Efficient Approaches for Solving Systems of Nonlinear Time-Fractional Partial Differential Equations

In this work, we present a modified generalized Mittag–Leffler function method (MGMLFM) and Laplace Adomian decomposition method (LADM) to get an analytic-approximate solution for nonlinear systems of partial differential equations (PDEs) of fractional-order in the Caputo derivative. We apply the MGMLFM and LADM on systems of nonlinear time-fractional PDEs. Precisely, we consider some important fractional-order nonlinear systems, namely Broer–Kaup (BK) and Burgers, which have found major significance because they arise in many physical applications such as shock wave, wave processes, vorticity transport, dispersal in porous media, and hydrodynamic turbulence. The analysis of these methods is implemented on the BK, Burgers systems and solutions have been offered in a simple formula. We show our results in figures and tables to demonstrate the efficiency and reliability of the used methods. Furthermore, our outcome converges rapidly to the given exact solutions.

Numerical simulation of thermal behavior in a naturally ventilated greenhouse

Inside a greenhouse, during the day, the temperature rises very quickly, while the plants have to face temperatures that rise to more than 35[Formula: see text]C. The plant closes its pores to limit sweating and stops growing. As soon as it gets hot, it is therefore necessary to ventilate the greenhouse. In this context, this research aims to investigate the behavior of the natural ventilation on the internal climate of the tunnel greenhouse, which contains two openings in the roof. The effect of the position of the openings on heat transfer is considered, thus promoting photosynthesis and plant growth. The vorticity transport equation, the Poisson equation and the energy equation are discretized by using the finite volume method. Two-dimensional simulations that described laminar flows in a steady state were carried out. Flows are studied for a range of parameters: the Rayleigh number, Ra, [Formula: see text], and three positions of opening ventilation. The results reveal that the ventilation through the top opening position allows the best creation of heat exchanges between the air inside the greenhouse and its atmosphere, which serves to conserve the plant under a favorable climate that allows its growth.

Verification and Validation of Large Eddy Simulation for Tip Clearance Vortex Cavitating Flow in a Waterjet Pump

In this paper, large eddy simulation (LES) was adopted to simulate the cavitating flow in a waterjet pump with emphasis on the tip clearance flow. The numerical results agree well with the experimental observations, which indicates that the LES method can make good predictions of the unsteady cavitating flows around a rotor blade. The LES verification and validation (LES V&V) analysis was used to reveal the influence of cavitation on the flow structures. It can be found that the LES errors in cavitating region are larger than those in the non-cavitating area, which is mainly caused by more complicated cavitating and tip clearance flow structures. Further analysis of the interaction between the cavitating and vortex flow by the relative vorticity transport equation shows that the stretching, dilatation and baroclinic torque terms have major effects on the generation and transport of vortex structure. Meanwhile the Coriolis force term and viscosity term also exacerbate the vorticity transport in the cavitating region. In addition, the flow loss characteristics of this pump are also revealed by the entropy production theory. It is indicated that the tip clearance flow and trailing edge wake flow cause the viscous dissipation and turbulent dissipation, and the cavitation can further enhance the instability of the flow field in the tip clearance.

Influence of different blade numbers on the performance of “saddle zone” in a mixed flow pump

In order to study the effect of different numbers of impeller blades on the performance of mixed-flow pump “saddle zone”, the external characteristic test and numerical simulation of mixed-flow pumps with three different impeller blade numbers were carried out. Based on high-precision numerical prediction, the internal flow field and tip leakage flow field of mixed flow pump under design conditions and stall conditions are investigated. By studying the vorticity transport in the stall flow field, the specific location of the high loss area inside the mixed flow pump impeller with different numbers of blades is located. The research results show that the increase in the number of impeller blades improve the pump head and efficiency under design conditions. Compared to the 4-blade impeller, the head and efficiency of the 5-blade impeller are increased by 5.4% and 21.9% respectively. However, the increase in the number of blades also leads to the widening of the “saddle area” of the mixed-flow pump, which leads to the early occurrence of stall and increases the instability of the mixed-flow pump. As the mixed-flow pump enters the stall condition, the inlet of the mixed-flow pump has a spiral swirl structure near the end wall for different blade numbers, but the depth and range of the swirling flow are different due to the change in the number of blades. At the same time, the change in the number of blades also makes the flow angle at 75% span change significantly, but the flow angle at 95% span is not much different because the tip leakage flow recirculates at the leading edge. Through the analysis of the vorticity transport results in the impeller with different numbers of blades, it is found that the reasons for the increase in the values of the vorticity transport in the stall condition are mainly impacted by the swirl flow at the impeller inlet, the tip leakage flow at the leading edge and the increased unsteady flow structures.

A High-Order Flux Reconstruction Method for 2-D Vorticity Transport

A compact high-order finite difference method on unstructured meshes is developed for discretization of the unsteady vorticity transport equations (VTE) for 2-D incompressible flow. The algorithm is based on the Flux Reconstruction Method of Huynh [1, 2], extended to evaluate a Poisson equation for the streamfunction to enforce the kinematic relationship between the velocity and vorticity fields while satisfying the continuity equation. Unlike other finite difference methods for the VTE, where the wall vorticity is approximated by finite differencing the second wall-normal derivative of the streamfunction, the new method applies a Neumann boundary condition for the diffusion of vorticity such that it cancels the slip velocity resulting from the solution of the Poisson equation for the streamfunction. This yields a wall vorticity with order of accuracy consistent with that of the overall solution. In this paper, the high-order VTE solver is formulated and results presented to demonstrate the accuracy and convergence rate of the Poisson solution, as well as the VTE solver using benchmark problems of 2-D flow in lid-driven cavity and backward facing step channel at various Reynolds numbers.

A Hybrid High-Order Vorticity-Based Eulerian and Lagrangian Vortex Particle Method, the 2-D Case

Hybrid Lagrangian-Eulerian solvers combine the convective and compactness advantages of vortex methods with the spatial anisotropy and boundary-resolving advantages of Eulerian methods to create flexible solvers capable of adequately capturing thin boundary layers while still maintaining wake vortex coherency for unsteady incompressible flow in complex geometries. The present paper details a new hybrid method which combines, in one open-source package, a novel, compact, high-order Eulerian scheme for vorticity transport to predict the flow in the near-boundary region with a grid-free, unremeshed, Lagrangian Vortex Particle Method (LVPM) for the off-boundary vorticity-containing region. This paper focuses on the hybridization of the two methods and demonstrates its effectiveness on two canonical benchmarks: flow in 2-D lid-driven cavity at Re = 1,000 and impulsively started flow over a circular cylinder at Re = 9,500. In each case, the hybrid method improves upon a pure LVPM and uses far fewer cells than a purely Eulerian scheme. In addition, the size of the associated Eulerian region is greatly reduced compared to previous hybrid methods.

Eulerian time-marching in Vortex-In-Cell (VIC) method: reconstruction of multiple time-steps from a single vorticity volume and time-resolved boundary condition

A data assimilation approach is proposed to enhance the dynamic range of the Vortex-In-Cell (VIC) method by simulating future- and past- instances. The VIC method mainly considers a vorticity field from which velocity and acceleration fields are calculated through Poisson equations, respectively bounded by prescribed conditions. In addition, a vorticity time derivative is also available by the vorticity transport equation. The proposed approach focuses on such already available data, i.e., the vorticity and its time derivative fields, for simulating additional instances and getting feedbacks from the corresponding measurement instances, e.g., particle image velocimetry (PTV). However, the self-simulated flow field can be depleted due to a lack of incoming information, which is out of the reconstruction domain at the source instance. To supply that kind of information and thus sustain the simulation, boundary conditions of the simulated instances are required and considered. As a result, the proposed approach can gather corrections from multiple PTV instances while optimizing a single vorticity volume and time-resolved boundary conditions. Since the boundary grid points are much smaller in number than that of the whole volume, one can expect an increased dynamic range. A former work, VIC# (Jeon et al. 2018), which supplements additional constraints and coarse-grid approximation to VIC+ (Schneiders and Scarano 2016), is selected as a 3D method to which the proposed 4D approach is applied. Two explicit Eulerian time-marching methods are tested as a simulation scheme: the forward Euler and the Runge-Kutta methods. A numerical assessment is conducted using the synthetic PTV data, whose ground truth is known, and returns reconstruction qualities based on the velocity and the identified vortical structures. Other practical features regarding convergence and computation complexity are also reported. To visually verify an improvement by the proposed approach, two kinds of time-resolved Shake-the-Box (STB) measurements, which were acquired in high-speed systems, are processed and discussed.

Numerical Analysis of the Effect of Cavitation on the Tip Leakage Vortex in an Axial-Flow Pump

To understand the effect of cavitation on the tip leakage vortex (TLV), turbulent cavitating flows were numerically investigated using the shear-stress transport (SST) k–ω turbulence model and the Zwart–Gerber–Belamri cavitation model. In this work, two computations were performed—one without cavitation and the other with cavitation—by changing the inlet pressure of the pump. The results showed that cavitation had little effect on the pressure difference between the blade surfaces for a certain cavitation number. Instead, it changed the clearance flow and TLV vortex structure. Cavitation caused the TLV core trajectory to be farther from the suction surface and closer to the endwall upstream of the blade. Cavitation also changed the vortex strength distribution, making the vortex more dispersed. The vortex flow velocity and turbulent kinetic energy were lower, and the pressure pulsation was more intense in the cavitating case. The vorticity transport equation was used to further analyze the influence of cavitation on the evolution of vortices. Cavitation could change the vortex stretching term and delay the vortex bending term. In addition, the vortex dilation term was drastically changed at the vapor–liquid interface.