Control Volume Method Research Articles

Numerical modeling of Liesegang structures formation process under an electric field influence

In the present paper the formation of Liesegang structures, i.e. the process of periodic deposition with the mutual diffusion of two reacting chemicals in the presence of an external constant electric field, is studied using numerical modeling. The mathematical model of the process consists of three differential equations of diffusion-reaction for the concentrations of the initial components and the resulting precipitate. The kinetics of sedimentation is described in accordance with the Ostwald’s supersaturation theory. The equations of the mathematical model in one-dimensional and two-dimensional statements were solved numerically using the control volume method using computer code written by the authors in the C ++ language. As a result of numerical simulation in the absence of an electric field, periodic structures were obtained formed of the precipitate, which qualitatively corresponds to the patterns observed in the experiments. It is shown that numerically obtained Liesegang rings satisfy the well-known laws: the ratio of the distances to neighbouring rings remains constant and there is a power dependence between the distances to the rings and the time of their formation. The influence of the ratio of the concentration of the starting substances and the electric field strength on the nature of the structures formed is investigated. It also has been shown that an increase in the electric field strength leads to an increase in the number of structures formed.

Mathematical model of the submerged jet for the cases of man-made spills

The paper considers a mathematical model of the spread of a multiphase submerged jet, the flow of which is associated with a well damage during the development of a field or pipeline during oil transport. The jet consists of oil droplets and gas bubbles, as well as water involved into the jet. The case is considered when the environmental conditions correspond to the conditions of a stable hydrate existence, because a hydrate shell begins to form and the gas bubble turns into a hydrate on the surface of the bubbles. The integral Lagrangian control volume method is used to describe the jet spread process. The thermo-physical characteristics of a multiphase submerged jet, depending on the vertical coordinate, as well as the trajectory of the jet, are obtained.

ОБТЕКАНИЕ ЦИЛИНДРА И АЭРОДИНАМИЧЕСКОГО ПРОФИЛЯ С УЧЕТОМ ЛАМИНАРНО-ТУРБУЛЕНТНОГО ПЕРЕХОДА

A simulation of the flow around a cylinder and a NACA 4412 airfoil was carried out using the γγ-Reθ laminar-turbulent transition model and without it. Numerical simulation was performed on the basis of the Reynolds-averaged Navier-Stokes equations (URANS), using the Spalart-Allmaras differential one-parameter turbulence model. The system of initial equations was written with respect to an arbitrary curvilinear coordinate system. The coordination of the pressure and velocity fields was carried out using the method of artificial compressibility modified to calculate non-stationary problems. The integration of the system of initial equations was carried out numerically using the control volume method. For convective flows, a countercurrent Rogers-Kwak approximation was used based on the third-order Roe scheme. In turbulence models, the TVD scheme with a third-order ISNAS flow limiter was used to approximate convective terms. Comparison of the calculation results for a flow past a cylinder is carried out using the laminar-turbulent transition model and without it. It is shown that at low Reynolds numbers, when the flow around a cylinder is laminar, and the track isturbulent use of the Spalart-Allmaras turbulence model leads to the development of a turbulent boundary layer on the cylinder and, consequently, to a change in the position of the separation point. The incorrect position of the separation point affects the pressure distribution in the bottom of the cylinder and, as a result, the integral aerodynamic characteristics. The use of the transition model allows one to adequately reconstruct the laminar separation near the leading edge of the airfoil with its subsequent reattachment. The use of the Spalart-Allmaras model alone leads to excessive generation of turbulent viscosity. It is shown that the use of the γ-Reθ laminar-turbulent transition model the qualitatively and quantitatively improves the results of numerical simulation. The results of numerical simulation of the flow around a circular cylinder and the NACA 4412 aerodynamic airfoil agree well with experimental data in a wide range of Reynolds numbers.

A control volume scheme using compact integrated radial basis function stencils for solving the Richards equation

A new control volume approach is developed based on compact integrated radial basis function (CIRBF) stencils for solution of the highly nonlinear Richards equation describing transient water flow in variably saturated soils. Unlike the conventional control volume method, which is regarded as second-order accurate, the proposed approach has high-order accuracy owing to the use of a compact integrated radial basis function approximation that enables improved flux predictions. The method is used to solve the Richards equation for transient flow in 1D homogeneous and heterogeneous soil profiles. Numerical results for different boundary conditions, initial conditions and soil types are shown to be in good agreement with Warrick’s semi-analytical solution and simulations using the HYDRUS-1D software package. Results obtained with the proposed method were far less dependent upon the grid spacing than the HYDRUS-1D finite element solutions.

Aerodynamics of the turbulent flow around a multi­element airfoil in cruse configuration and in takeoff and landing configuration

Numerical modeling of multi-element airfoil's aerodynamics employs the Reynolds averaged Navier-Stokes equations of incompressible environment, which are related via a single-parametric differential turbulence model by Spalart-Allmaras. The system of initial equations has been recorded with respect to an arbitrary curvilinear coordinate system. The pressure and velocity fields have been aligned by using an artificial compressibility method modified for calculating the nonstationary problems. The system of initial equations has been integrated numerically by applying a control volume method. The counter-flow Rogers-Kwak approximation has been used for convection flows, based on the Roe scheme of third-order accuracy. The turbulence models, in order to approximate the convection components, employed a TVD scheme with the third-order flow limiter ISNAS. The paper reports results from calculating a turbulent flow around a multi-element airfoil in a wide range of the angles of attack. The result of the current research is the performed analysis of the flow field around a multi-element airfoil, pressure coefficients, the lifting force, as well as the drag force. Physical features in a flow structure at flowing around the multi-element airfoil 30P30N have been identified. In the investigated range of the angles of attack, flowing around a airfoil in the takeoff and landing configuration is stationary in nature except for the regions where the flow is detached from sharp edges, such as the slat's inside part and a region in the tail part of the main profile. There are recirculation currents within these regions. With an increase in the angle of attack, dimensions of the detachable zone at the slat's inner surface decrease while remaining almost unchanged in the tail part of the main profile. At the top surface of the main profile there forms a jet of air, due to the acceleration of the flow between the slat and the leading edge of the main profile. The existence of a gap between the main profile and the flap leads to the interference of jet currents at the upper surface of the slat. It has been shown that the takeoff and landing configuration demonstrates the higher values of the lifting force coefficient than the cruise configuration, especially at large angles of attack. The calculation results agree well with the data by other authors.

Mathematical Modeling of the Processes of Heat and Mass Transfer of Hot Gas Jets with Fluid during Underwater Rocket Launches

A physical model is developed for math modeling of interphase interactions at the hot gas–water interface. The heat-and-mass transfer processes in two-phase flows appear at interfaces of phases. Profiles of the heat-and-mass transfer rates versus properties of phases and mode parameters are presented based on the Nusselt criterion equations. The hydrodynamic problem is solved numerically by integrating a set of equations of continuity, Navier–Stokes and energy equations using a control volume method (CVM). The conservation equations are supplemented with source terms: an equation of continuity—with a mass evaporation rate, an energy equation—with heat of interphase interactions. The computation results are compared with experimental data.

Профилирование камеры сгорания поршневого двигателя с непосредственным впрыскиванием бензина

The object of the study was a six-cylinder in-line engine for land transport system with direct gasoline supply and forced ignition. The problem of shaping the combustion chamber is solved using the numerical control volumes method in a three-dimensional formulation. Nonstationary equations of energy, motion, diffusion and continuity in the Reynolds form, supplemented by the k-ζ-f model of turbulence, are used as a basis for modelling the engine operation. To model fuel combustion, an extended coherent flame model (ECFM) was used. Calculations were performed using the AVL FIRE software. The processes of mixture formation were optimized by considering the current lines and velocity fields of a moving charge, taking into account the geometry of the combustion chamber and intake and exhaust ports. As a result, the efficiency of the engine increased and the combustion process became more stable in the part load modes employing different fuel supply laws.

Successive melting and solidification of paraffin–alumina nanomaterial in a cavity as a latent heat thermal energy storage

Latent heat thermal energy storage (LHTES) plays a main role in many industrial applications, especially in high-powered electronics cooling systems and providing the thermal energy demand when the energy supply is unavailable. In this study, the LHTES cycle process, including successive melting and solidification, investigates in a two-dimensional annular space of a square cavity filled with nanomaterial of paraffin–alumina as a nanoPCM. In the melting process, all sidewalls of the cavity are insulated. Meanwhile, a constant heat rate generates homogeneously within the central heat source. At the end of melting, the heat generation gets off, while a time-reducing temperature lower than the paraffin melting point imposes on the sidewalls, and then, solidification triggers. The numerical simulation was accomplished using control volume method and the governing equations solved using the SIMPLE algorithm. The enthalpy-porosity method was employed to model the phase-change front. The value of thermal conductivity and the viscosity of the nanofluid have been experimentally measured before the numerical modeling. In this study, the effect of volume fraction of nanoparticles (0–0.03) has been investigated on the successive melting and solidification rate for a constant Rayleigh number of 5.74 × 105. The results show that adding nanoparticles to the PCM equal to the volume fractions of 0.01 and 0.02 improves melting rate, but the nanofluid with the volume fraction of 0.03 represents a poor heat transfer rate during melting even weaker than those for nanofluid with the volume fraction of 0.01. It also observed that the nanomaterial with the volume fraction of φ = 0.03 represents the highest solidification rate. However, taking the overall performance of successive melting and solidification system into account, the nanofluid with the volume fraction of 0.02 remarked the most effective heat transfer rate in comparison with the other examined cases.

ИССЛЕДОВАНИЕ ТЕМПЕРАТУРНОГО ПОЛЯ В СЛОИСТОМ ПЛАСТЕ

Актуальность. В последнее время усиливается интерес к количественной интерпретации температурных измерений в скважине. Целями количественной интерпретации становятся индивидуальные дебиты жидкости из отдельных пластов, пластовое давление в отдельных пластах, дебит заколонного перетока, гидродинамические параметры пластов, характер изменения проницаемости пласта в прискважинной зоне. В качестве исходных данных для переходных процессов используются нестационарные поля давления и температуры в скважине, а для квазистационарных условий – распределение температуры по глубине. Особый интерес представляет определение дебитов притока из отдельных пластов и параметров прискважинной зоны пластов. Цель: оценить, насколько допущение об однородности пласта влияет на наблюдаемую температуру для некоторых сценариев работы скважины. Методы. Разработана и исследована численная модель, дискретизация осуществлена методом контрольных объемов, используется метод переменных направлений. Корректность численного решения проверена путем сравнения с известными аналитическими решениями. Результаты. Исследована задача о нестационарном температурном поле в слоистом пласте. Среднее отклонение решения для случая закачки может достигать 20 % и более, величина зависит от толщины (общего количества) пропластков. Среднее отклонение решения для случая закачки достигает максимум через 6–10 часов и далее медленно уменьшается со временем. Для случая добычи величина отклонения может достигать 100 %, максимум достигается в начале восстановления. Далее отклонение снижается, достигая через 24 часа величины порядка 40 % от первоначального отклонения. Задаваясь допустимой погрешностью решения в 10 % при моделировании восстановления в слоистых коллекторах, необходимо учитывать слоистость при ее величинах выше 15–25 % в зависимости от сценария работы скважины.

Probabilistic Control Volume Method for Evaluating the Effects of Notch Size and Loading Type on Fatigue Life

The probabilistic control volume method has great prospects in correlating the effects of specimen size, notch and loading type on fatigue life or fatigue strength. In this work, the effects of notch size and loading type on fatigue life are investigated by using the probabilistic control volume method. Rotating bending and axial loading fatigue tests are at first performed on the hourglass specimen, circumferential V-notch specimen and V-notch plate specimen of 30CrMnSiA steel. Experimental results indicate that the notch reduces the fatigue strength of specimens in terms of nominal stress amplitude while in terms of local stress amplitude, the notch specimen could endure higher fatigue strength. Then, the probabilistic control volume method is used to evaluate the effects of notch size and loading type on fatigue life. It is shown that the probabilistic control volume method correlates well the effects of notch size and loading type on fatigue life, even for the local stress of the notch root exceeding the yield stress of the material.

Mathematical Modeling of Canopy Forest Fire Spread in the Presence of Fire Breaks and Barriers

In this paper, we studied the spreading of canopy forest fires in the presence of fire breaks and barriers consisting of hardwood trees by mathematical modeling. Mathematically, this problem is reduced to solving Reynolds equations for a turbulent flow with allowance for chemical reactions. The control volume method was used to obtain a discrete analogue. Using numerical calculations, we obtained distributions of the fields of velocity, temperature, oxygen concentrations, volatile products of pyrolysis and combustion, and volume fractions of the condensed phase. The modeling allowed obtaining contours of the spread of canopy forest fires that depend on the stock and types of forest combustible materials, moisture content, wind speed and direction, etc. We were also able to determine the dependence of the size of fire breaks and barriers on the above parameters, at which a canopy fire stops spreading.

The development of a three-dimensional model of the ice growth process on aerodynamic surfaces

The object of research is the processes of hydroaerodynamics and heat and mass transfer that occur when icing the aerodynamic surfaces of aircraft during flight in adverse weather conditions. One of the problem areas in the development of software and methodological support that allows to simulate the icing processes, there are difficulties in the transition to solving the problem in three-dimensional formulation. As well as the presence of contradictions in existing methods in describing the physical picture and, accordingly, the thermodynamics of the process of ice growth.During the study, experimental and analytical methods were used to study the physical processes of ice growth on streamlined surfaces, based on a phased analysis of the interaction of supercooled droplets with the surface and their subsequent freezing at the wing edge. The proposed model of the process of ice growth is based on the use of the method of surface control volumes, based on the equations of continuity, conservation of momentum and energy. Based on the new experimental data obtained on the physics of icing, it is proposed to separate the processes of the formation of a bulk ice-water structure and subsequent complete freezing of this structure separately in the methodology for modeling ice growth. At the first stage of the fluid crystallization process, as part of the step on the icing time, the supercooled fluid contained in the droplets that fall on the streamlined surface passes into a state of thermodynamic equilibrium. That is, the latent heat of solidification released during the formation of an ice fraction in the ice-water structure will be equal to the internal heat required to heat the supercooled fluid from the temperature of the droplets to the temperature of the phase transition. At the second stage, the water contained in the ice-water structure will freeze due to heat loss by convection, evaporation, sublimation, thermal conductivity (minus the latent heat of solidification, kinetic and aerodynamic heating). It should be noted that the water that will freeze will also fetter the spatial ice structure. In this case, the method of successive approximations is applied to determine the direction of fluid movement along the streamlined surface.Compared with the well-known traditional methods, this approach makes it possible to take into account to a greater extent the real physical processes of icing of aerodynamic surfaces that are extremely complex for mathematical description.The results can be used to optimize the operation of anti-icing systems and determine ways to reduce energy costs during the operation of such systems.

An unstructured mesh control volume method for two-dimensional space fractional diffusion equations with variable coefficients on convex domains

In this paper, we propose a novel unstructured mesh control volume method to deal with the space fractional derivative on arbitrarily shaped convex domains, which to the best of our knowledge is a new contribution to the literature. Firstly, we present the finite volume scheme for the two-dimensional space fractional diffusion equation with variable coefficients and provide the full implementation details for the case where the background interpolation mesh is based on triangular elements. Secondly, we explore the property of the stiffness matrix generated by the integral of space fractional derivative. We find that the stiffness matrix is sparse and not regular. Therefore, we choose a suitable sparse storage format for the stiffness matrix and develop a fast iterative method to solve the linear system, which is more efficient than using the Gaussian elimination method. Finally, we present several examples to verify our method, in which we make a comparison of our method with the finite element method for solving a Riesz space fractional diffusion equation on a circular domain. The numerical results demonstrate that our method can reduce CPU time significantly while retaining the same accuracy and approximation property as the finite element method. The numerical results also illustrate that our method is effective and reliable and can be applied to problems on arbitrarily shaped convex domains.

Large eddy simulation of an axial pump with coupled flow rate calculation using the sharp interface immersed boundary method

PurposeTurbomachinery, including pumps, are mainly designed to extract/produce energy from/to the flow. A major challenge in the numerical simulation of turbomachinery is the inlet flow rate, which is routinely treated as a known boundary condition for simulation purposes but is properly a dependent output of the solution. As a consequence, the results from numerical simulations may be erroneous due to the incorrect specification of the discharge flow rate. Moreover, the transient behavior of the pumps in their initial states of startup and final states of shutoff phases has not been studied numerically. This paper aims to develop a coupled procedure for calculating the transient inlet flow rate as a part of the solution via application of the control volume method for linear momentum. Large eddy simulation of a four-blade axial hydraulic pump is carried out to calculate the forces at every time step. The sharp interface immersed boundary method is used to resolve the flow around the complex geometry of the propeller, stator and the pipe casing. The effect of the spurious pressure fluctuations, inherent in the sharp interface immersed boundary method, is damped by local time-averaging of the forces. The developed code is validated by comparing the steady-state volumetric flow rate with the experimental data provided by the pump manufacturer. The instantaneous and time-averaged flow fields are also studied to reveal the flow pattern and turbulence characteristics in the pump flow field.Design/methodology/approachThe authors use control volume analysis for linear momentum to simulate the discharge rate as part of the solution in a large eddy simulation of an axial hydraulic pump. The linear momentum balance equation is used to update the inlet flow rate. The sharp interface immersed boundary method with dynamic Smagorinsky sub-grid stress model and a proper wall model is used.FindingsThe steady-state volumetric flow rate has been computed and validated by comparing to the flow rate specified by the manufacturer at the simulation conditions, which shows a promising result. The instantaneous and time averaged flow fields are also studied to reveal the flow pattern and turbulence characteristics in the pump flow field.Originality/valueAn approach is proposed for computing the volumetric flow rate as a coupled part of the flow solution, enabling the simulation of turbomachinery at all phases, including the startup/shutdown phase. To the best of the authors’ knowledge, this is the first large eddy simulation of a hydraulic pump to calculate the transient inlet flow rate as a part of the solution rather than specifying it as a fixed boundary condition. The method serves as a numerical framework for simulating problems incorporating complex shapes with moving/stationary parts at all regimes including the transient start-up and shut-down phases.

Numerical investigation of the effect of second order slip flow conditions on interfacial heat transfer in micro pipes

Heat transfer that occurs in micro scale devices has a very important place among the engineering applications that cooling or heating. This heat transfer mechanism in devices having dimensions at micron level is a completely different problem in the macro level analysis. Therefore, in the calculations made, the flow events and heat transfer in micron scale pipes are calculated by using more realistic expressions. For this reason, in this study heat transfer in a circular micro pipe with wall and fluid conjugation for laminar rarefied gas flow in transient regime is investigated under the second order slip boundary conditions at the interface. Patankar’s control volume method is used here to solve the problem numerically. This analysis includes of axial conduction, viscous dissipation and rarefaction effects which are indispensable in micro-flow structure. From the results, it is seen that the values that are indicating heat transfer are excessively affected by wall thickness, viscous heating and gas rarefaction especially in transient regime.

The effects of porous wall as a novel flame stabilization method on flame characteristics in a premixed burner for CH4/air mixture by numerical simulation

In the present investigation, the effects of inner diameter and length of porous wall on flame stability limits, temperature distribution, methane conversion, and pressure drop in a premixed burner are numerically analyzed. The governing equations are solved by the control volume method, considering Re-Normalization Group k-ɛ for turbulence modeling and eddy dissipation concept for turbulence–chemistry interaction modeling. The simulations are performed for various porous walls with the inner diameter of 30, 40, and 50 mm and the length of 22, 44, and 66 mm. The results demonstrate that the increase in inner diameter of porous wall causes an increase in the lower flame stability limit and a decrease in the upper flame stability limit, gas and solid temperatures, pressure drop, and methane conversion. Also, the maximum solid and gas temperatures in the porous wall and methane conversion are related to the porous wall with 44 mm length. Furthermore, the methane conversion and pressure drop increase with the rise in the equivalence ratio. Finally, it can be said that the change in inner diameter of porous wall is more important than the length of porous wall in the studied phenomena.

Thermal-hydraulic model for axial piston pump with leakage and friction and its application

PurposeThis paper aims to carry out a thermal-hydraulic simulation model for pump and hydraulic system to predict the temperature increasing and pump performance. Based on the model, how to alleviate the temperature is introduced. Besides, the optimization of piston is carried out.Design/methodology/approachThis paper analyzes the heat generation in lubricating interfaces of the pump with energy conversion theory. The heat transfer inside the pump is analyzed with the control volume method. The simulation model is constructed in AMESim because of its operating friendly nature. The experiment is carried out to prove the validity and accuracy of the simulation model.FindingsTemperature has less effect on the mechanical loss of pump. However, it has a great impact on volumetric efficiency. To reduce the temperature on the piston surface, the size of the piston should be optimized.Originality/valueThis paper fulfills a novel thermal-hydraulic model to evaluate the temperature of the pump. Based on the model, the performance of the pump is determined and optimization is carried out.

The Effect of the Pipe Bending Angle on the Pressure Losses Vane Elbow Pipes

Pressure loss is one of the significant parameters in designing pipe bends. In this paper, the pressure distribution and pressure losses induced by turbulent flows in a circular cross-sectioned piping elbow with or without guide vane were simulated. The flow distribution in the piping elbow was simulated by the k- model using control volume method. The main objective of this study is to characterize the effect of changing the angle of pipe bend and Reynolds number on the flow separation of single-phase turbulent flow through numerical simulation. Results were validated by other experimental results and then loss coefficient was calculated in different angles from 45 to 135-degree pipe bend in various radius ratios with or without guide vane. Despite the fact that increasing pipe angle increased the pipe bend loss coefficient, using guide vane in the pipe elbow decreased this coefficient. In the radius ratio 1.5 with one guide vane, the loss coefficient of the pipe bends decreased by 50 percent in all degrees. Results revealed that the use of two vanes in pipe bend is more effective on the reduction of elbow pressure losses. Moreover, two guide vanes can decrease loss coefficient more than 50 percent. Also, the results indicated that loss coefficient decreased by increasing Reynolds number.

A New Approach to the Numerical Modeling of Pure Metal Solidification

Solidification and cooling processes proceeding in the metal domain can be described in different ways. One of them consists in the application of the Fourier-type equation in which the additional term (source function) controlling the solidification process is introduced. In this paper this type of energy equation is used, but for the phase change modeling the equation discussed is in some way transformed. Such a modification is possible if one considers the pure metal for which the solidification point is a constant value. The numerical model used at the stage of computations is based on the Control Volume Method. In the final part of the paper, examples of computations are shown.

Power-Law Fluid Flow in a T-Shaped Channel Under the Action of Assigned Pressure Drop

Plane non-Newtonian incompressible fluid flow in a T-shaped channel is considered. The effective viscosity is determined according to the Ostwald–de Waele law. The fluid moves under the action of the given pressure drop in the boundary sections of the channel. The no-slip condition is fulfilled on the solid walls. The problem is solved numerically by the control volume method with the use of the SIMPLE procedure. As a result of the investigation carried out, the characteristic flow regimes have been revealed. The influence of the basic parameters of the problem on the flow pattern has been demonstrated, and the dimensionless relations of flow characteristics have been constructed.