Closure Equations Research Articles

Sound Over-Approximation of Equational Reasoning with Variable-Preserving Rules Parameterized by Derivation Depth

Equational reasoning is one of the most intuitive and widely used types of symbolic reasoning. In this setting, the goal is to determine whether a given ground equation t=t' follows as a consequence of a set of equational axioms E using the process of replacing equals with equals. An equation t=t' is variable-preserving if each variable occurring in t occurs in t' and vice versa. Such equations capture essential algebraic properties, such as commutativity, associativity and distributivity, which arise in a wide variety of contexts. In this work, we show that for each fixed set E of variable-preserving equations, the set of ground equations derivable from E in depth at most d is soundly over-approximable in fixed-parameter tractable time. More specifically, we devise an algorithm that takes as input a set E of variable-preserving equations and a target ground equation t=t', and always halts with a YES or NO answer. 1) If equation t=t' can be derived from E in depth at most d, the algorithm always halts with a YES. 2) If equation t=t' does not belong to the equational closure of E, then the algorithm always halts with a NO. In other words, the set of YES instances contains the set of ground equations that can be deduced from E in depth at most d, and possibly other equations that require derivations of higher depth. However, this set contains no ground equation that is not in the equational closure of E. For this reason, the algorithm is sound. Our algorithm works in time f(d) * |t| * |t'|, where |t| and |t'| are the number of symbols in t and t' respectively, d is the depth parameter, and f(d) is a function whose growth depends only on d and on parameters associated with the equations in E.

Bayesian uncertainty estimation of adsorption closure models in the computational simulation of contaminant transport.

Identifying sources and understanding the evolution of the spread of contaminants in aquifers is critical and often results in a high-dimensional inverse problem. Computer contaminant transport models are developed by combining physical principles and phenomenological closure equations. However, such models suffer from considerable conceptual uncertainties, mainly due to using phenomenological closures to describe the contaminant adsorption in the media. In the present work, we investigated the impact of employing phenomenological state equations in the contaminant transport model to characterize the adsorption of contaminants in a porous media. Also, we adopted an embedded Bayesian error approach to understand the limits of using adsorption isotherms to describe the contaminant adsorption. We phrase it as a probabilistic error model and test the efficacy of a deep learning surrogate to replace the partial differential equations-based subsurface flow and contaminant transport model. The results show that the adsorption term plays a key role, yielding a higher level of uncertainty in the contaminant transport modeling, and the use of such a closure must be taken with care since the parameters are chosen to meet certain geochemical conditions.

A second-order direct Eulerian GRP scheme for ten-moment Gaussian closure equations with source terms

A second-order direct Eulerian GRP scheme for ten-moment Gaussian closure equations with source terms

Particle Sedimentation in a Fluid at Low Reynolds Number: A Generalization of Hindered Settling Described by a Two‐Phase Continuum Model

AbstractParticle‐fluid separation by settling is a ubiquitous process in Earth and Planetary Sciences. The settling velocity of particles is controlled by a balance between buoyancy and drag forces. Since the seminal work of George Gabriel Stokes, parameterizations for the reduction of particle velocities caused by viscous dissipation due to mutual interactions have been described by a non‐linear mapping between particle volume fraction and separation velocity. We argue that these parameterizations neglect important physical behavior at high particle volume fractions (>80% of the maximum packing) and are only appropriate when considering suspensions where the particle volume fraction does not evolve in space or time. We introduce a more general model that accounts for the energy dissipation caused by changes in local particle volume fraction, which introduces a new term similar to the compaction term at higher particle volume fraction. This term depends on a consolidation/compaction viscosity that measures the resistance to changes in solid volume fraction. We derive closure equations for this compaction viscosity under dilute and concentrated particle volume fraction limits. Numerical simulations show that the extended hindered settling model predicts two significant differences compared to traditional hindered settling. First, while the steepening of particle volume fraction fronts remains, a dynamic instability is also generated at the front. Second, the rate of growth and structure of a cumulate layer growing above a no‐flux boundary is affected by the compaction‐like term and predicts the trapping of a higher volume fraction of interstitial melt in a correspondingly thicker cumulate layer.

Combining integral equation closures with force density functional theory for the study of inhomogeneous fluids.

Classical density functional theory (DFT) is a powerful framework to study inhomogeneous fluids. Its standard form is based on the knowledge of a generating free energy functional. If this is known exactly, then the results obtained by using standard DFT or its alternative, recently developed version, force-DFT, are the same. If the free energy functional is known only approximately then these two routes produce different outcomes. However, as we show in this work, force-DFT has the advantage that it is also implementable without knowledge of the free energy functional, by using instead liquid-state integral equation closures. This broadens the range of systems that can be explored, since free energy functionals are generally difficult to approximate. In this paper we investigate the utility of using inhomogeneous integral equation closures within force-DFT thus demonstrating the versatility and accuracy of this approach.

Closure equationand higher-order moment relations in the Gauss-Hermite lattice Boltzmann method.

Moment methods are often used to solve transport problems involving the Boltzmann-BGK equation. Because the moment equationsare underdetermined, these methods require an additional "closure equation" that relates higher to lower-order moments. Here, we examine the closure equationand higher-order moment relations implicit in the lattice Boltzmann method (LBM) that use Gauss-Hermite quadrature for their discrete velocity sets. It is shown that the discrete-velocity-set itself defines the closure equationand higher-order moment relations, the precise forms of which are yet to be reported. The general formula we present facilitates the efficient computational evaluation of higher-order moments and provides insight into the operation of LBM that is anticipated to be useful in theoretical analyses of its performance. Derived formulas for two different velocity sets are validated against numerical implementations of the LBM for steady Couette flow.

Oriblock: The origami-blocks based on hinged dissection

Oriblock: The origami-blocks based on hinged dissection

A simple steady-state inflow model of the neutral and stable atmospheric boundary layer applied to wind turbine wake simulations

Abstract. Wind turbines are increasing in size and operate more frequently above the atmospheric surface layer, which requires improved inflow models for numerical simulations of turbine interaction. In this work, a steady-state Reynolds-averaged Navier–Stokes (RANS) model of the neutral and stable atmospheric boundary layer (ABL) is introduced. The model incorporates buoyancy in the turbulence closure equations using a prescribed Brunt–Väisälä frequency, does not require a global turbulence length-scale limiter, and is only dependent on two non-dimensional numbers. Assuming a constant temperature gradient over the entire ABL, although a strong assumption, leads to a simple and well-behaved inflow model. RANS wake simulations are performed for shallow and tall ABLs, and the results show good agreement with large-eddy simulations in terms of velocity deficit from a single wind turbine. However, the proposed RANS model underpredicts the magnitude of the velocity deficit of a wind turbine row for the shallow ABL case. In addition, RANS ABL models can suffer from numerical problems when they are applied as a shallow-ABL inflow model to large wind farms due to the low-eddy-viscosity layer above the ABL. The proposed RANS model inherits this issue, and further research is required to solve it.

A generalized kinetic theory of Ostwald ripening in porous media

A generalized kinetic theory of Ostwald ripening in porous media

High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms

High-order accurate positivity-preserving and well-balanced discontinuous Galerkin schemes for ten-moment Gaussian closure equations with source terms

Loss-Optimized Design of Magnetic Devices.

Maximizing efficiency, power density, and reliability stands as paramount objectives in the advancement of power electronic systems. Notably, the dimensions and losses of magnetic components emerge as primary constraints hindering the miniaturization of such systems. Researchers have increasingly focused on the design of loss minimization and size optimization of magnetic devices. In this paper, with the objective of minimizing the loss of magnetic devices, an optimal design method for the winding structure of devices is proposed based on the coupling relationship between the loss prediction model and the design variables. The method examines the decoupling conditions between the design variables and the loss model, deriving optimized design closure equations for the design variables. This approach furnishes a technical foundation for the miniaturized design of miniature apparatuses incorporating magnetic components, offering a straightforward and adaptable design methodology. The finite element method simulation results and experimental measurement data verify the accuracy of the prediction of the proposed method and the validity of the optimal design theory of device loss.

A simulation model for oil production by intermittent gas lift technology

The operating principle of the oil production by intermittent gas lift (IGL) technology and the mechanical processes occurring in it are complex with many different stages and many flows interacting with each other in each stage. This leads to many difficulties in calculating and optimizing oil well production system with IGL technology. This study develops a hydrodynamic simulation model of the flows occurring during oil production by IGL. By improving until now published hydrodynamic simulations the model in this work is developed based on changes/adjustments to be applicable for specific field conditions. Simulation of the IGL process is performed based on numerical solutions of the systems of conservation equations of mass and momentum and closure equations for each system component and each stage of the IGL process. The ordinary differential equations in the model are solved by the implicit Euler method. The simulation program has been developed and used in selecting a reasonable design of IGL technology for an oil well previously produced by the continuous gas lift (CGL). After the IGL system was installed and put into operation, the daily liquid production rate of the well increased by 91% and the measured data showed good agreement with the simulated results. The application result has initially demonstrated the applicability of the proposed model and the developed simulation program. Keywords: well; intermittent gas lift; simulation model; oil production.

Data-based instantaneous conditional progress variable dissipation rate modeling for turbulent premixed combustion

Data-based instantaneous conditional progress variable dissipation rate modeling for turbulent premixed combustion

Mobility analysis of tripod scissor structures using screw theory

Mobility analysis of tripod scissor structures using screw theory

Evaluation of Effects of Waterflooding-Induced Bilayer Fractures on Tight Reservoir Using Pressure-Transient Analysis Method

Summary Waterflooding will open natural fractures to form induced fractures, which differ from hydraulic fractures because the hydraulic fracture is filled with proppant but the induced fracture is not. Natural fractures are connected by waterflooding. However, because the waterflooding pressure is limited, induced fractures cannot run through the entire reservoir but instead form multiple parallel induced-fracture bands in the vertical direction. Currently, using conventional finite-conductivity methods to match field data will obtain unreasonable results, especially the half-length, conductivity of fracture, and reservoir permeability, which lead to the water breakthrough, which cannot be found in time. This paper presents the waterflooding-induced bilayer fracture (WIBF) model, considering induced-fracture dynamic closure (IDC), dynamic induced-fracture storage (DIS), and induced-fracture radial flow (IRF) effects. Two innovative flow regimes are interpreted, which are dynamic induced-fracture flow and early radial flow regimes. Five innovation parameters are introduced into the WIBF model to describe the IDC, DIS, and IRF effects. The WIBF model is calculated and solved by the Green equation and Newman product methods. Induced-fracture storage coefficient and half-length closure equations are derived to characterize the unique induced-fracture properties. Analytical and numerical methods verify the model’s accuracy. The WIBF model matches a type field case to prove its practicability. Results show that compared with the conventional finite-conductivity model, the proposed model matches the field case well and the interpreted parameters are consistent with the water injection profile and actual field data. The pressure derivative curve shows an early horizontal line, identified as a pressure response of bilayer-induced fractures. If the flow regime is misidentified as pseudoradial flow, some obtained parameters will be absurd, and permeability will be amplified many times. In conclusion, physical and mathematical models are established to describe induced fracture. Induced-fracture storage coefficient and half-length equations are derived. Model matching and equation calculation methods are mutually validated to improve the accuracy of the obtained parameters. Dynamic induced-fracture half-length is interpreted quantitatively to make the engineer take action before the water breakthrough. The model in this paper also provides some parameters for infilling well patterns or determining well spacing economically.

A study on the approximated angular averaging of distribution functions obtained from the Ornstein–Zernike theory for diatomic solutes consisting of fused Lennard-Jones particles immersed in a Lennard–Jones monatomic solvent

A study on the approximated angular averaging of distribution functions obtained from the Ornstein–Zernike theory for diatomic solutes consisting of fused Lennard-Jones particles immersed in a Lennard–Jones monatomic solvent

Necessary and sufficient conditions for exact closures of epidemic equations on configuration model networks

We prove that it is possible to obtain the exact closure of SIR pairwise epidemic equations on a configuration model network if and only if the degree distribution follows a Poisson, binomial, or negative binomial distribution. The proof relies on establishing the equivalence, for these specific degree distributions, between the closed pairwise model and a dynamical survival analysis (DSA) model that was previously shown to be exact. Specifically, we demonstrate that the DSA model is equivalent to the well-known edge-based Volz model. Using this result, we also provide reductions of the closed pairwise and Volz models to a single equation that involves only susceptibles. This equation has a useful statistical interpretation in terms of times to infection. We provide some numerical examples to illustrate our results.

Assessment of the interactions between soil–biosphere–atmosphere (ISBA) land surface model soil hydrology, using four closed-form soil water relationships and several lysimeters

Abstract. Soil water drainage is the main source of groundwater recharge and river flow. It is therefore a key process for water resource management. In this study, we evaluate the soil hydrology and the soil water drainage, simulated by the interactions between soil–biosphere–atmosphere (ISBA) land surface model currently used for hydrological applications from the watershed scale to the global scale, where parameters are generally not calibrated. This evaluation is done using seven lysimeters from two long-term model approach sites measuring hourly water dynamics between 2009 and 2019 in northeastern France. These 2 m depth lysimeters are filled with different soil types and are either maintained as bare soil or covered with vegetation. Four closed-form equations describing soil water retention and hydraulic conductivity functions are tested, namely the commonly used equations from Brooks and Corey (1966) and van Genuchten (1980), a combination of the van Genuchten (1980) soil water retention function with the Brooks and Corey (1966) unsaturated hydraulic conductivity function, and, for the very first time in a land surface model (LSM), a modified version of the van Genuchten (1980) equations, with a new hydraulic conductivity curve proposed by Iden et al. (2015). The results indicate good performance by ISBA with the different closure equations in terms of soil volumetric water content and water mass. The drained flow at the bottom of the lysimeter is well simulated, using Brooks and Corey (1966), while some weaknesses appear with van Genuchten (1980) due to the abrupt shape near the saturation of its hydraulic conductivity function. The mixed form or the new van Genuchten (1980) hydraulic conductivity function from Iden et al. (2015) allows the solving of this problem and even improves the simulation of the drainage dynamic, especially for intense drainage events. The study also highlights the importance of the vertical heterogeneity of the soil hydrodynamic parameters to correctly simulate the drainage dynamic, in addition to the primary influence of the parameters characterizing the shape of the soil water retention function.

Numerical Study on the Waterjet–Hull Interaction of a Free-Running Catamaran

Waterjet–hull interaction is the hot point and research focus in the research of waterjet-propelled crafts. This paper presents numerical studies on the interaction between a waterjet system and a catamaran. Numerical simulations of both bare hull and self-propulsion hull were carried out based on the URANS method. The SST k-ω model is selected for the closure of the URANS equations. The level set method together with the dynamic overset grid approach is used for the simulations. The body force model with the PI speed controller is used to simulate the rotational motion of the rotor in the simulations for the self-propulsion hull. Moreover, uncertainty analyses of the numerical method are conducted to verify the accuracy of the numerical solver. The numerical results of the bare hull and self-propulsion hull are compared in detail, such as the wave pattern, pressure distribution, hull attitude, and so on. The waterjet reduces the pressure on the hull surface near the stern and makes the height of the wave near the stern lower. This leads to a more violent change in hull attitude and the thrust deduction is positive, ranging from 0.1 to 0.2. The energy conversion is analyzed based on the ITTC recommended procedures, which shows the overall efficiency of the waterjet behind the hull is about 0.75~0.8 times the free stream efficiency.

Nonlinear Three-Dimensional Simulations of the Gradient Drift and Secondary Kelvin–Helmholtz Instabilities in Ionospheric Plasma Clouds

A newly developed three-dimensional electrostatic fluid model solving continuity and current closure equations aims to study phenomena that generate ionospheric turbulence. The model is spatially discretized using a pseudo-spectral method with full Fourier basis functions and evolved in time using a four-stage, fourth-order Runge Kutta method. The 3D numerical model is used here to investigate the behavior and evolution of ionospheric plasma clouds. This problem has historically been used to study the processes governing the evolution of the irregularities in the F region of the ionosphere. It has been shown that these artificial clouds can become unstable and structure rapidly (i.e., cascade to smaller scales transverse to the ambient magnetic field). The primary mechanism which causes this structuring of ionospheric clouds is the E×B, or the gradient drift instability (GDI). The persistence and scale sizes of the resulting structures cannot be fully explained by a two-dimensional model. Therefore, we suggest here that the inclusion of three-dimensional effects is key to a successful interpretation of mid-latitude irregularities, as well as a prerequisite for a credible simulation of these processes. We investigate the results of 2D and 3D nonlinear simulations of the GDI and secondary Kelvin–Helmholtz instability (KHI) in plasma clouds for three different regimes: highly collisional (≈200 km), collisional (≈300 km), and inertial (≈450 km). The inclusion of inertial effects permits the growth of the secondary KHI. For the three different regimes, the overall evolution of structuring of plasma cloud occurs on longer timescales in 3D simulations. The inclusion of three-dimensional effects, in particular, the ambipolar potential in the current closure equation, introduces an azimuthal “twist“ about the axis of the cloud (i.e., the magnetic field B). This azimuthal “twist” is observed in the purely collisional regime, and it causes the perturbations to have a non-flute-like character (k‖≠0). However, for the 3D inertial simulations, the cloud rapidly diffuses to a state in which the sheared azimuthal flow is substantially reduced; subsequently, the cloud becomes unstable and structures, by retaining the flute-like character of the perturbations (k‖=0).