Nozzle Flow Research Articles

Design and test of Kinect-based variable spraying control system for orchards.

Target detection technology and variable-rate spraying technology are key technologies for achieving precise and efficient pesticide application. To address the issues of low efficiency and high working environment requirements in detecting tree information during variable spraying in orchards, this study has designed a variable spraying control system. The system employed a Kinect sensor to real-time detect the canopy volume of citrus trees and adjusted the duty cycle of solenoid valves by pulse width modulation to control the pesticide application. A canopy volume calculation method was proposed, and precision tests for volume detection were conducted, with a maximum relative error of 10.54% compared to manual measurements. A nozzle flow model was designed to determine the spray decision coefficient. When the duty cycle ranged from 30% to 90%, the correlation coefficient of the flow model exceeded 0.95, and the actual flow rate of the system was similar to the theoretical flow rate. Field experiments were conducted to evaluate the spraying effectiveness of the variable spraying control system based on the Kinect sensor. The experimental results indicated that the variable spraying control system demonstrated good consistency between the theoretical spray volume and the actual spray volume. In deposition tests, compared to constant-rate spraying, the droplets under the variable-rate mode based on canopy volume exhibited higher deposition density. Although the amount of droplet deposit and coverage slightly decreased, they still met the requirements for spraying operation quality. Additionally, the variable-rate spray mode achieved the goal of reducing pesticide use, with a maximum pesticide saving rate of 57.14%. This study demonstrates the feasibility of the Kinect sensor in guiding spraying operations and provides a reference for their application in plant protection operations.

Direct Numerical Simulation of Atomization Characteristics of ECN Spray-G Injector: In-Nozzle Fluid Flow and Breakup Processes

AbstractIn order to sustain applications dealing with various liquid fuels in internal combustion engine (ICE), it is essential to make available prediction methodologies that allow an early evaluation of the potential usefulness of such fuels in terms of favourable mixture preparation process already in realistic configurations. Since the air-mixture formation and subsequent processes are predominantly governed by the fuel injection, a DNS based numerical investigation coupled with VOF as an interface tracking method is carried out in this paper to gain better insight on the fuel injection from an industrial injector "Spray G" configuration. Chosen from Engine Combustion Network (ECN), this is a gasoline direct injector (GDI) featuring 8-holes orifices and operating with high injection pressure (200 bar). Under consideration of the required computational cost associated with DNS, only the 1/8 of the nozzle geometry including one orifice is used. The numerical simulation is accomplished for the quasi-steady injection condition with nozzle needle fully opened. The obtained results are first validated with available experimental data for nozzle mass flow rate and spray spread angle showing a good agreement. Then, a detailed numerical analysis is provided for the in/near nozzle flow evolution especially for flow turbulence, primary and secondary atomization. Furthermore, droplet statistics in terms of droplet shape, and droplet size-velocity distribution together with a breakup regime map are reported. Finally, a 2-D data curation technique is proposed to extract the droplet statistics along selected planes and evaluated by direct comparison with three-dimensional droplet data, which may allow handling of the DNS data in more feasible and economical way especially for time series data with higher frequency. The comprehensive DNS data generated by this DNS-VOF approach enable not only to carry out detailed numerical analysis of in- and near-nozzle physical phenomena for which experimental data are still scarce, but also to provide a hint of more reliable injector boundary conditions useful for lower order spray injection method based on Lagrangian particle tracking.

ANALISIS ALIRAN PADA BUCKET TURBIN PELTON 1 KW DENGAN VARIASI SPEAR NOZZLE MENGGUNAKAN METODE CFD

Pelton turbine have been used for harvesting renewable energy by extracting kineticenergy of water into mechanical energy for more than 100 years. This characteristic makes thissystem remains potential to achieve Goals #12 SDGs. During the operation, the performance ofPelton Turbine is affected by several parameters: hydraulic power, water flow rate, bucketdesign, and nozzle. Among others, nozzle design is the most challenging since this part isresponsible to drive the water direction strike the buckets, converting impulse. The performanceof existing 1 kW Pelton Turbine equipped with 15mm Spear Nozzle is believed can be improvedby varying the nozzle diameter by using low-cost CFD simulation procedure. Ansys Fluent fromAnsys Student 2021 R1 is used in this research. Represented by two-buckets facing spear nozzle,three dimensional CFD simulation is conducted. The nozzle diameter is varied to 15 mm, 14 mm,and 13 mm from two configurations. The first and second configuration is done with water flowrate 3,3 kg/s and 1,3 kg/s respectively. The results show that 15 mm nozzle diameter can producehigher power as high as 594,08 Watt with the first configuration. Even though at the secondconfiguration this 15mm nozzle diameter did not produce the highest power, other parameterssuch as splashing water and erosion effect still show that the 15mm nozzle diameter performedbetter than others. This simulation results also consistent with experimental results whichperformed by using first configuration and 15mm nozzle diameter. The results also show that thesistem performance is much higher at higher water flow rates. Therefore, the development of thesystem can be continued to by using spear nozzle with diameter higher than 15mm.

Experimental and numerical study on liquid film cooling performance under heated wall condition

Liquid film cooling is broadly applied in the thermal protection of liquid rocket engine. An experimental system of liquid film cooling for a heated curved stainless-steel wall is built in this study to investigate the effects of jet mass flow rate, jet angle, and nozzle diameter on the cooling performance. The temperature distribution of the heated wall is demonstrated with infrared thermography technology and the flow spreading shape of the liquid film is captured with high-speed camera. A numerical liquid film cooling model based on the volume of fluid (VOF) coupled Level-set method is established to calculate the evaporative heat absorption and spreading shape of liquid film. The experimental study indicates that the peak film thickness in the hydraulic jump region increases with the jet mass flow rate, such as the peak film thickness increases from 265.15 to 632.25 μm at Q = 100–600 mL·min−1. Besides, a jet angle of 35 degree can maintain the maximum wall temperature drop of 29.4 °C. In addition, the simulated peak film thickness presents a minimum deviation of 5.2 % from the experimental data. The numerical model provides important reference for the study of liquid film cooling performance of the liquid rocket engine.

Array Formation Testing with Multiple Azimuthal and Axial Pressure Transducers

An early formation tester operating a “sink probe” that extracts fluids from the borehole, also measures pressures at its location; pressures are additionally collected at two passive sensors, one situated 180 deg away and the second about meter axially. These positions are not conducive to accurate permeability predictions, since pressure transient signals used for inverse analysis attenuate very rapidly. Closer probe spacings offer formation evaluation advantages at low mobilities because Darcy pressure dissipation is reduced. Logging applications for such testing tools include heterogeneity, anisotropy and layer characterization. Multiprobe tools with smaller transducer separations are ideal, but at present, analytically based hardware design and software interpretation methods are not available. Two design approaches are described in this paper, namely, testers with multiple probes that are displaced azimuthally, and those with axially displaced probes. For the former, we describe a new triple-probe array tester and related software models that support pressure transient analysis in transversely isotropic media. The numerical approach supports independently operable probes with different nozzle shapes, flow rates and start and stop times. This flexibility supports anisotropy and heterogeneity mapping ‘’circumferentially about the borehole. The latter class of tools, those formed by axial pressure arrays, extend conventional “dual probe tools” by including additional pressure probes axially along the same azimuth. These support improved pressure gradient analysis and detection of isolated zones in vertical wireline applications. Taken together, both array methods and their computational models support more effective job planning and inverse properties analysis. Comments are also offered on “hybrid multiprobe tools” consisting of both azimuthal and axial arrays. These math formulations and computing methods for both tool classes not only support pressure analysis, but also, for more effective hardware design. For example, what pump flow rates are required to provide a given depth of investigation? Analogous questions can be answered “on paper” prior to prototype building. Representative computed examples are presented demonstrating the versatility and capabilities of the new models. This article introduces techniques focusing on single-phase flow fundamentals. Applications to other multiprobe tools, together with multiphase extensions, dealing with coupled pressure and contamination models, nonlinear gas pumping, convergence acceleration, and in addition, inverse approaches and “big data” support, will be presented in a forthcoming 2024 book.

Numerical study of heterogeneous condensation in the de Laval nozzle to guide the compressor performance optimization in a compressed air energy storage system

Compressed air energy storage technology (CAES) has an enormous possibilities in terms of energy conversation, environmental protection, and economic benefits. Air compressor, as a core component, is of great significance for the CAES system efficiency. The impact of the non-equilibrium condensation of steam contained in the air on the blade erosion and thermal efficiency drop in the compressor is not to be missed. The compressor flow characteristics in blade-to-blade channels can be reproduced by the flow in the Laval nozzle, which can verify the accuracy of the condensation model based on similar flow characteristics and subcooling conditions in the non-equilibrium condensation flow. Firstly, the homogeneous condensation flow is numerically investigated by different condensation models, and the Blend model is used to investigate the heterogeneous condensation because of its higher accuracy. Secondly, the physical properties in the heterogeneous condensing flow are analyzed in detail by the particles number, radius, and types. The result presents that the foreign particles promote the heterogeneous condensation while the homogeneous condensation is weakened. Finally, the flow loss coefficients and thermal efficiency are calculated, and the guide to optimize compressor operation is proposed. It is concluded the heterogeneous condensation with liquid droplets as condensation nuclei can weaken the non-equilibrium condensation greatly and reach a higher thermal efficiency compared to the heterogeneous condensation with solid particles as condensation nuclei. The huge amount of air mass flow rate in compressors and expanders used in the CAES systems makes it difficult to filter out the solid particles and to dry air therefore, investigating the impact of steam condensation on the flow characteristics is recommended.

Viscosity measurement of molten alumina and zirconia using aerodynamic levitation, laser heating and droplet oscillation techniques

Reliable thermophysical properties of core melt (corium) are essential for the accurate prediction of the severe accident progression in light water reactors. Zirconia is one of the most important materials in corium. Despite the high interest in the viscosity of molten zirconia, few experimental data have been reported due to its high melting temperature and high vapor pressure. In the present study, the viscosity of molten zirconia was measured using aerodynamic levitation, laser heating and droplet oscillation techniques. A material sample was levitated by argon gas flow in a conical nozzle and then melted into a droplet by laser beams. The initial quiescent droplet was forced to oscillate by the excitation of a loudspeaker, and the viscosity was deduced based on the characteristics of the droplet damped oscillation after the loudspeaker was turned off. The viscosity of molten alumina was first measured for verification of the measurement system. Afterwards the viscosity of molten zirconia was measured. The results showed that the viscosity of molten zirconia at melting temperature (2988K) was 12.87 ± 1.03 mPa s and decreased with increasing temperature. The measurement uncertainties are within 21 %.

An investigation comparing various numerical approaches for simulating the behaviour of condensing flows in steam nozzles and turbine cascades

The process of condensation takes place within the low-pressure stages of the steam turbine when undergoing rapid expansion. The dry supercooled steam turns into very tiny liquid droplets, and the resulting wet steam mixture causes additional losses and maintenance problems. Accurate simulation of steam condensing flows helps to obtain accurate information about the thermodynamics of such flows and use this information in the design of highly efficient steam turbine stages. In this study, by using the available numerical models in commercial CFD codes, seven cases are considered for simulating steam condensing flows. The important difference between these cases is the use of different condensation models and different equations of state to calculate the thermodynamic properties of wet steam. The objective of this research is to identify an appropriate model that can be used to simulate the flow of steam condensation effectively. In recent studies, it is still necessary for further improvement of the modelling methods because the numerical results did not fully match the experimental findings, underscoring the persisting uncertainties within this domain. The current study explores various condensation models and equations of state for calculating the thermodynamic properties of wet steam. This diversity of approaches is novel and allows for the comparison of different models to determine the most suitable one for effective flow simulation. The geometries of the IWSEP nozzle (International Wet Steam Experimental Project) and linear cascade of the last-stage rotor blades were considered. To identify the most appropriate model for the simulation of steam condensing flows, the numerical results are validated against in-house experiments. The results indicate that combining the Young droplet growth model with the Vukalovich and Young equations in cases 1 and 2 is not a suitable method for modelling steam condensing flows. However, cases 3 & 4, which are the combination of Gyarmathy and Fuchs-Sutugin droplet growth models with NIST real gas models, are the most appropriate ones. The findings of this research confirm the high sensitivity of the modelling of such flows, and emphasize that the necessary care should be taken in choosing the appropriate numerical model.

Computational fluid dynamics modeling of rapid pyrolysis of solid waste magnesium nitrate hydrate under different injection methods

This study developed a numerical model to efficiently treat solid waste magnesium nitrate hydrate through multi-step chemical reactions. The model simulates two-phase flow, heat, and mass transfer processes in a pyrolysis furnace to improve the decomposition rate of magnesium nitrate. The performance of multi-nozzle and single-nozzle injection methods was evaluated, and the effects of primary and secondary nozzle flow ratios, velocity ratios, and secondary nozzle inclination angles on the decomposition rate were investigated. Results indicate that multi-nozzle injection has a higher conversion efficiency and decomposition rate than single-nozzle injection, with a 10.3% higher conversion rate under the design parameters. The decomposition rate is primarily dependent on the average residence time of particles, which can be increased by decreasing flow rate and velocity ratios and increasing the inclination angle of secondary nozzles. The optimal parameters are injection flow ratio of 40%, injection velocity ratio of 0.6 and secondary nozzle inclination of 30 degrees corresponding to the maximum decomposition rate of 99.33%.

Aerodynamic sealing using a segmented recirculating horizontal air curtain for buoyancy-driven flows in doorways

Dealing with buoyancy-driven flow to reduce infiltration heat is challenging for buildings such as cold storages where the indoor-outdoor temperature differences are significant. In this study, a segmented recirculating horizontal air curtain (SRHAC) is proposed to respond to the vertical thermal buoyancy profile in doorways and recover the cold air entrained by the air curtain. Specifically, SRHAC is a planar turbulent impingement jet discharged through a vertically mounted rectangular nozzle. The discharge nozzle is divided into segments and the return vent recirculates the discharged air to the rear of the discharge nozzle. In this study, the aerodynamic sealing performance of the SRHAC is investigated by considering structural, operational, and ambient parameters. Orthogonal analysis reveals the flow rate, nozzle width, and discharge angle as significant factors affecting the thermal separation efficiency of the SRHAC. Increasing the flow rate and decreasing the nozzle width leads to an increase in the jet momentum flux, thereby enhancing the thermal separation efficiency, with a maximum efficiency of 75.9 %. Setting the discharge parameters in the three sections along the height improves the thermal separation efficiency by 16.5% points. The maximum efficiencies of two-, three-, and five-segment air curtains are 83.6 %, 87.3 %, and 90.3 %, respectively. The separation efficiency of the SRHAC remains relatively stable even when the indoor-outdoor temperature difference varies between 32 and 57 K. These results establish that the SRHAC is an efficient energy-saving device, with the ratio of reduced infiltration load to energy consumption as high as 110.9.

The Influence of External Flow Field on the Flow Separation of Overexpanded Single-Expansion Ramp Nozzle

Flow separation and transitions of separation patterns are common phenomena of nozzles working with a wide Mach range. The maximum thrust method is applied to design the single-expansion ramp nozzle (SERN) for specific operating conditions. The nozzle is used to numerically simulate the transition processes of separation patterns under the linear change in the external flow Mach number and the actual trajectory take-off condition of a rocket-based combined cycle (RBCC), to investigate the mechanism through which the external flow field influences the separation pattern transition during acceleration. The computational fluid dynamics (CFD) method is briefly introduced, followed by experimental validation. Then, the design procedure of SERN is described in detail. The simulation results indicate that as the external Mach number increases, the flow field in the nozzle undergoes transitions from RSS (ramp) to FSS, and finally exhibits a no-flow separation pattern. The rate at which the external Mach number varies has little effect on the transition principle of the nozzle flow separation patterns, but it has a significant effect on the critical Mach number of the transition points. The external flow field of the nozzle has an airflow accumulation effect during acceleration, which can delay the transition of the flow separation pattern.

Research on the Fire Extinguishing Efficiency of Low-Pressure Water Mist in Urban Underground Utility Tunnel Cable Fires

Low-pressure water mist fire extinguishing systems are a cost-effective and highly reliable option for fire protection. However, they have not yet seen widespread use in urban underground utility tunnels. To validate the fire extinguishing effectiveness of the system in cable fires within urban utility tunnels and to identify the key factors influencing its efficiency, a scaled-down test platform for low-pressure water mist fire extinguishing in utility tunnels was constructed, and a series of fire extinguishing tests was conducted. The test results demonstrate that low-pressure water mist can rapidly and effectively extinguish cable fires in utility tunnels, with the quickest fire extinguishing time of 7 s. Within 50 s of activating the system, the internal temperature of the tunnel can be reduced from 650 °C to 40 °C. Among the influencing factors, the pressure and nozzle flow coefficient have a significant impact on the fire extinguishing efficiency, while nozzle spacing has a relatively smaller effect. Thus, when the nozzle spacing meets the requirement of “no dead zones”, priority should be given to increasing the pressure and nozzle flow coefficient.

Research on Calculation Method of Annular Leakage Rate based on Variable Conservation Method

The annular leakage rate is the most important parameter to measure the failure of gas well barrier units. At present, there are two methods that can be applied to calculate the leakage rate of offshore gas wells: safety valve method and nozzle flow method. However, applying these two methods to the calculation of annular leakage rate has encountered a problem of poor adaptability. In order to enhance the accuracy of leakage rate calculation results and improve the reliability of judging gas well integrity through annular leakage rate, this article summarizes an annular rate calculation method based on variable conservation method.

Investigation of nozzle geometry and wall roughness effects on diesel injector flow

The flow and design of fuel injector nozzles have a considerable influence on the spray and combustion characteristics of a diesel engine. In-cylinder combustion, atomization, and primary breakdown are all highly influenced by the cavitation and turbulence in the fuel injector nozzle. In this paper, the effect of the nozzle geometry parameters, wall roughness parameters, and pressure difference on the swirl number, mass flow rate, turbulent kinetic energy, and vapor volume fraction is explored. U-type nozzle hole geometry, a well-known benchmark for the injector nozzle flow, is used to evaluate mesh independence and model validation. Large-eddy simulations are performed to provide a precise presentation of the flow structures and turbulent eddies inside the nozzle. Multiphase flow is studied using the mixture model, whereas cavitation is studied using the Schnerr–Sauer model based on the Rayleigh–Plesset equation. We find that the wall roughness parameters have an exciting impact on the discharge coefficient, swirl number, and vapor volume fraction. Due to the non-monotonic dependence of nozzle flow characteristics on the pressure difference and the wall roughness parameters, we can always find such values of these input parameters that render optimal nozzle flow characteristics. In this way, these parameters provide good control of spray formation and consequently on the quality and rate of combustion in the diesel engine.

Application of the method of insignificant divergences to diagnose the technical aircraft gas turbine engine state under the transient-state conditions of its operation

The article deals with issues related to the use of parametric information of the transient-state gas turbine engines (GTE) operation conditions for diagnosing their technical condition during the operation. A review of general approaches to computational algorithms for the recognition and classification of the condition applicable to aircraft GTE has been carried out. The significance of analytical models in modern algorithms for assessing the technical GTE condition is emphasized. The construction of a linearized mathematical model for the transient-state condition of the generalized-scheme aircraft GTE operation has been considered. It represents a system of equations analytically combining the relative parameter divergences measured during the engine operation with the relative divergences of unmeasured thermogasdynamic parameters and geometric gas-air flow duct parameters allowing for the technical condition of gas-air channel elements to be classified. A method for constructing mathematical and diagnostic engine models, using the transient response data, has been formulated. The capability of employing a method of insignificant divergences, used to build linear (linearized) mathematical and diagnostic GTE models for the steady-state conditions of its operation, has been demonstrated as well. It is shown that, despite the structural similarity of linear models of the steady and transient-state processes, diagnostics by means of the stated above processes is based on completely different principles – under the steady-state condition, the classification of a technical condition is determined by the variation in the value of the group of controlled responses, and under the transient-state condition, this operation is based on correlating the change in the transient-state behavior. To ensure the versatility of employing proposed methods regarding various GTE designs installed on modern civil aircraft, a generalized-design aircraft GTE model – a three-shaft bypass turbojet engine with mixing flows in a common jet nozzle, has been considered.

Enhancing Seawater Salinity through Spray-Assisted Evaporation: Investigating the Effect of Nozzle Sprayer Diameter and Feed Flowrate on Evaporation Rate using Multiple Regression Analysis

Salt production plays a vital role in life due to its essential minerals. Traditional salt production methods using direct solar rays are the most common in Indonesia. It needs a large area and several weeks extended to evaporate sea water into concentrated brine. It is very weather dependent. This study investigates the utilization of a spray-assisted evaporator to increase the salinity of seawater for salt production. The experimental setup consisted of a custom-designed spray evaporator that atomized seawater into fine droplets. A series of experiments were conducted to analyze the effect of different operational parameters, including feed flow rate (1000 ml/min; 1200 ml/min; 1400 ml/min; 1600ml/min; 1800ml/min) and diameter of nozzle sprayer (1 mm; 2 mm; 3 mm) on the average of evaporation rate. The results demonstrated that Higher spray flow rates and smaller droplet sizes positively correlated with increased evaporation rates, leading to higher salt concentrations in the collected brine. Optimizing the Impact of nozzle sprayer diameter and flow rate on the average evaporation rate utilizing multiple linear regression and deriving the Equation ,where x1 represents the seawater feed flow rate, and X2 represents the nozzle sprayer diameter.

A Wide Operating Range Ejector with Part Nested Nozzles for PEMFC Hydrogen Recirculation System

The ejector drives unreacted hydrogen from the anode to improve fuel utilization ratio and discharges redundant water to prevent flooding and shutdown in the proton exchange membrane fuel cell (PEMFC). However, the traditional fixed structure ejector cannot meet the recycling requirements in the whole dynamic working condition of the fuel cell. In this article, a part nested four-nozzle (PNFN) ejector is proposed to enhance the hydrogen recycling efficiency under variable working conditions of the PEMFC by restricting the nozzle flow as 10%, 20%, 20%, and 50% of the fuel cell-rated power, respectively. Systematical analyses are performed on the experimentally verified 3D model to study inner flow characteristics and performance under different nozzle running modes. The results indicate that the PNFN ejector satisfies the recirculation ratio requirements in the power range of 34–220 kW within the 7–9 bar suitable supply pressure. By comparing with traditional ejectors, the PNFN ejector has a wider working range and especially outputs better performance in the low power range.

Effect of applied electric and magnetic fields on surface roughness in minimum quantity lubrication turning using Fe3O4 nano-ferrofluid

PurposeFirst, the effect of magnetic field intensity and nano-ferrofluid concentrations on surface roughness was evaluated in magnetic minimum quantity lubrication (MMQL). Then, the effect of lubricant flow rate and nozzle position on surface roughness was investigated in MQL, MMQL, electrostatic MQL (EMQL) and electromagnetic MQL (EMMQL).Design/methodology/approachThis study examined the performance of MQL under magnetic and electric fields in turning AISI 304 stainless steel in terms of surface roughness and compared the results with those obtained from wet cutting and MQL turning operations. To prepare the nano-ferrofluid used in different states of MQL, Fe3O4 nanoparticles were added to the base fluid.FindingsThe results showed that the surface roughness under the EMMQL technique decreased by 36% and 49.4% on average compared with wet and MQL techniques, respectively. The lubrication technique affected the surface roughness by 90.2%, whereas it was 8.3% for the lubricant flow rate. EMQL and EMMQL techniques had no significant difference in their effects on surface roughness. In the innovative MMQL technique, the nano-ferrofluid concentration of 6% and magnetic field intensity of 93 G resulted in lower surface roughness of the workpiece relative to other counterparts.Originality/valueExamining previously published studies showed that using nano-ferrofluids under a magnetic field for cooling purposes in machining processes have less considered by researchers. This study applies an innovative method of lubrication under the concurrent effect of magnetic and electric fields, called EMMQL, to improve the efficiency of MQL in machining hard-to-cut materials. For comprehensively inspecting the newly presented method, the effects of several parameters, including the nano-ferrofluid concentration, magnetic field intensity, lubricant flow rate and position of lubricant spray nozzle, on the surface roughness of workpiece in turning of AISI 304 stainless steel are investigated.

Asymptotic analysis of transonic shocks in divergent nozzles with respect to the expanding angle

In this paper we study the asymptotic behavior of the transonic shock solutions in divergent nozzles as the expanding angle goes to zero. It is well-known that there exist infinite shock solutions for steady 1-D flows in a flat nozzle with the position of the shock front being arbitrary, while there exists a unique shock solution in an divergent nozzle as the pressure at the exit is given within an appropriate interval. By analyzing the asymptotic behavior of the shock solutions as the expanding goes to zero, we are also trying to figure out a criterion which may be used to select the physical one among all shock solutions in the flat nozzle. It finally turns out that the limit shock solution as the expanding angle goes to zero strongly depends on the asymptotic behavior of the receiver pressure, which is assumed to be a function of the expanding angle, imposed at the exit. In particular, as the Mach number of the flow at the entrance is larger than γ+32, the function of the receiver pressure can be set identical to the value of the pressure behind the shock front for the flat nozzle. Then the limit shock solution may be considered as the physical one for the flat nozzle, and the limit position of the shock front being the admissible position of the shock front. To show these results, one of the key steps is to establish quantitative estimates for the position of the shock front for the given pressure at the exit. To this end, a free boundary problem for the linearized Euler system will be proposed which gives an approximation of the shock solution, then a nonlinear iteration scheme could be carried out to approach the shock solution. Moreover, the error estimates between the exact shock solution and the approximation are also established, which give quantitative information on the position of the shock front.

Physical description of an isentropic flow in a Laval nozzle

The convergent-divergent nozzles have many industrial uses such as wind turbine, the blades of turbine engines, aircraft wings and screw propeller. The main objective of this study is to identify and evaluate the physical properties of the compressible flow of an ideal gas in a one-dimensional Laval nozzle in terms of its physical and geometrical parameter. This paper will enable us to identify and determine the causes of right shockwave phenomenon. After simplify of the energy balance equations of gas dynamics, Matlab™ was used as a tool for programming. The numerical method for the resolution is based on the iterative method with an error (E=10-5). Numerical results were compared to the modeling results of CFD ‘Fluent’. The results were satisfactory.