkin (DG) method and a diagonally implicit Runge-Kutta scheme on graphics processor units. We propose a Jacobian-free Newton-Krylov method to solve nonlinear systems arising from the discretization of the Navier-Stokes equations. The method successfully predicts the buffet onset, the buffet frequency, and turbulence statistics owing to the high-order DG discretization and an efficient mesh refinement for the laminar and turbulent boundary layers. A number of physical phenomena present in the experiment are captured in our simulation, including periodical low-frequency oscillations of shock wave in the streamwise direction, strong shear layer detached from the shock wave due to shock-wave/boundary-layer interaction and small-scale structures broken down by the shear-layer instability in the transition region, and shock-induced flow separation. The pressure coefficient, the root mean square of the fluctuating pressure, and the streamwise range of the shock wave oscillation agree well with the experimental data. The results suggest that the proposed method can accurately predict the onset of turbulence and buffet phenomena at high Reynolds numbers without a subgrid scale model or a wall model. [ABSTRACT FROM AUTHOR]

The current work presents a simulation framework for modeling high-speed flows with suspended solid particles using the point-particle discrete element method. The dispersed particle phase is treated in a Lagrangian manner, whereas the dynamics of the surrounding carrier fluid are determined by solving the compressible Navier-Stokes equations in the Eulerian frame. Additionally, a variable fidelity paradigm is adopted with the ability to model one-, two-, and four-way coupling between the dispersed and carrier phases depending on the desired levels of physical realism. Particle mesh-localization is performed using mesh-connectivity information, which also simplifies boundary conditions enforcement, parallelization, and the backcoupling of particle-induced source terms to the carrier fluid. A time-driven hard-sphere approach is employed to accelerate calculations for particle-particle collisions. The particle solution methodology in conjunction with the unstructured, finite-volume US3D flow solver is used to simulate a variety of multiphase flows. These include a series of verification tests and the first-of-its-kind numerical investigation into elevated levels of surface heat flux experienced by the Mars 2020 spacecraft if planetary entry through the Martian atmosphere were to occur during a major dust storm. [ABSTRACT FROM AUTHOR]

High-order spectral element methods are becoming more established for solving problems in computational fluid dynamics. In this paper, we introduce new stabilization parameters for the solution of compressible Navier-Stokes equations in conjunction with high-order spectral element methods. We previously defined a new set of stabilization parameters for Euler equations in a high-order spectral/hp framework. In this paper, we examine the definition of these stabilization parameters in the context of solutions of full Navier-Stokes equations, and we report good agreement with previously published results in the literature. We establish spectral L2 convergence of errors for Kovasznay flow. We then validate the definition of the stabilization parameter for Couette flow, flat plate, and compression corner problems. In a later example, we solve the flow past a cylinder and benchmark results with previously published results obtained with low-order methods and other approaches in compressible flow computations. We further solve hypersonic flow past a wedge and supersonic flow past a Flight Investigation of Re-Entry program (FIRE) entry capsule. We explore the definition of the shock-capturing parameter d based on the YZß parameter. The introduced definitions are found to provide excellent results for the full Navier-Stokes equations. [ABSTRACT FROM AUTHOR]

Drag decomposition has classically been undertaken using a far-field approach both from a wind-tunnel testing and computational fluid dynamics standpoint. It has been suggested by Schmitz ("Drag Decomposition Using Partial-Pressure Fields in the Compressible Navier-Stokes Equations," AIAA Journal, Vol. 57, No. 5, 2019, pp. 2030-2038) that a decomposition of static pressure into partial pressure fields can allow one to decompose a computational fluids dynamics solution in the near field. This paper covers the implementation process of partial pressure fields on the ONERA M6 wing at conditions relevant to commercial transport aircraft. Relations between partial pressure field decomposition, the classical far-field theory, and physical drag sources are discussed. It is demonstrated that the near-field partial pressure field decomposition is equivalent to classical far-field analysis. In transonic flow, a new method to compute wave drag using a combination of both partial pressure fields and the far-field method is demonstrated. [ABSTRACT FROM AUTHOR]

We consider nonlinear solvers for the incompressible, steady (or at a fixed time step for unsteady) Navier–Stokes equations in the setting where partial measurement data of the solution is available. The measurement data is incorporated/assimilated into the solution through a nudging term addition to the Picard iteration that penalized the difference between the coarse mesh interpolants of the true solution and solver solution, analogous to how continuous data assimilation (CDA) is implemented for time dependent PDEs. This was considered in the paper (Li et al. 2023), and we extend the methodology by improving the analysis to be in the L 2 norm instead of a weighted H 1 norm where the weight depended on the coarse mesh width, and to the case of noisy measurement data. For noisy measurement data, we prove that the CDA-Picard method is stable and convergent, up to the size of the noise. Numerical tests illustrate the results, and show that a very good strategy when using noisy data is to use CDA-Picard to generate an initial guess for the classical Newton iteration. [ABSTRACT FROM AUTHOR]

Turbulent flow over a wall-mounted cube presents an important canonical problem with applications in circuit design, urban building design, or aerodynamics. Flow over a cube placed in a fully developed turbulent channel flow has long been the preferred problem of investigation for researchers. However, a more relevant practical scenario is flow over a cube placed in a spatially evolving turbulent boundary layer (SETBL). In this work, wall-resolved large-eddy simulations of flow over a wall-mounted cube placed in an SETBL on a flat plate are performed at a Reynolds number of 19,600 based on the mean velocity and the inlet boundary-layer thickness. The incompressible Navier-Stokes equations are solved using a second-order finite-volume approach with a dynamic k -equation eddy-viscosity model. At this Reynolds number, the key nondimensional parameter governing this problem is the ratio of cube height to boundary-layer thickness, h/δo. This paper investigates the dependence of the turbulence wake characteristics on h/δo. Specifically, this paper analyzes the vortical structures and the separation and reattachment lengths. It is further examined how the turbulent kinetic energy budget varies with h/δo. It is found that the size of the horseshoe vortex roughly scales with the size of the cube, and that the decay of turbulent kinetic energy per unit area over streamwise distance in the near wake exhibits a power-law behavior. [ABSTRACT FROM AUTHOR]

Traditional proper orthogonal decomposition (POD)-Galerkin projection for reduced-order models (ROMs) of fluid flows is based on a fixed domain. The current method removes this limitation by considering a single combined domain of fluid and solid, whereas the original solid boundary conditions are reinforced by additional ROM terms. The combined domain requires a new inner product defined in the same combined domain to compute POD modes and the projection of equations. Solid motion is considered first as a continuous motion, then a decomposed motion, represented by a few solid modes to further reduce the computational cost. This new global approach was applied first on a two-dimensional direct numerical simulation (DNS) database of the flow past an oscillatory cylinder, and then on a three-dimensional DNS database of the flow past an oscillatory sphere. Last, the approach was applied on a high-resolution particle image velocimetry database from an experiment at higher Reynolds number of the flow past a pitching-up NACA0012 airfoil. The ROMs derived by the global POD-Galerkin projection approach, for all three moving-boundary cases, have shown adequate accuracy in the reconstruction of flow fields and the prediction of key aerodynamic features while keeping very low computational costs. [ABSTRACT FROM AUTHOR]

The influence of the flat-plate leading-edge shape on laminar-turbulent transition is investigated. Experiments were carried out at a Mach number of M8=5; a unit Reynolds number Re18 from 1.5×107 to 9×107 m-1; and a Reynolds number Reb8, based on the leading-edge thickness b, up to 7×105. Four leading-edge shapes were investigated: cylindrical, elliptical, flat face, and "smoothed" cylindrical. It has been shown that for Reb8<0.4×105, these shapes practically do not affect transition; whereas at large Reb8, the laminar run can be lengthened or shortened by the leading-edge shaping. The influence of bluntness thickness on the flow parameters at transition onset is analyzed by solving the Navier-Stokes and boundary-layer equations. [ABSTRACT FROM AUTHOR]

Multidimensional numerical simulations are performed to investigate the evolution and formation of unburned fuels for a shock-fuel jet interaction scenario. A full set of Navier-Stokes equations with detailed chemical mechanisms are solved, and the results are analyzed through the Lagrangian method with the goal of improving combustion efficiency in supersonic flows. The flame morphology of a two-dimensional (2-D) non-premixed reactive shock-bubble interaction is first simulated and studied, in which unburned hydrogen is found to prevent efficient combustion. By applying the Lagrangian particle tracking method, most of the unburned hydrogen wrapped into the primary vortex turns out to be initially located upon the symmetry line of the bubble. Motivated by the idea of breaking the primary vortex, this study designs a novel geometry of a concentric bubble, which improves combustion efficiency to 94.4% in contrast to a solid fuel bubble (74%) due to multivortex interaction and a thick bridge structure. With the consistency between qualitative and quantitative 2-D and three-dimensional (3-D) flow dynamics, the idea of a 2-D concentric-bubble configuration is effectively extended to a 3-D coaxial jet interacting with oblique shock despite the existence of Kelvin-Helmholtz instability. [ABSTRACT FROM AUTHOR]

Spectral proper orthogonal decomposition (SPOD) is a major modal analysis that extracts a dominant structure at each frequency from complicated flow data, such as the wavepacket hidden under turbulent jets. This study investigated the physical interpretation of subdominant multilobe structures in the SPOD modes. A high-speed schlieren movie that exhibited multilobe structures in the SPOD was analyzed using a wavelet-based conditional sampling and averaging (WB-CSA) method. The results of WB-CSA showed spatiotemporally localized wavepackets, which were formed intermittently and convected downstream through growth and decay. Such intermittent convection causes multilobe structures in the SPOD, as illustrated by simple synthetic data that model the results of WB-CSA. Therefore, multilobe structures, together with the most dominant single-lobe wavepackets, can be interpreted as representing the intermittent convection of wavepackets. [ABSTRACT FROM AUTHOR]

Thermal (or heat) explosion occurs in a reacting medium if the heat production due to an exothermic chemical reaction exceeds the heat loss through the boundary. In the mathematical approximation heat explosion is characterized by an unbounded temperature growth. If the reaction occurs in a liquid or gaseous medium, then a nonuniform temperature distribution can lead to natural convection. The interaction of heat explosion with natural convection can result in various regimes with a bounded temperature distribution (stationary, periodic, chaotic) and to a transition to heat explosion. The latter can be accompanied by a monotonic temperature growth or by temperature oscillations (oscillating heat explosion). This paper presents a review on mathematical modelling of heat explosion with natural convection in a homogeneous fluid and in a porous medium. [ABSTRACT FROM AUTHOR]

Abstract Herein we present a study on the long-time stability of finite element discretizations of a generalized class of semi-implicit second-order time-stepping schemes for the 2D incompressible Navier–Stokes equations. These remarkably efficient schemes require only a single linear solve per time-step through the use of a linearly-extrapolated advective term. Our result develops a class of sufficient conditions such that if external forcing is uniformly bounded in time, velocity solutions are uniformly bounded in time in both the L 2 and H 1 norms. We provide numerical verification of these results. We also demonstrate that divergence-free finite elements are critical for long-time H 1 stability. [ABSTRACT FROM AUTHOR]

In hypersonic flow calculations, the grid spacing next to the wall surface is one of the key factors in the prediction of surface heat transfer. The main tool to build the chemical nonequilibrium mathematical model is the system of the axisymmetric Navier-Stokes equations, the thermodynamic relations, and the chemical reaction model. The axisymmetric solver is developed by the implicit finite volume schemes and is validated by comparing the computational results with that in the literature. Combining the Kemp-Riddell empirical formula with the numerical results, a new criterion for the first grid spacing next to the wall surface is proposed by considering three factors such as the freestream density, the velocity, and the radius of a blunt nose. It can effectively predict the surface heat flux at the stagnation point. [ABSTRACT FROM AUTHOR]

Aerodynamic aspects of train shapes suitable for Vacuum Tube Train System are investigated in this paper. Three feasible geometries for the vacuum tube train system have been considered and modelled in three dimensions and have been computationally studied using the commercial software Ansys Fluent. Aerodynamic drag loads on these geometries have been calculated under different tube pressures and speeds of the train, which provide insight on various operating parameters that need to be considered while designing the vacuum tube train system. The present computational research shows that, the suitable vacuum pressure, and different shapes of head and tail of the train have significantly effects the drag force of the vacuum train in the tunnel. Overall, the elliptical train shape with a height to base ratio of 2:1 is more efficient for aerodynamic drag reduction of the vacuum tube train at the vacuum tube pressure of 1013.25 Pa. [ABSTRACT FROM AUTHOR]

The core of this paper is the group-theoretical approach, initiated in 2015 by one of the present authors in collaboration with Alexander Sorin that brings into the classical field of mathematical fluid-mechanics a brand new vision, allowing for a more systematic classification and algorithmic construction of Beltrami flows on torii R 3 / Λ where Λ is a crystallographic lattice. Here this new hydro-theory based on the focal idea of a Universal Classifying Group UG Λ is revised, reorganized, improved and extended. In particular we construct the so far missing UG Λ H e x for the hexagonal lattice and we advocate that, mastering the cubic and hexagonal instances of this group, we can cover all cases. The relation between the classification of Beltrami Flows with that of contact structures is enlightened. The recent developments about the framework of b -manifolds are considered and it is shown that the choice of the allowed critical surfaces for the b -deformation of a Beltrami field seems to be strongly related to the group-theoretical structure of the latter. This opens new directions of investigations about a group theoretical classification of critical surfaces. Apart from that the most promising research direction opened by the present work streams from the fact that the Fourier series expansion of a generic Navier-Stokes solution can be regrouped into an infinite sum of contributions W r , each associated with a spherical layer of quantized radius r in the momentum lattice. Each W r is the superposition of a Beltrami field W r + plus an anti-Beltrami field W r −. These latter have a priori exactly the same decomposition into irreps of the group UG Λ that are variously repeated on higher layers. This crucial property enables the construction of generic Fourier series with prescribed hidden symmetries as candidate solutions of the NS equations. Alternatively the Fourier series representation of known solutions can be analyzed from the point of view of such symmetries. As a further result of this research program a complete and versatile system of MATHEMATICA Codes named AlmafluidaNSPsystem has been constructed and is now available through the site of the Wolfram Community. The exact solutions presented in this paper have to be considered as an illustration of the new conceptions and ideas that have emerged and of what can be further done utilizing the computer codes as an instrument. The main message streaming from our constructions is that the more symmetric is the Beltrami Flow the highest is the probability of the on-set of chaotic trajectories. [ABSTRACT FROM AUTHOR]

Shell models of turbulence are representation of turbulence equations in Fourier domain. Various shell models and their existence theory along with numerical simulations have been studied earlier. One of the most suit- able shell model of turbulence is so called sabra shell model. The existence, uniqueness and regularity property of this model are extensively studied in [11]. We follow the same functional setup given in [11] and study control problems related to it. We associate two cost functionals: one ensures minimizing turbulence in the system and the other addresses the need of taking the flow near a priori known state. We derive optimal controls in terms of the solution of adjoint equations for corresponding linearized problems. Another interesting problem studied in this work is to establish feedback controllers which would preserve prescribed physical constraints. Since fluid equations have certain fundamental invariants, we would like to preserve these quantities via a control in the feedback form. We utilize the theory of nonlinear semi groups and represent the feedback control as a multi-valued feedback term which lies in the normal cone of the convex constraint space, under certain assumptions. Moreover, one of the most interesting result of this work is that we can design a feedback control with only finitely many modes, which is able to preserve the flow in the neighborhood of the constraint set. [ABSTRACT FROM AUTHOR]

We show that the asymptotics of solutions to stationary Navier Stokes equations in 4, 5 or 6 dimensions in the whole space with a smooth compactly supported forcing are given by the linear Stokes equation. We do not need to assume any smallness condition. The result is in contrast to three dimensions, where the asymptotics for steady states are different from the linear Stokes equation, even for small data, while the large data case presents an open problem. The case of dimension n = 2 is still harder. [ABSTRACT FROM AUTHOR]