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Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 145 records · Page 8
All sourcesNASA NTRSDOE OSTI
NASA NTRS1978

Interaction between step fuel injectors on opposite walls in a supersonic combustor model

Results are presented from an experimental investigation of perpendicular, hydrogen fuel injection and combustion from opposing walls in a scramjet combustor model using a longitudinally staged laterally inline step-injection configuration. The model represents a portion of the flow in the Langley integrated modular scramjet engine combustor operating at a flight Mach number of 7. When operating at a ratio of jet pressure to free-stream dynamic pressure of 3, the injectors produce a bulk equivalence ratio of unity. This investigation represents part of a continuing study of the modular engine fuel injectors and is specifically designed to eliminate the adverse lateral pressure gradient observed at the injector location in a previous test. Flow survey contours at three axial locations, ranging from one-third of the engine combustor length to the total engine combustor length, are used to determine mixing efficiency and fuel distribution. Wall static pressures are analyzed by using one-dimensional theory to determine the combustion efficiency. Results show a significant improvement over previous injector designs tested in this duct geometry.

Mcclinton, C. R.↗
NASA NTRS2003

Minimizing Segregation during the Controlled Directional Solidification of Dendric Alloys

Gravity-driven convection induced in the liquid by density gradients of temperature or composition disrupts uniform dendritic growth during controlled directional solidification and promotes severe macrosegregation. The solute-rich region about the dendrite tip appears to play a pivotal role in channel initiation. Allen and Hunt referred to this region as an "initial transient" or dynamic region constituting steep concentration gradients. Experimental investigation also point to the role the tip region plays in developing microstructure. Hellawell and co-workers showed that flow-through dendritic channels could be effectively disrupted, and segregation minimized, during the gradient freezing of bulk castings by rotating the melt through a slight angle with respect to Earth's gravity vector. Adapting this principle to controlled directional solidification, it has been shown" that segregation in dendritic alloys can be minimized, and properties improved, by processing the sample near horizontal in conjunction with a slow axial rotation of the crucible. It is postulated that the observed microstructural uniformity arises by maintaining the developing solute field about the dendrite tip. Solute rejected during vertical directional solidification will rise or sink parallel to the primary dendrite arms during axial rotation setting the stage for accumulation, instabilities, and segregation. In contrast, during horizontal growth, the rejected solute will sink or rise perpendicular to the primary dendrite. Now, in the presence of a slight axial rotation, solute that was initially sinking (or rising) will find itself above (or below) its parent dendrite, i.e., still about the tip region. The following is intended to experimentally demonstrate the viability of this concept in coordination with a model that gives predictive insight regarding solute distribution about growing dendrites. Alloys based on the lead-tin eutectic system were used in this study. The system is well characterized, the constituent metals are available in a very pure form, and the thermophysical properties are well known. During solidification of hypoeutectic alloys, e.g., 55 wt pct Pb, the primary dendrites reject the less dense tin, and for the hypereutectic alloys, e.g., 75 wt pct Sn, the primary dendrites reject denser lead. Alloys were prepared by melting appropriate amounts of lead and tin in a glass crucible after which the homogeneous liquid was sucked directly into 5-mm i.d. glass tubes. The sample tube, containing approximately 30 cm of alloy, was then mechanically driven into the directional solidification furnace assembly and positioned such that approx. 20 cm of the sample was remelted. Subsequently, directional solidification was initiated by withdrawing the sample through a water-cooled jacket at a constant growth velocity of 2 ,microns/s. After 5 to 6 cm of growth, the sample was quickly removed from the furnace and quenched in a water bath to preserve the solid-liquid interface. Samples were directionally solidified vertically upward, nearly horizontally, and some in conjunction with an applied axial rotation of the crucible. Temperature gradients at the solid-liquid interface were measured with an in-siru K-type thermocouple. Solidified samples were cut perpendicular and parallel to the growth direction and conventionally prepared for microscopic examination.

Grugel, Richard N.↗
DOE OSTI2024

Prandtl number effects on extreme mixing events in forced stratified turbulence

Relatively strongly stratified turbulent flows tend to self-organise into a ‘layered anisotropic stratified turbulence’ (LAST) regime, characterised by relatively deep and well-mixed density ‘layers’ separated by relatively thin ‘interfaces’ of enhanced density gradient. Understanding the associated mixing dynamics is a central problem in geophysical fluid dynamics. It is challenging to study LAST mixing, as it is associated with Reynolds numbers $Re := UL/\nu \gg 1$ and Froude numbers $Fr :=(2{\rm \pi} U)/(L N) \ll 1$ ( $U$ and $L$ being characteristic velocity and length scales, $\nu$ the kinematic viscosity and $N$ the buoyancy frequency). Since a sufficiently large dynamic range (largely) unaffected by stratification and viscosity is required, it is also necessary for the buoyancy Reynolds number $Re_{b} := \epsilon /(\nu N^{2}) \gg 1$ , where $\epsilon$ is the (appropriately volume-averaged) turbulent kinetic energy dissipation rate. This requirement is exacerbated for oceanically relevant flows, as the Prandtl number $Pr := \nu /\kappa = {O}(10)$ in thermally stratified water (where $\kappa$ is the thermal diffusivity), thus leading (potentially) to even finer density field structures. We report here on four forced fully resolved direct numerical simulations of stratified turbulence at various Froude ( $Fr=0.5, 2$ ) and Prandtl ( $Pr=1, 7$ ) numbers forced so that $Re_{b}=50$ , with resolutions up to $30\,240 \times 30\,240 \times 3780$ . We find that, as $Pr$ increases, emergent ‘interfaces’ become finer and their contribution to bulk mixing characteristics decreases at the expense of the small-scale density structures populating the well-mixed ‘layers’. However, extreme mixing events (as quantified by significantly elevated local destruction rates of buoyancy variance $\chi _0$ ) are always preferentially found in the (statically stable) interfaces, irrespective of the value of $Pr$ .

Mechanics↗
NASA NTRS2005

The Planetesimal Bow Shock Model for Chondrule Formation: More Detailed Simulations in the Near Vicinity of the Planetesimal

Gas dynamic shock waves in a low temperature nebula have been considered to be a leading candidate mechanism for providing the repetitive, short-duration heating events that are believed to have been responsible for the formation of chondrules in chondrites. It has been found, for example, that shocks with Mach numbers greater than 4 or 5 would be capable of rapidly melting 0.1-1 mm sized silicate particles as required by meteoritic data. Near the nebula midplane where chondrite parent bodies are believed to have formed, possible energy sources for generating multiple shocks include mass concentrations in a gravitationally unstable nebula, tidal interactions of proto-Jupiter with the nebula, and bow waves upstream of planetesimals scattered gravitationally into eccentric orbits by proto- Jupiter. In a recent study, we have found that chondrule precursors that are melted following passage through a planetesimal bow shock would likely cool at rates that are too rapid to be consistent with meteoritic evidence. However, that study was limited to the bowshock exterior to about 1.5 planetesimal radii (measured perpendicular to the symmetry axis) to avoid complications interior to this distance where large pressure gradients and lateral flow occur as the gas flows around the planetesimal. In this paper, we reconsider the planetesimal bow shock model and report more detailed numerical simulations of chondrule precursor heating, cooling, and dynamical histories in the near vicinity of a representative planetesimal.

Hood, Lon L.↗
DOE OSTI2024

Measurement of pressure gradients near the interface in the viscous fingering instability

The viscous fingering instability, which forms when a less-viscous fluid invades a more-viscous one within a confined geometry, is an iconic system for studying pattern formation. For both miscible and immiscible fluid pairs the growth dynamics change after the initial instability onset and the global structures, typical of late-time growth, are governed by the viscosity ratio. Here we introduce an experimental technique to measure flow throughout the inner and outer fluids. This probes the existence of a new length scale associated with the local pressure gradients around the interface and allows us to compare our results to the predictions of a previously proposed model for late-time finger growth. Published by the American Physical Society 2024

Physics↗
NASA NTRS1994

Nonlinear stability of oscillatory core-annular flow: A generalized Kuramoto-Sivashinsky equation with time periodic coefficients

In this paper the nonlinear stability of two-phase core-annular flow in a pipe is examined when the acting pressure gradient is modulated by time harmonic oscillations and viscosity stratification and interfacial tension is present. An exact solution of the Navier-Stokes equations is used as the background state to develop an asymptotic theory valid for thin annular layers, which leads to a novel nonlinear evolution describing the spatio-temporal evolution of the interface. The evolution equation is an extension of the equation found for constant pressure gradients and generalizes the Kuramoto-Sivashinsky equation with dispersive effects found by Papageorgiou, Maldarelli & Rumschitzki, Phys. Fluids A 2(3), 1990, pp. 340-352, to a similar system with time periodic coefficients. The distinct regimes of slow and moderate flow are considered and the corresponding evolution is derived. Certain solutions are described analytically in the neighborhood of the first bifurcation point by use of multiple scales asymptotics. Extensive numerical experiments, using dynamical systems ideas, are carried out in order to evaluate the effect of the oscillatory pressure gradient on the solutions in the presence of a constant pressure gradient.

Coward, Adrian V.↗
NASA NTRS2018

PIV and Rotational Raman-Based Temperature Measurements for CFD Validation in a Single Injector Cooling Flow

NASA's Turbulent Heat Flux (THX) task under the Transformational Tools & Technologies project is focused on acquiring benchmark temperature and velocity data in turbulent shear layers for validation of computational fluid dynamics (CFD) codes. Standard CFD turbulence models lack the ability to accurately calculate a number of fundamental flow phenomena, including the turbulent transport of heat. In nearly all production class Reynolds-averaged Navier-Stokes (RANS) CFD codes, a gradient-diffusion approximation is used whereby a constant turbulent Prantdl number, Prt, is used to relate an eddy viscosity calculated for the momentum terms, to the turbulent heat flux terms. The default in most CFD codes is Prt = 0.9, however this is not representative of all flows. As discussed in Yoder et al. (2015) and Reynolds (1974), Prt = 0.7 is felt to be more appropriate for jets. Recent work, as discussed in Yoder (2016a), has explored more complex formulations including variable turbulent Prantdl number models. PARTICLE IMAGE VELOCIMETRY; TEMPERATURE MEASURING INSTRUMENTS; FILM COOLING; HIGH TEMPERATURE GASES; COMPUTATIONAL FLUID DYNAMICS; PREDICTION ANALYSIS TECHNIQUES; TURBULENCE MODELS; RAMAN SPECTRA; VELOCITY; MACH NUMBER; GAS TURBINE ENGINES; NOZZLE FLOW; THERMOCOUPLES; LASER INDUCED FLUORESCENCE; COOLING FLOWS (ASTROPHYSICS); RAMAN SPECTROSCOPY; HEAT FLUX

Wernet, Mark P.↗
DOE OSTI2024

Effect of Wettability Gradient on the Scale Formation in Falling Film Flow

Mitigation of scale formation and performance degradation remains a vital challenge for falling film evaporators in various industries. In this work, an experimental study of falling film flow on a horizontal tube is conducted to investigate the effects of wettability gradients on thermal, hydraulic, and fouling behavior. It is revealed that certain hydrophobic coating patterns, such as strip, ring, and grid patterns, lead to unwetted heat transfer area, which results in decreased heat transfer compared to fully wetted plain tube. By adjusting the geometry and position of the wettability gradient, the hybrid coating demonstrates improved heat transfer performance. Based on the characteristics of horizontal tube falling film flow, impinging jet, thin film flow, and liquid retention at the tube bottom, a hybrid coating pattern is developed to improve surface wetting and mitigate the scaling coverage. It is revealed that scale deposition is regulated by wettability gradient. Crystals tend to be dense and compact in hydrophilic areas, while they appear scattered or even absent in hydrophobic regions, depending on the dimension of the hydrophobic area. While at the hydrophilic/hydrophobic boundary, a noticeable scale thickness step is observed, which raises the potential for self-cleaning. Furthermore, the balance of minimization of scaling layer coverage and maximization of wetting area requires an optimal design in coating dimensions, for which a systemic study of both flow dynamics and fouling characteristics on the falling film is necessary in the future.

42 ENGINEERING↗
DOE OSTI2021

Multiscale Assessment of Caprock Integrity for Geologic Carbon Storage in the Pennsylvanian Farnsworth Unit, Texas, USA

Leakage pathways through caprock lithologies for underground storage of CO2 and/or enhanced oil recovery (EOR) include intrusion into nano-pore mudstones, flow within fractures and faults, and larger-scale sedimentary heterogeneity (e.g., stacked channel deposits). To assess multiscale sealing integrity of the caprock system that overlies the Morrow B sandstone reservoir, Farnsworth Unit (FWU), Texas, USA, we combine pore-to-core observations, laboratory testing, well logging results, and noble gas analysis. A cluster analysis combining gamma ray, compressional slowness, and other logs was combined with caliper responses and triaxial rock mechanics testing to define eleven lithologic classes across the upper Morrow shale and Thirteen Finger limestone caprock units, with estimations of dynamic elastic moduli and fracture breakdown pressures (minimum horizontal stress gradients) for each class. Mercury porosimetry determinations of CO2 column heights in sealing formations yield values exceeding reservoir height. Noble gas profiles provide a “geologic time-integrated” assessment of fluid flow across the reservoir-caprock system, with Morrow B reservoir measurements consistent with decades-long EOR water-flooding, and upper Morrow shale and lower Thirteen Finger limestone values being consistent with long-term geohydrologic isolation. Together, these data suggest an excellent sealing capacity for the FWU and provide limits for injection pressure increases accompanying carbon storage activities.

58 GEOSCIENCES↗
DOE OSTI2026

Numerical Investigation of Fluid Flow and Space Charge in Liquid Argon Time Projection Chamber (LArTPC) Detectors

Overview This project focused on developing a high-fidelity numerical framework to simulate the multiphysics environment within Liquid Argon Time Projection Chamber (LArTPC) detectors. The primary objective was to characterize the complex interplay between ion transport, background fluid dynamics, and electric field distortions—a critical factor for the calibration and sensitivity of next-generation High Energy Physics experiments, such as DUNE. Technical Achievements The research successfully yielded a hybrid numerical space-charge solver utilizing a Cell-Centered Finite Volume Method (FVM) for ion transport coupled with a Finite Element Method (FEM) for electric potential. Key accomplishments include: • Verification & Validation: The 3-D solver was rigorously verified against 1-D analytical solutions, demonstrating high numerical accuracy in predicting space-charge-induced field deviations. • Field Distortion Analysis: 3D simulations revealed that space charge effects introduce significant non-uniformities in the electric field. Critically, the research identified that background LAr flow velocities, when comparable to ion drift velocities, markedly exacerbate these distortions. • Technology Transfer: The resulting source code and comprehensive user manuals were successfully transferred to collaborators at Fermilab, providing a portable computational tool for the broader scientific community. Challenges and Future Directions While the space-charge solver achieved all performance metrics, the integrated fluid dynamics modeling encountered convergence challenges stemming from the extreme 200-fold disparity in length scales between the detector's 37 mm inlet pipes and the 8-meter global domain. To address this, the project has identified a clear technical pivot toward Hierarchical Geometric Adaptive Mesh Refinement (HG-AMR). By implementing an h-type refinement strategy with hanging nodes, future iterations of this solver will be capable of resolving localized high-gradient inlet flows without the prohibitive computational costs of regular grids. This advancement, combined with data-driven uncertainty quantification based on MicroBooNE-style calibration, will enable the precise modeling of detector responses in large-scale cryogenic environments where direct measurement remains difficult. Impact The computational tools developed under this award provide a foundation for enhancing the energy resolution and spatial reconstruction of noble liquid detectors. By bridging the gap between theoretical fluid dynamics and experimental field calibration, this work supports the DOE’s mission to advance the frontiers of neutrino physics and dark matter detection.

42 ENGINEERING↗
DOE OSTI2021

Computational Fluid Dynamics (CFD) Simulations of Taylor Bubbles in Vertical and Inclined Pipes with Upward and Downward Liquid Flow

Summary Two-phase flow is a common occurrence in pipes of oil and gas developments. Current predictive tools are based on the mechanistic two-fluid model, which requires the use of closure relations to predict integral flow parameters such as liquid holdup (or void fraction) and pressure gradient. However, these closure relations carry the highest uncertainties in the model. In particular, significant discrepancies have been found between experimental data and closure relations for the Taylor bubble velocity in slug flow, which has been determined to strongly affect the mechanistic model predictions (Lizarraga-García 2016). In this work, we study the behavior of Taylor bubbles in vertical and inclined pipes with upward and downward flow using a validated 3D computational fluid dynamics (CFD) approach with level set method implemented in a commercial code. A total of 56 cases are simulated, covering a wide range of fluid properties, pipe diameters, and inclination angles: Eo ∈ [10, 700]; Mo ∈ [1×10–6, 5×103]; ReSL ∈ [–40, 10]; θ ∈ [5°, 90°]. For bubbles in vertical upward flows, the simulated distribution parameter, C0, is successfully compared with an existing model. However, the C0 values of downward and inclined slug flows where the bubble becomes asymmetric are shown to be significantly different from their respective vertical upward flow values, and no current model exists for the fluids simulated here. The main contributions of this work are (1) the relatively large 3D numerical database generated for this type of flow, (2) the study of the asymmetric nature of inclined and some vertical downward slug flows, and (3) the analysis of its impact on the distribution parameter, C0.

Engineering↗
NASA NTRS1984

Convergent-tapered annular seals - Analysis and testing for rotordynamic coefficients

A combined analytical-computational method has been developed to calculate the pressure field and dynamic coefficients for tapered high-pressure annular seals. Completely developed turbulent flow is assumed in both the circumferential and axial directions, according to Hirs' (1973) bulk-flow turbulent-lubrication equations. In a numerical experiment with the method, a short bearing approximation is used to derive an analytical expression for the first-order (dynamic) pressure gradient. This expression is integrated numerically to define dynamic coefficients of the seal. Numerical results of the integration are compared with previous results for straight and tapered seals. It is found that direct stiffness and leakage coefficients increase in the present seal, while the remaining coefficients decrease. An optimal taper angle is shown to exist with respect to: (1) the direct stiffness and (2) the ratio of direct stiffness to leakage. On the basis of the optimal taper angle calculations, stiffness increases on the order of 40 to 50 percent are predicted. Experimental results for three taper angles are presented, and are found to be in good agreement with the theoretical predictions.

Childs, D. W.↗
NASA NTRS1993

Natural and thermocapillary convection in multiple liquid layers

A numerical and experimental investigation is conducted of the fluid dynamics of immiscible liquid layers, as encountered in the liquid encapsulation of electronic melts. Attention is given to the convective flow in multiple liquid layers that are driven by temperature gradients that are either parallel or perpendicular to the liquid/liquid interfaces. The experimental and theoretical results obtained are noted not to be in agreement for differential side-heating of a three-layer system.

Prakash, A.↗
NASA NTRS2000

The Dynamics of Miscible Interfaces: A Space Flight Experiment

Experiments as well as accompanying simulations are described that serve in preparation of a space flight experiment to study the dynamics of miscible interfaces. The investigation specifically addresses the importance of both nonsolenoidal effects as well as nonconventional Korteweg stresses in flows that give rise to steep but finite concentration gradients. The investigation focuses on the flow in which a less viscous fluid displaces one of higher viscosity and different density within a narrow capillary tube. The fluids are miscible in all proportions. An intruding finger forms that occupies a fraction of the total tube diameter. Depending on the flow conditions, as expressed by the Peclet number, a dimensionless viscosity ratio, and a gravity parameter, this fraction can vary between approximately 0.9 and 0.2. For large Pe values, a quasi-steady finger forms, which persists for a time of O(Pe) before it starts to decay, and Poiseuille flow and Taylor dispersion are approached asymptotically. Depending on the specific flow conditions, we observe a variety of topologically different streamline patterns, among them some that leak fluid from the finger tip. For small Pe values, the flow decays from the start and asymptotically reaches Taylor dispersion after a time of O(Pe). Comparisons between experiments and numerical simulations based on the 'conventional' assumption of solenoidal velocity fields and without Korteweg stresses yield poor agreement as far as the Pe value is concerned that distinguishes these two regimes. As one possibility, we attribute this lack of agreement to the disregard of these terms. An attempt is made to use scaling arguments in order to evaluate the importance of the Korteweg stresses and of the assumption of solenoidality. While these effects should be strongest in absolute terms when steep concentration fronts exist, i.e., at large Pe, they may be relatively most important at lower values of Pe. We subsequently compare these conventional simulations to more complete simulations that account for nonvanishing divergence as well as Korteweg stresses. While the exact value of the relevant stress coefficients are not known, ballpark numbers do exist, and their use in the simulations indicates that these stresses may indeed be important. We plan to evaluate these issues in detail by means of comparing a space experiment with corresponding simulations, in order to extract more accurate Korteweg stress coefficients, and to confirm or deny the importance of such stresses.

Maxworthy, Tony↗
DOE OSTI2022

Zonal shear layer collapse and the power scaling of the density limit: old L-H wine in new bottles

Edge shear layer collapse causes edge cooling and aggravates radiative effects. This paper details on the microscopic dynamics of the emergence of power (Q) scaling of density limit (DL) from the shear layer collapse transport bifurcation scenario. The analysis is based on a novel 4-field model, which evolves turbulence energy, zonal flow energy, temperature gradient and density, including the neoclassical screening of zonal flow response. Bifurcation analysis yields power scaling of critical density for shear layer collapse as $n_{crit}\sim Q^{1/3}$. The favorable Q scaling of the DL emerges from the fact that the shear layer strength increases with Q, thus preventing shear layer collapse. This in turn reduces particle transport and improves particle confinement. RMP induced ambient stochastic fields degrade the shear layer by inducing decoherence in the Reynolds stress. As a result the particle transport increases and particle confinement degrades. This leads to the emergence of unfavorable stochastic field intensity ($b_{st}^{2}$) scaling of the critical density as $n_{crit}\sim(1+b_{st}^{2})^{-5/3}$. All fields, including zonal flow shear, exhibit hysteresis when the power (Q) is ramped cyclically across the bifurcation point. The hysteresis is due to dynamical delay in bifurcation on account of critical slowing down. Furthermore, the dynamical hysteresis here is fundamentally different from the hysteresis associated with the existence of bi-stable states.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2023

Elastic Flow Modeling for Hydropower Digital Twins

This report details the elastic, unsteady, one-dimensional flow equations that are used to model flow through a penstock in a hydropower facility. The elastic flow model is accurate even in cases of fast transients, long penstock length, and high gravitation head. The compressibility of the water and elasticity of the pipe walls are explicitly accounted for in the developed models so that the water hammer phenomena can be accurately captured. The elastic flow model is coupled to a mechanistic turbine model to resolve the dynamic feedbacks between the elastic water column and the turbine rotation rate. A finite volume method is employed to solve the governing equations, and the method is shown to have low numerical diffusion in sharp gradient phenomena, such as those encountered when simulating water hammer. The method is applied to single dimensional linear advection benchmark problem and numerical results are compared with analytic results. This is followed on by an application to a full hydropower system with a coupled turbine. The elastic flow model results are compared against an inelastic model.

13 HYDRO ENERGY↗
NASA NTRS1974

Analysis of dynamic stall using unsteady boundary-layer theory

The unsteady turbulent boundary layer and potential flow about a pitching airfoil are analyzed using numerical methods to determine the effect of pitch rate on the delay in forward movement of the rear flow reversal point. An explicit finite difference scheme is used to integrate the unsteady boundary layer equations, which are coupled at each instant of time to a fully unsteady and nonlinear potential flow analysis. A substantial delay in forward movement of the reversal point is demonstrated with increasing pitch rate, and it is shown that the delay results partly from the alleviation of the gradients in the potential flow, and partly from the effects of unsteadiness in the boundary layer itself. The predicted delay in flow-reversal onset, and its variation with pitch rate, are shown to be in reasonable agreement with experimental data relating to the delay in dynamic stall. From the comparisons it can be concluded (a) that the effects of time-dependence are sufficient to explain the failure of the boundary layer to separate during the dynamic overshoot, and (b) that there may be some link between forward movement of the reversal point and dynamic stall.

Scruggs, R. M.↗
NASA NTRS1990

A simple nonlinear model for the return to isotropy in turbulence

A quadratic nonlinear generalization of the linear Rotta model for the slow pressure-strain correlation of turbulence is developed. The model is shown to satisfy realizability and to give rise to no stable nontrivial equilibrium solutions for the anisotropy tensor in the case of vanishing mean velocity gradients. The absence of stable nontrivial equilibrium solutions is a necessary condition to ensure that the model predicts a return to isotropy for all relaxational turbulent flows. Both the phase space dynamics and the temporal behavior of the model are examined and compared against experimental data for the return to isotropy problem. It is demonstrated that the quadratic model successfully captures the experimental trends which clearly exhibit nonlinear behavior. Direct comparisons are also made with the predictions of the Rotta model and the Lumley model.

Sarkar, Sutanu↗