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DOE OSTI2016

Multi-Material Closure Model for High-Order Finite Element Lagrangian Hydrodynamics

We present a new closure model for single fluid, multi-material Lagrangian hydrodynamics and its application to high-order finite element discretizations of these equations [1]. The model is general with respect to the number of materials, dimension and space and time discretizations. Knowledge about exact material interfaces is not required. Material indicator functions are evolved by a closure computation at each quadrature point of mixed cells, which can be viewed as a high-order variational generalization of the method of Tipton [2]. This computation is defined by the notion of partial non-instantaneous pressure equilibration, while the full pressure equilibration is achieved by both the closure model and the hydrodynamic motion. Exchange of internal energy between materials is derived through entropy considerations, that is, every material produces positive entropy, and the total entropy production is maximized in compression and minimized in expansion. Results are presented for standard one-dimensional two-material problems, followed by two-dimensional and three-dimensional multi-material high-velocity impact arbitrary Lagrangian–Eulerian calculations. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Achievements and Ongoing Challenges for advanced CFD Boiling Closure Model Development using Physical Insights from Validation-Oriented High-Fidelity Boiling Experiment

The fundamental understanding of the boiling-associated heat transfer, including sliding bubble effect, is still challenging despite the intensive research for decades in both experimental and computational boiling research communities. One of the main difficulties of boiling experiment come from the fact that direct observation of the underlying principle through experiment is very difficult due to the high complexity of the boiling phenomenon. Also, since the heat transfer characteristics associated with boiling is the result of non-linear interaction of various physical parameters coupled to each other, it is challenging to find the relationship between the parameters for modeling. This paper discusses the authors’ past and ongoing boiling research under CASL (Consortium for Advanced Simulation of Light Water Reactors) program, including the efforts to overcome the difficulties of boiling measurement and model development. In particular, we focus on describing the major achievements, lessons learnt, and ongoing challenges for the CFD boiling closure model development based on the high-fidelity subcooled flow boiling experiment at Texas A&M University (TAMU). The major research achievements discussed through this paper include (i) novel boiling measurement strategy and high-fidelity validation data production, (ii) dedicated effort to identify the critical measurement issues in boiling experiment, (iii) new physical insight into sliding bubble heat transfer mechanisms, and (iv) new boiling closure model development. Discussion is also made on the ongoing challenges for the boiling closure model development and the future research plan based on the lessons learnt.

42 ENGINEERING↗

DOE OSTI2021

Initial use of Nek5000/Cardinal to improve closure models in Pronghorn

Heat transfer coefficient closure models for pebble bed reactors are built using a data-driven approach by leveraging the capabilities of an Evolutionary Algorithm entitled Particle Swarm Optimization (PSO). In the present work, the Computational Fluid Dynamics code nekRS was used in order to collect the high-fidelity flow data for a core with 1,568 pebbles. To characterize the heat transfer, multiple concentric regions were considered to extract the physical quantities of interest, e.g./ the Reynolds number. The PSO algorithm is employed as part of an inverse problem targeting determine what are the coefficients for a Nusselt number correlation to match the collected data. Such correlation should follow any given format that is defined a priori. Finally, two correlations are proposed, one with an implicit dependence on the pebbles’ wall temperatures and another expressed as a fully explicit correlation depending on the flow conditions and the position within the core. Anyway, given the generic nature of the proposed approach, correlations following different formats could be tested. Preliminary results for the high-fidelity simulation of a fast MSR core are presented. The target Reynolds number is currently 20K, with the expectation that this will increase, pending the availability of further computational resources. These simulations will be used to inform lower fidelity models, including a coarse CFD turbulence model in Pronghorn. Additionally, they will serve as a reference for the RANS models in Nek5000/NekRS.

22 GENERAL STUDIES OF NUCLEAR REACTORS↗

NASA NTRS1974

Application of an invariant second-order closure model to compressible turbulent shear layers

A second-order closure model for two-dimensional, compressible shear flows is investigated using the invariant modeling technique developed by Donaldson. The invariant model parameters were originally selected by comparison of model predictions with critical experimental data on basic incompressible flows. Additional modeling for compressible flows has been introduced. A number of different shear flows such as the free shear layer, flat plate boundary layer and a simulation of the mixing region of a chemical laser are computed. The model is consistent with first-order closure turbulence models for equilibrium flows and is further capable of predicting nonequilibrium flows that cannot be correctly solved by eddy viscosity models. The results are in generally good agreement with experimental measurements but suggest the need for inclusion of a Mach number dependent model of the pressure diffusion terms in order to adequately represent high speed compressible flows.

Varma, A. K.↗

NASA NTRS1999

Ocean Turbulence I: One-Point Closure Model Momentum and Heat Vertical Diffusivities

Since the early forties, one-point turbulence closure models have been the canonical tools used to describe turbulent flows in many fields. In geophysics, Mellor and Yamada applied such models using the 1980 state-of-the art. Since then, no improvements were introduced to alleviate two major difficulties: 1) closure of the pressure correlations, which affects the correct determination of the critical Richardson number Ri(sub cr) above which turbulent mixing is no longer possible and 2) the need to express the non-local third-order moments (TOM) in terms of lower order moments rather than via the down-gradient approximation as done thus far, since the latter seriously underestimates the TOMs. Since 1) and 2) are still being dealt with adjustable parameters which weaken the credibility of the models, alternative models, not based on turbulence modeling, have been suggested. The aim of this paper is to show that new information, partly derived from the newest 2-point closure model discussed, can be used to solve these shortcomings. The new one-point closure model, which in its simplest form is algebraic and thus simple to implement, is first shown to reproduce a variety of data. Then, it is used in a Ocean-General Circulation Model (O-GCM) where it reproduces well a large variety of ocean data. While phenomenological models are specifically tuned to ocean turbulence, the present model is not. It is first tested against laboratory data on stably stratified flows and then used in an O-GCM. It is more general, more predictive and more resilient, e.g., it can incorporate phenomena like wave-breaking at the surface, salinity diffusivity, non-locality, etc. One important feature that naturally comes out of the new model is that the predicted Richardson critical value Ri(sub cr) is Ri (sub cr approx. = 1) in agreement with both Large Eddy Simulations (LES) and empirical evidence while all previous models predicted Ri (sub cr approx. = 0.2) which led to a considerable underestimate of the extent of turbulent mixing and thus to an incorrect mixed layer depth. The predicted temperature and salinity profiles (vs. depth) are presented and compared with those of the Kolmogorov-Petruvsky-Piskunuv (KPP) model and Levitus data.

Canuto, V. M.↗

NASA NTRS1994

An alternative assessment of second-order closure models in turbulent shear flows

The performance of three recently proposed second-order closure models is tested in benchmark turbulent shear flows. Both homogeneous shear flow and the log-layer of an equilibrium turbulent boundary layer are considered for this purpose. An objective analysis of the results leads to an assessment of these models that stands in contrast to that recently published by other authors. A variety of pitfalls in the formulation and testing of second-order closure models are uncovered by this analysis.

Speziale, Charles G.↗

DOE OSTI2022

Machine learning moment closure models for the radiative transfer equation I: Directly learning a gradient based closure

In this paper, we take a data-driven approach and apply machine learning to the moment closure problem for the radiative transfer equation in slab geometry. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks. This new approach is consistent with the exact closure we derive for the free streaming limit and also provides a natural output normalization. Finally, a variety of benchmark tests, including the variable scattering problem, the Gaussian source problem with both periodic and reflecting boundaries, and the two-material problem, show both good accuracy and generalizability of our machine learning closure model.

Computer science↗

NASA NTRS1995

Working toward a three-dimensional fatigue closure model for surface cracks

The first reliable elastic fracture mechanics solutions for a surface crack in a plate were obtained by Newman and Raju. The authors, both from the Mechanics of Materials Branch at NASA-Langley, used a highly detailed finite element solution requiring substantial computational resources. Computers have since become more powerful and available; however, many important related problems remain computationally expensive. The problem of three-dimensional fatigue crack growth taking into account plasticity-induced crack closure is one such problem. It is the goal of this research to provide an efficient method to account for three-dimensional crack closure in fatigue. Newman developed a two-dimensional plasticity-induced crack closure model for center cracked specimens. This model requires iterations to determine both the contact solution at each growth step and the extent of the plastic zone at the crack tip. A three-dimensional version of this model would require obtaining these nonlinear variables all along the crack front. This model must be efficient so that repeated calculations can be performed during crack growth simulations. The highly versatile line spring model (LSM) with contact, fatigue, and plasticity will form the basis of the closure model. There are several required additions to past work to address the three-dimensional crack closure problem. Initially, these additions will include (1) an improved LSM to more accurately obtain the crack opening displacement, stress intensity factors, and elastic T-stress near the ends of the surface crack; (2) a method to determine the extent of the plastic zone all along the crack front; (3) a method to determine the contact zone given a perfectly plastic layer of material on the crack surfaces; (4) a method to determine the magnitude of the compressive contact stress; and (5) a way to implement the degree of constraint along the curved crack front. During the summer ASEE program an enhanced LSM was developed. A method similar to that of 'strip synthesis' first introduced by Fujimoto was used. Briefly, the crack opening displacements of 'slices' of the surface crack in a direction parallel to the plate surface are considered in addition to the standard LSM approach that makes use of springs obtained from slices perpendicular to the plate surface. This enhancement is necessary so that an accurate three-dimensional representation of quantities such as contact zone size, plastic zone size, stress intensity factors, T-stress, and crack opening displacement can be determined. By combining results of previous investigations with the LSM, the problem of three-dimensional crack closure will be addressed. In addition to a closure model, the enhanced LSM can be used for many other problems including interacting surface cracks and fatigue crack growth of a through crack with a curved crack front.

Joseph, Paul F.↗

NASA NTRS1991

Second-order closure models for supersonic turbulent flows

Recent work on the development of a second-order closure model for high-speed compressible flows is reviewed. This turbulent closure is based on the solution of modeled transport equations for the Favre-averaged Reynolds stress tensor and the solenoidal part of the turbulent dissipation rate. A new model for the compressible dissipation is used along with traditional gradient transport models for the Reynolds heat flux and mass flux terms. Consistent with simple asymptotic analyses, the deviatoric part of the remaining higher-order correlations in the Reynolds stress transport equations are modeled by a variable density extension of the newest incompressible models. The resulting second-order closure model is tested in a variety of compressible turbulent flows which include the decay of isotropic turbulence, homogeneous shear flow, the supersonic mixing layer, and the supersonic flat-plate turbulent boundary layer. Comparisons between the model predictions and the results of physical and numerical experiments are quite encouraging.

Speziale, Charles G.↗

DOE OSTI2016

Fluid simulation of tokamak ion temperature gradient turbulence with zonal flow closure model

Nonlinear fluid simulation of turbulence driven by ion temperature gradient modes in the tokamak fluxtube configuration is performed by combining two different closure models. One model is a gyrofluid model by Beer and Hammett [Phys. Plasmas 3, 4046 (1996)], and the other is a closure model to reproduce the kinetic zonal flow response [Sugama et al., Phys. Plasmas 14, 022502 (2007)]. By including the zonal flow closure, generation of zonal flows, significant reduction in energy transport, reproduction of the gyrokinetic transport level, and nonlinear upshift on the critical value of gradient scale length are observed.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2015

Survey of Multi-Material Closure Models in 1D Lagrangian Hydrodynamics [Poster]

Accurately treating the coupled sub-cell thermodynamics of computational cells containing multiple materials is an inevitable problem in hydrodynamics simulations, whether due to initial configurations or evolutions of the materials and computational mesh. When solving the hydrodynamics equations within a multi-material cell, we make the assumption of a single velocity field for the entire computational domain, which necessitates the addition of a closure model to attempt to resolve the behavior of the multi-material cells’ constituents. In conjunction with a 1D Lagrangian hydrodynamics code, we present a variety of both the popular as well as more recently proposed multi-material closure models and survey their performances across a spectrum of examples. We consider standard verification tests as well as practical examples using combinations of fluid, solid, and composite constituents within multi-material mixtures. Our survey provides insights into the advantages and disadvantages of various multi-material closure models in different problem configurations.

42 ENGINEERING↗

NASA NTRS1992

Second order closure modeling of turbulent buoyant wall plumes

Non-intrusive measurements of scalar and momentum transport in turbulent wall plumes, using a combined technique of laser Doppler anemometry and laser-induced fluorescence, has shown some interesting features not present in the free jet or plumes. First, buoyancy-generation of turbulence is shown to be important throughout the flow field. Combined with low-Reynolds-number turbulence and near-wall effect, this may raise the anisotropic turbulence structure beyond the prediction of eddy-viscosity models. Second, the transverse scalar fluxes do not correspond only to the mean scalar gradients, as would be expected from gradient-diffusion modeling. Third, higher-order velocity-scalar correlations which describe turbulent transport phenomena could not be predicted using simple turbulence models. A second-order closure simulation of turbulent adiabatic wall plumes, taking into account the recent progress in scalar transport, near-wall effect and buoyancy, is reported in the current study to compare with the non-intrusive measurements. In spite of the small velocity scale of the wall plumes, the results showed that low-Reynolds-number correction is not critically important to predict the adiabatic cases tested and cannot be applied beyond the maximum velocity location. The mean and turbulent velocity profiles are very closely predicted by the second-order closure models. but the scalar field is less satisfactory, with the scalar fluctuation level underpredicted. Strong intermittency of the low-Reynolds-number flow field is suspected of these discrepancies. The trends in second- and third-order velocity-scalar correlations, which describe turbulent transport phenomena, are also predicted in general, with the cross-streamwise correlations better than the streamwise one. Buoyancy terms modeling the pressure-correlation are shown to improve the prediction slightly. The effects of equilibrium time-scale ratio and boundary condition are also discussed.

Zhu, Gang↗

NASA NTRS1991

Second-order closure models for supersonic turbulent flows

Recent work by the authors on the development of a second-order closure model for high-speed compressible flows is reviewed. This turbulence closure is based on the solution of modeled transport equations for the Favre-averaged Reynolds stress tensor and the solenoidal part of the turbulent dissipation rate. A new model for the compressible dissipation is used along with traditional gradient transport models for the Reynolds heat flux and mass flux terms. Consistent with simple asymptotic analyses, the deviatoric part of the remaining higher-order correlations in the Reynolds stress transport equation are modeled by a variable density extension of the newest incompressible models. The resulting second-order closure model is tested in a variety of compressible turbulent flows which include the decay of isotropic turbulence, homogeneous shear flow, the supersonic mixing layer, and the supersonic flat-plate turbulent boundary layer. Comparisons between the model predictions and the results of physical and numerical experiments are quite encouraging.

Speziale, Charles G.↗

DOE OSTI2013

Simulation of Alfv n frequency cascade modes in reversed shear discharges using a Landau-closure model

The dynamics of energetic particle destabilized Alfve n frequency sweeping modes in tokamak reversed-shear safety factor discharges are modelled using a new Landau-closure model that includes coupling to geodesic acoustic wave dynamics and closure relations optimized for energetic particle Alfve n mode resonances. Profiles and equilibria are based upon reconstructions of a DIII-D discharge (#142111) in which a long sequence of frequency sweeping modes were observed. This model (TAEFL) has recently been included in a verification and validation study of n = 3 frequency sweeping modes for this case along with two gyrokinetic codes, GTC and GYRO. This paper provides a more detailed documentation of the equations and methods used in the TAEFL model and extends the earlier calculation to a range of toroidal mode numbers: n = 2 to 6. By considering a range of toroidal mode numbers and scanning over a range of safety factor profiles with varying qmin, both up-sweeping frequency (reversed-shear Alfve n eigenmode) and down-sweeping frequency (toriodal Alfve n eigenmode) modes are present in the results and show qualitative similarity with the frequency variations observed in the experimental spectrograms.

shear-Alfven instabiltiy tokamak Landau-closure Al↗

NASA NTRS1992

Systematic study of Reynolds stress closure models in the computations of plane channel flows

The roles of pressure-strain and turbulent diffusion models in the numerical calculation of turbulent plane channel flows with second-moment closure models are investigated. Three turbulent diffusion and five pressure-strain models are utilized in the computations. The main characteristics of the mean flow and the turbulent fields are compared against experimental data. All the features of the mean flow are correctly predicted by all but one of the Reynolds stress closure models. The Reynolds stress anisotropies in the log layer are predicted to varying degrees of accuracy (good to fair) by the models. None of the models could predict correctly the extent of relaxation towards isotropy in the wake region near the center of the channel. Results from the directional numerical simulation are used to further clarify this behavior of the models.

Demuren, A. O.↗

NASA NTRS1993

Perspective - Systematic study of Reynolds stress closure models in the computations of plane channel flows

Demuren, A. O.↗

NASA NTRS1984

Computation and turbulence closure models for shear flows in rotating curved bodies

This paper is concerned with the effects of rotation and curvature on the turbulence closure model. The k-epsilon model that includes the effects of curvature has been modified to include the anisotropy in turbulence. A space-marching algorithm which solves the parabolized Navier-Stokes equations is coupled with the turbulence closure model to predict the turbulent shear flow over a rotating cylinder and inside a rotating channel. The agreement between the predictions and available data is good. The rotation and curvature effects are predicted well.

Pouagare, M.↗

NASA NTRS1987

Second-order closure models for rotating turbulent flows

The physical properties of the commonly used second-order closure models are examined theoretically for rotating turbulent flows. Comparisons are made with results which are a rigorous consequence of the Navier-Stokes equations for the problem of fully-developed turbulent channel flow in a rapidly rotating framework. It is demonstrated that all existing second-order closures yield spurious physical results for this test problem of rotating channel flow. In fact, the results obtained are shown to be substantially more unphysical than those obtained from the simpler k-epsilon and k-l models. Modifications in the basic structure of these second-order closure models are proposed which can alleviate this problem.

Speziale, Charles G.↗