CFD Simulation Of Flow Research Articles

Hyperbolic Paraboloid Tensile Structure—Numerical CFD Simulation of Wind Flow in RWIND Software

Tensile membrane structures combine a prestressed roofing envelope material and supporting elements. To design these structures, there is a set of recommendations in the European Design Guide for Tensile Surface Structures and some other national standards. However, currently, there is no official standard related to the design process of tensile structures in the European Union. The structure studied in this project is considered as permanent roofing of an external testing device in the shape of a simple hyperbolic paraboloid without enclosing walls. Snow and wind loads were analyzed as the most critical types of loading in the location. Determining the value of the snow load is relatively simple according to the European standard. However, in the case of the wind load, this shape is not considered in the European standard and needs to be solved experimentally or by numerical simulation in a wind tunnel. The present contribution focuses on numerical analysis of the wind flow in RFEM software and simulation of the wind tunnel in RWIND software.

CFD Simulation of Temperature and Air Flow in a Natural Convection Solar Tunnel Dryer with a Bare Flat-Plate Chimney

Computational Fluid Dynamics simulation of a natural convection solar tunnel dryer with a bare flat-plate chimney is presented. The chimney’s function is to create airflow in the dryer through the buoyancy effect by re-heating the air coming from the drying unit and is therefore a major factor in the drying rate. CFD simulation was therefore employed to study the temperature and airflow in the dryer and determine the areas that could be improved upon. The design of the solar tunnel dryer geometry used in this simulation was done in SOLIDWORKS 2016 whereas the simulation of temperature distribution of airflow inside the dryer was performed using SOLIDWORKS 2016 flow simulation program in a steady-state regime. The boundary conditions were set using the obtained experimental data. The simulation results showed that the chimney losses heat, that there is air recirculation in the collector, that the airflow experiences some pressure loss as it moves from the drying chamber to the chimney, and that there is some reduction of the velocity in some parts of the dryer. The simulated and experimental collector efficiencies were found to be 33.09 and 37.63%, respectively, giving a mean relative deviation of collector temperature of 5.1%. To improve the performance of the dryer, insulating and glazing of the chimney is suggested as well as using a curved joint between the chimney and the drying chamber.

CFD simulation of a full-scale three-dimensional flow and temperature distribution inside a reactor under ocean conditions

A series of three-dimensional full-scale fluid flow simulations are performed based on an integrated thermal marine reactor under ocean conditions. The aim is to understand the effects of ship motions including rolling, heaving, and inclination on flow field and thermal–hydraulic performance. Both forced circulation and natural circulation are taken into account since the abilities of coolant to maintain its original motion under ocean conditions are different. The temperature distribution, velocity field, and flow rate distribution of a static reactor are presented first for comparison. After additional forces or accelerations are introduced, the flow field in the marine reactors subjected to different types of ocean motions is explored. Results show that the rolling motion would have more significant impacts on marine reactors in natural circulation concerning temperature and SG (steam generator) flow rate. In natural circulation, the SG flow rate fluctuation can reach 35 kg/s when the rolling angle increases to 45°, compared to the maximum fluctuation of 8 kg/s under forced circulation. The rolling motion also greatly affects the flow field and flow rate distribution at the core inlet. Obvious vortex flows can be observed in the annulus both for forced circulation and natural circulation. The core channel with the minimum flow rate appears to be randomly located for the reactor subjected to rolling motion. For reactors in heaving motion, the fluid velocity changes periodically, and the flow rate in the SG and core regions increases significantly with the increase of the heaving period and acceleration. As for inclined reactors, the average velocity in the core becomes smaller while the temperature increases as the inclination angle of the reactor increases.

Effect of drag force modeling on the flow of electrostatically charged particles

In CFD simulations of two-phase flows, accurate drag force modeling is essential for predicting particle dynamics. However, a generally valid formulation is lacking, as all available drag force correlations have been established for specific flow situations. In particular, these correlations have not been evaluated for particle-laden flows subjected to electrostatic forces. The paper reports the effect of drag force modeling on the flow of electrically charged particles. To this end, we implemented different drag force correlations to the open-source CFD tool pafiX. Then, we performed highly-resolved Direct Numerical Simulations (DNS) using the Eulerian–Lagrangian approach of a particle-laden channel flow with the friction Reynolds number of 180. The simulations generally revealed a strong influence of the precise drag correlation on particles in the near-wall region and a minor effect on the particles far from the walls. Due to their turbophoretic drift, particles accumulate close to the channel walls. For uncharged particles, the simulations show large deviations of the particle concentration profile in the near-wall region depending on the drag force correlation. Therefore, the disturbance of the flow surrounding a particle by a nearby wall or other particles is important for its drag. Driven by electrostatic forces, charged particles accumulate even closer to the wall. Contrary to the uncharged cases, when the particles carry a high charge (in our case one femto-coulomb), we found minor effects of drag force modeling on particle concentration profiles. In conclusion, for the investigated conditions, we propose to account for the effect of nearby particles and walls on the drag of low- or uncharged particles.

Computational study of gas flow characteristics when using intra-nozzle interceptors for thrust vector control

This article proposes the use of cylindrical nozzle interceptors (probes) as an alternative method of rocket engine thrust vector control, which can replace traditional thrust vector control systems. The objective of this paper was to perform a three-dimensional CFD simulation of the gas flow in a rocket engine nozzle with a rod interceptor taken out of the wall without secondary injection, analyzing the calculated flow data and estimating the lateral force. The simulations will be performed in steady-state mode. In the CFD simulations, velocity distribution contours were obtained on the symmetry plane for all positions and values of the interceptor height.

Scaling analysis as a tool to validate CFD simulation of a lubricant flow in the bearing housing of a gas turbine

• The effects of rotor speed on the temperature of the lubricant oil are studied. • CFD model of oil flow and heat transfer in the bearing chamber is developed. • Scaling analysis is used to simplify and nondimensionalize the governing equations. • Scaling analysis is used to replace experimental data to validate CFD results. • An equation to compute viscous dissipation energy is obtained by Scaling analysis. This work concentrates on the journal bearing chamber of a gas turbine engine, to explain the impacts of the rotational speed of the rotor on the velocity, pressure and temperature distribution in the bulk oil. In the first step, a CFD model of fluid flow and heat transfer in the bulk oil of the bearing chamber was developed via COMSOL 6. Next, and in a novel approach, scaling analysis was employed to verify the CFD simulation results. Scaling analysis, as a systematic method for non-dimensionalizing a set of equations, was used to simplify the governing equations based on the specific conditions of the physics. Afterward, the velocity, pressure and temperature profiles attained from the CFD simulation were compared to this set of simplified equations. In general, it was confirmed that the CFD model developed in the current article is physically valid and follows the main descriptive equations governing the entire physics of the system. Scaling analysis was effectively used to confirm the simulation results where insufficient experimental results were available, due to the geometrical and operational restrictions in the test rig. Temperature, pressure and velocity were the factors considered in this study and the average of simulation and scaling results for each factor were compared. Additionally, the average point-to-point error between the simulation and scaling results for each factor was reported as another basis for the comparison. In summary, scaling analysis in this work is presented as a straightforward method to substitute experimental measurements in order to validate CFD simulation results.

Design and analysis of energy efficient wind turbine blades

Inspired by humpback whale flippers, this study presents the effect of leading-edge tubercles on the wind turbine blade. Force models for lift and drag are developed on the blade which have an important role in the wind turbine performance. The study compares the CFD simulation of fluid flow over blade with leading edge tubercles and the conventional blade. Present work also covers strength and random vibration analysis for both the blades. Furthermore, different tubercle blades have been compared for lift and drag coefficients by varying tubercle amplitude. Structural analysis shows tubercled blade is suitable from structural and strength point view. In tubercled blade overall lift to drag-coefficient ratio increased up to 8.5, whereas in conventional this ratio is about 1.7 which is found to be very low. This ratio clearly indicates that in the presence of tubercles on the blade, performance improving even in lower wind speed areas. Overall tubercled blade found to be more suitable for modern wind turbines.

The critical role of scale resolution in CFD simulation of gas-solid flows: A heat transfer study using CFD-DEM-IBM method

The surface-to-bed heat transfer coefficient is an important engineering parameter for quantifying the heat transfer capability of fluidized beds. In this study, computational fluid dynamics-discrete element method-immersed boundary method (CFD-DEM-IBM method) is used to simulate the velocity and temperature fields around the immersed tube in two fluidized beds, the heat transfer coefficients of fluid-wall, particle–wall, and particle–fluid-wall are then analyzed and compared with experimental data. It is shown that in order to quantitatively predict the surface-to-bed heat transfer coefficient without using empirical correlations, the thermal boundary layer of gas phase that is of the order of particle diameter needs to be explicitly resolved by refining the fluid grid around the immersed tube up to 1/16 of particle diameter, whereas an empirical correlation is necessary to correctly calculate it in the state-of-the-art CFD studies which have used coarse grids. Present study highlights the critical role of scale resolution (specifically, the explicit resolution of the thermal boundary layer) in the study of surface-to-bed heat transfer in gas–solid fluidized beds.

Water Movement Direction through an Anion Exchange Membrane Fuel Cell Exerted By Gas Humidity and Flow Rate: A Simulation Study

The growing demand for clean energy generation has been urged due to the threat of climate change by greenhouse emissions (1 kg of CO2 per 1 kW h of generated electricity). In 2021, the electricity generation rose to 1530 TW h, 62% coming from fossil fuels [1]. To lower this significant fossil fuels usage, the development and use of alternative energy sources and green fuels have reached a formidable level of interest. Renewable solar and wind energy sources are currently available; however, those appear to be limited by natural fluctuations in energy production relegated to day- and night-time wheatear or geographical regions [2]. Electrochemical energy storage and conversion technologies can be employed to minimize production variations as an alternative to this limitation. The anion-exchange membrane fuel cell (AEMFC) is an excellent alternative for energy generation due to its air CO2-free emissions, low-cost electrocatalyst used at the cathode, and industrial-scale membrane production [3]. The design of a fuel cell involves several tightly interrelated factors, such as electrode and membrane materials, reactor geometry, and characterization of the reaction environment in terms of the momentum, mass and heat transport, electrode kinetics, and current distribution, to find the best operational conditions to reach a suitable energy efficiency [4]. In either case, the present work deals with designing a hybrid flow-field single-fuel cell. The monopolar anode plate is a multiple-serpentine flow field that benefits the water drainage. On the other hand, the cathode plate is an interdigitated leaf-type flow field that might ensure an efficient gas reactant distribution over the entire electrode geometric area.A two-phase flow CFD simulation is performed to analyze the influence of the flow field patterns on the mass transport processes in the electrodes. Moreover, the water movement direction through the membrane as a function of the operational conditions is presented. The commercial code COMSOL Multiphysics was used to solve the transport equations through the finite element method.In the experimental field, the water movement direction between the anodic and cathodic sides shows the capability to favor the electro-osmotic drag (EOD) or the diffusion water transport mechanism under different gas flow rates and humidities, as shown in Figure 1a. Accordingly, the results demonstrated that the EOD predominance was promoted when low symmetric flow rates (125 cm3 min-1 for anode and cathode) were used in the AEMFC. On the contrary, the water diffusion dominance appears under asymmetric flow rates between anode and cathode (125 and 250 cm3 min-1, respectively). The different tested relative humidities do not influence the water direction once the flow rates are given. The AEMFC performance was affected by the water transport mechanism, giving lower peak power densities for EOD-dominated systems (35.2 mW cm-2, in Figure 1b) than those obtained by diffusion (62.8 mW cm-2, in Figure 1c). The most suitable operating condition is for water diffusion dominance at non-saturated streams, where a decrease in performance is experienced as the flow becomes saturated (at higher humidifier temperature). A close agreement between simulation and experimental results was obtained regarding current-potential curves and water movement through the membrane dictated by electro-osmotic drag and water diffusion.

CFD simulation of flow and mixing characteristics in a stirred tank agitated by improved disc turbines

To reduce the power consumption and improve the mixing performance in stirred tanks, two improved disc turbines namely swept-back parabolic disc turbine (SPDT) and staggered fan-shaped parabolic disc turbine (SFPDT) are developed. After validation of CFD model with experimental results, CFD simulations are carried out to study the flow pattern, mean velocity, power consumption, pumping capacity and mixing efficiency of the improved and traditional impellers in a dished-bottom tank under turbulent flow conditions. The results indicate that compared with the commonly used parabolic disc turbine (PDT), the power number of proposed SPDT and SFPDT impellers is reduced by 43% and 12%, and the pumping efficiency is increased by 68% and 13%, respectively. Furthermore, under the same power consumption (0−700 W·m −3 ), the mixing performance of both SPDT and SFPDT is also superior to that of Rushton turbine and PDT.

Experience of Patient-Specific CFD Simulation of Blood Flow in Proximal Anastomosis for Femoral-Popliteal Bypass

Femoral artery bypass surgery needs postoperative monitoring due to the high complication risks after bypass. Numerical simulation is an effective tool to help solve this task. This work presents the experience of patient-specific CFD simulation of blood flow in proximal anastomosis for femoral-popliteal bypass, including patient follow-up after bypass surgery. Six cases of proximal anastomosis of femoral-popliteal bypass 3–30 months after surgery were studied. A repeated study was performed for four patients to monitor geometric and hemodynamic changes. The blood flow structure variety in proximal anastomoses and the blood flow dynamics during the cardiac cycle are described in detail using CFD simulation. Special attention is paid to time-average wall shear stresses (TAWSS) and oscillatory shear index (OSI) distributions. Low and oscillatory wall shear stresses were registered in the graft downstream from the suture, especially in case of low inlet flow. It was shown that the postoperative geometry changes led to significant hemodynamic changes; thereby, neointima has grown in areas with initially low and oscillatory wall shear stresses.

On the existence of adverse pressure gradient in decelerated swirling flow and its suppression by water jet injection

The paper revisits the theory originally published by Hall that explains the presence of adverse pressure gradient at the axis of a diffuser with swirling flow. It presents the fields of the axial pressure derivative obtained by CFD simulations of flow in a swirl generator, which support the theory. The last part brings cautionary results regarding the suppression of the backflow caused by this adverse pressure gradient by water jet injection. On a case with a strong swirl intensity leading to a strong vortex rope, it is shown that the injected jet may be deflected to the outer walls and rotate with the outer swirling flow, eventually leading to stronger pressure pulsations than in the original state. Further investigation shows that high enough diameter of the jet is needed to ensure its stability.

Aerodynamic Benefits by Optimizing Cycling Posture

An approach to aerodynamically optimizing cycling posture and reducing drag in an Ironman (IM) event was elaborated. Therefore, four commonly used positions in cycling were investigated and simulated for a flow velocity of 10 m/s and yaw angles of 0–20° using OpenFoam-based Nabla Flow CFD simulation software software. A cyclist was scanned using an IPhone 12, and a special-purpose meshing software BLENDER was used. Significant differences were observed by changing and optimizing the cyclist’s posture. Aerodynamic drag coefficient (CdA) varies by more than a factor of 2, ranging from 0.214 to 0.450. Within a position, the CdA tends to increase slightly at yaw angles of 5–10° and decrease at higher yaw angles compared to a straight head wind, except for the time trial (TT) position. The results were applied to the IM Hawaii bike course (180 km), estimating a constant power output of 300 W. Including the wind distributions, two different bike split models for performance prediction were applied. Significant time saving of roughly 1 h was found. Finally, a machine learning approach to deduce 3D triangulation for specific body shapes from 2D pictures was tested.

From μCT data to CFD: an open-source workflow for engineering applications

The generation of high-quality volume meshes out of µCT data can be a challenging task that is mandatory to perform CFD simulations of fluid flows within or around scanned objects. Despite the growing relevance of CT-based CFD simulations, there is still a lack of a standardized, free-to-use workflow. In this work, an open-source based workflow is presented, which covers all steps from the CT image stack to a volume mesh in the end. The software packages ImageJ, ParaView, and Blender are used for surface reconstruction and processing of µCT data; resulting meshes are evaluated regarding geometric parameters, surface fidelity, and used memory. For volume meshing, the mesh generator snappyHexMesh implemented in OpenFOAM is used. A detailed description of this workflow is provided in the Supplementary Materials. We demonstrate how potential pitfalls and shortcomings can be avoided and how surface smoothing is especially important to preserve the surface area of scanned objects. We use CT data of a 10 ppi open cell foam to illustrate that data decimation can reduce the required time for the volume meshing by up to 45% and RAM up to 75%. The resulting meshes are used to simulate flow fields and heat transport with a surface heat source. The resulting temperature fields show that differences in the surface area and recesses in an unsmoothed mesh can affect the outcome of heat transport simulations, highly relevant for reactive CFD simulations. The results highlight the quality of our workflow’s outcome, which can help engineers that rely on CT reconstruction processes, also for other applications.

CFD Simulation of Particle-Laden Flow in a 3D Differentially Heated Cavity Using Coarse Large Eddy Simulation

Particulate flow in closed space is involved in many engineering applications. In this paper, the prediction of particle removal is investigated in a thermally driven 3D cavity at turbulent Rayleigh number Ra = 109 using Coarse Large Eddy Simulation (CLES). The depletion dynamics of SiO2 aerosol with aerodynamic diameters between 1.4 and 14 µm is reported in an Euler/Lagrange framework. The main focus of this work is therefore to assess the effect of the subgrid-scale motions on the prediction of the particulate flow in a buoyancy driven 3D cavity flow when the mesh resolution is coarse and below optimal LES standards. The research is motivated by the feasibility of modeling more complex particulate flows with reduced CPU cost.The cubical cavity of 0.7 m side-length is set to have a temperature difference of 39 K between the two facing cold and hot vertical walls. As a first step, the carrier fluid flow was validated by comparing the first and second-moment statistics against both previous well-resolved LES and experimental databases [Kalilainen (J. Aero Sci. 100:73–87, 2016); Dehbi (J. Aero. Sci. 103:67–82, 2017)]. First moment Eulerian statistics show a very good match with the reference data both qualitatively and quantitatively, whereas higher moments show underprediction due to the lesser spatial resolution. In a second step, six particle swarms spanning a wide range of particle Stokes numbers were computed to predict particle depletion. In particular, predictions of 1.4 and 3.5 µm particles were compared to LES and available experimental data. Particles of low inertia i.e. dp < 3.5 µm are more affected by the SGS effects, while bigger ones i.e. dp = 3.5–14 µm exhibit much less grid-dependency. Lagrangian statistics reported in both qualitative and quantitative fashions show globally a very good agreement with reference LES and experimental databases at a fraction of the CPU power needed for optimal LES.

Investigation into Influence of Wall Roughness on the Hydraulic Characteristics of an Axial Flow Pump as Turbine

Pump as turbine (PAT) is a factual alternative for electricity generation in rural and remote areas where insufficient or inconsistent water flows pose a threat to local energy demand satisfaction. Recent studies on PAT hydrodynamics have shown that its continuous operations lead to a progressive deterioration of inner surface smoothness, serving the source of near-wall turbulence build-up, which itself depends on the level of roughness. The associated boundary layer flow incites significant friction losses that eventually deteriorate the performance. In order to study the influence of wall roughness on PAT hydraulic performance under different working conditions, CFD simulation of the water flow through an axial-flow PAT has been performed with a RNG k-ε turbulence model. Study results have shown that wall roughness gradually decreases PAT’s head, efficiency, and shaft power. Nevertheless, the least wall roughness effect on PAT hydraulic performance was experienced under best efficiency point conditions. Wall roughness increase resulted in the decrease of axial velocity distribution uniformity and the increase of velocity-weighted average swirl angle. This led to a disorderly distribution of streamlines and backflow zones formation at the conduit outlet. Furthermore, the wall roughness impact on energy losses is due to the static pressure drop on the blade pressure surface and the increase of turbulent kinetic energy near the blade. Further studies on the roughness influence over wider range of PAT operating conditions are recommended, as they will lead to quicker equipment refurbishment.

Uncertainty Quantification of Multivariate Correlated Output in the Simulation of Airfoil Flow

Due to the limitation of human cognition, there are uncertainties in the turbulence model closure coefficients that cannot be ignored. The key to assess model objectively is quantifying the uncertainty of model coefficients scientifically. In the CFD simulation of flow around the airfoil, wall pressure distribution is a multivariate correlated quantity. This paper developed a surrogate model method based on artificial neural network to characterize the mapping relationship between multivariate input and multivariate output. In order to reduce the output dimension and modelling difficulty, this paper use Proper Orthogonal Decomposition (POD) method. An RAE2822 benchmark case under transonic conditions is used to study the uncertainty of Spalart-Allmaras (SA) turbulence model coefficients and verify the method developed by this paper.

Study of the phase resonance phenomenon in Francis turbine

In hydraulic machinery, the phenomenon of rotor-stator interaction will produce periodic pressure waves in the stationary cascade. These pressure waves have the same phase, and will spread and superimpose in the volute. When the pressure waves propagate and superpose, the pressure pulsation with large amplitude will be formed. This phenomenon is called phase resonance, which will threaten the safe and stable operation of the machinery. Therefore, it is necessary to study the rotor-stator interference phenomenon and analyze the risk of phase resonance. In this paper, a high head Francis turbine with phase resonance in engineering is analyzed, which is also verified by the existing theoretical analysis. The phase resonance in the turbine is analyzed by using one-dimensional hydro acoustic model and three-dimensional compressible flow CFD simulation method, and the results are consistent with the actual engineering phenomenon. In order to study the measures to improve the phase resonance phenomenon, this paper carries out the simulation calculation from changing the rotation speed, wave speed and guide vane-runner blade number combination. The results show that adjusting the above parameters can reduce the amplitude of large pressure pulsation caused by the phase resonance in different degrees, and changing the number combination of guide vanes and runner blades is the most effective and feasible measure.

Simulation of an interdigitated flow channel assembled in a proton exchange membrane Fuel Cell (PEMFC)

This short communication deals with designing an interdigitated flow channel fitted in a PEMFC. The flow plate created here through CFD simulations consists of interdigitated channels based on leaf veins with an active area of 50 cm2. CFD simulations of fluid flow, mass transport, and current distribution were conducted using the finite element method (FEM). Experimental and CFD simulations were carried out to characterize the operating conditions applied to the cathode of a PEMFC using different O2–H2O mixtures. Flow velocities ranged from 3 to 5 m/s, while the humidity and temperature were between 80 and 100% and 323–343 K, respectively. A typical pressure drop around 400 Pa was observed over the flow velocities studied in the flow channel. The linear flow velocities found in the gas diffusion layer (GDL) domain were between 2 × 10−7 to 0.98 m/s. The experimental polarization curve showed close agreement with CFD simulations for the current density interval of 0–0.5 A/cm2, where no limitations to mass transport affect the fuel cell responses. The current model and flow channels configuration will be on continuous development considering water transport through the membrane and future application on Anion Exchange Membrane Fuel Cells (AEMFC).

Improving Liquid Film Thickness Uniformity of Semiconductor Etching Equipment Using Flow Field Visualization and CFD Simulation

With the rapid development of semiconductor-based products, semiconductor and integrated circuit (IC) design industries have become indicators of global technological advance in this field. As IC structure becomes smaller and smaller, improved process technology and equipment are necessary to maintain and improve wafer yield rates and quality. Among them, the etching technology in the process has a close relationship with the size of IC structure. According to extensive research and analyses put forward by predecessors, the etching process and methods that affect the yield rates and quality of wafers have been well characterized. This study investigates the spray etching method of the single nozzle and double nozzle. Using computational fluid dynamics software developed by Ansys Fluent, the fluid is injected from the center as the starting point at the same flow rate and speed, and according to 2, 4, and 8 seconds of a different cycle and nozzle forms, moving back and forth to supply liquid on the surface of the 300 mm disc. It is assumed that fluid properties do not change with temperature. Additionally, the chemical reaction and thermal reaction of the etching process are not considered. Changes in the surface flow field resulting from different nozzles and cycle conditions are observed. This study involved setting up a set of flow field visualization systems, replacing the etching liquid with colored water, using a digital camera to capture the relationship between the gray index and the thickness of the water film, and calculate the film thickness. The average film thickness deviation ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${t}_{avg}$ </tex-math></inline-formula> ) and variable quantity ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${dt}$ </tex-math></inline-formula> ) were characterized and used as a means to compare performance. The results show that the average film thickness deviation of the double nozzles is higher than the single nozzle, and variable quantity ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${dt}$ </tex-math></inline-formula> ) is lower than that of the single nozzles. In general, the double nozzles increase the thickness and stability of the liquid film thickness on the surface of the disc.