Fluid Dynamic Processes Research Articles

Modeling of Explosive Pingo-like Structures and Fluid-Dynamic Processes in the Arctic Permafrost: Workflow Based on Integrated Geophysical, Geocryological, and Analytical Data

Understanding the mechanisms responsible for the origin, evolution, and failure of pingos with explosive gas emissions and the formation of craters in the Arctic permafrost requires comprehensive studies in the context of fluid dynamic processes. Properly choosing modeling methods for the joint interpretation of geophysical results and analytical data on core samples from suitable sites are prerequisites for predicting pending pingo failure hazards. We suggest an optimal theoretically grounded workflow for such studies, in a site where pingo collapse induced gas blowout and crater formation in the Yamal Peninsula. The site was chosen with reference to the classification of periglacial landforms and their relation to the local deformation pattern, according to deciphered satellite images and reconnaissance geophysical surveys. The deciphered satellite images and combined geophysical data from the site reveal a pattern of periglacial landforms matching the structural framework with uplifted stable permafrost blocks (polygons) bounded by eroded fractured zones (lineaments). Greater percentages of landforms associated with permafrost degradation fall within the lineaments. Resistivity anomalies beneath pingo-like mounds presumably trace deeply rooted fluid conduits. This distribution can be explained in terms of fluid dynamics. N–E and W–E faults, and especially their junctions with N–W structures, are potentially the most widely open conduits for gas and water which migrate into shallow sediments in the modern stress field of N–S (or rather NEN) extension and cause a warming effect on permafrost. The results obtained with a new workflow and joint interpretation of remote sensing, geophysical, and analytical data from the site of explosive gas emission in the Yamal Peninsula confirm the advantages of the suggested approach and its applicability for future integrated fluid dynamics research.

Laser induced spark ignition of a gaseous methane–oxygen model rocket combustor

The spatio-temporal evolution of laser induced spark ignition of non-premixed gaseous methane and oxygen is investigated. The reactants are injected into an optically accessible combustion chamber from an oxidizer centered shear-coaxial injector. High-speed schlieren imaging and measurements of laser energy deposited are used to characterize ignition behavior at various spark locations throughout the chamber. A spatial map of ignition probability is generated from tests at multiple axial and radial locations from the point of propellant injection. The rate of pressure rise from successful tests reveals two modes of ignition dependent on the location of the laser-induced breakdown, direct and indirect, based on laser-induced breakdown occurring within reactant jet or in the recirculation region, respectively. Physical processes occurring over multiple timescales throughout the direct ignition method are first discussed. Ignition outcomes associated with the indirect method are determined by the hydrodynamic ejection protruding from the laser spark, the behavior of which is explored in detail using flow statistics extracted from images taken at a single spark location. The spatial–temporal progression of this jet is found to be dependent on deposited laser energy, leading to a relationship between the amount of energy added to the flow and ignition outcome. General bounds for spark location, deposited laser energy, and ejection behavior are identified to predict ignition outcomes. Cases that are an exception from these bounds are investigated in detail to understand the cause of unique behavior. These cases lie within regions of the variable space that are susceptible to stochastic elements of the flow.Novelty and Significance Statement: This work investigates laser induced spark ignition of O2 and CH4 issuing from a shear-coaxial injector in a model rocket combustor. Probability of ignition as a function of deposited energy and spatial location is studied with high-speed schlieren imaging and pressure measurements. Two distinct ignition modes that have previously not been identified, direct and indirect, are dependent on the laser deposition location with respect to the reactant jet. For indirect cases, where laser deposition is at a distance from the reactants, the hydrodynamic jet ejection generated from the breakdown of the plasma plays a critical role on ignition probability. The behavior of the ejection jet is characterized to provide a phenomenological description of the ignition process. The results of this research show the important interaction between chemical, fluid dynamic, and thermodynamic processes and their impact on laser ignition.

Deep learning models for air quality forecasting based on spatiotemporal characteristics of data

The distribution of air-borne pollutants is governed by complex fluid dynamics processes involving convection and diffusion. The process is further affected by the characteristics of emission sources, meteorological parameters, socioeconomic factors, and land use patterns. Compared to deterministic and probabilistic air quality forecasting methods, data driven modeling of air quality parameters can address the large degree of freedom in air quality influencing parameters as well as offer interpretability and understanding of air pollutants' distribution at an increased spatial and temporal resolutions. This study focuses on the citywide prediction of air quality index (AQI) based on observations of pollutant concentrations, meteorological parameters, and spatiotemporal data. The study area includes Ansan city in South Korea, which has been observed as a hotspot for high concentrations of particulate matter. The air quality and meteorological were collected from 16 monitoring stations located in Ansan city. A detailed spatiotemporal analysis was performed to investigate the correlation between AQI records at the air quality monitoring stations. Based on strong spatiotemporal correlations observed between stations, several deep learning (DL) models were proposed, and their performance was investigated for different scenarios. It was observed that the selection of appropriate DL models should be based on (1) understanding of the underlying fluid dynamics process that control pollutant distribution and (2) spatiotemporal characteristics of data. Additionally, the complexity of DL models does not always guarantee the accuracy of the forecasts, and simple models can give good performance if the predictors are selected carefully to reflect the underlying physical process.

The dynamics of champagne cork popping revisited through high-speed schlieren imaging and computational fluid dynamics simulations

Cork popping represents a rich and complex fluid dynamics process, involving up to three phases (liquid, gas, and solid), three main chemical compounds (ethanol, water, CO2), and a moving cork gradually opening the bottle and blocking the fast progression of the expanding gas. In this work, we used high-speed, high-sensitivity schlieren imaging setups to provide a fresh perspective on this dynamical phenomenon. Our experimental results are systematically compared and interpreted on the basis of new computational fluid dynamics simulations. Our combined experimental and numerical works confirm the establishment of two supersonic expansions during cork popping from a champagne bottle.

Computational Fluid Dynamics Process for Front Windshield Mist Deposition and Its Subsequent Demisting

<div>A vehicle’s heating, ventilation, and air-conditioning system plays a dual role in passenger thermal comfort and safety. The functional aspects of safety include the front windshield demist and deicing feature of the system. The thin-film mist is a result of condensation of water vapor on the inner side of the windshield, which occurs at low ambient temperatures or high humidity. This mist deposition depends on the air saturation pressure at the front windshield. Indian regulation AIS-084 defines the experimental setup for testing, which encompasses both the mist deposition and its subsequent demist process. This regulation mandates testing, which occurs at a later stage of product development. This performance validation can be performed using a three-dimensional computational fluid dynamics approach.</div> <div>Current work summarizes the simulation process for both the mist deposition and the subsequent demisting phenomenon. The complexity of the flow physics is captured via the transient multiphase fluid flow phenomenon subjected to buoyancy effects. This phenomenon is simulated using the Eulerian wall film approach. The wall film deposition of the mist is modeled via species transport. Further, the near-wall thermal effects of surface conduction and heat transfer are simulated by modeling shell conduction layers. Vapor diffusion and relative humidity inside the cabin are modeled using a user-defined function. The process correlation is achieved for two categories of vehicles to establish the process efficacy and robustness. Moving ahead, a design of experiments (DOE) is planned to mitigate the need to simulate mist deposition. The DOE is planning to incorporate deviations in both the input airflow conditions and the imposed ambient conditions. The results from DOE point toward factors that might cause deviations in simulation results with respect to the test. Importantly, the study concludes that the uniform initial thickness assumption of the mist layer can be used for subsequent demist analysis.</div>

Constructing Rainfall Threshold for Debris Flows of a Defined Hazardous Magnitude

Debris flow can cause damage only when its discharge exceeds the drainage capacity of the prevention engineering. At present, most rainfall thresholds for debris flows mainly focus on the initiation of debris flow and do not adequately consider the magnitude and drainage measures of debris flows. These thresholds are likely to initiate numerous warnings that may not be related to hazardous processes. This study proposes a method for calculating the rainfall threshold that is related to a defined level of debris flow magnitude, over which certain damage may be caused. This method is constructed by using the transient rainfall infiltration analysis slope stability model (TRIGRS) and the fluid dynamics process simulation model (MassFlow). We first use the TRIGRS model to analyze slope stability in the study area and obtain the distribution of unstable slopes under different rainfall conditions. Afterward, the MassFlow model is employed to simulate the movement process of unstable slope units and to predict the depositional processes at the mouth of the catchment. Lastly a rainfall threshold is constructed by statistically analyzing the rainfall conditions that cause debris flows flushing out of the given drainage ditch. This method is useful to predict debris flow events of a hazardous magnitude, especially for areas with limited historical observational data.

Levenberg-Marquardt technique analysis of thermal and concentration storage in cone-disk apparatus with neural network-enhancement

An artificial intelligence-driven neural network with the Levenberg-Marquardt method (NN-LMM) has been developed to investigate the thermal and mass transfer characteristics of a liquid flowing through the conical gap in a cone-disk apparatus. The disk and cone can either remain static or rotate at changing angular speeds, with discussion for heat transfer influenced by cosmic emission. The Rosseland resemblance is employed to compute heat diffusion in this study. To monitor variations in mass removal on the exterior, thermophoresis effects are appropriated into consideration. By applying appropriate transformations, the partial differential equations (PDEs) governing this problem are converted into typical ones. A reference dataset for NN-LMM is generated, covering various influential model variations, simulating scenarios using the Lobatto III-A numerical method. The purpose of employing the Levenberg-Marquardt technique in the analysis of thermal and concentration storage in a cone-disk apparatus, enhanced by neural networks, lies in optimizing the modeling and prediction accuracy of complex fluid dynamics and heat transfer phenomena. This technique, coupled with neural network enhancement, aims to address challenges associated with traditional analytical methods by offering a robust and efficient approach to modeling intricate processes in fluid dynamics and heat transfer. The novelty of utilizing the Levenberg-Marquardt technique, along with neural network enhancement, lies in its ability to effectively capture and simulate the dynamic interactions between thermal and concentration storage in the cone-disk apparatus. By leveraging the power of neural networks to learn complex patterns and relationships from data, combined with the optimization capabilities of the Levenberg-Marquardt algorithm, this approach offers improved accuracy and predictive capabilities compared to conventional methods. It is analyzed that the mass transport field for Nt decreases as temperature ratios increase. This phenomenon occurs because the thermophoretic force becomes more pronounced at higher temperatures, causing more particles to move towards the apparatus. Further it is seen that stationary cone and a rotating disk exhibit enhanced heat transport with increasing radiation parameter values. This suggests that elevating the radiation parameter leads to more efficient heat transfer in this specific model. Highest flow intensity observed around the cone. This reference data is subjected to testing, validation, and training processes to fine-tune the approximate solution to meet the desired results. The accuracy, stability, capacity, and robustness of NN-LMM are verified through mean squared error (MSE)-based fitness curves, regression plots, error histograms and absolute error assessments. A comparative analysis demonstrates the accuracy of the proposed solver, with absolute errors in the range of 10-9 to 10-6 for all influential parameters results.

Experimental and Numerical Investigation of Effect of Inclination on sCO2 Heat Transfer in a Circular Pipe

Abstract Supercritical carbon dioxide (sCO2) can be utilized as a working fluid in various thermal systems including large-scale power cycles; portable power production units, centralized coolant systems, and standalone cooling devices. However, the lack of accurate prediction tools such as heat transfer coefficient correlations and insufficient research studies about the mechanisms controlling heat transfer processes are hindering its practical realization for key energy and cooling systems. The overall objective of this study is to extend fundamental knowledge about heat transfer and fluid dynamic processes in conduits pertinent to sCO2 with an emphasis on flow direction and inclination effects. This paper presents the study on effects of gravity, buoyancy on sCO2 flow at temperatures near and away from the pseudo-critical temperature. The experimental setup consists of a high temperature and pressure sCO2 heat transfer loop and flow testing facility. Recently, researched sCO2 heat exchangers can have tubes oriented at different angles such as 45 deg or 90 deg to horizontal. For optimized design of efficient and cost-effective turbomachinery components utilizing sCO2 as the heat transfer fluid, an understanding of convective heat transfer inside a tube/pipe is equally as important as external heat transfer. This paper presents an experimental and numerical study on sCO2 heat transfer at various inclinations with angles ranging from 0 deg (horizontal) to 90 deg (vertical) along with upward and downward flow direction with different inlet temperatures. Thermocouple-based temperature measurement is utilized at multiple locations within the test section axially and circumferentially to study the temperature distributions on the tube surface. Computational fluid dynamics (CFD) simulations have been performed using ANSYS Fluent to complement experimental data. The CFD and experiment have been analyzed against the well-known Gnielinski Nusselt number correlation.

Projection–Subtraction X-ray Imaging Scheme for Studying Fast Fluid-Dynamics Processes in Porous Media

Projection–Subtraction X-ray Imaging Scheme for Studying Fast Fluid-Dynamics Processes in Porous Media

Computational Fluid Dynamics and Its Potential Applications for the ENT Clinician.

This article is an examination of computational fluid dynamics in the field of otolaryngology, specifically rhinology. The historical development and subsequent application of computational fluid dynamics continues to enhance our understanding of various sinonasal conditions and surgical planning in the field today. This article aims to provide a description of computational fluid dynamics, the methods for its application, and the clinical relevance of its results. Consideration of recent research and data in computational fluid dynamics demonstrates its use in nonhistological disease pathology exploration, accompanied by a large potential for surgical guidance applications. Additionally, this article defines in lay terms the variables analyzed in the computational fluid dynamic process, including velocity, wall shear stress, area, resistance, and heat flux.

Web Application for Mathematical Modeling of Unsteady Oil Flow in Porous Medium

Introduction. The analysis and interpretation of hydrodynamic research data are based on theoretical models and computational algorithms. Despite the demand for this topic, numerous issues related to unsteady fluid flows in oil reservoirs still require solutions. Therefore, mathematical setting of problems related to the account of unsteady fluid flow, the development of effective numerical methods and algorithms, their solution using modern web technologies are pressing. This study is aimed at developing a web application for mathematical modeling of the process of fluid filtration in single- porosity reservoirs when conducting a hydrodynamic study at a production well.Materials and Methods. To solve the problem, the methods of continuum mechanics and computational mathematics were applied. A model of oil flow in a single-porosity reservoir was presented. Python and JavaScript programming languages were used in the development of the application. The calculation results were stored in a relational database implemented using PostgreSQL tools.Results. A new web application has been developed for modeling the oil filtration process in single- porosity reservoirs. It is applicable for studying fluid dynamic processes and can be used to predict flowrates, production and calculation of optimal well operation modes.Discussion and Conclusion. The developed web application provides for building pressure and temperature fields in the reservoir near a flowing and shut-in production well and at various distances from it. This information makes it possible to urgently assess the duration of hydrodynamic studies, as well as to regulate the operation of wells. The application can be deployed in an existing network infrastructure and use all the functionality by connecting to a remote server. It is optimized for use on various platforms and has broad prospects for further development.

A two-phase pure slurry model for planetary cores: one-dimensional solutions and implications for Earth's F-layer

We develop and analyse a continuum model of two-phase slurry dynamics for planetary cores. Mixed solid–liquid slurry regions may be ubiquitous in the upper cores of small terrestrial bodies and have also been invoked to explain anomalous seismic structure in the F-layer at the base of Earth's liquid iron core. These layers are expected to influence the dynamics and evolution of planetary cores, including their capacity to generate global magnetic fields; however, to date, models of two-phase regions in planetary cores have largely ignored the complex fluid dynamics that arises from interactions between phases. As an initial application of our model, and to focus on fundamental fluid dynamical processes, we consider a non-rotating and non-magnetic slurry comprised of a single chemical component with a temperature that is tied to the liquidus. We study one-dimensional solutions in a configuration set up to mimic Earth's F-layer, varying gravitational strength$R$, the solid/liquid viscosity ratio$\lambda _{\mu }$and the interaction parameter$K$, which measures friction between the phases. We develop scalings describing behaviour in the limit$R \gg 1$and$\lambda _{\mu } \gg 1$, which are in excellent agreement with our numerical results. Application to Earth's core, where$R \sim 10^{28}$and$\lambda _{\mu } \sim 10^{22}$, suggests that a pure iron slurry F-layer would contain a mean solid fraction of at most 5 %.

Fluid Dynamics of Airtanker Firefighting

Airtanker firefighting is the most spectacular tool used to fight wildland fires. However, it employs a rudimentary large-scale spraying technology operating at a high speed and a long distance from the target. This review gives an overview of the fluid dynamics processes that govern this practice, which are characterized by rich and varied physical phenomena. The liquid column penetration in the air, its large-scale fragmentation, and an intense surface atomization give shape to the rainfall produced by the airtanker and the deposition of the final product on the ground. The cloud dynamics is controlled by droplet breakup, evaporation, and wind dispersion. The process of liquid deposition onto the forest canopy is full of open questions of great interest for rainfall retention in vegetation. Of major importance, but still requiring investigation, is the role of the complex non-Newtonian viscoelastic and shear-thinning behavior of the retardant dropped to stop the fire propagation. The review describes the need for future research devoted to the subject.

Prospects for the oil and gas potential of the zone of abnormally high reservoir pressures in the Yamalo-Nenets Autonomous Okrug assuming deep fluid dynamics

In Western Siberia, the zone of abnormally high reservoir pressures (AHRP) covers an area of more than 500 thousand km2 in the north of the basin. It begins with a clay layer above the Achimov formation of Neocomian sandy-silty formation, covers the Achimov formation, Upper–Lower Jurassic, Triassic and partially Paleozoic and is subject to tectonic control, which indicates the deep origin of this phenomenon. Gas-pressure, or gas-dynamic, theory of AHRP, proposed by K.A. Anikiev in the 70s of the 20th century, allows us to assess the prospects for oil and gas content of the AHRP zone higher than it is commonly-accepted. Analysis of the results of previously completed geological exploration work on the deep horizons of the Yamalo-Nenets Autonomous Okrug indicates that their relatively low efficiency (50–60%) is associated with the insufficiently high quality of well operations, primarily cementing, which is also due to the influence of AHRP. In all wells drilled to deep horizons, direct signs of oil and gas potential were obtained and cementing defects were identified. It is concluded that deep fluid-dynamic processes (active, pressure degassing of the Earth’s interior) are responsible both for the saturation of reservoir rocks with hydrocarbons and for the dynamics of their filling (ultra-high pressures and velocities), which determine the main characteristics of reservoir rocks. Recognition of a deep source of hydrocarbons will not only make it possible to fundamentally increase the resource base of the AHRP zone, but will also require a revision of ideas about the formation and structure of hydrocarbon deposits in this zone and the petrophysical substantiation of their models. However, to realize the unique hydrocarbon potential of the AHRP zone, it is necessary, first of all, to improve the quality of deep wells construction and appropriate information content.

Multimaterial fiber as a physical simulator of a capillary instability

Capillary breakup of cores is an exclusive approach to fabricating fiber-integrated optoelectronics and photonics. A physical understanding of this fluid-dynamic process is necessary for yielding the desired solid-state fiber-embedded multimaterial architectures by design rather than by exploratory search. We discover that the nonlinearly complex and, at times, even chaotic capillary breakup of multimaterial fiber cores becomes predictable when the fiber is exposed to the spatiotemporal temperature profile, imposing a viscosity modulation comparable to the breakup wavelength. The profile acts as a notch filter, allowing only a single wavelength out of the continuous spectrum to develop predictably, following Euler-Lagrange dynamics. We argue that this understanding not only enables designing the outcomes of the breakup necessary for turning it into a technology for materializing fiber-embedded functional systems but also positions a multimaterial fiber as a universal physical simulator of capillary instability in viscous threads.

Efficient Solvers for Nonlinear Partial Differential Equations in Computational Fluid Dynamics

Improving the computational efficiency of solving nonlinear partial differential equations (PDEs) is a key area of focus for this research, which has the ability to significantly impact computational fluid dynamics (CFD) simulations. The complexities of fluid flow phenomena can be difficult for traditional methods to handle, leading to simulations that are resource-intensive and computationally slow. The urgent requirement for improved computational tools capable of quickly and precisely simulating complex fluid dynamics scenarios is the driving force behind the significance of this research. New approaches are needed to solve nonlinear partial differential equations (PDEs) in computational fluid dynamics (CFD), which are characterized by turbulent flows, fluid-structure interactions, and high computing resource intensity. Adaptive mesh optimization and high-performance computing architectures are combined in the AMO-HPI technique, which provides a strategic solution. Adaptive Mesh Optimization based on High-Performance Integration (AMO-HPI) is suggested in this research as a way to improve the accuracy of solutions by focusing computational resources on regions of interest through adaptive mesh refinement and coarsening. Making effective use of shared-memory parallelism and message-passing interfaces (MPI) to divide up computational tasks. Improvements in engineering system performance, reliability, and safety can be achieved by the effective simulation of fluid dynamics processes, which opens up new design areas that were previously unreachable. When compared to other methods, AMO-HPI performs better in simulations due to its optimal resource utilization and decreased processing time. In addition to improving CFD methodology, this research could have far-reaching effects on engineering design processes in several industries, which would be great for innovation and sustainable development.

Swirling Fluid Reduces the Bounce of Partially Filled Containers.

Certain spatial distributions of water inside partially filled containers can significantly reduce the bounce of the container. In experiments with containers filled to a volume fraction ϕ, we show that rotation offers control and high efficiency in setting such distributions and, consequently, in altering bounce markedly. High-speed imaging evidences the physics of the phenomenon and reveals a rich sequence of fluid-dynamics processes, which we translate into a model that captures our overall experimental findings.

Benchmarking a new TH2M implementation in OGS-6 with regard to processes relevant for nuclear waste disposal

In this paper, thermo-hydro-mechanically (THM) coupled processes triggered during the construction, operation and closure of a deep geological repository for heat generating, high level radioactive waste are discussed based on a generic disposal concept. For this purpose, we are using the numerical non-isothermal two-phase–two-component flow in deformable porous media (TH2M) implementation (Grunwald et al. in Geomech Geophys Geo-energy Geo-resour, 2022) in the open-source software OpenGeoSys (Bilke et al. in Transport Porous Media 130(1):337–361, 2019, https://doi.org/10.1007/s11242-019-01310-1). THM coupled effects covered in this work focus on single and two-phase-flow phenomena, gas and heat generation as well as poro-elastic medium deformation. A suitable set of benchmarks covering aforementioned THM-effects, devised in the scope of the BenVaSim benchmarking project (Lux et al. in Synthesis report. BenVaSim—International Benchmarking for Verification and Validation of TH2M Simulators with Special Consideration of Fluid Dynamical Processes in Radioactive Waste Repository Systems. Tech. rep., 2021, https://doi.org/10.13140/RG.2.2.28998.34887) is chosen and one additional benchmark is presented, allowing for the demonstration and comparison of the OGS-6 TH2M implementation against results obtained by other well-established codes used in the field. Apart from the code comparison, the benchmarks also serve as means to analyze THM coupled processes in a repository based on very simplified geometries. Therefore, they can help to improve the process understanding, but any quantitative results should not be interpreted as predictions of the behaviour of a real repository. The results obtained in this work agree well with the results presented by the project partners in BenVaSim—both in single phasic, fully liquid saturated cases and in partially saturated two phase regions. Hence, the suitability of the OGS-6 TH2M implementation for the application in the field of radioactive waste management, supporting the safety case and analyzing the integrity of the geological and geotechnical barrier systems is demonstrated. Finally, a detailed discussion of observed phenomena in the benchmarks increases our understanding and confidence in the prediction of the behaviour of TH2M coupled systems in the context of deep geological radioactive waste disposal.

Hydrophobic ionic liquid-modified graphene via fluid-dynamic process for ion-to-electron transducers for all-solid-state potentiometric sensors

Hydrophobic ionic liquid-modified graphene via fluid-dynamic process for ion-to-electron transducers for all-solid-state potentiometric sensors

Investigation of the do minant Rossiter modal tones at the locked-on state

Rossiter resonance noise, which is flow driven cavity noise involving fluid dynamics processes, was investigated in a low-speed wind tunnel. The experiments include near-field and far-field acoustic field measurements of the rectangular cavity at different depths as well as velocity measurements of the shear layer. The rectangular cavity geometry was also designed to enable the Rossiter mode to interact with the first-order depth resonant mode in a locked-on state. The experimental results showed the variation of frequency and overall sound pressure level (OASPL) when the self-sustained oscillation was in the locked-on state. When the self-sustained oscillatory mode is in the unlocked state, the frequency of the self-sustained oscillatory mode can be predicted well by using the Rossiter formula, whereas the frequency of the dominant Rossiter mode in the locked-on state will be predicted inconsistently. Moreover, the corresponding selection of the dominant mode also depends on the resonant mode, and there are local variations in the OASPL. The results indicate that the ratio of the resonant mode frequency to the self-sustained oscillatory fundamental frequency, Rd, plays a key role in the locked-on state. In this paper, the Rossiter model is modified using the parameter Rd for the prediction of the dominant mode frequency in the locked-on state. The difference between the current formula and previous studies is that the variation of the phase delay coefficient γ with velocity is considered, and γ is related to Rd. Finally, the accuracy of the formulation is verified with experimental data from different models.