Boundary conditions for probability density function transport equations in fluid mechanics.

Reactive flow simulations using large-eddy simulations (LES) require modelling of sub-filter fluctuations. Although conserved scalars like mixture fraction can be represented using a beta-function, the reactive scalar probability density function (PDF) does not follow an universal shape. A one-point one-time joint composition PDF transport equation can be used to describe the evolution of the scalar PDF. The high-dimensional nature of this PDF transport equation requires the use of a statistical ensemble of notional particles and is directly coupled to the LES flow solver. However, the large grid sizes used in LES simulations will make such Lagrangian simulations computationally intractable. Here we propose the use of a Eulerian version of the transported-PDF scheme for simulating turbulent reactive flows. The direct quadrature method of moments (DQMOM) uses scalar-type equations with appropriate source terms to evolve the sub-filter PDF in terms of a finite number of delta-functions. Each delta-peak is characterized by a location and weight that are obtained from individual transport equations. To illustrate the feasibility of the scheme, we compare the model against a particle-based Lagrangian scheme and a presumed PDF model for the evolution of the mixture fraction PDF. All these models are applied to an experimental bluff-body flame and the simulated scalar and flow fields are compared with experimental data. The DQMOM model results show good agreement with the experimental data as well as the other sub-filter models used.

A large eddy simulation based on filtered vorticity transport equation has been coupled with filtered probability density function transport equation for scalar field, to predict the velocity and passive scalar fields. The filtered vorticity transport has been formulated using diffusion-velocity method and then solved using the vortex method. The methodology has been tested on a spatially growing mixing layer using the two-dimensional vortex-in-cell method in conjunction with both Smagorinsky and dynamic eddy viscosity subgrid scale models for an anisotropic flow. The transport equation for filtered probability density function is solved using the Lagrangian Monte-Carlo method. The unresolved subgrid scale convective term in filtered density function transport is modelled using the gradient diffusion model. The unresolved subgrid scale mixing term is modelled using the modified Curl model. The effects of subgrid scale models on the vorticity contours, mean streamwise velocity profiles, root-mean-square velocity and vorticity fluctuations profiles and negative cross-stream correlations are discussed. Also the characteristics of the passive scalar, i.e. mean concentration profiles, root-mean-square concentration fluctuations profiles and filtered probability density function are presented and compared with previous experimental and numerical works. The sensitivity of the results to the Schmidt number, constant in mixing frequency and inflow boundary conditions are discussed. Copyright © 2005 John Wiley & Sons, Ltd.

A statistical description of turbulent diffusion flame holes

LES-based Eulerian PDF approach for the simulation of scramjet combustors

The composition fields in turbulent reacting flows are affected by turbulent transport (macromixing), molecular diffusion (micromixing), and chemical reactions. In the joint velocity‐composition probability density function transport equation the highly nonlinear macromixing and chemical reaction terms appear in closed form. This is a considerable advantage over second moment closure methods. Micromixing on the other hand requires modelling and especially for turbulent combustion accurate mixing models are crucial. In this paper we present an approach to model the mixing of scalars, e.g. species mass fractions or temperature, based on considering one‐dimensional parameterized scalar profiles (PSP). (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

The objective of the present work is twofold: (1) extend the coupled Monte Carlo probability density function (PDF)/computational fluid dynamics (CFD) computations to the modeling of turbulent spray flames, and (2) extend the PDF/SPRAY/CFD module to parallel computing in order to facilitate large-scale combustor computations. In this approach, the mean gas phase velocity and turbulence fields are determined from a standard turbulence model, the joint composition of species and enthalpy from the solution of a modeled PDF transport equation, and a Lagrangian-based dilute spray model is used for the liquid-phase representation. The PDF transport equation is solved by a Monte Carlo method, and the mean gas phase velocity and turbulence fields together with the liquid phase equations are solved by existing state-of-the-art numerical representations. The application of the method to both open as well as confined axisymmetric swirl-stabilized spray flames shows good agreement with the measured data. Preliminary estimates indicate that it is well within reach of today's modern parallel computer to do a realistic gas turbine combustor simulation within a reasonable turnaround time. The article provides complete details of the overall algorithm, parallelization, and other numerical issues related to coupling between the three solvers.

In this paper, a special integrable negative-order Calogero–Bogoyavlenskii–Schiff equation (nCBS) in fluid mechanics is studied by means of the symmetry reduction method and consistent tanh expansion method. The Painlevé integrability is investigated to confirm the compatibility conditions. This integrable nCBS equation has been transformed into different reduction equations and the corresponding invariant solutions with arbitrary functions are obtained. The corresponding structures of the invariant solutions for the nCBS equation are also shown graphically. At last, new types of soliton–cnoidal interaction solutions for the nCBS equation are presented through the consistent tanh expansion method on the basis of the truncated Painlevé expansion.

Mixed lump-stripe, bright rogue wave-stripe, dark rogue wave-stripe and dark rogue wave solutions of a generalized Kadomtsev–Petviashvili equation in fluid mechanics

Of current interest, the paper commented [i.e., Eur. Phys. J. Plus (2020) 135:272] has investigated the $$(3+1)$$ -dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation in ocean dynamics, fluid mechanics and plasma physics. However, the story that the paper commented has caused could be made more complete. Hereby, for the same equation, via the Bell polynomials, we work out the same bilinear form as that in the paper commented, but the ways of derivation are different. With symbolic computation, we also construct two auto-Backlund transformations for the same equation with some solitons. Our results are with respect to the amplitude or elevation of the relevant wave in ocean dynamics, fluid mechanics and plasma physics.

Due to the growing concern on the issues of global warming, climate change, and energy shortage, more rigorous requirements are imposed on the energy conversion process of combustion. Due to the relative safety and convenience in transportation and storage, liquid fuels are commonly used in many practical combustion systems such as diesel engines, gas turbines and industrial furnaces. In these combustion devices, turbulent spray flows are involved. The character of spray dispersion, evaporation, mixing and combustion of fuel droplets strongly determines the performance of these systems with respect to the combustion efficiency, stability, and emissions. Therefore, an improved understanding of turbulent spray combustion and the development of predictive models are needed for a better design of more efficient combustion systems. The present work focuses on the development of a computational methodology based on the transported joint probability density function (PDF) method for the modeling and simulation of two-phase turbulent spray flows without and with chemical reactions. In the non-reacting situation, the dependent variables of the joint PDF include the gas velocity and the mixture fraction. For the simulation of reacting spray flows, a three-variate joint PDF transport equation is derived and modeled in order to account for the pre-vaporization effects and partially premixed regime in turbulent spray flames. The dependent variables include the gas phase mixture fraction, the reaction progress variable and gas enthalpy. Detailed combustion chemistry is considered through an extended spray flamelet model by including a reaction progress variable in addition to the classical formulation. The dilute spray is simulated using a Lagrangian discrete parcel method for the description of droplet motion, heating and evaporation. The infinite conductivity model with consideration of non-equilibrium effects based on the Langmuir-Knudsen law is considered. The spray evolution and flame structures in the frame of the polydisperse reacting spray flows are investigated. Numerical results are compared with experimental data provided by Prof. Masri at the University of Sydney, Australia, and the test cases include three different turbulent non-reacting acetone spray flows in air and turbulent spray flames with the liquid fuel ethanol. For the acetone spray flows, computational results generally show good agreement with experimental data in terms of the droplet size and mean velocity distribution, as well as the liquid volume flux. The results show that the inflow liquid mass loading hardly affects the droplet diameter distribution, whereas the inlet turbulence level has a pronounced effect. The tendency of droplet accumulation near the jet centerline is found with a somewhat overprediction of liquid volume flux at downstream locations. A more sophisticated turbulence model is expected to eliminate this discrepancy. Moreover, the local joint PDF of the gas velocity and the mixture fraction is analyzed. A linear correlation of the gas velocity and the mixture fraction exists close to the nozzle exit outside the main spray jet, and no regions are found where statistical independence prevails. For the simulation of spray flames, computations with the newly developed spray flamelet/progress variable approach and with the previous standard spray flamelet formulation are carried out and compared with the experiments by Prof. Masri at the University of Sydney. A good agreement between the computations with the new formulation and the experiments for gas temperature and droplet size and velocity is achieved. The major spray and combustion properties are correctly captured using this new formulation, which is compared with an unphysically attached flame near the nozzle exit predicted by the previous model. A partially premixed combustion prevails in this piloted turbulent spray flame. Due to the prevaporization of the ethanol droplets near the nozzle exit, a lean premixed gas mixture is found at the inner side of spray jet. Moving downstream, the lean-sided diffusion flame is promoted towards the inner fuel-rich side by heating up the inner premixed core that is controlled by the droplet evaporation. Additionally, it is observed that in the far-field region, the diffusion flame becomes the dominant combustion mode. In summary, an efficient computational model based on the transported joint PDF method is developed to two-phase turbulent spray flows. The combined transported joint PDF and a newly proposed spray flamelet/progress variable approach shows an improved performance in the prediction of complex turbulent spray flames, and new insights on the local flame structure influenced by evaporating sprays are obtained.

We present a fluid---structure interaction (FSI) modeling method based on using the deforming-spatial-domain/stabilized space---time (DSD/SST) method for the fluid mechanics part and a finite difference (FD) method for the structural mechanics part. As the structural mechanics model, we focus on the thin-shell model. The fluid mechanics equations with moving boundaries are solved with the DSD/SST method and the thin-shell structural mechanics equation is solved with a FD method, with partitioned coupling between the two parts. The coupling of the DSD/SST and FD solvers makes sure that the boundary conditions on the fluid-structure interface at the end of each time step are matched between the fluid and the structure. A hanging plate in vacuum under gravitational force is performed to validate the structure solver. In addition, a pitching plate in a uniform flow is simulated to validate the FSI solver. The present results are in reasonable agreement with data predicted by other methods.

The Euler-Taylor-Galerkin flux-corrected transport (ETG-FCT) algorithm for the numerical solution of particle transport equations is described, based on the method developed by Lohner to solve conservation equations in fluid mechanics, and its application is extended to gas discharge problems. To improve the efficiency of computing and reduce numerical error, the nonuniform triangular mesh method is introduced, and the continuity equation is solved by ETG-FCT. The new contributions in this paper include the development of the ETG scheme and its application to rod-plate gap streamer discharge in 50~50% SF6/N2gas mixtures problems. Results are obtained: the spatial distributions of electron densities, positive ion densities, negative ion densities, photoelectron densities, and the electric field, respectively. The velocities of streamer propagations and the radius of streamer obtained from the proposed model are in good agreement with experimental and simulation results in literature. The results also prove that the ETG-FCT method is valid.

This paper deals with the solutions of linear inhomogeneous time-fractional partial differential equations in applied mathematics and fluid mechanics. The fractional derivatives are described in the Caputo sense. The fractional Green function method is used to obtain solutions for time-fractional wave equation, linearized time-fractional Burgers equation, and linear time-fractional KdV equation. The new approach introduces a promising tool for solving fractional partial differential equations.

Under investigation in this paper is the (2 + 1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt equation, which can be utilized to describe certain nonlinear phenomena in fluid mechanics. We obtain the higher-order lump, breather and hybrid solutions, and analyze the effects of the constant coefficients $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ in that equation on those solutions, since the higher-order lump solutions are generalized via the long-wave limit method, and since the higher-order breather solutions and hybrid solutions composed of the solitons, breathers and lumps are derived. With the help of the analytic and graphic analysis, we get the following: (1) amplitudes of the humps and valleys of the first-order lumps are related to $$h_1$$ , $$h_2$$ , $$h_4$$ and $$h_5$$ , proportional to $$h_4$$ while inversely proportional to $$h_2$$ . Velocities of the first-order lumps are proportional to $$h_4$$ . The second-order lumps describe the interaction between the two first-order lumps, which is elastic since those lumps keep their shapes, velocities and amplitudes unchanged after the interaction. Effects of $$h_2$$ and $$h_4$$ on the second-order lumps are graphically illustrated. (2) Amplitudes of the first-order breathers are proportional to $$h_2$$ . Interaction between the breather waves is graphically presented. Effects of $$h_2$$ and $$h_1$$ on the amplitudes and shapes of the second-order breathers are graphically discussed. (3) Elastic interactions are graphically illustrated, between the first-order breathers and one solitons, the first-order lumps and one solitons, as well as the first-order breathers and first-order lumps. Also graphically illustrated, amplitudes of all those three kinds of hybrid solutions are inversely proportional to $$h_2$$ , and velocity of the one soliton is positively correlated to $$h_4$$ .

This paper investigates the quasi-periodic breathers of the (2+1)-dimensional Hirota equation, which can describe the wave phenomena in nonlinear optics and fluid mechanics. By combining the theta function and Hirota bilinear method, the quasi-periodic breather solutions are constructed. The solvability problem of the quasi-periodic breathers is transformed into a least squares problem, which can be solved via a new algorithm called the global Levenberg-Marquardt (LM) method. Different types of quasi-periodic breathers are obtained, including quasi-periodic Kuznetsov-Ma breather, quasi-periodic Akhmediev breather as well as regular quasi-periodic breather. By introducing an analysis method related to the characteristic line, the dynamical characteristics of the quasi-periodic breathers are derived. In addition, the data-driven quasi-periodic breathers are studied by using the physics-informed neural network-adaptive residual distribution optimization (PINN-ARDO) method.