Search results for: coda wave velocity

Authors: Ning Chang, Zelong Yuan, Yunpeng Wang, Jianchun Wang

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The utilization of Large Eddy Simulation (LES) has been extensive in turbulence research. LES concentrates on resolving the significant grid-scale motions while representing smaller scales through subfilter-scale (SFS) models. The deconvolution model, among the available SFS models, has proven successful in LES of engineering and geophysical flows. Nevertheless, the thorough investigation of how sub-filter scale dynamics and filter anisotropy affect SFS modeling accuracy remains lacking. The outcomes of LES are significantly influenced by filter selection and grid anisotropy, factors that have not been adequately addressed in earlier studies. This study examines two crucial aspects of LES: Firstly, the accuracy of direct deconvolution models (DDM) is evaluated concerning sub-filter scale (SFS) dynamics across varying filter-to-grid ratios (FGR) in isotropic turbulence. Various invertible filters are employed, including Gaussian, Helmholtz I and II, Butterworth, Chebyshev I and II, Cauchy, Pao, and rapidly decaying filters. The importance of FGR becomes evident as it plays a critical role in controlling errors for precise SFS stress prediction. When FGR is set to 1, the DDM models struggle to faithfully reconstruct SFS stress due to inadequate resolution of SFS dynamics. Notably, prediction accuracy improves when FGR is set to 2, leading to accurate reconstruction of SFS stress, except for cases involving Helmholtz I and II filters. Remarkably high precision, nearly 100%, is achieved at an FGR of 4 for all DDM models. Furthermore, the study extends to filter anisotropy and its impact on SFS dynamics and LES accuracy. By utilizing the dynamic Smagorinsky model (DSM), dynamic mixed model (DMM), and direct deconvolution model (DDM) with anisotropic filters, aspect ratios (AR) ranging from 1 to 16 are examined in LES filters. The results emphasize the DDM’s proficiency in accurately predicting SFS stresses under highly anisotropic filtering conditions. Notably high correlation coefficients exceeding 90% are observed in the a priori study for the DDM’s reconstructed SFS stresses, surpassing those of the DSM and DMM models. However, these correlations tend to decrease as filter anisotropy increases. In the a posteriori analysis, the DDM model consistently outperforms the DSM and DMM models across various turbulence statistics, including velocity spectra, probability density functions related to vorticity, SFS energy flux, velocity increments, strainrate tensors, and SFS stress. It is evident that as filter anisotropy intensifies, the results of DSM and DMM deteriorate, while the DDM consistently delivers satisfactory outcomes across all filter-anisotropy scenarios. These findings underscore the potential of the DDM framework as a valuable tool for advancing the development of sophisticated SFS models for LES in turbulence research.

Keywords: deconvolution model, large eddy simulation, subfilter scale modeling, turbulence

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Authors: K. Aravinth, C. T. Vignesh

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The usage of pulse tube cryocoolers is significantly increased mainly due to the advantage of the absence of moving parts. The underlying idea of this project is to optimize the design of pulse tube, regenerator, a resonator in cryocooler and analyzing the thermo-acoustic oscillations with respect to the design parameters. Computational Fluid Dynamic (CFD) model with time-dependent validation is done to predict its performance. The continuity, momentum, and energy equations are solved for various porous media regions. The effect of changing the geometries and orientation will be validated and investigated in performance. The pressure, temperature and velocity fields in the regenerator and pulse tube are evaluated. This optimized design performance results will be compared with the existing pulse tube cryocooler design. The sinusoidal behavior of cryocooler in acoustic streaming patterns in pulse tube cryocooler will also be evaluated.

Keywords: acoustics, cryogenics, design, optimization

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Authors: A.Kouidri, B.Madani

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The present work deals with the study of the heat transfer in a rectangular channel equipped with a metallic foam. The tested metallic foam sample is made from copper with 20 PPI (Pore per Inch Linear) and 93% of porosity and the working fluid used is the n-pentane. In the present work the independent variables are the velocity in the range from 0.02 to 0.06 m/s and a boiling heat flux rate varying between 30 and 70 kW/m2. The heat transfer coefficient is presented versus boiling heat flux, vapor quality and superheat ΔTsat. The thermal results are compared to those found for a plain tube for the same conditions. The comparison with the plain tube shows that the insert of a metallic foam enhances the heat transfer coefficient by a factor between 1.3 and 3.

Keywords: boiling, metallic foam, heat transfer, plate heat exchanger

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Authors: Mohammed F. Alhaddad

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Vortex induced Vibrations (VIV) is a phenomenon that occurs as a result of a flow passing by a bluff body. This phenomenon has been mainly studied to be suppressed to prevent fatigue and instability in offshore platforms. In 2006, some studies were conducted to maximize VIV instead of suppressing it, as these studies claimed that VIV is a potential method of generating energy. The aim of this paper is to identify factors for maximizing oscillation amplitude generated by VIV in order to enhance the energy harnessed through this method. The experimental study in this paper will examine the effect of oscillating cylinder diameter, surface roughness, the location of surface roughness with respect to the centerline of the oscillating cylinder and the velocity on the oscillation amplitude of the used module.

Keywords: energy, generation, generating, vibration, vortex.

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Authors: Yamileth P. Herrera, Ronald R. Gutierrez, Carlos, Pacheco-Bustos

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This research aims at the numerical modeling of a rectangular contact tank in order to improve the hydrodynamic behavior and the retention time of the water to be treated with the disinfecting agent. The methodology to be followed includes a hydraulic analysis of the tank to observe the fluid velocities, which will allow evidence of low-speed areas that may generate pathogenic agent incubation or high-velocity areas, which may decrease the optimal contact time between the disinfecting agent and the microorganisms to be eliminated. Based on the results of the numerical model, the efficiency of the tank under the geometric and hydraulic conditions considered will be analyzed. This would allow the performance of the tank to be improved before starting a construction process, thus avoiding unnecessary costs.

Keywords: contact tank, numerical models, hydrodynamic modeling, residence time

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Authors: Mehdi Fuladipanah

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Stepped spillway includes several steps from the crest to the toe. The steps of stepped spillway could cause to decrease the energy with making energy distribution in the longitude mode and also to reduce the outcome speed. The aim of this study was to stimulate the stepped spillway combined with stilling basin-step using Fluent model and the turbulent superficial flow using RNG, K-ε. The free surface of the flow was monitored by VOF model. The velocity and the depth of the flow were measured by tail water depth by the numerical model and then the dissipated energy was calculated along the spillway. The results indicated that the stilling basin-step complex may cause energy dissipation increment in the stepped spillway. Also, the numerical model was suggested as an effective method to predict the circular and complicated flows in the stepped spillways.

Keywords: stepped spillway, fluent model, VOF model, K-ε model, energy distribution

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Authors: Ghasem Yazadani, Hamidreza Ahmadi, Masoud Ahmadi, Sajad Rezazadeh

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In this paper with designing and proposing a compound of a heating and cooling system has been tried to show effect of this system on preventing esophagus cancer that can be caused by hot and cold drinks such as tea, coffee and ice water. This system has been simulated mechanically by fluent software and also has been validated by experimental way and a comprehensive result has been presented. Both of solution ways show that this system can reduce or increase temperature of drink to safe very dramatically and it can be a huge step toward consuming drinks safely and also it can be efficient about time issues. The system consists of a temperature sensor and an electronic controller that has a computer program to act automatically this task. Also this system has been presented after many different simulations and has been tried to find the best one in the point view of velocity of heating and cooling.

Keywords: fluent, heat transfer, controller, esophagus cancer

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Authors: Badar Muneer, Waseem Shabir, Faisal Karim Shaikh

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Substrate Integrated waveguide based 6-element array of Planar Inverted F antenna (PIFA) has been presented and analyzed parametrically in this paper. The antenna is fed with coupled transverse slots on a plate laminated waveguide cavity to ensure wide bandwidth and simplicity of feeding network. The two-layer structure has one layer dedicated for feeding network and the top layer dedicated for radiating elements. It has been demonstrated that the presented feeding technique for feeding such class of array antennas can be far simple in structure and miniaturized in size when it comes to designing large phased array antenna systems. A good return loss and standing wave ratio of 2:1 has been achieved while maintaining properties of typical PIFA.

Keywords: feeding network, laminated waveguide, PIFA, transverse slots

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Authors: Noha Al Majid, Said Mazer, Moulhime El Bekkali, Catherine Algani, Mahmoud Mehdi

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A low noise figure (NF) and high linearity V-band Low Noise Amplifier (LNA) is reported in this article. The LNA compromises a three-stage cascode configuration. This LNA will be used as a part of a WPAN (Wireless Personal Area Network) receiver in the millimeter-wave band at 60 GHz. It is designed according to the MMIC technology (Monolithic Microwave Integrated Circuit) in PH 15 process from UMS foundry and uses a 0.15 μm GaAs PHEMT (Pseudomorphic High Electron Mobility Transistor). The particularity of this LNA compared to other LNAs in literature is its very low noise figure which is equal to 1 dB and its high linearity (IIP3 is about 22 dB). The LNA consumes 0.24 Watts, achieving a high gain which is about 23 dB, an input return loss better than -10 dB and an output return loss better than -8 dB.

Keywords: low noise amplifier, V-band, MMIC technology, LNA, amplifier, cascode, pseudomorphic high electron mobility transistor (PHEMT), high linearity

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Authors: Jan Busch, Peter Nyhuis

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Considering the challenges of short product life cycles and growing variant diversity, cost minimization and manufacturing flexibility increasingly gain importance to maintain a competitive edge in today’s global and dynamic markets. In this context, an aerodynamic part feeding system for high-speed industrial assembly applications has been developed at the Institute of Production Systems and Logistics (IFA), Leibniz Universitaet Hannover. The aerodynamic part feeding system outperforms conventional systems with respect to its process safety, reliability, and operating speed. In this paper, a multi-objective optimisation of the aerodynamic feeding system regarding the orientation rate, the feeding velocity and the required nozzle pressure is presented.

Keywords: aerodynamic feeding system, genetic algorithm, multi-objective optimization, workpiece orientation

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Authors: R. O. Ocaya, J. J. Terblans

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The laws of Newtonian mechanics allow ab-initio molecular dynamics to model and simulate particle trajectories in material science by defining a differentiable potential function. This paper discusses some considerations for the coding of ab-initio programs for simulation on a standalone computer and illustrates the approach by C language codes in the context of embedded metallic atoms in the face-centred cubic structure. The algorithms use velocity-time integration to determine particle parameter evolution for up to several thousands of particles in a thermodynamical ensemble. Such functions are reusable and can be placed in a redistributable header library file. While there are both commercial and free packages available, their heuristic nature prevents dissection. In addition, developing own codes has the obvious advantage of teaching techniques applicable to new problems.

Keywords: C language, molecular dynamics, simulation, embedded atom method

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Authors: Humanyun Zahir, Irtsam Ghazi

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This report outlines the basic installation and operation of magnetic inductive flow velocity sensors on large underground cooling water pipelines. Research on the effects of cathodic protection as well as into other factors that might influence the overall performance of the meter are presented in this paper. The experiments were carried out on an immersion type magnetic meter specially used for flow measurement of cooling water pipeline. An attempt has been made in this paper to outline guidelines that can ensure accurate measurement related to immersion type magnetic meters on underground pipelines.

Keywords: magnetic induction, flow meter, Faraday's law, immersion, cathodic protection, anode, cathode, flange, grounding, plant information management system, electrodes

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Authors: O. W. Lawal, L. O. Ahmed, Y. K. Ali

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The unsteady MHD Poiseuille flow of a third grade fluid between two parallel horizontal nonconducting porous plates is studied with heat transfer. The two plates are fixed but maintained at different constant temperature with the Joule and viscous dissipation taken into consideration. The fluid motion is produced by a sudden uniform exponential decaying pressure gradient and external uniform magnetic field that is perpendicular to the plates. The momentum and energy equations governing the flow are solved numerically using Maple program. The effects of magnetic field and third grade fluid parameters on velocity and temperature profile are examined through several graphs.

Keywords: exponential decaying pressure gradient, MHD flow, Poiseuille flow, third grade fluid

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Authors: Misarah Abdelaziz, L Djenidi, Mergen H. Ghayesh, Rey Chin

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Measurements of a turbulent boundary layer over 3D sinusoidal roughness are performed for friction Reynolds numbers ranging from 650 < Reτ < 2700. This surface was fabricated by a Multicam CNC Router machine of an acrylic sheet to have an amplitude of k/2 = 0.8 mm and an equal wavelength of 8k in both streamwise and spanwise directions, a 0.6 mm stepover and 12 mm ball nose cutter was used. Single hotwire anemometry measurements are done at one location x=1.5 m downstream at different freestream velocities under zero-pressure gradient conditions. As expected, the roughness causes a downward shift on the wall-unit normalised streamwise mean velocity profile when compared to the smooth wall profile. The shift is increasing with increasing Reτ, 1.8 < ∆U+ < 6.2. The coefficient of friction is almost constant at all cases Cf = 0.0042 ± 0.0002. The results show a gradual reduction in the inner peak of profiles with increasing Reτ until fully destruction at Reτ of 2700.

Keywords: hotwire, roughness, TBL, ZPG

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Authors: Mengqi Zhu, Chang Nyung Kim

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This study numerically investigates three-dimensional liquid-metal (LM) magnetohydrodynamic (MHD) flows in a misaligned duct under a uniform magnetic field. The duct consists of two misaligned horizontal channels (one is inflow channel, the other is outflow channel) and one central vertical channel. Computational fluid dynamics simulations are performed to predict the behavior of the MHD flows, using commercial code CFX. In the current study, a case with Hartmann number 1000 is considered. The electromagnetic features of LM MHD flows are elucidated to examine the interdependency of the flow velocity, current density, electric potential, pressure drop and Lorentz force. The results show that pressure decreases linearly along the main flow direction.

Keywords: CFX, liquid-metal magnetohydrodynamic flows, misaligned duct, pressure drop

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Authors: Barenten Suciu

Abstract:

In this paper, effectiveness of a machine design process is evaluated on the basis of the specific output calculus. Concretely, a screw-worm gear mechanical transmission is designed by using the classical and the 3D-CAD methods. Strength analysis and drawing of the designed parts is substantially aided by employing the SolidWorks software. Quality of the design process is assessed by manufacturing (printing) the parts, and by computing the efficiency, specific load, as well as the specific output (work) of the mechanical transmission. Influence of the stroke, travelling velocity and load on the mechanical output, is emphasized. Optimal design of the mechanical transmission becomes possible by the appropriate usage of the acquired results.

Keywords: mechanical transmission, design, screw, worm-gear, efficiency, specific output, 3D-printing

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Authors: A.Zitouni, S.Bentata, B.Bouadjemi, T.Lantri, W. Benstaali, Z.Aziz, S.Cherid, A. Sefir

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Using the first-principles method, we investigate the structural, electronic, and magnetic properties of the diluted magnetic semiconductors CdCoTe and CdMnTe in the zinc blende phase with 12.5% of Cr. The calculations are performed by a developed full potential augmented plane wave (FP-L/APW) method within the spin density functional theory (DFT). As exchange–correlation potential, we used the new generalized gradient approximation GGA. Structural properties are determined from the total energy calculations and we found that these compounds are stable in the ferromagnetic phase. We discuss the electronic structures, total and partial densities of states and local moments. Finally, CdCoTe and CdMnTe in the zinc-blend phase show the half-metallic ferromagnetic nature and are expected to be potential materials for spintronic devices.

Keywords: DFT, GGA, band structures, half-metallic, spintronics

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Authors: Hariti Rafika, Fekih Malika, Saighi Mohamed

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Heat transfer by natural convection in storage tanks for LNG is extremely related to heat gains through the walls with thermal insulation is not perfectly efficient. In this paper, we present the study of natural convection in the unsteady regime for natural gas in aware phase using the fluent software. The gas is just on the surface of the liquid phase. The CFD numerical method used to solve the system of equations is based on the finite volume method. This numerical simulation allowed us to determine the temperature profiles, the stream function, the velocity vectors and the variation of the heat flux density in the vapor phase in the LNG storage tank volume. The results obtained for a general configuration, by numerical simulation were compared to those found in the literature.

Keywords: numerical simulation, natural convection, heat gains, storage tank, liquefied natural gas

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Authors: Arash Mirabdolah Lavasani, Taher Maarefdoost

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Effect of blockage ratio on heat transfer from non-circular tube is studied experimentally. For doing this experiment a suction type low speed wind tunnel with test section dimension of 14×14×40 and velocity in rage of 7-20 m/s was designed. The blockage ratios varied between 1.5 to 7 and Reynolds number based on equivalent diameter varies in range of 7.5×103 to 17.5×103. The results show that by increasing blockage ratio from 1.5 to 7, drag coefficient of the cam shaped tube decreased about 55 percent. By increasing Reynolds number, Nusselt number of the cam shaped tube increases about 40 to 48 percent in all ranges of blockage ratios.

Keywords: wind tunnel, non-circular tube, blockage ratio, experimental heat transfer, cross-flow

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Authors: A. Shishkin, V. Mironovs, D. Goljandin, A. Korjakins

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The goal of this work is to develop sustainable and durable ceramic cellular structures using widely available natural resources- clay and milled waste glass. Present paper describes method of obtaining clay ceramic foam (CCF) with addition of milled waste glass in 5, 7 and 10 wt% by direct foaming with high speed mixer-disperser (HSMD). For more efficient clay and waste glass milling and mixing, the high velocity disintegrator was used. The CCF with 5, 7, and 10 wt% were obtained at 900, 950, 1000 and 1050 °C firing temperature and they have demonstrated mechanical compressive strength for all 12 samples ranging from 3.8 to 14.3 MPa and porosity 76-65%. Obtained CCF has compressive strength 14.3 MPa and porosity 65.3%.

Keywords: ceramic foam, waste glass, clay foam, glass foam, open cell, direct foaming

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Authors: Y. J. Song, Q. S. Xu, X. C. Wang, H. Wang, C. Q. Li

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The catalyst of copper oxide loaded on HZSM-5 was developed for nitrous oxide (N₂O) direct decomposition. The kinetic of nitrous oxide decomposition was studied for CuO/HZSM-5 catalyst prepared by incipient wetness impregnation method. The external and internal diffusion of catalytic reaction were considered in the investigation. Experiment results indicated that the external diffusion was basically eliminated when the reaction gas mixture gas hourly space velocity (GHSV) was higher than 9000h⁻¹ and the influence of the internal diffusion was negligible when the particle size of the catalyst CuO/HZSM-5 was small than 40-60 mesh. The experiment results showed that the kinetic of catalytic decomposition of N₂O was a first-order reaction and the activation energy and the pre-factor of the kinetic equation were 115.15kJ/mol and of 1.6×109, respectively.

Keywords: catalytic decomposition, CuO/HZSM-5, kinetic, nitrous oxide

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Authors: Brian W. Mandikiana

Abstract:

Homestead food production has the potential to alleviate hunger, improve health and nutrition for children and adults. This article examines the relationship between self-reported hunger and homestead food production using the ordered probit model. A sample of households participating in homestead food production was drawn from the first wave of the South African National Income Dynamics Survey, a nationally representative cross-section. The sample selection problem was corrected using an ordered probit model with sample selection approach. The findings show that homestead food production exerts a positive and significant impact on children and adults’ ability to cope with hunger and malnutrition. Yet, on the contrary, potential gains of homestead food production are threatened by shocks such as crop failure.

Keywords: agriculture, hunger, nutrition, sample selection

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Authors: Huseyin Gokberk, Shian Gao

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CFD analysis of the relationship between the coefficients of lift and drag with respect to the angle of attack on a NACA 2412 wing section of an RC plane is conducted. Both the 2D and 3D models are investigated with the turbulence model. The 2D analysis has a free stream velocity of 10m/s at different AoA of 0°, 2°, 5°, 10°, 12°, and 15°. The induced drag and drag coefficient increased throughout the changes in angles even after the critical angle had been exceeded, whereas the lift force and coefficient of lift increased but had a limit at the critical stall angle, which results in values to reduce sharply. Turbulence flow characteristics are analysed around the aerofoil with the additions caused due to a finite 3D model. 3D results highlight how wing tip vortexes develop and alter the flow around the wing with the effects of the tapered configuration.

Keywords: CFD, turbulence modelling, aerofoil, angle of attack

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Authors: Muhammad Sohail Khan, Rehan Ali Shah

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The present paper applies the optimal homotopy perturbation method (OHPM) and the optimal homotopy asymptotic method (OHAM) introduced recently to obtain analytic approximations of the non-linear equations modeling the flow of polymer in case of wire coating of a corotational Maxwell fluid. Expression for the velocity field is obtained in non-dimensional form. Comparison of the results obtained by the two methods at different values of non-dimensional parameter l₁₀, reveal that the OHPM is more effective and easy to use. The OHPM solution can be improved even working in the same order of approximation depends on the choices of the auxiliary functions.

Keywords: corotational Maxwell model, optimal homotopy asymptotic method, optimal homotopy perturbation method, wire coating die

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Authors: Matheus Fernando Pereira, Varese Salvador Timoteo

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In this work, we have solved the two-dimensional advection-diffusion equation numerically for a spatially dependent solute dispersion along non-uniform flow with a pulse type source in order to make a systematic study on the influence of medium heterogeneity, initial flow velocity, and initial dispersion coefficient parameters on the solutions of the equation. The behavior of the solutions is then investigated as we change the three parameters independently. Our results show that even though the parameters represent different physical features of the system, the effect on their variation is very similar. We also observe that the effects caused by the parameters on the concentration depend on the distance from the source. Finally, our numerical results are in good agreement with the exact solutions for all values of the parameters we used in our analysis.

Keywords: advection-diffusion equation, dispersion, numerical methods, pulse-type source

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Authors: Yue Yan, Chang Nyung Kim

Abstract:

The numerical investigation of the three-dimensional liquid-metal (LM) magnetohydrodynamic (MHD) flows in a curved duct with flow channel insert (FCI) is presented in this paper, based on the computational fluid dynamics (CFD) method. A uniform magnetic field is applied perpendicular to the duct. The interdependency of the flow variables is examined in terms of the flow velocity, current density, electric potential and pressure. The electromagnetic characteristics of the LM MHD flows are reviewed with an introduction of the electric-field component and electro-motive component of the current. The influence of the existence of the FCI on the fluid flow is investigated in detail. The case with FCI slit located near the side layer yields smaller pressure gradient with stable flow field.

Keywords: curved duct, flow channel insert, liquid-metal, magnetohydrodynamic

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Authors: Prashant Sharma, Jyotsna Dutta Majumdar

Abstract:

The present study concerns development of a composite thermal barrier coating consisting of a mixture of La2O3 and YSZ (with 8 wt.%, 32 wt.% and 50 wt.% 50% La2O3) by plasma spray deposition technique on a CoNiCrAlY based bond coat deposited on Inconel 718 substrate by high velocity oxy-fuel deposition (HVOF) technique. The addition of La2O3 in YSZ causes the formation of pyrochlore (La2Zr2O7) phase in the inter splats boundary along with the presence of LaYO3 phase. The coefficient of thermal expansion is significantly reduced from due to the evolution of different phases and structural defects in the sprayed coating. The activation energy for TGO growth under isothermal and cyclic oxidation was increased in the composite coating as compared to YSZ coating.

Keywords: plasma spraying, oxidation resistance, thermal barrier coating, microstructure, X-ray method

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Authors: O. Arbouche, Y. Benallou, K. Amara

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We report a first-principles study of structural, electronic and magnetic properties of ternary plumbides (rare earth-transition metal-Plumb) Dy2Ni2Pb crystallizes with the orthorhombic structure of the Mn2AlB2 type (space group Cmmm), were studied by means of the full-relativistic version of the full-potential augmented plane wave plus local orbital method within the frame work of spin-polarized density functional theory (SP-DFT). The electronic exchange-correlation energy is described by generalized gradient approximation (GGA). We have calculated the lattice parameters, bulk modulii and the first pressure derivatives of the bulk modulii, total densities of states and magnetic properties. The calculated total magnetic moment is found to be equal to 9.52 μB.

Keywords: spin-polarized, magnetic properties, Dy2Ni2Pb, Density functional theory

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Authors: S. Ghafoori, A. Al-Harbi, B. Al-Ajmi, A. Al-Shaalan, A. Al-Ajmi, M. Ali Juma

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Seismic methods play an important role in the exploration for hydrocarbon reservoirs. However, the success of the method depends strongly on the reliability of the measured or predicted information regarding the velocity of sound in the media. Speed of sound has been used to study the thermodynamic properties of fluids. In this study, experimental data are reported and analyzed on the speed of sound in toluene and octane binary mixture. Three-factor three-level Box-Benhkam design is used to determine the significance of each factor, the synergetic effects of the factors, and the most significant factors on speed of sound. The developed mathematical model and statistical analysis provided a critical analysis of the simultaneous interactive effects of the independent variables indicating that the developed quadratic models were highly accurate and predictive.

Keywords: experimental design, octane, speed of sound, toluene

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Authors: Jurriaan Gillissen

Abstract:

This work is concerned with stabilizing the numerical integration of the Navier-Stokes equation (NSE), backwards in time. Applications involve the detection of sources of, e.g., sound, heat, and pollutants. Stable reverse numerical integration of parabolic differential equations is also relevant for image de-blurring. While the literature addresses the reverse integration problem of the advection-diffusion equation, the problem of numerical reverse integration of the NSE has, to our knowledge, not yet been addressed. Owing to the presence of viscosity, the NSE is irreversible, i.e., when going backwards in time, the fluid behaves, as if it had a negative viscosity. As an effect, perturbations from the perfect solution, due to round off errors or discretization errors, grow exponentially in time, and reverse integration of the NSE is inherently unstable, regardless of using an implicit time integration scheme. Consequently, some sort of filtering is required, in order to achieve a stable, numerical, reversed integration. The challenge is to find a filter with a minimal adverse affect on the accuracy of the reversed integration. In the present work, we explore an adjoint gradient method (AGM) to achieve this goal, and we apply this technique to two-dimensional (2D), decaying turbulence. The AGM solves for the initial velocity field u0 at t = 0, that, when integrated forward in time, produces a final velocity field u1 at t = 1, that is as close as is feasibly possible to some specified target field v1. The initial field u0 defines a minimum of a cost-functional J, that measures the distance between u1 and v1. In the minimization procedure, the u0 is updated iteratively along the gradient of J w.r.t. u0, where the gradient is obtained by transporting J backwards in time from t = 1 to t = 0, using the adjoint NSE. The AGM thus effectively replaces the backward integration by multiple forward and backward adjoint integrations. Since the viscosity is negative in the adjoint NSE, each step of the AGM is numerically stable. Nevertheless, when applied to turbulence, the AGM develops instabilities, which limit the backward integration to small times. This is due to the exponential divergence of phase space trajectories in turbulent flow, which produces a multitude of local minima in J, when the integration time is large. As an effect, the AGM may select unphysical, noisy initial conditions. In order to improve this situation, we propose two remedies. First, we replace the integration by a sequence of smaller integrations, i.e., we divide the integration time into segments, where in each segment the target field v1 is taken as the initial field u0 from the previous segment. Second, we add an additional term (regularizer) to J, which is proportional to a high-order Laplacian of u0, and which dampens the gradients of u0. We show that suitable values for the segment size and for the regularizer, allow a stable reverse integration of 2D decaying turbulence, with accurate results for more then O(10) turbulent, integral time scales.

Keywords: time reversed integration, parabolic differential equations, adjoint gradient method, two dimensional turbulence

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