One-dimensional Flow Research Articles

Mixcoatl Software (Part 2): Verification of 1D Fluid Flow Methods Coupled to 3D FEM Thermal Diffusion Models

The features and equations of MixcoatlTM are presented in Part 1 of this two-paper series with no discussion on physics verification. The verification of Mixcoatl specifically for prismatic fuel designs/shell-and-tube–like designs is presented here. Furthermore, conclusions are made on the uncertainties that are extendable to the results of other codes that utilize a one-dimensional (1D) flow assumption with Nusselt correlations. Demonstration of commonly used scenarios for reactor designs is explored, and applicable ranges/deviations from the 1D flow assumption are investigated.

Multi-Disciplinary Analysis of Working Fluids on Thermal Performance of the High-Power Diesel Engine System

Multi-disciplinary analysis was performed to analyze and investigate the thermal performance during transient operation of a 2 L diesel engine system with two different cooling systems. The multi-disciplinary model consisted of the engine thermal management system (ETMS) comprising a zero-dimensional engine model that can simulate the engine performance, and a one-dimensional flow model for cooling and lubrication systems with a controller. By deploying this approach, we were able to model different physical domains, including mechanical for the engine and the dynamometer and thermodynamic for the heat exchangers. Therefore, the thermal performance of the ETMS could be numerically predicted and analyzed. To develop the ETMS model, the physical properties, the heat transfer model, and the pressure drop were modeled. The base fluid, a 50/50 mixture of water and ethylene glycol (EG), and an Al2O3 nanofluid with a 1.5% volume ratio were modeled based on the thermodynamic properties such as density, dynamic viscosity, thermal conductivity, and specific heat. Nanofluid, with its higher thermal conductivity and higher heat transfer coefficient, absorbed more heat from the combustion chamber through the water-jacket in the engine block. Therefore, the oil temperature for the nanofluid was effectively 2.5 °C less than for the base fluid following the step-load condition. Simulation results showed the better effect of nanofluid on thermal performance. The total flow rate of nanofluid decreased by 2.2 L/min, although the flow rate through the radiator with nanofluid increased by 0.81 L/min to obtain greater heat dissipation. Eventually, the piston and the liner temperatures with the nanofluid were drastically reduced by 7.55 and 8 °C, respectively, compared to those of the base fluid. Finally, when nanofluids was applied in automotive cooling systems, the temperature of the piston decreased by 7.3 °C due to the reduced overall thermal resistance from combustion chambers to outside air. The effect of working fluid on the diesel engine system could be predicted through the multi-disciplinary model.

Comparison of one-dimensional and three-dimensional glottal flow models in left-right asymmetric vocal fold conditions.

While the glottal flow is often simplified as one-dimensional (1D) in computational models of phonation to reduce computational costs, the 1D flow model has not been validated in left-right asymmetric vocal fold conditions, as often occur in both normal and pathological voice production. In this study, we performed three-dimensional (3D) and 1D flow simulations coupled to a two-mass model of adult male vocal folds and compared voice production at different degrees of left-right stiffness asymmetry. The flow and acoustic fields in 3D were obtained by solving the compressible Navier-Stokes equations using the volume penalization method with the moving vocal fold wall as an immersed boundary. Despite differences in the predicted flow pressure on vocal fold surface between the 1D and 3D flow models, the results showed reasonable agreement in vocal fold vibration patterns and selected voice outcome measures between the 1D and 3D models for the range of left-right asymmetric conditions investigated. This indicates that vocal fold properties play a larger role than the glottal flow in determining the overall pattern of vocal fold vibration and the produced voice, and the 1D flow simplification is sufficient in modeling phonation, at least for the simplified glottal geometry of this study.

Motion of dark solitons in a non-uniform flow of Bose-Einstein condensate.

We study motion of dark solitons in a non-uniform one-dimensional flow of a Bose-Einstein condensate. Our approach is based on Hamiltonian mechanics applied to the particle-like behavior of dark solitons in a slightly non-uniform and slowly changing surrounding. In one-dimensional geometry, the condensate's wave function undergoes the jump-like behavior across the soliton, and this leads to generation of the counterflow in the background condensate. For a correct description of soliton's dynamics, the contributions of this counterflow to the momentum and energy of the soliton are taken into account. The resulting Hamilton equations are reduced to the Newton-like equation for the soliton's path, and this Newton equation is solved in several typical situations. The analytical results are confirmed by numerical calculations.

Free-Windmilling Speed Prediction Based on Aerodynamic Similarity of an Axial Wide-Chord Fan

This paper explores the windmilling operational boundary and aerodynamic characteristics of an axial wide-chord fan rotor according to the zero-torque characteristic. The interesting results show the radial profiles of the aerodynamic parameters at the rotor exit nearly collapse to the same curve for different windmilling speeds, which indicate that windmilling flow has a strong similarity. Meanwhile, there are three different operating modes along the blade span. The formation mechanism and evolution process of the tip leakage flow at the windmill are also analyzed. The reversal of the pressure difference will generate two kinds of tip leakage flows in opposite directions. A clear blockage zone appears near the blade tip under the joint action of the two kinds of tip leakage flows. Finally, this study proposes a mathematical model to predict the windmilling speed of a high-bypass-ratio turbofan based on the aerodynamic similarity of windmilling flow and the one-dimensional semiempirical windmilling mass flow function. This can be applied to the early stages of high-bypass-ratio turbofan development and can evaluate safety characteristics during engine windmilling operations.

One-dimensional MHD flows with cylindrical symmetry: Lie symmetries and conservation laws

A recent paper considered symmetries and conservation laws of the plane one-dimensional flows for magnetohydrodynamics in the mass Lagrangian coordinates. This paper analyzes the one-dimensional magnetohydrodynamics flows with cylindrical symmetry in the mass Lagrangian coordinates. The medium is assumed inviscid and thermally non-conducting. It is modeled by a polytropic gas. Symmetries and conservation laws are found. The cases of finite and infinite electric conductivity need to be analyzed separately. For finite electric conductivity σ(ρ,p) we perform Lie group classification, which identifies σ(ρ,p) cases with additional symmetries. The conservation laws are found by direct computation. For cases with infinite electric conductivity variational formulations of the equations are considered. Lie group classifications are obtained with the entropy treated as an arbitrary element. A variational formulation allows to use the Noether theorem for computation of conservation laws. The conservation laws obtained for the variational equations are also presented in the original (physical) variables.

Similarity Analysis on Two-Dimensional Hypersonic Inlets Unstart

The prediction of unstart boundaries of hypersonic inlets is significant to the engine control. To correct the previous conclusions based on the one-dimensional and inviscid flow assumption, the present paper numerically investigates the unstart characteristics of two-dimensional contraction ducts in viscous flow that vary with multiple controlling variables. Based on the newly defined dimensionless cowl length and effective internal contraction ratio, the similarity law of unstart is discovered. The universal empirical equation of unstart boundary for different cowl lengths, cowl lip heights, entrance boundary-layer thicknesses, unit Reynolds numbers, and Mach numbers is proposed, which can simplify the design of two-dimensional hypersonic inlets. With varying the dimensionless cowl length, three different types of short-cowl, transitional, and long-cowl unstarts are identified, owing to the different evolution mechanisms of separation region. The unstart displays a separation region growing under the streamwise pressure difference enhanced by the shock system or flow choke. In addition, the wall pressure criteria for unstart are correlated to the characteristic separation scale at unstart critical state and the dimensionless cowl length, which facilitates the unstart detection or early warning. It is observed that the Kantrowitz limit is close to the unstart boundaries for very short cowl and the isentropic limit is close to the unstart boundaries for very long cowl.

Characteristics of one-dimensional bubbly flow interfacial area transport in horizontal narrow rectangular channel

The experiments of horizontal adiabatic air-water flow in narrow rectangular channel with a cross-sectional area of 68 mm × 1.9 mm were conducted under an atmosphere pressure condition in order to study bubbly flow interfacial area transport phenomenon. The liquid film thickness was obtained by PCB liquid film sensor. The results show that the liquid film thickness can be correlated with the product of Capillary number Ca and jg/jl, so a correlation for liquid film thickness was proposed. The effect of liquid film thickness on IAC and void fraction was also evaluated. The results show that the average error caused by ignoring liquid film thickness is about 6.1% and 10.1%, respectively. The lateral profiles and axial direction development of the void fraction, interfacial area concentration, bubble Sauter mean diameter were obtained by high-speed camera at three axial locations of Z/D = 63, 241 and 510. The simplified one-dimensional, steady-state, one-group interfacial area transport equation for an adiabatic bubbly flow with six sets of IAC source and sink models was evaluated by the current experimental data. The evaluation results showed that the models of Yao and Morel (2004) and Shen and Hibiki (2013) performed better with mean prediction errors of 12.13% and 18.92%, respectively.

Quadrature-based moment methods for kinetic plasma simulations

Quadrature-based moment methods (QBMM) are applied to rarefied gas flow and plasma physics problems represented by the Boltzmann equation or the Vlasov-Poisson system of equations. QBMM are a computationally advantageous alternative to both direct Eulerian and Lagrangian particle solvers, able to provide a noise-free solution to the Boltzmann equation and capture non-equilibrium velocity distribution functions (VDF) without excessive computational cost. To provide a closure of the moment equations, the VDF is assumed to be represented by the sum of weighted kernel functions that are placed at given velocity abscissas. Three QBMM approaches, QMOM (Dirac delta kernels), HyQMOM (hyperbolic QMOM), and EQMOM (Gaussian kernels), are applied to canonical one-dimensional rarefied gas flow and plasma physics problems. A new algorithm for EQMOM that is able to treat pathological VDFs and provide control over large abscissas for more than two nodes is developed. QMOM and EQMOM are suitable for problems where the flow does not have shock-like features (for QMOM) or a discontinuity in the VDF (for EQMOM) and where the VDF is far from non-equilibrium and features strong phase-mixing. HyQMOM is generally applicable to all test problems and yields good representations of both the moments and the VDF. HyQMOM also provides significant wall-time improvement (factor of 10–100) compared to other direct Eulerian solvers.

A New Mapped WENO Method for Hyperbolic Problems

In this study, a new family of rational mapping functions gRM(ω;k,m,s) is introduced for seventh-order WENO schemes. gRM is a more general family of mapping functions, which includes other mapping functions such as gM and gIM as special cases. The mapped WENO scheme WENO-IM(2,0.1), which uses gIM, performs excellently at fifth order but rather poorly at seventh order. The reason for this loss of accuracy was found to be the over-amplification of very small weights by the mapping process, which can be traced back to the large slope of gIM at ω = 0. For m > 1, gRM can be designed to have a unit slope at ω = 0, which will preserve small weights with little to no amplification. It has been demonstrated through several one-dimensional linear advection test cases that the mapped WENO scheme WENO-RM(6,3,2 × 103), which uses the mapping function gRM(ω;6,3,2 × 103), outperforms both WENO-M and WENO-IM(2,0.1) at seventh order. The proposed scheme also performs better at a number of one-dimensional inviscid gas flow problems compared to other popular WENO schemes such as the WENO-Z scheme.

Upscaling unsaturated flows in vertically heterogeneous porous layers

Characterising interfacial and unsaturated flows in heterogeneous porous layers is of both fundamental and practical interest. Under the assumption of vertical gravitational–capillary equilibrium, we present a theoretical model to describe one-dimensional flows in a porous layer with vertical variations in average pore size, porosity, intrinsic permeability and capillary pressure jump between invading and displaced fluids. The model leads to asymptotic solutions for the saturation distribution and outer envelope of the invading fluid, and for the background pressure drop across the porous layer. Eight dimensionless parameters are recognised after appropriate non-dimensionalisation of the governing equations, the influence of which is demonstrated through a series of example calculations. In particular, four asymptotic regimes are identified, representing unconfined sharp-interface flows, confined sharp-interface flows, unconfined unsaturated flows and confined unsaturated flows. Finally, in the context of flow upscaling, analytical solutions are derived for the effective relative permeability curves on the basis of exact solutions of the saturation field and interface shape, shedding light on the subtle influence of competition between injection/pumping and gravitational forces, wetting and capillary effects, viscosity contrast between the invading and displaced fluids and vertical heterogeneity of the porous layer.

The interaction of rarefaction waves for a system of granular flow equations

In this paper, we consider the interaction problem of two centered rarefaction waves for the one-dimensional granular flow equations, which originates from the interaction and impact of two avalanche flows, mud flows or tsunamis in reality. This interaction problem is essentially a Goursat-type boundary value problem in mathematical theory. When there is no vacuum in the initial data, we show that no vacuum appears in the interaction region, while the vacuum occurs when the collision time is sufficiently large. Based on the characteristic method, the a priori estimates are derived to establish the existence and uniqueness of smooth solutions for the Goursat problem on the whole interaction region.

Iterative feedback design of de Laval nozzle

The cold gas propulsion through de Laval nozzles is a convenient way to enable position and attitude manoeuvres of groundbased satellite simulators. A parametric model of a de Laval nozzle is established base on the theory of one-dimensional isentropic flow, and the methodology of Model-Based Systems Engineering (MBSE) is adopted to guide the iterative feedback design process. In the process of design, the effect of design factors on the nozzle performance are evaluated, such as nozzle shape, length of convergent and divergent section, inlet and outlet diameters, and throat diameter. Simulation and experimental results demonstrate the improvement of the nozzle performance through iterative feedback design.

Influence of transient flow during infiltration and isotropic/anisotropic matric suction on the passive/active lateral earth pressures of partially saturated soils

In this study, the influence of transient flow on the active and passive lateral earth pressures of variably saturated backfills is thoroughly evaluated employing the lower bound theorems of the finite element limit analysis with second-order cone programming. In order to account for the tempo-spatial variations of suction stress within the soil stratum during infiltration, a closed-form solution derived for one-dimensional transient flow through variably saturated porous media is adopted. The unsaturated ground condition is simulated by incorporating the well-established unified effective stress approach into the soil yield function. The soil medium is considered to be both isotropic and anisotropic, due primarily to the various distributions of suction stress along different directions within the partially saturated porous medium. In order to model the soil suction stress anisotropy, an iterative procedure is employed by differentiating between the mobilized suction stress within the horizontal and vertical planes. In general, the influence of the transient flow condition is magnified for cohesive soils compared with granular backfills. In either case, such an effect is more noticeable for the active lateral earth pressure as compared to the passive counterpart. In the anisotropic condition, while the suction stress anisotropy within the soil medium has fairly negligible influence on the contribution of transient flow to the lateral earth pressure coefficient in the active state, the influence of elapsed infiltration time on the passive earth pressure coefficient turns out to be more at play at lower anisotropy ratios.

Integrated water quality modeling in a river-reservoir system to support watershed management

Water resources planning and management are dependable on an adequate integration of physical, chemical, biological, and socio-economic realities of multiple water users. The dynamic of water quantity and quality in rivers are affected by several conditions, such as land use, soil characteristics, and meteorological and hydrological processes. Among these, the presence of hydraulic structures, such as dams and reservoirs, are often responsible for hydrodynamic and geomorphological alterations, leading to impacts on water quality and ecological behavior. In this context, this study presents the combination of a solution of the one-dimensional flow and transport and fate of contaminants in rivers (SihQual model) with a continuously stirred tank reactor (CSTR) to represent the reservoir. The first model solves the hydrodynamic and water quality equations under unsteady state, while the second approach considers the reservoir as a complete mixed system with inputs that vary over time. The main goal is to provide an integrated analysis for planning and management in a watershed where water has multiple purposes of use. The case study is the Iguaçu watershed, where the main river is affected by several dams for hydroelectric power generation. The simulation covers 542 km of the Iguaçu River and the Foz do Areia reservoir (flooded area of 139.5 km2), using data from 12 monitoring stations. The region also has issues of water quality impairment and water scarcity events due to deforestation, and urban and agricultural activities, exemplifying challenges throughout the world. Results show that the integrated models can reproduce the expected variability in different systems, although calibration challenges arise in multiscale modeling. The data indicate that, overall, the lentic environment is able to deplete organic matter and phosphorus, in comparison to levels in the fluvial flow. Nonetheless, experiments show that the river-reservoir system may be highly sensible to external and internal changes, such as water availability throughout time and pollution from the main tributary, as well as outlet discharges and transformation processes in the water column, leading to a possibility of critical events. Therefore, the study highlights how planning and managing actions in the watershed can benefit from an integrated river-reservoir analysis.

Study on non-monotonic pressure-drop of supercritical n-decane with pyrolysis in heated channels

The non-monotonic relation between pressure-drop and flow-rate leading to the flow instability of fuel with pyrolysis was investigated. A characteristic pressure-drop model of n-decane coupled pyrolysis reactions under supercritical pressure was established. The model includes one-dimensional flow control equation, species equation, RK-PR equation of state, high-pressure viscosity model and a reaction mechanism consisting of 10 species and 15 reactions. The non-isothermal friction factor formula with wall correction coefficient fitted by experimental data was used in the model. To verify the model, the pressure-drops at different flow-rates were measured in a horizontal stainless steel tube with the inner diameter of 1.93 mm under different back pressures (3, 4 and 5 MPa). The results show that the model can accurately capture the non-monotonicity of characteristic pressure-drop. Under 3 MPa condition, the characteristic pressure-drop curve of n-decane has a pseudo-critical negative slope (PCNS) region and a reaction negative slope (RNS) region caused by the pyrolysis. With the increase of pressure, the PCNS region disappears but the RNS region still exists. The high pressure makes the onset of RNS region appear earlier due to the accelerated initial reaction rate. The total pressure-drop is separated and analyzed based on the model mathematically. In the PCNS region, the derivative of acceleration pressure-drop to mass flux is negative, whereas the derivative of friction pressure-drop to mass flux is positive. In the RNS region, the two derivatives are both negative because of the sharp decrease in density and the increase in friction factor caused by the pyrolysis reaction. The two parallel channels’ instability with constant total mass flow rate is analyzed based on the characteristic pressure-drop curves for reacting and non-reactive flow. The results indicate that the fuel pyrolysis reaction not only expands the range where the flow excursion may occur but also makes it difficult for the flow-rate to reach stable state again after the flow excursion caused by pseudo-critical instability.

Numerical analysis of coal size influencing the performance and slag flow characteristics of gasifier based on a comprehensive model

A comprehensive char gasification model based on random pore model with correction factor η is established to depict the char conversion process for entrained-flow gasification. The new comprehensive model uses advanced correlations for char gasification process parameters, including the correlations of a simplified multicomponent molecular diffusion, the evolution of particle size and density, char specific surface area and char thermal deactivation. A one-dimensional slag flow model coupled with the particle capture model is used to describe the deposition, flow and heat transfer process of the molten slag on the wall of HNCERI (Huaneng Clean Energy Research Institute) gasifier. The species mole fractions, carbon conversion efficiency, outlet temperature and the wall heat loss of the gasifier under the baseline are compared with the operating data. The simulation results indicate that the comprehensive model has sufficient accuracy to model the multiphase reactions and slag flow characteristics of HNCERI gasifier. The effect of coal size on exit gas temperature, detailed char heterogeneous reactions, gasification performance and slag flow characteristics of the gasifier are studied. Results show that, as coal size increases from 10 μm to 320 μm, the volume average rate of C + CO2 drops from 0.0095 to 0.0035 kmol/m3·s and C + H2O increases from 0.0064 to 0.012 kmol/m3·s at the first stage, and the carbon conversion efficiency at the first stage decreases from 99.72 % to 93.42 % while that at the second stage increases from 65 % to 93 %. The increase of coal size with values<80 μm has none significant influence on the formation of slag in the gasifier. With coal size further increasing, the formation of slag is highly promoted and thereby results in a rapid increasing thickness of the solid and liquid slag at the slag tap hole. An optimum coal size might be<80 μm for the first stage while an appropriate increase of coal size at the second stage is beneficial to improve carbon conversion efficiency.

The Riemann problem for one-dimensional dusty gas dynamics with external forces

This article concerns the study of the Riemann Problem (RP) for a non-homogeneous system governing the one-dimensional compressible flow of dusty gas, where the external forces are assumed to be continuous function of time. The elementary wave solutions such as rarefaction wave, shock wave, and contact discontinuity are rigorously determined. The effect of dust particles on the density and flow velocity and shock speed is examined, and their consequences on the solution of the Riemann Problem are discussed. Moreover, we discussed the condition for the existence of vacuum appears in the solution of the Riemann Problem.

Study on supercritical CO2 critical flow through orifices under power cycle operating conditions

For prediction of critical flow occurring under supercritical carbon dioxide (S-CO2) condition, one-dimensional (1-D) analytical critical flow models are evaluated in this study. Specifically, homogeneous equilibrium model (HEM) and a newly proposed non-equilibrium model are evaluated. The new non-equilibrium model adopts Moody’s slip ratio for mechanical non-equilibrium and a newly proposed correlation of supercooling for thermal non-equilibrium. For evaluation of the models over various thermodynamic states, S-CO2 critical flow experiment through an orifice is conducted to supplement and expand a range of previously published S-CO2 critical flow experiment data. From the evaluation results with the expanded S-CO2 critical flow experiment database, the new non-equilibrium model shows better prediction performance than HEM in terms of the prediction of critical mass flux and critical pressure. For practical applications, discharge coefficients for each 1-D analytical critical flow model are suggested for the orifice geometry.

Numerical analysis of deformation and failure mechanism of metal truss structure of spacecraft under re-entry aerothermodynamic environment

A coupled aerodynamic-thermoelastic analysis framework is presented to simulate the deformation and failure mechanism of metal truss structure for spacecraft under re-entry aerothermodynamic environment. The three-dimensional dynamic thermo-mechanical coupling finite element algorithm is developed by computable modeling of thermoelasticity and heat conduction equations to solve the thermo-mechanical coupling response of spacecraft structure. The aerodynamic boundary conditions such as temperature, heat flux and pressure distribution are computed and provided by the gas-kinetic unified algorithm (GKUA) based on the Boltzmann model equation. The linear interpolation method is designed and employed to couple the boundary conditions between the external aerodynamic flow field and the solid structure interface. The present thermo-mechanical coupling algorithm is verified on one-dimensional and two-dimensional transient heat flow problems with analytical solutions, and good consistency is achieved. The reliability of the GKUA for solving the boundary conditions of the external flow field and the dynamic thermo-mechanical coupling algorithm for the thermal response behavior of the internal structure material is verified by the consistency calculation of the internal and external temperature distribution. Then, the aerodynamic-thermoelastic analysis framework is further applied to study the deformation behavior of structural thermo-mechanical responses and failure mechanism of a hollow sphere under re-entry aerothermodynamic environment with emphasis placed on the influence of flight speeds, rarefied effects, algorithm schemes and damping effects. The numerical results obtained by the presented algorithm including the internal temperature, thermal stress and deformation distribution law are compared and analyzed. Finally, the deformation and failure mechanism of spacecraft structure during re-entry process are solved and analyzed by the present algorithm framework. The displacement, temperature and stress distribution contours of the structure are presented to make a preliminary understanding and evaluation on the process of spacecraft deformation and failure disintegration. It is indicated that the present coupled aerodynamic-thermoelastic analysis framework can provide a reliable, applicable and efficient platform for the thermo-mechanical coupling responses and failure mechanism of re-entry spacecraft in the end of life.