Transport Equation Model Research Articles

Amplification factor transport equation modelling of Mack mode disturbances in hypersonic boundary layers

In the hypersonic boundary layer, the streamwise instability modes are mainly the Mack first mode and the Mack second mode. After the Mach number exceeds 4.0, the Mack second mode becomes the dominant instability mode for boundary layer transition. Currently, there is no complete analytical method to establish an amplification factor transport equation for the second mode. In order to improve the application of the amplification factor transport (AFT) model in hypersonic boundary layer transition prediction, this paper employs boundary layer similarity solutions to conduct stability analysis on the second mode and introduces a function to calculate the growth rate of disturbances. Additionally, a transport equation for the amplification factor of the second mode is formulated. For the non-local variables appearing in the equation, a local calculation method is provided. Combined with the intermittency factor transport equation, a transition prediction model for the second mode is formed. Hypersonic flat plate, wedge, and flared wedge are selected for model validation. The computed results show good agreement with standard stability analysis or experimental results, demonstrating the rationality of the transition model and its high prediction accuracy and reliability.

Development of a diffusive simplified double P[formula omitted] approximation to the neutron transport equation

The neutron transport equation models the distribution of the neutron power inside a reactor core. The complexity of integrating this equation implies the use of some angular approximations. Discrete ordinates and spherical harmonics equations (PN equations) are commonly used to this purpose, but even when these approaches are used, the resulting set of equations requires high computational demands. An alternative is to develop simplified formulations of them, such as the simplified spherical harmonics equations (SPN equations), which provide accurate results. The spherical harmonic equations are based on the regularity of solution in terms of the angular dependence. However, in some reactor configurations, the angular fluxes can be discontinuous. For that, the double spherical harmonics equations, where angular fluxes are expanded in terms of two sets of Legendre polynomials, were studied. In this work, a simplified treatment of them is developed that leads to a diffusive set of equations, the simplified double spherical harmonics equation (SDPN equations). Numerical results show that the SDPN approximation provides faster and more accurate results than the SPN equations for some problems with strong variations of the angular neutron flux.

Sea and brackish water desalination through a novel PVDF-PTFE composite hydrophobic membrane by vacuum membrane distillation

The present study aims to evaluate the performance of porous hydrophobic Polyvinylidene fluoride − Polytetrafluoroethylene (PVDF-PTFE) composite membranes for desalination by vacuum membrane distillation (VMD) technique. The effect of operating parameters such as feed NaCl concentration (10,000 to 40,000 mg/L), feed temperature (50 °C to 80 °C), and downstream pressure (80 to 120 mmHg) on water permeation rate was studied. The increase in feed temperature enhanced the water permeation rate due to a rise in driving force across the membrane. For a constant downstream pressure of 80 mmHg, feed temperature of 80 °C and feed flow rate of 80 L/h, the membrane exhibited a maximum water flux of 3 kg/m2h with 99.86% salt rejection when aqueous NaCl concentration of 10,000 mg/L was charged as feed. Membrane characterization was performed using various analytical tools to determine physico-chemical properties such as pore size, structural elucidation, thermal stability, crystallinity, and hydrophobicity of the membrane material. Further, a temperature and concentration polarization coefficient-based analysis was performed by solving the mass and heat transport model equations using MATLAB software. The proposed research study promotes the application of VMD for recovering potable water from highly saline sea/brackish water and alleviates brine disposal issues.

To the theory of turbulent dynamo in magnetised weakly compressible astro-geophysical objects

We discuss a phenomenological self-consistent model of a turbulent magnetized conducting fluid for mean flow and magnetic field in geophysical and astrophysical contexts. This model is designed to study spirality and dynamo effect in the study of astro-geophysical magnetic fields. It is believed that the very existence of cosmic magnetic fields is largely explained by the chirality of background turbulence, the simplest measure of which is the non-vanishing helicity of small-scale turbulence. The mean velocity and magnetic fields are modelled using averaged MHD -equations and phenomenological differential equations for the four bulk turbulence quantities (hydro magnetic turbulent energy, magnetic and cross helicity and residual helicity). These model equations, combined with the mean-field equations, allow the most complete construction of a self-consistent model of the turbulent dynamo. These problems and open questions are considered, especially those concerning the model transport equations for the determination of the alpha effect in astrophysical magnetic fields. The presented work is mainly aimed at modelling rotation effects in hydro magnetic turbulent flows of spiral space and thermonuclear plasma.

Towards energy filtering in Mg2X-based composites: Investigating local carrier concentration and band alignment via SEM/EDX and transient Seebeck microprobe analysis

A comprehensive understanding of multi-phase materials requires the characterization of transport properties at the micro/nano-scale. Optimized alignments of the conduction bands and valence bands at the interfaces of multi-phase materials can enhance the properties of thermoelectric materials by filtering out undesired charge carriers and is a promising path towards high performance thermoelectric devices. Here, a micro-scale characterization technique using transient Seebeck microprobe analysis, scanning electron microscopy with energy-dispersive X-ray spectroscopy, and electronic transport modelling based on the Boltzmann transport equation modeling is introduced to assess the band diagram and estimate the band offset in multi-phase materials. This characterization technique is applied to a composite of magnesium silicide-based materials. This material class is prone to form self-assembling nano-structured composites driven by a miscibility gap in the solid solutions series. Our analysis reveals changes in carrier concentration upon composite formation and considerable band offsets for both conduction and valence band when synthesizing nominally undoped composites. Employing Bi as dopant we show that Bi exhibits a preference for the Sn-rich phase, changing the carrier concentration differently in the Sn-rich matrix phase compared to the Si-rich secondary phase, effectively altering the band alignment of the composite. This demonstrates that our approach can be utilized to measure and manipulate the band offset within the composite to achieve optimal thermoelectric performance.

Realizability-preserving time-stepping for the differential Reynolds stress turbulence models

In the context of incompressible turbulence modelling, differential Reynolds stress models allow a better representation of turbulence anisotropy than classical linear eddy viscosity models and are less expensive than large eddy simulations in terms of computational cost. They rely on a modelled transport equation for the Reynolds stress tensor, which is by definition a covariance tensor and must remain symmetric positive semi-definite. However, the solution of the modelled and discretized transport equation may not preserve these mathematical properties. In the present article, a methodology to time-split Reynolds stress R__ terms in the transport equation is proposed using a decomposition theorem which proves the realizability of R__ provided realizable initial conditions. Realizability-preserving time-steppings are designed for several differential Reynolds stress models. The numerical schemes are presented in a finite volume context but may be used with others space discretization schemes. The realizability, precision and robustness of the numerical scheme is verified through the anisotropy time evolution in homogeneous shear-flow simulations with various initial conditions.

Numerical Study on Heat and Mass Transfer of Evaporated Binary Zeotropic Mixtures in Porous Structure

This study investigates the heat and mass transfer characteristics of a binary mixture (R134a/R245fa) evaporated in a porous medium. The Eulerian model coupled with the multiphase VOF model and species transport equations is employed to establish a multi-component evaporation model. The effects of heat flux ranging from 200 kW/m2 to 500 kW/m2, porosity ranging from 0.4 to 0.6, and mass fraction ratios (R134a/R245fa) of 3:7, 5:5, and 7:3 are explored. The results indicate that an increase in heat flux contributes to an increase in the evaporation rate. For the overall evaporation rate, the evaporation rates of R134a and R245fa improve by 11.3%, 6.9%, and 16.3%, respectively, while the maximum improvement in heat transfer coefficient is only 1.4%. The maximum evaporation rate is achieved at intermediate porosity in the porous medium, and the highest heat transfer coefficient is obtained at a porosity of 0.4. With the increase in mass fraction, the evaporation rate of the corresponding species also increases, while the overall evaporation rate and heat transfer coefficient remain almost unchanged.

Numerical Simulation of Flashing Flows in a Converging–Diverging Nozzle with Interfacial Area Transport Equation

Flashing flows of initially sub-cooled water in a converging–diverging nozzle is investigated numerically in the framework of the two-fluid model (TFM). The thermal non-equilibrium effect of phase change is considered by an interfacial heat transfer model, while the pressure jump across the interface is ignored. The bubble size distribution induced by nucleation, bubble growth/shrinkage, coalescence, and breakup is described based on the interfacial area transport equation (IATE) and constant bubble number density model (CBND), respectively. The results are compared with the experimental data. Satisfactory prediction of the axial pressure distribution along the nozzle as well as the flashing inception, is achieved by the TFM-IATE coupling method. It was also found that the vapor production in the diverging section was overpredicted, and the radial gas volume fraction distribution deviated from the experiment. The radial diameter profiles exhibit opposite patterns at the nozzle throat and near the outlet, and similar trends can be observed for the superheated degree. A poly-disperse method is suggested to be introduced to describe the evolution of interfacial area concentration.

Numerical Exploration of Low Altitude Rocket Plume in Aircraft Vicinity

Low altitude plume of a solid rocket motor is numerically explored. The effects of free stream Mach number, altitude and presence of launch platform wing on plume shape are studied. Three-dimensional Reynolds Averaged Navier-Stokes (RANS) equations along with k-ε turbulence model and species transport equations are solved using commercial CFD software CFX-11. Thermochemical parameters (Mach number, pressure, temperature and propellant hot gas mass fraction) are analysed to characterize the plume. Mixing layer boundaries, its thickness and the attainment of self-similarity are evaluated to study the plume structure. With the increase of free stream Mach number, the computed plume diameter reduces. Centerline pressure, temperature and radial expansion of the plumes increases with the increase in altitude. Plume is found to be elongated and narrowed with relatively warm core region due to the presence of platform wing.

Theory and Transport of Nearly Incompressible Magnetohydrodynamic Turbulence: High Plasma Beta Regime

Nearly incompressible magnetohydrodynamic (NI MHD) theory for β ∼ 1 (or β ≪ 1) plasma has been developed and applied to the study of solar wind turbulence. The leading-order term in β ∼ 1 or β ≪ 1 plasma describes the majority of 2D turbulence, while the higher-order term describes the minority of slab turbulence. Here, we develop new NI MHD turbulence transport model equations in the high plasma beta regime. The leading-order term in a β ≫ 1 plasma is fully incompressible and admits both structures (flux ropes or magnetic islands) and slab (Alfvén waves) fluctuations. This paper couples the NI MHD turbulence transport equations with three fluid (proton, electron, and pickup ion) equations, and solves the 1D steady-state equations from 1–75 au. The model is tested against 27 yr of Voyager 2 data, and Ulysses and NH SWAP data. The results agree remarkably well, with some scatter, about the theoretical predictions.

Finite/fixed-time stabilization of a chain of integrators with input delay via PDE-based nonlinear backstepping approach

In this paper, we present a general approach to studying the problem of finite-time and fixed-time stabilization of a chain of integrators with input delay. To accomplish this, we first reformulate the chain of integrators with input delay as a cascade ODE–PDE system (i.e., a cascade of a linear transport partial differential equation (PDE) with the chain of integrators) where the transport equation models the effect of the delay on the input. Next, we use a nonlinear infinite-dimensional backstepping transformation to convert the cascade system to a suitable target system that is chosen to be finite-time or fixed-time stable. We perform the stability analysis on the target system by means of classical non-asymptotic concepts and tools such as the linear homogeneity and “generalized KL” functions. Then, we use the inverse transformation to transfer back the stability property to the closed-loop system. Finally, we give some characterizations of finite/fixed time predictor-based controllers followed by numerical simulations.

Comparison of bubbles interaction mechanisms of two-group Interfacial Area Transport Equation model

This work deals with 3D simulations of complex bubbly, cap-bubbly and churn regimes exhibiting bubbles of different shapes and with broad bubble size distribution. The first contribution of this work is to investigate and compare several bubble interaction mechanisms of coalescence and fragmentation for the 2-Group Interfacial Area Transport Equation (IATE) model. For two of these models, this is the first time their performances are assessed within a CFD code. The second contribution is to propose and assess a novel model of fragmentation and coalescence. Finally, a validation versus experimental data on three different configurations and three different regimes is performed. In particular, the interaction mechanisms are analyzed for one specific regime. The implementation of the two-group IATE model has been systematically performed in the 3D NEPTUNE CFD code.

A Priori Direct Numerical Simulation Assessment of MILD Combustion Modelling in the Context of Reynolds Averaged Navier–Stokes Simulations

A priori Direct Numerical Simulation (DNS) assessment of mean reaction rate closures for reaction progress variable in the context of Reynolds Averaged Navier–Stokes (RANS) simulations has been conducted for MILD combustion of homogeneous (i.e., constant equivalence ratio), methane-air mixtures. The reaction rate predictions according to statistical (e.g., presumed probability density function), phenomenological (e.g., eddy-break up (EBU), eddy dissipation concept (EDC) and the scalar dissipation rate (SDR) based approaches), and flame surface description (e.g., Flame Surface Density) based closures are compared. The performance of the various reaction rate closures has been assessed by comparing the models’ predictions to the corresponding quantities extracted from DNS data. It has been found that the usual presumed probability density function (PDF) approach using the beta-function predicts the PDF of the reaction progress variable in homogenous mixture MILD combustion throughout the flame brush for all cases considered here provided that the scalar variance is accurately predicted. The accurate estimation of scalar variance requires the solution of a modelled transport equation, which depends on the closure of Favre-averaged SDR. A linear relaxation based algebraic closure for the Favre-averaged SDR has been found to capture the behaviour of the Favre-averaged SDR in the current homogenous mixture MILD combustion setup. It has been found that the EBU, SDR and FSD-based mean reaction rate closures do not adequately predict the mean reaction rate of the reaction progress variable for the parameter range considered here. However, a variant of the EDC closure, with model coefficients expressed as functions of micro-scale Damköhler and turbulent Reynolds numbers, has been found to be more successful in predicting the mean reaction rate of reaction progress variable compared to other modelling methodologies for the range of turbulence intensities and dilution levels considered here.

Experimental and computational investigation of the vortical structures generated from a blended-wing-body UAV model

The current study presents a combination of experimental and computational investigations of a scaled-down Blended Wing Body (BWB) Unmanned Aerial Vehicle (UAV) model. The BWB model features highly swept wings. The experiments are conducted in a subsonic windtunnel in the Laboratory of Fluid Mechanics and Turbomachinery, in the Aristotle University of Thessaloniki. More specifically, surface oil flow visualization and Laser-Doppler Anemometry (LDA) are employed to investigate the vortical structures emanating from the leading edge and the wingtips of the model and provide the corresponding velocity and vorticity profiles. Using the windtunnel measurement data as a reference point, both steady-state and transient computational fluid dynamics (CFD) analyses are also conducted. The flow around the BWB scaled-down model is modeled by adopting four turbulence models, three of them based on the concept of the eddy-viscosity and one, on the Reynolds-stress transport equation modeling. The modeling results are compared to the experimental measurements, emphasizing on the streamwise and spanwise velocity profiles, axial vorticity profiles and separation patterns. The results of this study provide insight on the vortical phenomena dominating the flowfield over and around BWB configurations and conclusions concerning the tradeoffs between accuracy and computational efficiency are drawn.

On the concentration fluctuation moments of gas species in turbulent combustion

For exploring the influences of chemical reaction on the turbulent fluctuations of gas species concentrations, the reaction source terms in the transport equations for the concentration fluctuation moments are analyzed. It is found that the reaction source terms taken from the equations for the fuel-oxygen, fuel, and oxygen concentration fluctuation moments have a fixed relationship among them. The reaction source terms are not pure production or dissipation terms and they play a redistribution role in these equations. The algebraic concentration moment (ACM)-probability density function (PDF) turbulent combustion model is evaluated. It indicates that the reaction source terms in the modeled transport equations and the resultant explicit algebraic expressions for the concentration fluctuation moments all satisfy the fixed relationship. The reaction source terms serve as the redistribution terms in the explicit expressions for the concentration fluctuation moments.

Reacting flow analysis in scramjet engine: effect of mass flow rate of fuel and flight velocity

Abstract The objective of the study is realizing the effect of fuel mass flow rate and flight speed on combustion in scramjet engine. DLR conical strut based scramjet combustor configuration was chosen and simulated the chemical reaction between the air and hydrogen fuel. A slot of size 40 mm × 0.295 mm provided at the center of the strut to injected hydrogen fuel from the rare side in to the downstream flow. ICEM CFD software is used for the generation of structured elements in computational domain for three dimensional flow analyses. Standard k-epsilon turbulence model and species transport equation is used in ANSYS fluent solver. The predicted temperature, velocity distribution along the axial length was compared with the experimental results and validated. The temperature distribution at different Mach numbers and mass flow rate reveals that the peak temperature increased with the flight speed and inlet fuel mass flow rate. The peak temperature noticed at the center of the combustor is around 3500 K at a flight speed of Mach 4. The predicted variation of temperature, pressure, velocity in the combustor and the flow structure for reacting flow facilitate good understanding of the combustion process in scramjet combustor.

Reacting flow analysis in scramjet engine: effect of mass flow rate of fuel and flight velocity

Abstract The objective of the study is realizing the effect of fuel mass flow rate and flight speed on combustion in scramjet engine. DLR conical strut based scramjet combustor configuration was chosen and simulated the chemical reaction between the air and hydrogen fuel. A slot of size 40 mm × 0.295 mm provided at the center of the strut to injected hydrogen fuel from the rare side in to the downstream flow. ICEM CFD software is used for the generation of structured elements in computational domain for three dimensional flow analyses. Standard k-epsilon turbulence model and species transport equation is used in ANSYS fluent solver. The predicted temperature, velocity distribution along the axial length was compared with the experimental results and validated. The temperature distribution at different Mach numbers and mass flow rate reveals that the peak temperature increased with the flight speed and inlet fuel mass flow rate. The peak temperature noticed at the center of the combustor is around 3500 K at a flight speed of Mach 4. The predicted variation of temperature, pressure, velocity in the combustor and the flow structure for reacting flow facilitate good understanding of the combustion process in scramjet combustor.

Non-equilibrium extension of the explicit algebraic subgrid-scale stress model with application to turbulent channel flow at low Reynolds numbers

An extension of the explicit algebraic subgrid-scale (SGS) stress model (EASSM) of Marstorp et al. [J. Fluid Mech. 639, 403–432 (2009)] is proposed to account for the non-local equilibrium between the production, P, and viscous dissipation, ϵ, of the SGS turbulent kinetic energy in its formulation. The original derivation of the EASSM uses the equilibrium assumption P=ϵ, whereas in the current new derivation, a cubic algebraic equation is extracted for P/ϵ from the modeled transport equation for the SGS stress anisotropy, which can be solved to improve the EASSM predictions with less than 4% additional computational costs per time step. The performance of the extended EASSM is assessed in large-eddy simulation of plane turbulent channel flow at Reτ=587 and 179 for a wide range of resolutions, where deviations from the local equilibrium assumption are expected at the vicinity of the walls. The enhanced EASSM formulation lowers resolution dependence of the model predictions and improves its predictions at low resolutions. The improvements affected major one-point turbulence statistics of interest, such as the wall shear stress, mean velocity, Reynolds stresses, and SGS dissipation as well as the two-point velocity correlations and premultiplied spanwise spectra of the streamwise velocity. The predicted mean P/ϵ also reasonably agrees with the filtered direct numerical simulation data.

Charge–velocity correlation transport equations in gas–solid flow with triboelectric effects

This paper is dedicated to the development of particle charge and velocity second-order moment transport equations for monodisperse particles in gas flow with a tribocharging effect. The full transport equations for the particle charge–velocity covariance and the charge variance are derived in the framework of the kinetic theory of granular flow assuming that the electrostatic interaction does not modify the collision dynamics. The collision integrals are solved without presuming the form of the electric part for the particle probability density function. The full second-order transport equation model is tested in a one-dimensional periodic domain. The results show that this model is able to capture more important physical mechanisms that are neglected by simple algebraic models proposed in the past. An in-depth analysis of the transport equations is also performed. This study reveals that, for sufficiently small covariance characteristic destruction time scales, the transient and third-order moments terms can be safely neglected. In addition, two different reduced-order models are proposed: a more general algebraic model that takes into account the variance effect and a semi-algebraic model that only resolves a transport equation for the charge variance coupled with an algebraic model for the covariance. The former could, however, lead to non-physical predictions in many cases, while the latter can be a suitable alternative only for a sufficiently small interparticle collision time. Finally, a simple chart based on test case simulations is proposed to show under which conditions a semi-algebraic model could be considered as a suitable alternative.

Verification of a Pressure-Based Compressible Flow Solver

This paper presents Solution Verification exercises with the pressure-based compressible flow solver ReFRESCO for five test cases available in the NASA Turbulence Modeling Resource: the two-dimensional flows over a flat plate, a bump-in-channel, a DSMA661 airfoil and a multi-element airfoil and the three-dimensional flow of a bump-in-channel. Simulations are performed with the Favre-averaged continuity and Navier-Stokes equations using the Spalart & Allmaras turbulence model. ReFRESCO results are compared with reference data from density-based compressible flow solvers (CFL3D and FUN3D). two aspects of the implementation of the turbulence model are addressed: the calculation of the distance to the wall and the discretization scheme used in the convective terms of the turbulence model transport equation. Results of this study show perfect consistency with the reference data for the test cases that are not affected by the determination of the distance to the wall.