Three-field Model Research Articles

Effect of Local Thermal Nonequilibrium on the Stability of the Flow in a Vertical Channel Filled With Nanofluid Saturated Porous Medium

Abstract The stability of nanofluid flow in a vertical channel packed with a porous medium is examined for the local thermal nonequilibrium state of the fluid, particle, and solid–matrix phases. The effects of Brownian motion along with thermophoresis are incorporated in the nanofluid model. The Darcy–Brinkman model for the flow in a porous medium and three-field model, each representing the fluid, particle, and solid–matrix phases separately, for the temperature is used. A normal mode analysis is used to obtain the eigenvalue problem for the perturbed state, which is then solved using the Chebyshev spectral collocation technique. The critical Rayleigh number and corresponding wavenumber are presented graphically for the effect of different local thermal nonequilibrium parameters. It is noticed that the influence of local thermal nonequilibrium (LTNE) parameters on convective instability is significant.

Role of the pedestal current on the stability of non-ideal ballooning modes

On the basis of a three-field flute-reduced magnetohydrodynamic model, which mainly describes the edge instabilities by shielding a major part of the J × B force in the flute reduction, we study the stability of ballooning modes in the edge pedestal, highlighting the role of an equilibrium parallel current gradient. This effect, which is designated as the current gradient driven (CGD) term in this paper, is shown to have an influence on the stability of finite-n pedestal ballooning modes due to the existence of a highly localized bootstrap current. An analysis in the ideal limit shows that the CGD term destabilizes the ballooning modes regardless of the sign of its gradient, especially near the stability boundaries. An inclusion of the finite Larmor radius (FLR) effect via ion diamagnetic flow and finite resistivity results in a coupling of the FLR effect and the current gradient. In this particular regime where the deviation from the ideal stability is considerable, this coupling effect is shown to dominate stability in intermediate n (20<n≤40) modes. Here, n is the toroidal mode number. Stability analyses using a series of model pedestal equilibria indicate that an increase in a bootstrap current can move the most unstable position from the pedestal middle to the bottom and the negative gradient of the bootstrap current at the pedestal bottom leads to further destabilization of intermediate n modes.

Inflation and scale-invariant R2 gravity

In scale-invariant models of fundamental physics, mass scales are generated by spontaneous symmetry breaking. In this work, we study inflation in scale-invariant $R^2$ gravity, in which the Planck mass is generated by a scalar field, which is responsible for spontaneous breaking of scale--symmetry. If the self-interactions of the scalar field are non-zero, a cosmological constant is generated, which can be potentially quite large. To avoid fine-tuning at late times, we introduce another scalar field which drives the classical cosmological constant to zero during inflation. Working in the Einstein-frame, we find that due to a conserved Noether current the corresponding three-field inflationary model (consisting of the two scalar fields plus the scalaron) becomes effectively a two-field model. The prize to be paid for introducing the field which cancels the classical cosmological constant at the end of inflation is that the running of the spectral index and the running of the running can be quite large due to entropy perturbations during inflation, making the model testable with future cosmological experiments.

Improvement of the heat transfer enhancement model considering the droplet-wall heat transfer downstream of the flow blockage in the reflood phase

When a flow blockage is formed by fuel swelling in the reflood phase of a large-break loss-of-coolant-accident (LOCA) in a pressurized water reactor (PWR), flow separation and reattachment occur due to the flow area change downstream of the blockage. These phenomena increase turbulence and vorticity in the downstream, resulting in the enhancement of heat transfer by steam and liquid droplets. The existing heat transfer enhancement model has considered the enhancement factor for the single-phase heat transfer only. For the assessment of the existing model, it was implemented in the CUPID code in which a two-fluid three-field model is applied. The existing model was evaluated using the FEBA flow blockage test without flow bypass. As a result, the existing model greatly over-predicted the rod temperature downstream of the blockage. To improve this, a new enhancement model was proposed that considers the wall heat transfer by the droplet deposition. The heat transfer enhancement factor was derived based on the data from the abrupt pipe expansion experiments that measured heat and mass transfer coefficients after flow separation point. For accurate calculation of the amount of droplets approaching the fuel rod, an improved droplet deposition model was developed using the droplet deposition experimental data for dispersed droplet and annular-mist flow. The new model implemented in the CUPID code was evaluated using the FEBA flow blockage tests with different pressure, flooding rate, and blockage ratio. It was shown that the new model gave good predictions for the temperature of the early cooled rod bundle downstream of the blockage.

Validation of mechanistic dryout and rewetting model based on the three-field model with single-tube experiments

ABSTRACT A mechanistic model for dryout and rewetting in boiling water reactors, is necessary in safety evaluations since it could be commonly applicable to various fuel assemblies and wider ranges of thermal-hydraulic conditions. A dryout and rewetting mechanistic model based on the three-field model in the COolant Boiling in Rod Arrays-Three Field (COBRA-TF) code was validated in this study. The dryout and rewetting criteria were introduced into COBRA-TF with the critical liquid film thickness correlation proposed by Chun et al., taking into account the disappearance and reformation of a liquid film. The calculated results were compared with not only the experimental results on critical powers and rewetting velocities measured in single-tube experiments, but also those calculated with the original model in COBRA-TF. It was shown that the model had a good predictive capability for the dryout. Although the dependencies of the calculated rewetting velocities on wall surface temperatures were qualitatively consistent with the experimental results, there were quantitatively significant differences between the experimental and calculated results. The sensitivity analyses on the rewetting predictive capability showed a critical liquid film thickness model for the rewetting, which would consider the rewetting front behaviors, is necessary for an improvement of the rewetting prediction.

Nonlinear simulation of edge localized mode with pressure profile modified by pellet injection through a BOUT++ three-field MHD model

Abstract A BOUT++ three-field MHD model is extended and applied to study the effect of the pedestal pressure profile modified by the pellet injection from the low field side on the nonlinear behaviors of ELM. The linear simulation shows that the growth rates of all n modes can be enhanced regardless of the deposition location and are more affected by the pellet atoms depositing in the middle of the steep pedestal region. From the nonlinear simulation, an ELM triggering threshold for the rate of the modified pressure over the unperturbed one at the most affected position is demonstrated when the pellet deposits its atoms on the pedestal top, which supports the observation in the non-fuel pellet pacing experiments on EAST. In contrast, when the pellet deposits its atoms within the steep pedestal region, the ELM size is increased at low and high rates but decreased in an intermediate range; compared with the deposition mainly on the pedestal top, fewer perturbed atoms introduced to the steep pedestal region could cause a larger ELM size, which can somehow explain the probabilistic behavior of ELM triggering by small pellet injection on EAST.

The Tensor Diffusion approach for simulating viscoelastic fluids

In this paper, the novel Tensor Diffusion approach for the numerical simulation of viscoelastic fluids is introduced based on the idea, that the extra-stress tensor in the momentum equation of the flow model can be replaced by the product of the strain-rate tensor and a (nonsymmetric) tensor-valued viscosity. As potential advantage (which can be demonstrated for fully developed channel, resp., pipe flows), this approach offers the possibility to reduce the full nonlinear viscoelastic three-field model to a generalized Tensor Stokes problem, avoiding the need of considering a separate stress tensor in the solution process. Moreover, an (artificial) diffusive operator is introduced into the momentum equation in the case of “no solvent” viscoelastic fluids, which is related to the physics of the problem leading to an improved numerical behaviour w.r.t. approximation and convergence properties of the iterative solvers. After validating the Tensor Diffusion approach for fully developed channel flow configurations, it is evaluated in the context of more general two-dimensional flow settings of benchmarking character, too, taking into account a symmetrized formulation. Numerical simulations for several prototypical flow configurations illustrate the mathematical and numerical properties of this new approach and can be viewed as proof-of-concept for further development.

Numerical simulation of impact of supersonic molecular beam injection on edge localized modes

A three-field model with the impact of supersonic molecular beam injection (SMBI) based on the BOUT++ code is built to simulate edge localized modes (ELMs). Different parameters of SMBI are explored to find an optimal SMBI scenario for ELM mitigation. The linear simulations show that the growth rate of peeling-ballooning mode is reduced by SMBI. The reduction amplitude of the growth rate is increased when the amplitude or width of SMBI is increased, and when SMBI is deposited at the top, bottom and middle of the pedestal, the reduction amplitude increases successively. The nonlinear simulations show that the ELM size is reduced by SMBI. The reduction amplitude of the ELM size is increased when the amplitude or width of SMBI is increased, and when SMBI is deposited at the bottom, top and middle of the pedestal, the reduction amplitude increases successively. Surface-averaged pressure profiles and filamentary structures are analyzed when the ELMs erupt. Deep deposition of SMBI such as at the top and middle of the pedestal reduces the inward collapse amplitude of the pressure profiles, which can improve the confinement efficiency during ELMs. Shallow deposition of SMBI such as at the middle and bottom of the pedestal reduces the outer extent of the filamentary structures, which can slow down the erosion of plasma-facing components caused by ELMs. In conclusion, shallow deposition of SMBI with sufficient amplitude and width can meet the needs of ELM mitigation.

Ice crystal growth in the freezing desalination process of binary water-NaCl system

In this study, ice crystal growth in the freezing desalination process of binary water-NaCl system is investigated. The phase field method is used to conduct simulation and predict dendrite growth behaviour during the crystallization of sea ice. An experimental setup focusing on the directional crystallization of binary water-NaCl solutions on the horizontal wall is established to verify the e theoretical model. The results show that the crystal with single nucleus obtains obvious six-fold symmetry, while in the competitive crystallization of multiple crystal nuclei, spindle growth is inhibited. Brine channels and salt cells are formed in the directional crystallization, seawater consists of higher Mg2+and SO42− get more difficult in freezing desalination. The degree of subcooling significantly affects the salt cell concentration at the roots, the solid phase ratio, and the height of the planar crystal, while the heat flux significantly affects dendrite growth rate. The experimental average tip growth rate is 9.15 × 10−5 m/s, and the experimental average tip radius is 5.16 × 10−6 m. Peclet number of both simulation and experiment have the same order of magnitudes, with 0.1–0.2 and 0.3–0.6, respectively, which suggests the three-field coupling model established is reasonable for desalination simulation.

Monolithic and splitting solution schemes for fully coupled quasi-static thermo-poroelasticity with nonlinear convective transport

This paper concerns monolithic and splitting-based iterative procedures for the coupled nonlinear thermo-poroelasticity model problem. The thermo-poroelastic model problem we consider is formulated as a three-field system of PDE’s, consisting of an energy balance equation, a mass balance equation and a momentum balance equation, where the primary variables are temperature, fluid pressure, and elastic displacement. Due to the presence of a nonlinear convective transport term in the energy balance equation, it is convenient to have access to both the pressure and temperature gradients. Hence, we introduce these as two additional variables and extend the original three-field model to a five-field model. For the numerical solution of this five-field formulation, we compare six approaches that differ by how we treat the coupling/decoupling between the flow and/from heat and/from the mechanics, suitable for varying coupling strength between the three physical processes. The approaches have in common a simultaneous application of the so-called L-scheme, which works both to stabilize iterative splitting as well as to linearize nonlinear problems, and can be seen as a generalization of the Undrained and Fixed-Stress Split algorithms. More precisely, the derived procedures transform a nonlinear and fully coupled problem into a set of simpler subproblems to be solved sequentially in an iterative fashion. We provide a convergence proof for the derived algorithms, and demonstrate their performance through several numerical examples investigating different strengths of the coupling between the different processes.

Robust block preconditioners for poroelasticity

In this paper we develop and analyze several preconditioners for the linear systems arising from discretized poroelasticity problems. The preconditioners include one block preconditioner for the two-field Biot model and several preconditioners for the classical three-field Biot model. We manage to analyze these different preconditioners under a same theoretical framework and show that all of them are uniformly optimal with respect to material and discretization parameters. Numerical tests demonstrate the robustness of these preconditioners.

Research on the Electromagnetic-Heat-Flow Coupled Modeling and Analysis for In-Wheel Motor

In this paper, a 15 KW in-wheel motor (IWM) is taken as the research object, and the coupling factors among the electromagnetic field, temperature field and flow field are analyzed, and the strong and weak coupling factors between the three fields are clarified, and by identifying the strong and weak coupling factors between the three fields, a three-field coupling analysis model for IWM with appropriate complexity is established, and the validity of the model is verified. In a certain driving condition, the electromagnetic field, temperature field and flow field characteristics of IWM are analyzed with the multi-field coupling model. The result shows that, after the IWM runs 8440 s under driving conditions, in this paper, the IWM electromagnetic torque of the rated working condition is 134.2 Nm, and IWM the electromagnetic torque of the peak working condition is 451.36 Nm, and the power requirement of the motor can be guaranteed. The highest temperature of the IWM is 150 °C, which does not exceed the insulation grade requirements of the motor (155 °C), the highest temperature of the permanent magnet (PM) is 65.6 °C, and it does not exceed the highest operating temperature of the PM, and ensures the accurate calculation of components loss and the temperature of the motor. It can be found, through research, that the electromagnetic torque difference between unidirectional coupling and bidirectional coupling is 3.2%, the maximum temperature difference is 7.98% in the three-field coupling analysis of IWM under rated working conditions. Therefore, it is necessary to consider the influence of coupling factors on the properties of motor materials when analyzing the electromagnetic field, temperature field and flow field of IWM; it also provides some reference value for the simulation analysis of IWM in the future.

Development of a three-field mechanistic model for dryout prediction in annular flow

Critical heat flux (CHF) is one of the most important thermal criteria for nuclear power plants. It has traditionally been evaluated using look-up tables or empirical correlations. In this paper an annular film dryout (AFD) mechanistic model has been developed based on the interaction of three fields in annular flow region, i.e. the liquid film, entrained droplets and vapor core. The model describes the mass and momentum conservation equations of three fields together with a series of constitutive relations. The effect of some constitutive correlations (the entrainment and deposition of droplets and the onset of annular flow) on the prediction accuracy of the model is studied. The results are compared with CHF experimental data in flow boiling. Fairly good agreements are observed for the CHF in circular tubes with uniform and axially non-uniform heating, as well as rectangular channels with uniform heating. Through coupling with the subchannel analysis method, the model is used to predict the dryout-type CHF in the rod bundles with a good precision.

Simulation and Analysis of Flexible Piezoelectric Beam under Karmen Vortex Street Effect

Recent years, researchers have gradually increased their research efforts on energy harvesters for low power applications. Among them, many piezoelectric energy harvesters have been widely used because of their simple structures, low energy consumption and easy miniaturization. In order to study the energy harvesting with a flexible piezoelectric beam deformed by vortex-induced vibration, this paper takes polyvinylidene fluoride as the piezoelectric beam, and relies on ANSYS simulation software, establishes a fluid-solid-piezoelectric three-field coupling model, and calculates the energy harvesting capacity of piezoelectric beam. The results show that the positive output voltage is 59.763V and the negative output voltage is -67.633V.

Beyond bubbly two-phase flow investigation using a CFD three-field two-fluid model

Modeling and simulation study of industrial scale beyond-bubbly flows, like cap-bubbly, cap-turbulent and churn-turbulent, using the two-fluid model, has been limited due to the wide range of interfacial length scales involved and lack of physics-based models for bubble hydrodynamics and interaction mechanisms for such flow conditions. One important feature of beyond bubbly flow is the existence of a wide range of bubble sizes and shapes and thus different transport characteristics. The accuracy of two-fluid model predictions is strongly dependent on the constitutive relations used for the two-fluid model. The objectives of this proposed work are the following: select a set of physics-based constitutive relations and implement these models into CFX; evaluate these models for a wide range of test conditions in cap-turbulent and churn-turbulent flows. The interfacial structure is dynamically evaluated using bubble interaction mechanisms (coalescence and break up) from the two-group Interfacial Area Transport Equation implemented as source and sink terms into ANSYS CFX. The assessments for the validity of the models were carried out in comparison to data collected at Purdue University for two-phase flow in a narrow, rectangular duct. Non-uniform inlet conditions were applied in order to evaluate phase diffusion models in CFX. A new turbulence-induced bubble collision diffusion model for the two-group bubbles system has been implemented in ANSYS CFX. The CFD results predict the measured data to within the expected error, correctly capturing the transverse redistribution of phases due to diffusion for both uniform and non-uniform inlet phase distributions. This illustrates the importance of bubble collision phenomena in correctly predicting phase distributions in a channel.

Multidimensional Hydraulic Solver Using Structured and Unstructured Meshes for the System Thermal-Hydraulic Code SPACE

The system thermal-hydraulic code SPACE adopts a multidimensional two-fluid, three-field model to simulate two-phase-flow phenomena encountered during various anticipated transients and postulated accidents of pressurized water reactors. The applicable mesh systems include structured/staggered and unstructured/collocated ones. The staggered mesh system is based on the orthogonal hexahedral shape of cells and their surrounding faces, but it is generalized to describe not only multidimensional Cartesian meshes but also cylindrical meshes and one-dimensional pipe flow networks. The unstructured/collocated mesh system is used to represent more complex geometry using hexahedron, tetrahedron, pyramid, or prism shapes of cells. The structured/staggered mesh system hydraulic solver and the unstructured/collocated mesh system hydraulic solver are merged into a unified version of SPACE so that those hydraulic solvers can analyze simultaneously a complicated system comprising several structured and unstructured mesh blocks. In this paper, the governing equations, mesh systems, and numerical formulations for SPACE are introduced, and the application results are presented for several conceptual problems including the connection of heterogeneous mesh blocks.

Assessment of the CUPID Code for Bubbly Flows in Horizontal Pipes

Two-phase flow in a horizontal pipe has a pronounced feature; that is, two-phase-flow parameters are highly nonsymmetric because gravity is perpendicular to the mean flow direction. Thus, three-dimensional analysis is necessary for the accurate prediction of two-phase flow in a horizontal pipe, such as the hot leg and cold leg of a pressurized water reactor and the pressure tubes in a CANDU reactor. In this study, we simulated bubbly flows in horizontal pipes using the CUPID code, which adopts a two-fluid, three-field model for two-phase flow. In the preliminary calculations, it was found that the particle-averaged two-fluid momentum equation, rather than the standard two-fluid momentum equation, predicts a physically reasonable slip ratio and nondrag forces, except turbulent dispersion forces have negligible effects on the radial void distribution when the particle-averaged two-fluid momentum equation is used. Based on the results, we selected the physical models and computational mesh for subsequent code assessment using various bubbly flow experiments in horizontal pipes. The turbulent dispersion force model was improved to take into account the large void fraction change at the top. The results of the code assessment show good predictions for the axial pressure drop, liquid velocity, and turbulent kinetic energy profile and predict reasonably well the effects of jl and jg on two-phase-flow parameters. However, additional studies are needed for more accurate prediction of the nonsymmetric distribution of gas velocity and turbulent kinetic energy.

An efficient three-field mixed finite element model for the linear analysis of composite beams with deformable shear connection

In this paper, we develop a new and efficient finite element for the linear static analysis of composite beams with deformable shear connection. We adopt a 3-field mixed approach, based on the Hu-Washizu principle, combined with the enhanced strain concept. Our proposal includes the possibility of systematically choosing interpolating functions of increasing order for certain generalized stresses and for the enhanced strains. Another distinctive feature of our approach is the fact that only three generalized stresses are directly approximated. As in many mixed formulations, the degrees of freedom associated with the approximation of generalized stresses and enhanced strains can be condensed out at the element level at negligible cost, leading to discrete systems involving only the displacement degrees of freedom. For benchmarking purposes, a conventional displacement-based conforming finite element is also briefly derived. Several illustrative examples demonstrate the mixed element’s ability to perform very well on the coarsest of meshes – often consisting of a single finite element –, even when the material data exhibits a jump discontinuity in its interior, in sharp contrast with the displacement-based conforming element. This is particularly true when it comes to the estimation of generalized stresses, often the variables of most interest to designers.

Theory of ITG turbulent saturation in stellarators: Identifying mechanisms to reduce turbulent transport

A three-field fluid model that allows for general three-dimensional equilibrium geometry is developed to describe ion temperature gradient turbulent saturation processes in stellarators. The theory relies on the paradigm of nonlinear transfer of energy from unstable to damped modes at comparable wavelength as the dominant saturation mechanism. The unstable-to-damped mode interaction is enabled by a third mode that for dominant energy transfer channels primarily serves as a regulator of the nonlinear energy transfer rate. The identity of the third wave in the interaction defines different scenarios for turbulent saturation with the dominant scenario depending upon the properties of the 3D geometry. The nonlinear energy transfer physics is quantified by the product of a turbulent correlation lifetime and a geometric coupling coefficient. The turbulent correlation time is determined by a three-wave frequency mismatch, which at long wavelength can be calculated from the sum of the linear eigenfrequencies of the three modes. Larger turbulent correlation times denote larger levels of nonlinear energy transfer and hence smaller turbulent transport. The theory provides an analytic prediction for how 3D shaping can be tuned to lower turbulent transport through saturation processes.

Impact of E × B shear flow on low-n MHD instabilities.

Recently, the stationary high confinement operations with improved pedestal conditions have been achieved in DIII-D [K. H. Burrell et al., Phys. Plasmas 23, 056103 (2016)], accompanying the spontaneous transition from the coherent edge harmonic oscillation (EHO) to the broadband MHD turbulence state by reducing the neutral beam injection torque to zero. It is highly significant for the burning plasma devices such as ITER. Simulations about the effects of E × B shear flow on the quiescent H-mode (QH-mode) are carried out using the three-field two-fluid model in the field-aligned coordinate under the BOUT++ framework. Using the shifted circular cross-section equilibriums including bootstrap current, the results demonstrate that the E × B shear flow strongly destabilizes low-n peeling modes, which are mainly driven by the gradient of parallel current in peeling-dominant cases and are sensitive to the Er shear. Adopting the much more general shape of E × B shear ([Formula: see text]) profiles, the linear and nonlinear BOUT++ simulations show qualitative consistence with the experiments. The stronger shear flow shifts the most unstable mode to lower-n and narrows the mode spectrum. At the meantime, the nonlinear simulations of the QH-mode indicate that the shear flow in both co- and counter directions of diamagnetic flow has some similar effects. The nonlinear mode interaction is enhanced during the mode amplitude saturation phase. These results reveal that the fundamental physics mechanism of the QH-mode may be shear flow and are significant for understanding the mechanism of EHO and QH-mode.