Closure Problem Research Articles

Numerical Investigation of the Performance of Linear Interpolation Schemes Coupled Finite Volume Method in the Analysis of Confined Convection-Diffusion Turbulent Flow Field

Numerical solutions are never exact due to errors emanating from the scheme used in discretizing the governing equations and the flow domain. For convection-diffusion flow, the magnitude of these errors varies depending on the scheme used to interpolate the nodal values of the flow quantities to the interfaces. An interpolation scheme that minimizes these errors would give results that are consistent to experimental results. This paper documents the performance of three linear interpolation schemes; upwind differencing, central differencing scheme and the hybrid scheme in obtaining temperature profiles for a convection-diffusion turbulent flow field. To eliminate the enormous scales inherent in turbulent flow, the field variables present in the governing equations are decomposed into a mean and a fluctuating component and averaged. The closure problem was solved using the turbulence model. The resulting equations are discretized using the robust finite volume discretization technique. The discretized equations are solved using a segregated pressure-based algorithm. The results revealed that the central difference interpolation scheme generate temperature profiles that were consistent with experimental results of Ampofo and Karayiannis, (2003).

Turbulent plane Poiseuille flow

The understanding of fully developed turbulence remains a major unsolved problem of statistical physics. A challenge there is how to use what we understand of this problem to build-up a closure method, that is to express the time-averaged turbulent stress tensor as a function of the time-averaged velocity field u(x). We have shown that the closure problem is strongly restricted due to constraints on the time-averaged quantities, and to scaling laws derived from the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations. It implies that the turbulent stress is a non-local function in space of the time-averaged velocity u(x), involving an integral kernel, an extension of classical Boussinesq theory of turbulent viscosity. We treat one of the simplest possible physical situation, the turbulent Poiseuille flow between two parallel plates. In this case, the integral kernel takes a simple form leading to full analysis of the time-averaged turbulent flow. In the limit of a very large Reynolds number, one has to match a viscous boundary layer near the walls bounding the flow and an outer solution in the bulk of the flow, a non-trivial asymptotic analysis because of logarithms. Besides the boundary layers close to the walls, there is another “inner” boundary layer near the center plane of the flow. Our expression for the turbulent stress tensor yields ultimately the complete structure of the boundary layer, including in locations where viscosity becomes important.

휴대용 수력 터빈 이노마드 우노의 성능향상을 위한 최적설계

In this study, an optimization design was performed to improve the hydrodynamic performances of an ultra-small turbine based on a 3-D Reynolds-averaged Navier-Stokes (RANS) analysis. The runner of the turbine was selected as the computational domain and optimization object. The locations of the inlet, outlet and external interface of the computational domain were set to 3 times, 6 times and 3 times the diameter of the runner, respectively. For the turbulent closure problem in the RANS analysis, the shear stress transport (SST) turbulence model was used, and a grid dependency test was performed by applying the grid convergence index (GCI) based on the Richardson extrapolation method considering the grid discretization error. Four design variables related to the geometry of the runner blade were selected to maximize the power coefficient. The radial basis neural network (RBNN) was used as the surrogate model and trained to improve prediction accuracy. The optimization results obtained by using machine learning method showed highly accurate prediction values, and compared with the reference design, the optimum design provided improved hydraulic performances.

Explicit physics-informed neural networks for nonlinear closure: The case of transport in tissues

In upscaling methods, closures for nonlinear problems present a well-known challenge. While a number of theoretical methods have been proposed for handling such closures, nonlinearities still remain a significant obstacle for many problems. In this work, we use a combination of formal upscaling and data-driven machine learning for explicitly closing a nonlinear transport and reaction process in multiscale tissues. The classical effectiveness factor model is used to formulate the macroscale reaction kinetics. We train a multilayer perceptron network using training data generated by direct numerical simulations over microscale examples. Once trained, the network is used in an algorithm for numerically solving the upscaled (coarse-grained) differential equation describing mass transport and reaction in two example tissues. The network is described as being explicit in the sense that the network is trained using macroscale concentrations and gradients of concentration as components of the feature space rather than incorporating them as part of a constraint in the optimization process.Network training and solutions to the macroscale transport equations were computed for two different tissues. The two tissue types (brain and liver) exhibit markedly different geometrical complexity and spatial scale (cell size and sample size). The upscaled solutions for the average concentration are compared with numerical solutions derived from the microscale concentration fields by a posteriori averaging. There are three outcomes of this work of particular note.1.Our overall approach results in an upscaled nonlinear PDE. The PDE is closed using a neural network, and our approach results in the definition of the classical effectiveness factor for effecting closure.2.We identify particular source terms for the closure problem that are important for representing the structure of the closure. These source terms involve macroscale concentrations and their gradients. We adopt these source terms to use as explicit features in the learning algorithm. We find the trained networks that include the macroscale source terms generate models that are able to predict the correction factor with increased fidelity over those that do not.3.We find that the trained network exhibits good generalizability, and it is able to predict the effectiveness factor with high fidelity for realistically-structured tissues despite the significantly different scale and geometrical complexity of the two example tissue types.This latter result emphasizes our purposeful connection between conventional averaging methods with the use of machine learning for closure; this contrasts with some machine learning methods for upscaling where the exact form of the macroscale equation remains unknown.

Angular Modeling of the Components of Net Radiation in Agricultural Crops and Its Implications on Energy Balance Closure

Efficient water management in agricultural crops is necessary to increase productivity and adapt to climate change. Evapotranspiration (ET) data are key in determining the water requirements of crops and set efficient irrigation schedules. Estimating ET at the regional scale (for example, in irrigation districts) is a technically complex task that has been tackled by using data acquired by remote sensors on satellites that can be validated with scaled up field measurements when area sources are matched. Energy and matter flux measurements using the eddy covariance (EC) technique are challenging due to balance closure issues, claimed to be due to the different footprints of the energy-balance components. We describe net radiometer footprints in terms of the sun-sensor geometry to characterize the bidirectional distribution functions of albedo and thermal emissions. In this context, we describe a one-parameter model of the components of net radiation that can be calibrated with a single data point. The model was validated in an experiment with five agricultural crops (bean, sorghum, chickpea, safflower, and wheat) at Valle del Yaqui, in Sonora, Mexico, using different sun-sensor geometry configurations. The results from the experimental fits were satisfactory (R2 > 0.99) and support the use of the model for albedo and radiative (surface) temperature in order to estimate net radiation. The analysis of the implications regarding a mismatch among footprints of the components of the energy balance showed that net radiometer fluxes are overestimated most of the time, implying that the closure problem could be solved by using a similar footprint as the aerodynamic components of the energy balance.

Covariant approach to relativistic large-eddy simulations: The fibration picture

Models of turbulent flows require the resolution of a vast range of scales, from large eddies to small-scale features directly associated with dissipation. As the required resolution is not within reach of large scale numerical simulations, standard strategies involve a smoothing of the fluid dynamics, either through time averaging or spatial filtering. These strategies raise formal issues in general relativity, where the split between space and time is observer dependent. To make progress, we develop a new covariant framework for filtering/averaging based on the fibration of spacetime associated with fluid elements and the use of Fermi coordinates to facilitate a meaningful local analysis. We derive the resolved equations of motion, demonstrating how "effective" dissipative terms arise because of the coarse-graining, and paying particular attention to the thermodynamical interpretation of the resolved quantities. Finally, as the smoothing of the fluid-dynamics inevitably leads to a closure problem, we propose a new closure scheme inspired by recent progress in the modelling of dissipative relativistic fluids, and crucially, demonstrate the linear stability of the proposed model.

Momentum transport in the free fluid-porous medium transition layer: one-domain approach

In this work, we consider the momentum transport of a incompressible fluid in a like Beavers and Joseph (1967) system. For this purpose, in the context of the volume averaging method, we use a one-domain approach (ODA). Thus, the momentum generalized transport equations (GTE), which are written in terms of position-dependent effective medium coefficients, are valid everywhere in the system and contains two Brinkman corrections in addition to a Darcy’s term. The ODA predictions are tested against the results obtained from averaging the local profiles resulting from pore-scale simulations. One of the key points for solving the ODA remains on the prediction of the permeability, which in this work is obtained either by solving the associated local closure problem or from pore-scale profiles. Our analysis shows that the GTE for momentum transport accurately predicts the average velocity profiles everywhere in the system. To this end, the first and the second Brinkman’s corrections, as well as a position-dependent intrinsic permeability tensor in Darcy’s term must be included.

Group analysis of the one-dimensional Boltzmann equation. Invariants and the problem of moment system closure

Group analysis of the one-dimensional Boltzmann equation. Invariants and the problem of moment system closure

A Mechanical Model for Radial Inertial Impact Switch

The traditional inertial impact switch based on experience design often appears in the problem of early closure within the threshold. To solve this problem, a new method of inertial switch design is presented in this paper. Firstly, the force analysis of inertia switch is carried out and a mechanical model is established. Through the model, it is concluded that the closing threshold of inertia switch is related to the spring resistance, the mass and center of mass of contact pole, the spring radius of contact pole and the horizontal projection of the minimum distance between the spring and the reference point. After that, the mechanical model is verified by simulation, and the correctness of the mechanical model is proved. The experimental results show that the proposed model can effectively solve the problem of premature closing of the inertial switch within the threshold and improve the pass rate of the product.

Simulating mitigation scenarios for natural and artificial inlets closure through validated morphodynamic models

The coastline of the Nile Delta extends between Alexandria and Port Said and spans approximately 240 km. It experienced accelerated erosion since the construction of Aswan High Dam (AHD) where more than 1.2 billion EGP were spent to mitigate the imbalance in sediment transport patterns as AHD blocks most of the sediments reaching the Mediterranean coast. The current study investigates various combinations of interventions to mitigate the complex situation around natural and artificial tidal inlets namely Rosetta promontory and El-Burullus fishing harbor inlet. After various previous interventions in both sites, the problem of structures stability, inlets closure, and erosion in the nearby beaches still exists. The study will utilize already developed, calibrated and field validated morphodynamic models for the two areas of interest. Out of the past and present tested scenarios, the study comes up with the optimum combination of interventions at the two inlets. For Rosetta promontory, the third scenario (one-kilometer jetty in the western direction, a 0.4 kilometer one at the eastern direction, another three groins at the eastern direction, and two groins at the western direction) has the least net sedimentation volume at the inlet (2206.58 m3) and the least change in depth at the critical sections. On the other hand, the simulations indicated that SC3 (extension (90 m) of west jetty + 5 east 200 m groins + 2 west groins (200 m)) produced ideal results in terms of the El-Burullus harbor entrance siltation, the erosion in front of seawall and the accretion at the west of breakwater; However, it caused some slight erosion at the shoreline west of the breakwater. The successful concluded results of this study could be adopted with appropriate modifications for other Deltas worldwide.

Group analysis of the one-dimensional Boltzmann equation. Invariants and the problem of moment system closure

Завершено исследование вопроса о принципиальной возможности замыкания моментной системы кинетического уравнения посредством инвариантных соотношений, полученных методами группового анализа. Для одномерного уравнения получен положительный ответ, найдено инвариантное замыкание моментной системы и обнаружен ряд новых феноменов.

Machine learning based Reynolds averaged simulation of backward-facing step flows at different Reynolds numbers

To address the closure problem of Reynolds-averaged Navier–Stokes in numerical simulations of turbulence, the method of solving Reynolds-averaged Navier–Stokes equations based on artificial neural network is introduced in this paper. We establish the nonlinear mapping relationship between the average flow field and the steady-state eddy viscosity field. The machine learning (ML) surrogate model for the shear stress transport turbulence model is constructed. The solution process of replacing the original turbulence model equations with the predicted field variables is realized by coupling the ML algorithm with the CFD solver. The classical backward facing step problem is selected in our study to verify the simulation accuracy of the surrogate model. The comparative analysis is carried out on the six backward facing step flows simulations at different Reynolds numbers. The results of simulations show that the testing flows with the Reynolds numbers closest training datasets Reynolds numbers can obtain the best simulation accuracy. Then for the Reynolds number that is lower than the training datasets, the simulation accuracy will decrease as the Reynolds number decreases. On the contrary, the simulation accuracy of the test flow will increase as the Reynolds number increases. These results indicate the feasibility of the ML surrogate model to simulate at higher Reynolds number. It shows the great potential of applying ML algorithms to Reynolds-averaged Navier–Stokes simulation (RANS) turbulence model and also provides a new idea for industrial simulations of turbulent flows.

Numerical Modelling of Heat Transfer in Fine Dispersive Slurry Flow

Slurry flows commonly appear in the transport of minerals from a mine to the processing site or from the deep ocean to the surface level. The process of heat transfer in solid–liquid flow is especially important for the long pipeline distance. The paper is focused on the numerical modelling and simulation of heat transfer in a fine dispersive slurry, which exhibits yield stress and damping of turbulence. The Bingham rheological model and the apparent viscosity concept were applied. The physical model was formulated and then the mathematical model, which constitutes conservative equations based on the time average approach for mass, momentum, and internal energy. The slurry flow in a pipeline is turbulent and fully developed hydrodynamically and thermally. The closure problem was solved by taking into account the Boussinesque hypothesis and a suitable turbulence model, which includes the influence of the yield shear stress on the wall damping function. The objective of the paper is to develop a new correlation of the Nusselt number for turbulent flow of fine dispersive slurry that exhibits yield stress and damping of turbulence. Simulations were performed for turbulent slurry flow, for solid volume concentrations 10%, 20%, 30%, and for water. The mathematical model for heat transfer of the carrier liquid flow has been validated. The study confirmed that the slurry velocity profiles are substantially different from those of the carrier liquid and have a significant effect on the heat transfer process. The highest rate of decrease in the Nusselt number is for low solid concentrations, while for C > 10% the decrease in the Nusselt number is gradual. A new correlation for the Nusselt number is proposed, which includes the Reynolds and Prandtl numbers, the dimensionless yield shear stress, and solid concentration. The new Nusselt number is in good agreement with the numerical predictions and the highest relative error was obtained for C = 10% and Nu = 44.3 and is equal to −12%. Results of the simulations are discussed. Conclusions and recommendations for further research are formulated.

Primary geochemical haloes and alteration zoning applied to gold exploration in the Zarshuran Carlin-type deposit, northwestern Iran

The Zarshuran gold deposit as sediment-hosted Carlin-type has a total reserve of 120 million metric tonnes and an average grade of 3.5 g/t Au. The dominant hydrothermal alterations associated with gold mineralization are argillic and silicic as well as decalcification of the carbonate host rocks. The exploration geochemistry was applied to characterize the mineralization in the subsurface and peripheries of the open-pit mine at Zarshuran in order to enhance the proved reserves of the gold ores. A sampling network of a 10 × 10 m grid (n = 1720) was applied to almost unweathered surface outcrops to determine the primary geochemical haloes in the study area. Initially, the cut-off grades utilized in the exploratory data analysis were applied to identify the threshold values of the trace elements with lognormal distribution. Then, the central log ratio transformation was used to determine the robustness against outliers to characterize the closure problems with compositional data during Principle Component Analysis (PCA). The supra-ore and near-ore geochemical haloes are recognized by the indicator elements like As-Sb-Hg-Te-Au-Ag-Fe-Mn-Ba-F which are associated with mineral assemblages such as arsenical pyrite, orpiment, realgar, cinnabar, barite, and fluorite. The sub-ore geochemical haloes are characterized by index element association like Cu-Mo-Bi-W-Sn-Co-Ni-V-Cr. The ores at Zarshuran are enriched in Au, As, Sb, Ag, Te, Tl, Zn, Pb, and F but depleted in Cu, W, and Sn, as were observed in many other Carlin-type gold deposits. The enrichment factor of indicator elements (except S, Ag, and Sb) gradually decreases in value from the highly altered and mineralized zones (in proximity of the mine) toward the zones with least alteration and mineralization (in the distal parts of the mine). According to PCA analysis, two main types of ore mineralization occurred at Zarshuran, (1) Au mineralization associated with As, Sb, and Hg, and (2) Cu-Zn-Pb base-metal mineralization related to Ag. The ore-forming index was used to determine the primary axial haloes in both shallow (<−80 m) and deep (>−120 m) levels. The results of this study show that the main ore body may extend eastwardly and downwardly to greater depths, as was verified by the data obtained from diamond drill cores in the eastern part of the mine.

Changes in Quality of Life After Secondary Closure of Palatal Defects: Prosthetic Obturation Versus Surgical Reconstruction.

The closure of palatal defects after tumor resection or irradiation is performed with either a prosthesis or autogenous tissue; however, there are no clear criteria regarding selection of the method. Thus, this study aimed to investigate the real-world situation and problems of palatal closure using prostheses, and examined patient opinion on how palatal closure using autogenous tissue improved their postoperative quality of life (QOL). In 5 patients whose palatal defects resulted from treatment for head and neck cancer and were closed with a prosthesis, the palate was closed secondarily with autogenous tissue; a questionnaire on daily life was administered pre- and post-operatively. Functional improvements in terms of speech and eating were achieved in all and in 4 of 5 cases, respectively. In all cases, the QOL was better for palatal closure with autogenous tissue than with the prosthesis. As postoperative QOL was considered to be better when reconstructing the palate with autogenous tissue than with the prosthesis, we recommend to actively select autogenous tissue for palate reconstruction.

Turbulence in a wedge: The case of the mixing layer

The ultimate goal of a sound theory of turbulence in fluids is to close in a rational way the Reynolds equations, namely to express the tensor of turbulent stress as a function of the time average of the velocity field. Based on the idea that dissipation in fully developed turbulence is by singular events resulting from an evolution described by the Euler equations, it has been recently observed that the closure problem is strongly restricted, and that it implies that the turbulent stress is a non local function in space of the average velocity field, a kind of extension of classical Boussinesq theory of turbulent viscosity. This leads to rather complex nonlinear integral equation(s) for the time averaged velocity field. This one satisfies some symmetries of the Euler equations. Such symmetries were used by Prandtl and Landau to make various predictions about the shape of the turbulent domain in simple geometries. We explore specifically the case of mixing layer for which the average velocity field only depends on the angle in the wedge behind the splitter plate. This solution yields a pressure difference between the two sides of the splitter which contributes to the lift felt by the plate. Moreover, because of the structure of the equations for the turbulent stress, one can satisfy the Cauchy-Schwarz inequalities, also called the realizability conditions, for this turbulent stress. Such realizability conditions cannot be satisfied with a simple turbulent viscosity.

Flow near porous media boundaries including inertia and slip: A one-domain approach

This work addresses the macroscopic modeling of flow near porous media boundaries. This includes the vicinity with a fluid channel (i.e., a fracture), another rigid porous medium, or an impervious non-deformable solid. The analysis is carried out for one-phase, steady, incompressible, inertial, and isothermal flow of a Newtonian fluid, considering slip effects at the solid–fluid interfaces. A one-domain approach is proposed, employing a simplified version of the volume averaging method, while conceiving the system as two homogeneous regions separated by an inter-region. The upscaling procedure yields a closed macroscopic model including a divergence-free average (filtration) velocity for the mass balance equation and a unique momentum equation having a Darcy structure. The latter involves apparent permeability tensors that are constant in the homogeneous regions and position-dependent in the inter-region. All the permeability tensors are determined from the solution of coupled closure problems that are part of the developments. The derived model is validated by comparisons with direct numerical simulations in several two-dimensional configurations, namely, two porous media of contrasted properties in direct contact or separated by a fracture, the boundaries being either flat or wavy and a porous medium in contact with a flat or corrugated solid wall or separated from the latter by a fluid layer. The simplicity and versatility of the derived model make it an interesting alternative to existing one- and two-domain approaches developed so far.

The Development and Application of a Kinetic Theory for Modeling Dispersed Particle Flows

Abstract This Freeman Scholar article reviews the formulation and application of a kinetic theory for modeling the transport and dispersion of small particles in turbulent gas-flows. The theory has been developed and refined by numerous authors and now forms a rational basis for modeling complex particle laden flows. The formalism and methodology of this approach are discussed and the choice of closure of the kinetic equations involved ensures realizability and well posedness with exact closure for Gaussian carrier flow fields. The historical development is presented and how single-particle kinetic equations resolve the problem of closure of the transport equations for particle mass, momentum, and kinetic energy/stress (the so-called continuum equations) and the treatment of the dispersed phase as a fluid. The mass fluxes associated with the turbulent aerodynamic driving forces and interfacial stresses are shown to be both dispersive and convective in inhomogeneous turbulence with implications for the build-up of particles concentration in near wall turbulent boundary layers and particle pair concentration at small separation. It is shown how this approach deals with the natural wall boundary conditions for a flowing particle suspension and examples are given of partially absorbing surfaces with particle scattering and gravitational settling; how this approach has revealed the existence of contra gradient diffusion in a developing shear flow and the influence of the turbulence on gravitational settling (the Maxey effect). Particular consideration is given to the general problem of particle transport and deposition in turbulent boundary layers including particle resuspension. Finally, the application of a particle pair formulation for both monodisperse and bidisperse particle flows is reviewed where the differences between the two are compared through the influence of collisions on the particle continuum equations and the particle collision kernel for the clustering of particles and the degree of random uncorrelated motion (RUM) at the small scales of the turbulence. The inclusion of bidisperse particle suspensions implies the application to polydisperse flows and the evolution of particle size distribution.

Probability theory of active suspensions

A new approach to studying active suspensions is presented. They exhibit a specific behavior pattern, sometimes referred to as active turbulence. Starting from first principles, we establish a description for an active suspension, consisting of a Newtonian fluid and active Janus particles. The fluid phase is described by Navier–Stokes equations and the particles by Newton–Euler equations. A level set approach is used to separate the two phases, well-known from the representation of sharp interfaces in various numerical schemes. By introducing the multi-point probability density function (PDF)-approach known from hydrodynamic turbulence, we obtain a hierarchical ordered infinite set of linear statistical equations. However, the equations for the K-point PDF depend on the K + 1 and K + 2-point PDF, exposing the closure problem of active turbulence. As all statistical moments can be formed from the PDF, the latter set of equations already includes every statistical model for an active suspensions. To illustrate this, we derive the Eulerian spatial averaging theory from the hierarchy of multi-point PDF-equations.

Local equilibrium approximation in free turbulent flows: Verification through the method of differential constrains

AbstractWe present a full version of the results obtained in Grebenev et al [Doklady Physics 47(7), 518–521 (2002)] wherein the closure formula, that is, the local equilibrium approximation of second‐order moments for modeling free turbulent flows was justified by the method of differential constrains. The proposed analysis provides us a point of view from the modern theory of symmetry analysis on the closure problem in turbulence. Specifically, closure relationships in the physical space are interpreted as the (differential) equations of invariant sets (manifolds) in a phase‐space. We demonstrate how this concept can be applied for verification of the local equilibrium approximations (LEA) of second‐order moments. With this, we obtain the equivalence of LEA and vanishing the Poisson bracket for the defect of the longitudinal velocity component and of the turbulent energy. Numerical experiments carried out in a far turbulent wake confirm this conclusion.