Channel Of Rectangular Cross-section Research Articles

Analysis of the limits of applicability of photomultiplier as part of a laser Doppler anemometer

This paper considers the use of laser Doppler anemometry methods for non-contact measurement of the velocity of a moving medium in various phase states. It notes the specifi city of conditions of application of laser Doppler anemometers, in that in a number of cases there are restrictions on the use of photomultipliers. In order to find an alternative to the classic photomultiplier of a laser Doppler anemometer, three types of photomultipliers were investigated: classic electric vacuum; silicon avalanche multipixel; with microchannel plate. Physical experiments were conducted to measure the velocity of aero- and hydrodynamic fl ows by laser Doppler anemometer with photomultipliers of three types. The velocities of a rotating glass disk, aerodynamic fl ow in a channel of rectangular cross-section, and aerosol flow, as well as hydrodynamic flooded jet, were measured. The obtained experimental data were analyzed according to the criteria for assessing the quality (efficiency) of operation and finding the limits of applicability of the specified photomultipliers. Based on the physical properties of the studied photomultipliers, their efficiency indicators in various experiments are explained. The boundaries of the effective use of these photomultipliers in a wide class of physical experiments are determined. The results obtained are important for researchers involved in the study of aero- and hydrodynamic flows, where accurate and reliable measurements of flow velocity are required. The choice of an optimal photomultiplier tube for the laser anemometer allows increasing the accuracy and ease of obtaining experimental data and reducing the cost of the final equipment.

Experimental and numerical study on the hydrothermal performances of a mirror Y-shaped convergent mini-channel heat sink

To enhance the hydrothermal performances of conventional mini-channel heat sinks, this paper experimentally and numerically investigates the impacts of channel convergence α on the pressure drop (ΔP), the maximum temperature (Tmax), the convective heat transfer coefficient (hs), and the coefficient of performance (COP) of a mirror Y-shaped convergent mini-channel heat sink (MYCMCHS) with a rectangular channel cross-section when the overall channel volume is fixed. The results show that ΔP of MYCMCHS initially decreases and subsequently increases with the growth of the channel convergence α; however, Tmax declines but hs grows with the drop of α. The different dependences of hydraulic and thermal performances on the channel convergence result in the COP of the MYCMCHS exhibiting an initial increase followed by a subsequent decrease with the rise of α. The numerical results further indicate that at the inlet volume flow rates of 0.05, 0.1, and 0.15 L/min, the maximum COP achieved at α = 0.7 rises 19.91 %, 20.57 %, and 22.59 % respectively in comparison to the COP of the heat sink without channel convergence (α = 1). This work validates the performance advantages of channel convergence and provides a possibility to enhance the hydrothermal performances of heat sink.

Condensation of refrigerant R450A inside a mini-channel of rectangular cross section with an internal longitudinal groove

The increase in jeopardizing the natural life and environmental resources together with high rates of energy consumption are still in focus of international agencies and scientific communities. For this issue, global awareness has been oriented to uncover alternatives to harmful species and to magnify effectiveness of thermal systems. For the present study, the refrigerant R450A was investigated as a potential substitute to R134a during condensation inside two mini-channels which are rectangular cross section channel (SRC) and a modified channel (MRC). The latter was characterized by an internal longitudinal groove. Wall subcooling, vapor superheat, and mass velocity were considered while the condensing temperature was kept constant. Numerical CFD simulation was realized for the condensation inside the channel using ANSYS-CFX 15.0 with additional user defined expressions to describe condensate layer thickness inside the channel, which in turn was utilized for the condensation heat transfer coefficient (HTC). Predicted data was compared with experimental one in terms of HTC and flow patterns. The predicted data of HTC showed an average deviation by about 6.6% compared to experimental data for R134a.The refrigerant R450A resulted in lower condensation HTC than R134a. The MRC resulted in an average enhancement in the HTC by about 55.7% and a reduction in the pressure drop by about 10.4% compared to the SRC. The enhancement of the MRC in HTC was remarkable at high degree of superheat, low mass velocity, and low wall subcooling. Accordingly, the novel MRC represents a successful potential design for condensation of refrigerants in mini-channels.

Numerical Analysis on the Flow Depth Behaviour in Dividing Open-Channel for Trapezoidal and Rectangular Cross-Sectional Channel

Open-channel with dividing flow refers to any side water withdrawals from streams, rivers or main channel. It has become an important issue, especially when measuring the flow rate and depth of water as part of environmental management schemes and many practical projects such as the system of irrigation and drainage, water and wastewater treatment plants and other water resources projects. Although the behaviour of water flow has been studied by the previous researchers, most studies have only been carried out in simulation and lab experiment for a straight prismatic main channel. The mathematical research has not yet been fully understood, particularly in relation to the behaviour of water flow. Therefore, in this study, the modified general equations of dividing open-channel flow for rectangular and trapezoidal in the form of non-linear polynomial equations are proposed to determine the upstream flow depth in rectangular and trapezoidal cross-sectional channel. The computations of water flow depth influenced by Froude number and discharge ratio are performed with Newton-Raphson procedure in Maple software. The numerical solution shows the good agreement with the experimental data as reported in the literature. When the discharge ratio is low, the flow depth in trapezoidal and rectangular channels exhibits similarity. However, as the discharge ratio increases, the flow depth in the rectangular channel surpasses that in the trapezoidal channel. Furthermore, as the Froude number increases, the rectangular channel experiences a greater flow depth in comparison to the trapezoidal channel.

Experimental study of three-phase helium mixtures in confined channels

An experimental campaign was conducted to investigate heat and mass transfer phenomena in superfluid helium (He II) in two rectangular cross-section channels of high aspect ratios and different thickness resembling the space between steel collars in the LHC superconducting magnets. The experiments consisted of clamped heat flux tests at atmospheric pressure, in which a heater strip suddenly releases a constant heat load into the channel that is open to a helium bath on one side. The difference in thickness between the two channels allowed exploring the effect of the geometrical confinement on the propagation of both phase change fronts: i) the He II-He I λ-transition front; ii) the He I-vapour first-order transition front.The observations show that, in the thinner channel, it is possible to distinguish different behaviours of the phase fronts depending on the extent of the heat flux. For increasing heat flux values, the λ-front speed successively increases sharply, decreases, and increases weakly. This sequence is determined by the presence of the vapour film, which either diminishes the He II-He I transformation rate by lowering the heat transfer or pushes the λ-front while expanding. In the thicker channel, the intermediate behaviour is absent as the level of confinement is lower and the He I phase never expands considerably along the highest dimension of the channel.

Method to fabricate round cross-sectional channel using thermal expansion of air for passive flow regulators.

We present a novel method for fabricating round cross-sectional channels for autoregulatory pressure regulators. The application of previously reported methods in multilayered soft lithography using three-dimensional printed molds is challenging. Herein, we used a thermal expansion technique to create round cross-sectional channels in replica layers using air in the cavity space of a master mold. The width and height of the round channel in the replica could be adjusted in the range of 80-300 and 3-57µm, respectively, by varying the precuring time of the replica in the gel state and adjusting the cavity size of the master mold. We successfully fabricated a pressure regulator with a round cross-sectional channel, achieving a constant output pressure at a low threshold input pressure. Our device exhibited superior performance, with a constant output pressure at a threshold input pressure of less than 77%, compared to a device with a rectangular cross-sectional channel. Our method has significant potential for application in the fabrication of integrated microfluidic systems with round cross-sectional channels.

Experimental study of DDT run-up distance and detonation wave velocity deficit for stoichiometric hydrogen-oxygen mixture in micro spiral channels

The determination of the deflagration-to-detonation transition (DDT) run-up distance LDDT is crucial in both the design of pulse detonation engines (PDEs) and safety protection. In this paper, the LDDT of stoichiometric hydrogen–oxygen premixed mixtures in millimeter-scale spiral channels were studied experimentally and high-speed photography was employed to capture the flame propagation process. The mixtures in three different rectangular cross-sectional channel widths and various initial pressures were tested. The present results show that the LDDT decreases with increasing the initial pressure, and with increasing the channel width, the larger LDDT is observed. When fitted with the LDDT ∝ P0−m relationship, the values of m obtained in this study ranged from 0.997 to 4.222. The critical value of 0.1 for the equivalent area divergence induced by the boundary layer ξ is proposed to distinguish the behaviors between micro and macro channels, regardless of the type of mixture and channel configuration. Additionally, an empirical formula considering the influence of the boundary layer is obtained, which allows for the quantitative prediction of LDDT. Because of the influence of curvature effect, the smaller the channel width, the greater the prediction deviation. Due to the boundary layer effect, the momentum and heat dissipation are more pronounced in smaller tubes, resulting in a loss of velocity for the detonation wave. It is suggested that the detonation wave velocity increases as the channel width and initial pressure increase. Fay's model was employed to estimate the theoretical velocity deficit, and the experimental results were in good agreement with the theoretical predictions. The current findings are valuable for the field of micro PDE and may help the assessment of the potential risk of hydrogen detonation in micro systems.

Permanent Magnet Pump for Aluminum Transport in a Linear Channel

Electromagnetic systems designed for liquid metal transportation offer a wide range of promising applications within the metallurgical industry. Among these systems, permanent magnet pumps stand out due to their remarkable advantages in processing liquid aluminum. These pumps enable the generation of a vigorous flow of liquid metal, all while maintaining a non-contact approach by bypassing the insulating walls. In this study, we investigate the flow of liquid metal in a rectangular cross-section channel created by a permanent magnet pump. To comprehensively analyze the flow characteristics and assess the technology's scalability, we employ order of magnitude evaluations alongside an experimental model. Ultrasound Doppler velocimetry is utilized to measure the velocity distribution, which reveals an intriguingly asymmetric flow pattern. Furthermore, additional measurements are conducted to determine the integral flow rate and pressure difference. Remarkably, the numerical simulations align closely with the experimental results, demonstrating good agreement in the p-Q curves. These findings provide substantial evidence supporting the remarkable potential of this technology in various applications, such as aluminum degassing, particle agglomerate dispersion, and efficient transportation.

Development of Ultrasound Phantom Made of Transparent Material: Feasibility of Optical Particle Image Velocimetry.

The need for ultrasound flow phantoms to validate ultrasound systems requires the development of materials that can clearly visualize the flow inside for measurement purposes. A transparent ultrasound flow phantom material composed of poly(vinyl alcohol) hydrogel (PVA-H) with dimethyl sulfoxide (DMSO) and water solution manufactured using the freezing method and mixed with quartz glass powder to exhibit scattering effects is proposed. To achieve transparency of the hydrogel phantom, the refractive index (RI) was changed to match that of the glass by modifying the PVA concentration and the ratio of DMSO to water in the solvent. The feasibility of optical particle image velocimetry (PIV) was verified by comparing an acrylic rectangular cross-section channel with a rigid wall. After the feasibility tests, an ultrasound flow phantom was fabricated to conduct ultrasound B-mode visualization and Doppler-PIV comparison. The results revealed that the PIV measured through PVA-H material exhibited 0.8% error in the measured maximum velocity compared with PIV through the acrylic material. B-mode images are similar to real tissue visualization with a limitation of a higher sound velocity, when compared with human tissue, of 1792 m/s. Doppler measurement of the phantom revealed approximately 120% and 19% overestimation of maximum and mean velocities, respectively, compared with those from PIV. The proposed material possesses the advantage of the single-phantom ability to improve the ultrasound flow phantom for validation of flow.

Experimental investigation on the DDT run-up distance and propagation characteristics of detonation wave in a millimeter-scale spiral channel filled with hydrogen-air mixture

An experimental study on flame acceleration (FA) of hydrogen–air premixed gas in a millimeter-scale Archimedean spiral channel is presented in this paper. Experiments were performed in a transparent rectangular cross-section channel and the flames were recorded by high-speed photography. The flame propagation characteristics under different equivalent ratios and different initial pressures was compared. Results show that the minimum LDDT (DDT run-up distance) are obtained at stoichiometric ratio. With increasing the initial pressure, the smaller LDDT is observed. Moreover, Due to the combined effects of the chemical reaction, wall dissipation, and geometric curvature effects, the flame velocity showed a significant oscillation pattern. Decreasing initial pressure and keeping away from the stoichiometric ratio increasingly favour the low velocity mode. The current conclusions of this study are helpful to the development of new pre-detonator and evaluate the danger of hydrogen detonation in micro systems.

Determination of pressure in a flat channel of a solar water heater when water is freezing

The article considers the problem of modeling a flat channel of a rectangular section of a solar water heater filled with water when it freezes to estimate the pressure in the channel. It is assumed that when water freezes, due to the density difference between the solid and liquid phases, the volume it occupies in the channel changes, and under the influence of the pressure difference between the external environment and the internal region, the channel walls are deformed. The theory of elastic cylindrical bending of a two-layer plate is used for modeling. Calculation relations are obtained for determining the pressure in the channel depending on the thickness of the freezing ice. An example of calculating the pressure in a channel of rectangular cross section when water freezes is given.

Численное моделирование интенсификации теплообмена в каналах термоэлектрической генераторной установки

This article discusses some options for improving the efficiency of an experimental thermoelectric generator set designed to study, design and create power plants of this type, during direct operation of which there are no fuel combustion processes intended for power supply to industrial consumers located at a considerable distance from the centralized power supply, and namely, objects of production, distribution and transportation of natural gas, a schematic diagram of the mentioned power plant is shown, the principle of operation and the main advantages of its application are described. Improving the efficiency of the installation in the present study consists in intensifying heat transfer in profiled channels of rectangular cross section, through which flows of gaseous coolant with different temperatures are supplied. Several variants of the possible heat transfer intensification are shown, which consist in the implementation of intensifiers inside the channels in the form of transverse solid partitions, alternately changing the direction of the coolant movement. In the presented study, numerical simulation of several variants of heat transfer intensification with different geometric dimensions of the flow section of profiled channels and transverse baffles installed in these channels was carried out. In conclusion, the results of the numerical simulation are shown, the effectiveness of the methods used is evaluated, of which the most optimal one is selected.

Experimental thermohydraulic characterisation of flow boiling and condensation in additive manufactured plate-fin heat exchanger

In this paper, an experimental thermohydraulic characterisation of flow boiling and condensation in additive manufactured plate-fin heat exchanger was conducted. An original experimental setup was developed allowing the operation of the heat exchanger in both boiling and condensation modes. This heat exchanger mainly consists of two identical rectangular cross-section channels, each of hydraulic diameter 3 mm, with n-pentane and water used as refrigerant and heat transfer fluid, respectively. An experimental metrology, based on the infrared thermography, was implemented in order to determine the temperature distribution of the heat exchanger external walls. Based on this temperature distribution, the mean temperature profiles of the n-pentane and water were determined, allowing the identification of three main flow zones: the subcooled liquid zone, the two-phase (boiling/condensation) zone and the superheated vapour zone. These mean temperature profiles were then coupled to a developed thermal model in order to determine the local thermal parameters, such as heat flux, vapour quality and heat transfer coefficient. The total pressure drop in the two-phase zone was finally determined. All the obtained results were analysed and compared with data available in the literature.

Effect of PVD-coated chromium on the subcooled flow boiling performance of nuclear reactor cladding materials

We elucidate the separate effect of a thin Cr coating deposited by physical vapor deposition (PVD) on the subcooled flow boiling performance of zircaloy-4. First, we run flow boiling experiments on prototypical zircaloy-4 surfaces mimicking the scratch pattern and surface roughness of nuclear reactor claddings. Then, we PVD-coat a 0.3 µm thick chromium layer on the same exact surface and repeat the same flow boiling investigations. All experiments are run using deionized water at atmospheric pressure, flowing on a 1 × 3 cm2 rectangular cross section channel at a rate of 1000 kg/m2/s and a subcooling of 10 K. We measure the average temperature of the boiling surface at increasing surface heat fluxes, covering a wide range of heat transfer regimes, from single-phase forced convection to the boiling crisis. We also record high-speed videos of the boiling process, which we postprocess to measure bubble nucleation site density, growth time, departure diameter and frequency. The surface analysis reveals that, while the chromium coating does not seem change the surface roughness and morphology, it improves surface wettability. However, it decreases the critical heat flux. The chromium coating causes an increase of nucleation temperature, bubble departure diameter and growth time, and a reduction of the nucleation site density. The concurrence of these observations indicates that a size reduction of the nucleation sites, conformally covered by the chromium coating, may be the cause of the boiling performance deterioration. We confirm this hypothesis repeating the same analysis on a FeCrAl sample prepared and tested using the same protocol as the zircaloy-4 sample, but with a different initial surface texture.

Reinforced Metal Bipolar Plate Structure with Stiffening Rib Channel for Proton Exchange Membrane Fuel Cell

Clean energy has become a concern because fossil fuels are the primary cause of climate change and air pollution. A proton exchange membrane fuel cell (PEMFC) is an environmentally friendly vehicle's alternative energy source. Bipolar plates, gaskets, gas diffusion layers, and a membrane electrode assembly (MEA) constitute a PEMFC. The bipolar plate accounts for a large portion of the fuel cell stack cost. Metallic bipolar plates are utilized in vehicle fuel cell stacks because of its low fabrication cost, high manufacturability, mechanical strength, and low weight. Despite the benefits, a thin metal plate's major drawbacks are deformation and corrosion resistance.Stamping metallic plates form trapezoidal cross-section channels, whereas machined graphite bipolar plates can form rectangular cross-section channels. These trapezoidal channels make dimensional errors such as draft angle and fillet radius. Besides, the spring back effect makes shape errors in the manufacturing process. Many kinds of research are conducted to understand the impact of channel tolerances on the performance of PEMFCs. Mainly, inhomogeneous compression in the gas diffusion layer results from channel deformation caused by unevenly distributed clamping force and shape inaccuracy of the metal bipolar plates. High contact resistance and non-uniform reactant gas distribution would result from inhomogeneous compression of GDL.A reinforced metal bipolar plate structure with a stiffening rib channel is proposed in this work to reduce bipolar plate deformation and improve gas transfer. Stiffening ribs are imprinted on the top of the channel. The geometric parameters of the stiffening ribs are shown in Fig. 1. Four geometric design parameters are selected in this study, the distance between the ribs d1, length of the rib l1, height of the rib l2, width of the rib l3. The stainless steel with a thickness of 0.1mm, 50mm×50mm active area, is used. The influence of stiffening ribs on mechanical and fluid characteristics is investigated through numerical analyses and experiments. The static structural analysis illustrated the deformation of the bipolar plates and GDL under the clamping force. The stiffening rib design improves the structure's mechanical strength, lowering bipolar plate deformation. When the endplates clamp the stack, the stiffening rib supports more stress to prevent deformation. The flow distribution in the flow field channel and GDL was then visualized using computational fluid analysis. Stiffening ribs increase the velocity of the reactant. Enhanced under rib convection makes reactant gas evenly distributed. In addition, stiffening ribs causes a convective flow beneath the rib, which improves PEMFC performance. Finally, single-cell studies proved that the modeling results are valid. Under the same operating conditions with the same geometry to the model, electrochemical reactions are investigated to decouple the leading cause of the resistances. The ohmic resistance decreases with the robust structural stiffening ribs, and the mass transport resistance decreases with the well-distributed reactants flow. Figure 1

Type and Size Filler Grain Selection During Abrasive Flow Machining of Non-Ferrous Alloys Rectangular Parts

Abrasive flow machining is one of the promising technologies for finishing rectangular channels, which are characterized by a constant increase in accuracy and reduction of surface roughness. During modeling of the contact interaction of grains in the corners of rectangular part cross-section in the intended abrasive flow machining, the key features of the size and filler grain selection were fixed. In the proposed spherical grain model, the cutting edge is represented by an arc of a circle that fits into the angle of the cross-section of the part. It is confirmed that if the grain size exceeds the value of the inner radius of the rounded corner of the part, an untreated zone appears, with the area calculated using the CAD system on a straight, blunt and sharp angle. It is proved that the abrasive flow machining with a filler grain size not exceeding the value of the doubled radius of the inner rounding of the rectangular cross section is carried out. The physical properties, the shape of the cutting edges and the applicability of fillers, electrolytically produced corundum, silicon carbide and synthetic diamond were analyzed. The shape of a diamond grain, in comparison with the fillers mentioned above, shows a more rounded shape of the crystal faces, which causes greater microhardness and abrasive ability. The revealed features confirm the exclusion of a possible phenomenon, caricature in the process of post-processing. It is recommended to use abrasive grains of a certain grain size for uniform pretreatment of a rough surface, followed by polishing with an ointment-like diamond paste of channels of small transverse rectangular cross-section. Using as a filler a salve-like paste containing diamond grains allows to reduce medium viscosity and machining corners of rectangular section. Abrasive flow machining with medium viscosity with synthetic heat-resistant rubber as the base and a salve-like paste containing diamond grains is used for finishing machining of non-ferrous parts. The use of working media with diamond pastes will allow the internal channels of blanks made of various non-ferrous alloys to be treated with an abrasive flow. For the minimum cross sections of rectangular pipe blanks presented in this article, it is proposed to use the working medium of the following weight composition: synthetic heat-resistant rubber - 40 %, salve-like diamond paste ASN 60/40 PTO - 60 %. A number of general recommendations are proposed for further research.

Parametric analysis of entropy generation in turbulent convective forced flow in spiral channels at constant wall temperature

This paper discuses parametric analysis of entropy generation in a forced convective turbulent flow of an incompressible fluid at a constant wall temperature through spiral channel of rectangular cross section of flow path has been carried out in terms of mass flow rates, temperature differences between fluid flow and temperature of the wall of the passage surface, width of the spiral passages and the shape geometry, as well as the types of fluids. Air is used as the working fluid, and in addition, water and oil are used in the parametric study. The competition between heat transfer enhancement and entropy generation has been addressed. Comparison between entropy generation profiles for different passage widths and shapes have been reported. It is found that mass flow rate and inlet fluid temperature have a considerable impact on entropy generation and hence on heat transfer enhancement. The findings show that the narrowing passage consumes more pump power to overcome viscous dissipation. Regarding the channel shape, it was found that the rectangular channel has the highest Bejan number (Be), while the circular spiral channel requires more pump power than the others. The data obtained indicate that the selection of the most suitable configuration and the best flow conditions becomes a critical task. Also, secondary flow due to centrifugal force and curvature has a significant effect on increasing heat transfer and entropy generation.

Preliminary measurements of the thermal field in a rectangular channel by the Planar Laser-Induced Fluorescence method

This paper describes the preliminary measurements of the temperature field in water channel by the Planar Laser-Induced Fluorescence (pLIF) method in the one-colour mode using commercial LaVision system. The aim of the work is to find suitable parameters of the experiment (concentration of fluorescent dye, laser power, arrangement of the experiment, etc.) for measuring the temperature field in a closed glass channel of rectangular cross-section with a heated bottom-wall and to verify an applicability of this method in general. The intention is to carry out measurements in the transverse plane of the duct such that the specific thermos-convective structures of the Rayleigh-Bénard-Poiseuille convection, i.e. mixed convection, are detected. For simplicity, the initial measurements were performed in the natural convection regime, i.e. the water in the channel was not driven by the circulation pump. In this case, a larger temperature gradient was created near the wall compared to the force convection case when a cold liquid flows continuously into the experimental area. The presented work serves as a methodological guide for future measurements and provides valuable experience for the design of a new channel for investigation of the thermal flow instabilities.

Double phase transition numerical modeling of superfluid helium for fixed non-uniform grids

A thermo-fluid dynamic numerical model for the simulation of the first-order and second-order phase transitions in superfluid helium (He II) is presented. The model is based on a Finite Volume Method algorithm for transient multidimensional problems. The method is also intended to simulate the conjugate heat transfer between helium and solids, which required the implementation of customized boundary conditions to replicate the Kapitza resistance and superfluid partial slip. The implemented governing equations for He II constitute an advanced version of an existing single-fluid model, which is herein newly derived with a more generalized conductive heat power law. The vapor-He I phase change is addressed through an explicit calculation of the volume fractions of the helium mixture and by introducing surface tension forces. An algorithm for fixed non-uniform orthogonal grids is conceptualized to deal with the lambda transition. The model is validated against data collected during clamped heat flux experiments in a rectangular cross-section channel of a high aspect ratio. The experiments were conducted both above and below the lambda temperature at atmospheric pressure with the channel in horizontal and vertical positions. The comparison with the data showed satisfactory agreement in the temperature profiles.

THREE-DIMENSIONAL PROBLEM OF FLOW OF VISCOUS LIQUID THROUGH AN OPEN CHANNEL IN THE FORMULATION OF A NUMERICAL SOLUTION

The article considers the statistically stationary incompressible flow of a viscous liquid in an open inclined channel of rectangular cross-section with large Reynolds numbers. A threedimensional numerical model with a periodic boundary condition along the axis of the main flow is proposed. Three-layer boundary conditions are placed on the wall and near the channel wall, and on the free surface - conditions of non-zero or zero tangential voltage. To solve the problem, the predictor-corrector schemes of the method of fractional steps by horizontal coordinates with a cosine-oidal spectral function by vertical coordinate are used. A general algorithm for solving the problem with an isoperimetrically optimized cross-section parameter is compiled.