Distributed Source Term Research Articles

Source Term-Based Turbulent Flow Simulation on GPU with Link-Wise Artificial Compressibility Method

We present a GPU-based turbulent flow simulation by link-wise artificial compressibility method (LW-ACM). The standard implementations of the lattice Boltzmann method are limited by memory requirements due to the nature of the distribution functions employed. LW-ACM avoids the need to store the density distribution function via the use of a hybrid of LBM and finite difference method. This method, previously used only for simple cases without inlet/outlet boundary conditions, is here extended for general-purpose 3D turbulent flow via the introduction of the synthetic eddy method (SEM) as a distributed source term into the channel. A channel flow is performed to validate the implementation in this paper. Experimental results demonstrate performance on a single GPU of up to 11237 MLUPS and 4656 MLUPS in single and double precision, respectively, amongst the fastest results reported to date, demonstrating the practical opportunities this approach can offer for systematic evaluation of complex turbulent flow.

An Eulerian Single-Phase Transport Model for Solid Fission Products in the Molten Salt Fast Reactor: Development of an Analytical Solution for Verification Purposes

Nuclear reactor modeling has been shifting, over the last decades, towards full-core multiphysics analysis due to the ever-increasing safety requirements and complexity of the designs of innovative systems. This is particularly true for liquid-fuel reactor concepts such as the Molten Salt Fast Reactor (MSFR), given their strong intrinsic coupling between thermal-hydraulics, neutronics and fuel chemistry. In the MSFR, fission products (FPs) are originated within the liquid fuel and are carried by the fuel flow all over the reactor core and through pumping and heat exchange systems. Some of FP species, in the form of solid precipitates, can represent a major design and safety challenge, e.g., due to deposition on solid boundaries, and their distribution in the core is relevant to the design and safety analysis of the reactor. In this regard it is essential, both for the design and the safety assessment of the reactor, the capability to model the transport of solid FPs and their deposition to the boundary (e.g., wall or heat exchanger structures). To this aim, in this study, models of transport of solid FPs in the MSFR are developed and verified. An Eulerian single-phase transport model is developed and integrated in a consolidated multiphysics model of the MSFR based on the open-source CFD library OpenFOAM. In particular, general mixed-type deposition boundary conditions are considered, to possibly describe different kinds of particle-wall interaction mechanisms. For verification purposes, analytical solutions for simple case studies are derived ad hoc based on the extension of the classic Graetz problem to linear decay, distributed source terms and mixed-type boundary conditions. The results show excellent agreement between the two models, and highlight the effects of decay and deposition phenomena of various intensity. The resulting approach constitutes a computationally efficient tool to extend the capabilities of CFD-based multiphysics MSFR calculations towards the simulation of solid fission products transport.

A new and consistent well model for one-phase flow in anisotropic porous media using a distributed source model

A new well model for one-phase flow in anisotropic porous media is introduced, where the mass exchange between well and a porous medium is modeled by spatially distributed source terms over a small neighborhood region. To this end, we first present a compact derivation of the exact analytical solution for an arbitrarily oriented, infinite well cylinder in an infinite porous medium with anisotropic permeability tensor in R3, for constant well pressure and a given injection rate, using a conformal map. The analytical solution motivates the choice of a kernel function to distribute the sources. The presented model is independent from the discretization method and the choice of computational grids. It can be applied to porous media with full anisotropic permeability tensors. In numerical experiments, the new well model is shown to be consistent and robust with respect to rotation of the well axis, rotation of the permeability tensor, and different anisotropy ratios. Finally, a comparison with a Peaceman-type well model suggests that the new scheme leads to an increased accuracy for injection (and production) rates for arbitrarily-oriented pressure-controlled wells.

A Discretization-Independent Distributed Well Model

Summary In this paper, we present a new well model for reservoir simulation. The proposed well model relates the volumetric flow rate and the bottomhole pressure (BHP) of the well to the reservoir pressure through a spatially distributed source term that is independent of the numerical method and the discrete mesh used to solve the flow problem. This is in contrast to the widely used Peaceman–type well models, which are inherently tied to a particular numerical discretization by the definition of an equivalent well radius. The proposed distributed well model does not require the calculation of an equivalent well radius. Hence, it can be readily applied to finite–difference, finite–volume (FV), or finite–element discretizations on arbitrarily unstructured meshes, which also makes it an attractive option for mesh–adaptation schemes. The new well model is demonstrated on a steady-state single-phase flow problem and an unsteady two-phase flow problem, using a conventional FV method and a high–order discontinuous Galerkin (DG) method. The distributed well model produces error–convergence behaviors that are very similar to the Peaceman well model on uniform structured meshes, but its applicability to high–order discretizations and mesh–adaptation schemes allows for higher convergence rates and more cost-efficient solutions, especially on adapted unstructured meshes.

Avoiding stick slip vibrations in drilling through startup trajectory design

A distributed model of a drill string with a collars section is presented with Coulomb friction as a distributed source term. This model is capable of replicating stick slip oscillations as caused by the reduction in friction from static to dynamic. We design a feed-forward startup trajectory for initiating rotation of the drill string that effectively avoids the stick slip limit cycle. The trajectory design is performed using the differential flatness of the bit angular velocity, and by treating the reduction from static to dynamic friction as an estimated disturbance to be canceled, thus conforming to the canonical 3-DOF controller design for tracking and disturbance rejection. A simulation study illustrates the feasibility of the approach.

Fan Performance Scaling With Inlet Distortions

Applications such as boundary-layer-ingesting (BLI) fans and compressors in turboprop engines require continuous operation with distorted inflow. A low-speed axial fan with incompressible flow is studied in this paper. The objectives are to (1) identify the physical mechanisms which govern the fan response to inflow distortions and (2) determine how fan performance scales as the type and severity of inlet distortion varies at the design flow coefficient. A distributed source term approach to modeling the rotor and stator blade rows is used in numerical simulations in this paper. The model does not include viscous losses so that changes in diffusion factor are the primary focus. Distortions in stagnation pressure and temperature as well as swirl are considered. The key findings are that unless sharp pitchwise gradients in the diffusion response, strong radial flows, or very large distortion magnitudes are present, the response of the blade rows for strong distortions can be predicted by scaling up the response to a weaker distortion. In addition, the response to distortions which are composed of nonuniformities in several inlet quantities can be predicted by summing up the responses to the constituent distortions.

Benchmarking of Industrial Stick-Slip Mitigation Controllers

The present paper evaluates how three different top-drive feedback controllers influence the occurrence of a stick-slip limit cycle in a rotating drill string. The considered controllers are: 1) The industry standard stiff, high-gain controller, 2) SoftTorque, and, 3) ZTorque. The evaluation is performed as a simulation study on a distributed model of a drill string with a collar section and Coulomb friction as a distributed source term. This model is capable of replicating stick-slip oscillations as caused by the reduction in friction from static to dynamic, and have been shown to yield a good match with field data. The simulation study is summarized as a map for each controller indicating the existence and amplitude of oscillations, parametrized in the key friction parameters.

Torsional vibrations with bit off bottom: Modeling, characterization and field data validation

A distributed model of a drill string is presented, with Coulomb stiction as a distributed source term, to investigate the effect of borehole inclination and borehole friction on the incidence of torsional vibrations along a drill-string. To isolate the effect of this distributed friction, only cases where the bit is off bottom and with no axial movement are considered, and consequently a purely torsional model is used. The model is used to study the stick slip limit cycle, as caused by distributed Coulomb friction, and the limit-cycle period and amplitude dependence on the friction parameters is derived. This enables the qualitative limit-cycle behavior to be characterized as inertia or stick dominated, and examples of this characterization is validated with the field data. For comparison, high frequency field data from two deviated wells from surface and downhole sensors are considered. Time-series are isolated where angular rotation is restarted after a connection, with the bit off bottom and before axial motion is re-initiated, to make the data consistent with the model assumptions. A close match is obtained between recorded and modeled downhole data, however, in some cases either the initial torsional strain in the drillstring is not known or axial motion is initiated, which violates model assumptions, causing a mismatch.

Low-Order Modeling of Nonlinear High-Frequency Transversal Thermoacoustic Oscillations in Gas Turbine Combustors

This paper analyzes transversal thermoacoustic oscillations in an experimental gas turbine combustor utilizing dynamical system theory. Limit-cycle acoustic motions related to the first linearly unstable transversal mode of a given 3D combustor configuration are modeled and reconstructed by means of a low-order dynamical system simulation. The source of nonlinearity is solely allocated to flame dynamics, saturating the growth of acoustic amplitudes, while the oscillation amplitudes are assumed to always remain within the linearity limit. First, a reduced order model (ROM) which reproduces the combustor's modal distribution and damping of acoustic oscillations is derived. The ROM is a low-order state-space system, which results from a projection of the linearized Euler equations (LEE) into their truncated eigenspace. Second, flame dynamics are modeled as a function of acoustic perturbations by means of a nonlinear transfer function. This function has a linear and a nonlinear contribution. The linear part is modeled analytically from first principles, while the nonlinear part is mathematically cast into a cubic saturation functional form. Additionally, the impact of stochastic forcing due to broadband combustion noise is included by additive white noise sources. Then, the acoustic and the flame system is interconnected, where thermoacoustic noncompactness due to the transversal modes' high frequency (HF) is accounted for by a distributed source term framework. The resulting nonlinear thermoacoustic system is solved in frequency and time domain. Linear growth rates predict linear stability, while envelope plots and probability density diagrams of the resulting pressure traces characterize the thermoacoustic performance of the combustor from a dynamical systems theory perspective. Comparisons against experimental data are conducted, which allow the rating of the flame modes in terms of their capability to reproduce the observed combustor dynamics. Ultimately, insight into the physics of high-frequency, transversal thermoacoustic systems is created.

Symplectic spatial integration schemes for systems of balance equations

A method to generate geometric pseudo-spectral spatial discretization schemes for hyperbolic or parabolic partial differential equations is presented. It applies to the spatial discretization of systems of conservation laws with boundary energy flows and/or distributed source terms. The symplecticity of the proposed spatial discretization schemes is defined with respect to the natural power pairing (form) used to define the port-Hamiltonian formulation for the considered systems of balance equations. The method is applied to the resistive diffusion model, a parabolic equation describing the plasma dynamics in tokamaks. A symplectic Galerkin scheme with Bessel conjugated bases is derived from the usual Galerkin method, using the proposed method. Besides the spectral and energetic properties expected from the symplecticity of the method, it is shown that more accurate approximation of eigenfunctions and reduced numerical oscillations result from this choice of conjugated approximation bases. Finally, the obtained numerical results are validated against experimental data from the tokamak Tore Supra facility.

Retraction statement: Adaptive distributed parameter and input estimation in linear parabolic PDEs

Summary In this paper, we discuss the on-line estimation of distributed source term, diffusion, and reaction coefficients of a linear parabolic partial differential equation using both distributed and interior-point measurements. First, new sufficient identifiability conditions of the input and the parameter simultaneous estimation are stated. Then, by means of Lyapunov-based design, an adaptive estimator is derived in the infinite-dimensional framework. It consists of a state observer and gradient-based parameter and input adaptation laws. The parameter convergence depends on the plant signal richness assumption, whereas the state convergence is established using a Lyapunov approach. The results of the paper are illustrated by simulation on tokamak plasma heat transport model using simulated data. Copyright © 2016 John Wiley & Sons, Ltd.

Nonuniform Multiconductor Transmission Line Analysis by a Two-Step Perturbation Technique

A two-step perturbation technique to model nonuni- form multiconductor transmission lines in the frequency domain is presented. In this method, nonuniformities are treated as per- turbations with respect to the nominal uniform multiconductor line. Starting from the Telegrapher's equations and applying two consecutive perturbations steps, at each step, we obtain second- order ordinary differential equations with distributed source terms. Solving these equations together with the appropriate boundary conditions provides the sought-for voltages and cur- rents along the interconnect structure. The method is validated by means of a frequency domain analysis of a ten-conductor microstrip line with random uniformities, confirming its accu- racy and efficiency. Additionally, the time-domain accuracy and efficiency is demonstrated by means of a high-speed packaging nonuniform interconnect with six signal conductors.

Production of particulate Composition B during simulated weathering of larger detonation residues

Explosives and energetics continue to be prominent contaminants on many military installations. This research was undertaken to understand the extent to which microscale (10's of μm) particles are produced when macroscale residues are weathered by artificial precipitation. Initial experiments, in which artificial rainwater was applied drip-wise to single chunks of Composition B detonation residues from multiple heights, confirmed that microscale particles were produced during precipitation-driven aging, with 30% of the explosive mass collected detected as particulate Composition B (e.g., particles >0.45μm in diameter). Follow-on experiments, during which multiple cm-sized residue chunks were subjected to realistic simulated precipitation, demonstrated an initial large pulse of particulate Composition B, followed by sustained production of microscale particles that represented 15–20% of recovered explosives. These findings indicate that the effective footprint of detonation residues likely increases as particulates are produced by the production and spreading of microscale particles across the soil surface. Combined with results published elsewhere that microscale particles can move into porous media to become a distributed source term, these findings point to the need for inclusion of these processes in explosive contaminant fate and transport modeling.

LES-pdf simulation of a spark ignited turbulent methane jet

A spark ignited turbulent jet flame is simulated using Large Eddy Simulation (LES) in conjunction with the filtered probability density function (pdf) equation approach, which is solved using the Eulerian stochastic field method. The spark energy deposition is mimicked by a gaussian distributed source term in the enthalpy equation. A dynamic model for the sub-grid stresses together with a simple gradient diffusion approximation for the scalar fluxes is adopted. The chemistry is represented by a global 4-step chemistry involving seven species. The different stages of the ignition sequence, namely the flame kernel growth, the triple flame propagation against the flow and the stabilisation were in good agreement with the experimental data.

On the generation of exact solutions for evaluating numerical schemes and estimating discretization error

In this paper we further develop the Method of Nearby Problems (MNP) for generating exact solutions to realistic partial differential equations by extending it to two dimensions. We provide an extensive discussion of the 2D spline fitting approach which provides C k continuity (continuity of the solution value and its first k derivatives) along spline boundaries and is readily extendable to higher dimensions. A detailed one-dimensional example is given to outline the general concepts, then the two-dimensional spline fitting approach is applied to two problems: heat conduction with a distributed source term and the viscous, incompressible flow in a lid-driven cavity with both a constant lid velocity and a regularized lid velocity (which removes the strong corner singularities). The spline fitting approach results in very small spline fitting errors for the heat conduction problem and the regularized driven cavity, whereas the fitting errors in the standard lid-driven cavity case are somewhat larger due to the singular behaviour of the pressure near the driven lid. The MNP approach is used successfully as a discretization error estimator for the driven cavity cases, outperforming Richardson extrapolation which requires two grid levels. However, MNP has difficulties with the simpler heat conduction case due to the discretization errors having the same magnitude as the spline fitting errors.

Some results on the accuracy of an edge‐based finite volume formulation for the solution of elliptic problems in non‐homogeneous and non‐isotropic media

AbstractThe numerical simulation of elliptic type problems in strongly heterogeneous and anisotropic media represents a great challenge from mathematical and numerical point of views. The simulation of flows in non‐homogeneous and non‐isotropic porous media with full tensor diffusion coefficients, which is a common situation associated with the miscible displacement of contaminants in aquifers and the immiscible and incompressible two‐phase flow of oil and water in petroleum reservoirs, involves the numerical solution of an elliptic type equation in which the diffusion coefficient can be discontinuous, varying orders of magnitude within short distances. In the present work, we present a vertex‐centered edge‐based finite volume method (EBFV) with median dual control volumes built over a primal mesh. This formulation is capable of handling the heterogeneous and anisotropic media using structured or unstructured, triangular or quadrilateral meshes. In the EBFV method, the discretization of the diffusion term is performed using a node‐centered discretization implemented in two loops over the edges of the primary mesh. This formulation guarantees local conservation for problems with discontinuous coefficients, keeping second‐order accuracy for smooth solutions on general triangular and orthogonal quadrilateral meshes. In order to show the convergence behavior of the proposed EBFV procedure, we solve three benchmark problems including full tensor, material heterogeneity and distributed source terms. For these three examples, numerical results compare favorably with others found in literature. A fourth problem, with highly non‐smooth solution, has been included showing that the EBFV needs further improvement to formally guarantee monotonic solutions in such cases. Copyright © 2008 John Wiley & Sons, Ltd.

Active shielding model for hyperbolic equations

The problem of active shielding (AS) in application to hyperbolic equations is analysed. According to the problem, two domains effecting each other via distributed source terms are considered. It is required to implement additional sources nearby the common boundary of the domains in order to isolate one domain from the action of the other domain. It is important to note that the total field of the original sources is only known. In the paper, the theory of difference potentials is applied to the system of hyperbolic equations for the first time. It allows one to obtain a one-layer AS not requiring any additional computations. Local one-layer and two-layer AS sources are obtained for an arbitrary hyperbolic system. The solution does not require either the knowledge of the Green's function or the specific characteristics of the sources and medium. The optimal one-layer AS solution is derived in the case of free space. In particular, the results are applicable to the system of acoustics equations. The questions related to a practical realization including the mutual situation of the primary and secondary sources, as well as the measurement point, are discussed. The active noise shielding can be realized via a one-layer source term requiring the measurements only at one layer nearby the domain shielded.

Readdressing the Issue of Thermally Significant Blood Vessels Using a Countercurrent Vessel Network

A physiologically realistic arterio-venous countercurrent vessel network model consisting of ten branching vessel generations, where the diameter of each generation of vessels is smaller than the previous ones, has been created and used to determine the thermal significance of different vessel generations by investigating their ability to exchange thermal energy with the tissue. The temperature distribution in the 3D network (8178 vessels; diameters from 10 to 1000 microm) is obtained by solving the conduction equation in the tissue and the convective energy equation with a specified Nusselt number in the vessels. The sensitivity of the exchange of energy between the vessels and the tissue to changes in the network parameters is studied for two cases; a high temperature thermal therapy case when tissue is heated by a uniformly distributed source term and the network cools the tissue, and a hypothermia related case, when tissue is cooled from the surface and the blood heats the tissue. Results show that first, the relative roles of vessels of different diameters are strongly determined by the inlet temperatures to those vessels (e.g., as affected by changing mass flow rates), and the surrounding tissue temperature, but not by their diameter. Second, changes in the following do not significantly affect the heat transfer rates between tissue and vessels; (a) the ratio of arterial to venous vessel diameter, (b) the diameter reduction coefficient (the ratio of diameters of successive vessel generations), and (c) the Nusselt number. Third, both arteries and veins play significant roles in the exchange of energy between tissue and vessels, with arteries playing a more significant role. These results suggest that the determination of which diameter vessels are thermally important should be performed on a case-by-case, problem dependent basis. And, that in the development of site-specific vessel network models, reasonable predictions of the relative roles of different vessel diameters can be obtained by using any physiologically realistic values of Nusselt number and the diameter reduction coefficient.

A viscous-acoustic boundary element formulation based on analytical-numerical matching

Analytical-numerical matching (ANM) is an analysis scheme that combines a low-resolution global numerical solution with a high-resolution local analytical solution and a matching solution to form a uniformly valid composite solution. The application of ANM to harmonically oscillating bodies in a fluid leads to a novel reformulation of boundary element problems in acoustics and aeroacoustics. The singular kernal encountered in the integral equation is handled simply within the analytical local solution. The numerical implementation utilizes a smoothed Green’s function solution to the governing equation with a distributed source term. Because the singular behavior has been removed from the numerical aspect of the problem, the approach converges rapidly and exhibits insensitivity to node (control point) location. The method allows low-resolution numerics to be combined with analytical corrections to obtain high accuracy solutions using a robust calculation scheme. The method has recently been extended to include viscous effects within the local solution. The appropriately modified global solution remains irrotational and can be expressed in terms of a smoothed potential. Sample results are shown for radiation from plates up to high frequencies. Ongoing research on ANM BEM is described, including work to include nonlinear convection within the viscous flow effects.

Nonlinear convection-dispersion models with a distributed pollutant source I: Direct initial boundary value problems

This paper deals with the modeling and solution of a class of nonlinear direct problems related to a transport diffusion model with a source term. Specifically, the first part of the paper deals with the derivation of a class of transport and diffusion models (with a distributed source term) in one space dimensions with variable properties along the channel and nonlinear decay term. The second part with simulations, that is the approximation to the solution of nonlinear initial boundary value problems by generalized collocation methods. The third part develops a critical analysis mainly addressed to research perspectives on the solution of inverse problems related to the identification of the source term.