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Aerodynamic and Aeroacoustic Analysis of Low Reynolds Number Propellers Using Higher-Order RANS Transition Turbulence ModelingPisharoti, Naina 05 June 2024 (has links)
The advent of advanced vehicle concepts involving Urban Air Mobility (UAM) and small Unmanned Aerial Systems (sUAS) has brought about a new class of rotorcraft technology which operate predominantly in low-Reynolds ($Re$) number regimes. In such regimes, the flow experiences complex boundary layer phenomena like laminar separation, flow transition and reattachment. These effects are known to greatly alter the flow at and near the rotor wall, influencing its aerodynamic performance as well as the noise generated. Capturing these effects in our computational models is necessary to further our understanding of rotor aerodynamics and acoustics.
The current study has introduced a novel RANS transition turbulence model, SSG/LRR-$\omega$-$\gamma$, that is capable of modeling different modes of transition involving natural, bypass, separation-induced and crossflow transition. The model framework uses a Reynolds stress transport model, SSG/LRR-$\omega$, as the base turbulence formulation and is coupled with Menter's $\gamma$ transition model. It was validated using a number of canonical cases that exhibited different transition mechanisms and the model performed equivalently or better than existing state-of-the-art transition models. It is worthy to note that the proposed model was able to perform well in three-dimensional flows, demonstrated using the case of a prolate spheroid. This underscores the capability of Reynolds stress models to accurately capture complex flow curvatures, improving upon the capabilities of linear eddy viscosity models.
The transition model, integrated into OpenFOAM, was then employed to analyze two different UAV propellers. The rotor flow was examined using a URANS simulation with an overset grid. The objective was twofold: firstly, to validate the predictions generated by the proposed model for low-Reynolds number (low-$Re$) rotors, and secondly, to evaluate its effectiveness across a range of operating conditions. Comparisons were drawn against established fully turbulent and transition models. The analysis showed that transition models in general tended to be consistent in their predictions and less sensitive to changing operating conditions when compared to fully turbulent models. They also demonstrated the ability to accurately predict the mechanisms leading to separation and transition. Further, the proposed transition model demonstrated superior capability in capturing detailed flow features, particularly in the wake, compared to other fully turbulent and transition models, which is attributed to its Galilean invariant framework.
To leverage the boundary layer information obtained from the proposed model, a semi-empirical broadband noise prediction method was implemented. This method utilized boundary layer data predicted by URANS simulations to estimate blade self-noise. An evaluation of the fully turbulent $k$-$\omega$ SST model and the proposed transition model revealed that both exhibited reasonable accuracy at lower rotor advance ratios. However, the transition model performed better at higher advance ratios. It was also observed that CFD-based approaches provided superior prediction accuracy compared to lower-fidelity aerodynamic models in the context of blade self-noise prediction Finally, the proposed aerodynamic and acoustic computational framework was applied to a design case study of swept propellers to understand the advantages of blade sweep on rotor aerodynamics and noise. A qualitative analysis of the flow suggested that the swept rotor exhibited lower levels of blade wake interaction compared to the unswept geometry, in line with the experimental observations. / Doctor of Philosophy / Advanced vehicle concepts such as air taxis for Urban Air Mobility (UAM) and other multi-copter applications like drone delivery, reconnaissance, etc. are emerging sectors in aviation that have garnered great industrial as well as academic interest. However, since these vehicles are expected to fly at low altitudes within urban settings, noise mitigation is of particular interest to improve their public acceptance. The vehicle configurations in these applications predominantly comprise of rotorcraft which operate at low Reynolds ($Re$) numbers and tip speeds. These operating conditions introduce complex phenomena like flow transition and separation within the boundary layer that significantly alter their aerodynamic as well as aeroacoustic performance. The current work proposes a novel transition turbulence model that improves prediction of these complex boundary layer mechanisms in low-$Re$ propellers compared to the state-of-the-art. Furthermore, this work establishes a fast broadband noise prediction method by leveraging the detailed flow data from the transition model. The focus of this method is on modeling those propeller noise sources that are directly influenced by the aforementioned boundary layer phenomena (blade self-noise). The noise prediction study revealed that transition models yield consistent predictions across different operating conditions. Finally, a brief design study is conducted using the proposed aerodynamic and acoustic framework to assess the flow dynamics and possible noise mitigation capabilities of a swept propeller.
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Machine Learning and Field Inversion approaches to Data-Driven Turbulence ModelingMichelen Strofer, Carlos Alejandro 27 April 2021 (has links)
There still is a practical need for improved closure models for the Reynolds-averaged Navier-Stokes (RANS) equations. This dissertation explores two different approaches for using experimental data to provide improved closure for the Reynolds stress tensor field. The first approach uses machine learning to learn a general closure model from data. A novel framework is developed to train deep neural networks using experimental velocity and pressure measurements. The sensitivity of the RANS equations to the Reynolds stress, required for gradient-based training, is obtained by means of both variational and ensemble methods. The second approach is to infer the Reynolds stress field for a flow of interest from limited velocity or pressure measurements of the same flow. Here, this field inversion is done using a Monte Carlo Bayesian procedure and the focus is on improving the inference by enforcing known physical constraints on the inferred Reynolds stress field. To this end, a method for enforcing boundary conditions on the inferred field is presented. The two data-driven approaches explored and improved upon here demonstrate the potential for improved practical RANS predictions. / Doctor of Philosophy / The Reynolds-averaged Navier-Stokes (RANS) equations are widely used to simulate fluid flows in engineering applications despite their known inaccuracy in many flows of practical interest. The uncertainty in the RANS equations is known to stem from the Reynolds stress tensor for which no universally applicable turbulence model exists. The computational cost of more accurate methods for fluid flow simulation, however, means RANS simulations will likely continue to be a major tool in engineering applications and there is still a need for improved RANS turbulence modeling. This dissertation explores two different approaches to use available experimental data to improve RANS predictions by improving the uncertain Reynolds stress tensor field. The first approach is using machine learning to learn a data-driven turbulence model from a set of training data. This model can then be applied to predict new flows in place of traditional turbulence models. To this end, this dissertation presents a novel framework for training deep neural networks using experimental measurements of velocity and pressure. When using velocity and pressure data, gradient-based training of the neural network requires the sensitivity of the RANS equations to the learned Reynolds stress. Two different methods, the continuous adjoint and ensemble approximation, are used to obtain the required sensitivity. The second approach explored in this dissertation is field inversion, whereby available data for a flow of interest is used to infer a Reynolds stress field that leads to improved RANS solutions for that same flow. Here, the field inversion is done via the ensemble Kalman inversion (EKI), a Monte Carlo Bayesian procedure, and the focus is on improving the inference by enforcing known physical constraints on the inferred Reynolds stress field. To this end, a method for enforcing boundary conditions on the inferred field is presented. While further development is needed, the two data-driven approaches explored and improved upon here demonstrate the potential for improved practical RANS predictions.
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Hydrodynamic Design of Highly Loaded Torque-neutral Ducted Propulsor for Autonomous Underwater VehiclesPawar, Suraj Arun 24 January 2019 (has links)
The design method for marine propulsor (propeller/stator) is presented for an autonomous underwater vehicle (AUV) that operates at a very high loading condition. The design method is applied to Virginia Tech Dragon AUV. It is based on the parametric geometry definition for the propulsor, use of high-fidelity CFD RANSE solver with the transition model, construction of the surrogate model, and multi-objective genetic optimization algorithm. The CFD model is validated using the paint pattern visualization on the surface of the propeller for an open propeller at model scale. The CFD model is then applied to study hydrodynamics of ducted propellers such as forces and moments, tip leakage vortex, leading-edge flow separation, and counter-rotating vortices formed at the duct trailing edge. The effect of variation of thickness for stator blades and different approaches for modeling the postswirl stator is presented. The field trials for Dragon AUV shows that there is a good correlation between expected and achieved design speed under tow condition with the designed base propulsor. The marine propulsor design is further improved with an objective to maximize the propulsive efficiency and minimize the rolling of AUV. The stator is found to eliminate the swirl component of velocity present in the wake of the propeller to the maximum extent. The propulsor designed using this method (surrogate-based optimization) is demonstrated to have an improved torque balance characteristic with a slight improvement in efficiency than the base propulsor design. / Master of Science / The propulsion system is the critical design element for an AUV, especially if it is towing a large payload. The propulsor for towing AUVs has to provide a very large thrust and hence the propulsor is highly loaded. The propeller has to rotate at very high speed to produce the required thrust and is likely to cavitate at this high speed. Also at this high loading condition, the maximum ideal efficiency of the propulsor is very less. Another challenge is the induced torque from the propeller on AUV that can cause the rolling of an AUV which is undesirable. This problem can be addressed by installing the stator behind the propeller that will produce torque in the opposite direction of the propeller torque. In this work, we present a design methodology for marine propulsor (propeller/stator) that can be used in AUV towing a large payload. The propulsor designed using this method has improved torque characteristics and has the efficiency close to 80 % of the ideal efficiency of ducted propeller at that loading condition.
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Computational Fluid Dynamics Analysis in Support of the NASA/Virginia Tech Benchmark ExperimentsBeardsley, Colton Tack 23 June 2020 (has links)
Computational fluid dynamics methods have seen an increasing role in aerodynamic analysis since their first implementation. However, there are several major limitations is these methods of analysis, especially in the area of modeling separated flow. There exists a large demand for high-fidelity experimental data for turbulence modeling validation. Virginia Tech has joined NASA in a cooperative project to design and perform an experiment in the Virginia Tech Stability Wind Tunnel with the purpose of providing a benchmark set of data for the turbulence modeling community for the flow over a three-dimensional bump. This process requires thorough risk mitigation and analysis of potential flow sensitivities. The current study investigates several aspects of the experimental design through the use of several computational fluid dynamics codes.
An emphasis is given to boundary condition matching and uncertainty quantification, as well as sensitivities of the flow features to Reynolds number and inflow conditions. Solutions are computed for two different RANS turbulence models, using two different finite-volume CFD codes. Boundary layer inflow parameters are studied as well as pressure and skin friction distribution on the bump surface. The shape and extent of separation are compared for the various solutions. Pressure distributions are compared to available experimental data for two different Reynolds numbers. / Master of Science / Computational fluid dynamics (CFD) methods have seen an increasing role in engineering analysis since their first implementation. However, there are several major limitations is these methods of analysis, especially in the area of modeling of several common aerodynamic phenomena such as flow separation. This motivates the need for high fidelity experimental data to be used for validating computational models. This study is meant to support the design of an experiment being cooperatively developed by NASA and Virginia Tech to provide validation data for turbulence modeling. Computational tools can be used in the experimental design process to mitigate potential experimental risks, investigate flow sensitivities, and inform decisions about instrumentation. Here, we will use CFD solutions to identify risks associated with the current experimental design and investigate their sensitivity to incoming flow conditions and Reynolds number. Numerical error estimation and uncertainty quantification is performed. A method for matching experimental inflow conditions is proposed, validated, and implemented. CFD data is also compared to experimental data. Comparisons are also made between different models and solvers.
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Hybrid RANS-LES closure for separated flows in the transitional regimeHodara, Joachim 27 May 2016 (has links)
The aerodynamics of modern rotorcraft is highly complex and has proven to be an arduous challenge for computational fluid dynamics (CFD). Flow features such as massively separated boundary layers or transition to turbulence are common in engineering applications and need to be accurately captured in order to predict the vehicle performance. The recent advances in numerical methods and turbulence modeling have resolved each of these issues independent of the other. First, state-of-the-art hybrid RANS-LES turbulence closures have shown great promise in capturing the unsteady flow details and integrated performance quantities for stalled flows. Similarly, the correlation-based transition model of Langtry and Menter has been successfully applied to a wide range of applications involving attached or mildly separated flows. However, there still lacks a unified approach that can tackle massively separated flows in the transitional flow region. In this effort, the two approaches have been combined and expended to yield a methodology capable of accurately predicting the features in these highly complex unsteady turbulent flows at a reasonable computational cost. Comparisons are evaluated on several cases, including a transitional flat plate, circular cylinder in crossflow and NACA 63-415 wing. Cost and accuracy correlations with URANS and prior hybrid URANS-LES approaches with and without transition modeling indicate that this new method can capture both separation and transition more accurately and cost effectively.
This new turbulence approach has been applied to the study of wings in the reverse flow regime. The flight envelope of modern helicopters has increased significantly over the last few decades, with design concepts now reaching advance ratios up to μ = 1. In these extreme conditions, the freestream velocity exceeds the rotational speed of the blades, and a large region of the retreating side of the rotor disk experiences reverse flow. For a conventional airfoil with a sharp trailing edge, the reverse flow regime is generally characterized by massive boundary layer separation and bluff body vortex shedding. This complex aerodynamic environment has been utilized to evaluate the new hybrid transitional approach. The assessment has proven the efficiency of the new hybrid model, and it has provided a transformative advancement to the modeling of dynamic stall.
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A FILTER-FORCING TURBULENCE MODEL FOR LARGE EDDY SIMULATION INCORPORATING THE COMPRESSIBLE "POOR MAN'S" NAVIER--STOKES EQUATIONSStrodtbeck, Joshua 01 January 2012 (has links)
A new approach to large-eddy simulation (LES) based on the use of explicit spatial filtering combined with backscatter forcing is presented. The forcing uses a discrete dynamical system (DDS) called the compressible ``poor man's'' Navier--Stokes (CPMNS) equations. This DDS is derived from the governing equations and is shown to exhibit good spectral and dynamical properties for use in a turbulence model. An overview and critique of existing turbulence theory and turbulence models is given. A comprehensive theoretical case is presented arguing that traditional LES equations contain unresolved scales in terms generally thought to be resolved, and that this can only be solved with explicit filtering. The CPMNS equations are then incorporated into a simple forcing in the OVERFLOW compressible flow code, and tests are done on homogeneous, isotropic, decaying turbulence, a Mach 3 compression ramp, and a Mach 0.8 open cavity. The numerical results validate the general filter-forcing approach, although they also reveal inadequacies in OVERFLOW and that the current approach is likely too simple to be universally applicable. Two new proposals for constructing better forcing models are presented at the end of the work.
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A New Approach for Turbulent Simulations in Complex GeometriesIsrael, Daniel Morris January 2005 (has links)
Historically turbulence modeling has been sharply divided into Reynolds averaged Navier-Stokes (RANS), in which all the turbulent scales of motion are modeled, and large-eddy simulation (LES), in which only a portion of the turbulent spectrum is modeled. In recent years there have been numerous attempts to couple these two approaches either by patching RANS and LES calculations together (zonal methods) or by blending the two sets of equations. In order to create a proper bridging model, that is, a single set of equations which captures both RANS and LES like behavior, it is necessary to place both RANS and LES in a more general framework.The goal of the current work is threefold: to provide such a framework, to demonstrate how the Flow Simulation Methodology (FSM) fits into this framework, and to evaluate the strengths and weaknesses of the current version of the FSM. To do this, first a set of filtered Navier-Stokes (FNS) equations are introduced in terms of an arbitrary generalized filter. Additional exact equations are given for the second order moments and the generalized subfilted dissipation rate tensor. This is followed by a discussion of the role of implicit and explicit filters in turbulence modeling.The FSM is then described with particular attention to its role as a bridging model. In order to evaluate the method a specific implementation of the FSM approach is proposed. Simulations are presented using this model for the case of separating flow over a "hump" with and without flow control. Careful attention is paid to error estimation, and, in particular, how using flow statistics and time series affects the error analysis. Both mean flow and Reynolds stress profiles are presented, as well as the phase averaged turbulent structures and wall pressure spectra. Using the phase averaged data it is possible to examine how the FSM partitions the energy between the coherent resolved scale motions, the random resolved scale fluctuations, and the subfilter quantities.The method proves to be qualitatively successful at reproducing large turbulent structures. However, like other hybrid methods, it has difficulty in the region where the model behavior transitions from RANS to LES> Consequently the phase averaged structures reproduce the experiments quite well, and the forcing does significantly reduce the length of the separated region. Nevertheless, the recirculation length is signficantly too large for all cases.Overall the current results demonstrate the promise of bridging models in general and the FSM in particular. However, current bridging techniques are still in their infancy. There is still important progress to be made and it is hoped that this work points out the more important avenues for exploration.
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A CFD investigation on the flow around a low aspect ratio vertical cylinder: modeling free surface and turbulent effects. / Uma investigação do escoamento ewm torno de cilindro vertical de baixa razão de aspecto através da dinâmica dos fluídos computacional: modelamento de efeitos de superfície livre e de turbulência.Lopes, Pedro Paludetto Silva de Paula 22 January 2019 (has links)
The fow around bluff bodies is an essential topic in fuid dynamics. This fow is characterized by large vortical fow regions separating from the surface of the bluff body, and they cause oscillating drag and lift forces on the structure. The fow around an infinite long cylinder is a well-known case being studied in the literature. However, a cylinder with low aspect-ratio piercing the free surface has not been studied much although such an arrangement can be found in many floating offshore structures. In this thesis the results of CFD calculations are presented for a fixed, free surface piercing cylinder with an aspect-ratio L/D equals to 2. The Reynolds number was equal to 4:3 x 104 indicating that the flow is in the sub-critical turbulent regime. An extensive methodology of verication and validation is followed to investigate the reliability of the results. To investigate the effect of the free surface on the calculated hydrodynamic loads, two approaches have been considered: a double-body symmetry condition and an interface capturing Volume-of-Fluid (VoF) method. Additionally, two turbulence models are investigated: a two-equation turbulence model; a non-linear Explicit Algebraic Reynolds Stress Model (EARSM); and the Improved Delayed Detached Eddy Simulation (IDDES) turbulence model. The results are presented in terms of integral results (drag and lift coefficients) and flow visualizations. Based on the results of the cases in which the free surface was modeled as a double body symmetry boundary condition, it is concluded that the model is not suitable for this type of flow as the model damps out the flow dynamics due to over-production of eddy-viscosity. Hence, the characteristic oscillating lift forces are not captured using this turbulence model. However, this turbulence model showed good agreements regarding the flow fields in comparison with experimental PIV measurements. Results of the case modeled with EARSM turbulence model shows better agreement with the experimental results compared with the turbulence model. In the cases where the free-surface is considered, results with the EARSM turbulence model show similar results for the drag forces whereas the lift uctuations were one order of magnitude smaller, compared with the double body case. Lastly, the results using the IDDES turbulence model and free-surface VoF modeling are shown to produce the best comparison with the experimental results, regarding both, integral values and flow field results. / O escoamento ao redor de corpos rombudos é um tópico essencial na dinâmica de fluidos. O escoamento é caracterizado por regiões com grande vorticidade que se separam do corpo e causam oscilações das forças de arrasto e sustentação sobre a estrutura. O escoamento ao redor de cilindros longos é um tema que tem sido extensivamente estudado com muitos trabalhos encontrados na literatura. Entretanto, o cilindro com baixa razão de aspecto perfurante à superfície livre é um caso pouco estudado, apesar desta estrutura ser encontrada em várias estruturas oceânicas flutuantes. Esta dissertação apresenta cálculos numéricos para o escoamento ao redor de um cilindro fixo, que trespassa a superfície livre com razão de aspecto L/D igual a 2. O problema é estudado em um regime subcrítico de turbulência, com número de Reynolds igual a 4:3 x 104. Uma vasta metodologia de verificação e validação foi seguida para avaliar a confiabilidade dos resultados obtidos numericamente. Para investigar os efeitos da superfície livre nas cargas hidrodinâmicas, duas abordagens s~ao consideradas: condição de simetria de duplo corpo e um método de captura de interface Volume of Fluid. Além disso, dois modelos de turbulência foram investigados: o modelo não linear Explicit Algebraic Reynolds Stress Model (EARSM), e o modelo de turbulência Improved Delayed Detached Eddy Simulation (IDDES). Os resultados relacionados aos coeficientes de arrasto e sustentação são apresentados a partir de análise estatística, complementados através de ilustrações que permitem visualizar os campos de escoamento e pressão. Com base nos resultados de casos em que a superfície livre é modelada com uma condição de contorno de simetria, conclui-se que o modelo de turbulência não é adequado para este tipo de escoamento, pois o modelo amortece a dinâmica do escoamento devido à superprodução de viscosidade turbulenta. Consequentemente, as oscilações na força de sustentação não são capturadas usando este modelo. Entretanto, resultados dos campos médios do escoamento mostram-se concordantes com imagens experimentais obtidas com técnicas de PIV - Particle Image Velocimetry. Resultados do caso modelado com o modelo de turbulência EARSM mostram melhores concordâncias na comparação dos parâmetros estatísticos com experimentos do que o modelado com o modelo EARSM. Nos casos em que a superfície livre é modelada com o método VoF, o modelo de turbulência EARSM mostra resultados semelhantes para o arrasto, enquanto as flutuações da sustentação apresentam-se uma ordem de grandeza menores, quando comparadas ao caso de duplo corpo. Resultados usando o modelo de turbulência IDDES e VoF apresentam melhores comparações aos resultados experimentais, tanto para os parâmetros estatísticos quando para as visualizações do escoamento.
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Computational fluid dynamics (CFD) simulations of aerosol in a u-shaped steam generator tubeLongmire, Pamela 15 May 2009 (has links)
To quantify primary side aerosol retention, an Eulerian/Lagrangian approach was
used to investigate aerosol transport in a compressible, turbulent, adiabatic, internal,
wall-bounded flow. The ARTIST experimental project (Phase I) served as the physical
model replicated for numerical simulation. Realizable k-ε and standard k-ω turbulence
models were selected from the computational fluid dynamics (CFD) code, FLUENT, to
provide the Eulerian description of the gaseous phase.
Flow field simulation results exhibited: a) onset of weak secondary flow
accelerated at bend entrance towards the inner wall; b) flow separation zone
development on the convex wall that persisted from the point of onset; c) centrifugal
force concentrated high velocity flow in the direction of the concave wall; d) formation
of vortices throughout the flow domain resulted from rotational (Dean-type) flow; e)
weakened secondary flow assisted the formation of twin vortices in the outflow cross
section; and f) perturbations induced by the bend influenced flow recovery several pipe diameters upstream of the bend. These observations were consistent with those of
previous investigators.
The Lagrangian discrete random walk model, with and without turbulent
dispersion, simulated the dispersed phase behavior, incorrectly. Accurate deposition
predictions in wall-bounded flow require modification of the Eddy Impaction Model
(EIM). Thus, to circumvent shortcomings of the EIM, the Lagrangian time scale was
changed to a wall function and the root-mean-square (RMS) fluctuating velocities were
modified to account for the strong anisotropic nature of flow in the immediate vicinity of
the wall (boundary layer). Subsequent computed trajectories suggest a precision that
ranges from 0.1% to 0.7%, statistical sampling error. The aerodynamic mass median
diameter (AMMD) at the inlet (5.5 μm) was consistent with the ARTIST experimental
findings. The geometric standard deviation (GSD) varied depending on the scenario
evaluated but ranged from 1.61 to 3.2. At the outlet, the computed AMMD (1.9 μm) had
GSD between 1.12 and 2.76. Decontamination factors (DF), computed based on
deposition from trajectory calculations, were just over 3.5 for the bend and 4.4 at the
outlet. Computed DFs were consistent with expert elicitation cited in NUREG-1150 for
aerosol retention in steam generators.
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A Dynamical Systems Approach Towards Modeling the Rapid Pressure Strain CorrelationMishra, Aashwin A. 2010 May 1900 (has links)
In this study, the behavior of pressure in the Rapid Distortion Limit, along with its
concomitant modeling, are addressed. In the first part of the work, the role of pressure in
the initiation, propagation and suppression of flow instabilities for quadratic flows is
analyzed. The paradigm of analysis considers the Reynolds stress transport equations to
govern the evolution of a dynamical system, in a state space composed of the Reynolds
stress tensor components. This dynamical system is scrutinized via the identification of
the invariant sets and the bifurcation analysis. The changing role of pressure in quadratic
flows, viz. hyperbolic, shear and elliptic, is established mathematically and the
underlying physics is explained. Along the maxim of "understanding before prediction", this allows for a deeper insight into the behavior of pressure, thus aiding in its modeling.
The second part of this work deals with Rapid Pressure Strain Correlation modeling in
earnest. Based on the comprehension developed in the preceding section, the classical
pressure strain correlation modeling approaches are revisited. Their shortcomings, along
with their successes, are articulated and explained, mathematically and from the
viewpoint of the governing physics. Some of the salient issues addressed include, but are not limited to, the requisite nature of the model, viz. a linear or a nonlinear structure,
the success of the extant models for hyperbolic flows, their inability to capture elliptic
flows and the use of RDT simulations to validate models. Through this analysis, the
schism between mathematical and physical guidelines and the engineering approach, at
present, is substantiated. Subsequently, a model is developed that adheres to the classical
modeling framework and shows excellent agreement with the RDT simulations. The
performance of this model is compared to that of other nominations prevalent in
engineering simulations. The work concludes with a summary, pertinent observations
and recommendations for future research in the germane field.
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