Correct numerical simulation of a two-phase coolant

Kroshilin, A. E.; Kroshilin, V. E.

doi:10.1134/s004060151602004x

Cited by 4 publications

(6 citation statements)

References 2 publications

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“…Some of the phase velocities are equal and two-fold. As we can see from (5) equal phase velocity values are the roots of (5). Consequently excluding the case when a root is equal to identical phase velocity values equations ( 5) and ( 6) have the same roots.…”

Section: Hyperbolicity and Characteristics Of A N-velocity Mixturementioning

confidence: 89%

“…Nowadays three methods for solution of the named problem are known [2][3][4]. The first one is to introduce additional stabilizing terms into the momentum equations [5]. The second way is to take into consideration the difference of phase pressures [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

“…Nevertheless (see for example [3]) this case (N > 2) can widely occur naturally and in different areas of human activities, for example in thermal and nuclear power engineering, chemical, metallurgical, petroleum industries and others. In particular in the oil and gas branch of industry we need to calculate five-velocity currents [3][4][5][6][7][8][9]: the flow of three liquids (oil, water and hydraulic fracturing liquid), proppant balls and reservoir gases. In this article we investigate hyperbolicity and characteristics of the relevant system of differential equations.…”

Section: Introductionmentioning

confidence: 99%

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A drift model N of a velocity mixture and the evolution of a discontinuous solution of a two-velocity drift model

Kroshilin¹,

Kroshilin

2022

MathMon

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At present, to describe the two-velocity flow of a multiphase mixture, either a two-fluid model is used; or a drift model using simplified momentum equations that do not take into account inertial forces. The corresponding system of equations for a two-fluid model with the same phase pressure without special, postulated, stabilizing terms is non-hyperbolic. This can lead to difficulties in finding a solution. Accounting for the difference in phase pressure in this model can lead to instability of solutions, which can also complicate the search for a solution. The drift model, which is the subject of this work, does not have this shortcoming. The case of N phase velocities (N > 2), considered in this article, has not been studied so far, although it has a large area of practical applications, for example, a three-velocity flow of a dispersed-film flow in the energy and chemical industries, and others. Hyperbolicity and characteristics are investigated. The characteristic equation of the system, which has N-1 degree, is analyzed. It is proved that there is one eigenvalue between neighboring speeds. If the k phases have the same speed, then the k-1 eigenvalue is equal to that same speed. So, if there are no coinciding phase velocities, or the number of phases with the same speed does not exceed 2, then all eigenvalues are real and different, which means that the system is hyperbolic. If the number of phases with the same speed exceeds 2, then all eigenvalues are also real, but there are multiples among them. An additional study carried out for this case showed that the system is also hyperbolic. An analytical solution of the discontinuity decay problem for a three-velocity flow (N=3) is found. It is assumed that the speed difference between the first and second phases is much greater than the speed difference between the second and third phases. Without this assumption, the solution can be found numerically. For the case of two phase velocities (N = 2), an analytical solution is found that describes the transition from a continuous distribution to a jump. This solution describes the flow for a given dependence of the slip rate on the regime parameter.

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Section: Hyperbolicity and Characteristics Of A N-velocity Mixturementioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A drift model N of a velocity mixture and the evolution of a discontinuous solution of a two-velocity drift model

Kroshilin¹,

Kroshilin

2022

MathMon

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“…Если, же, например, с шагом м, нужно просчитать с, то максимальное значение k порядка , и применимо приближение (5). Погрешность начальных данных в конце такого расчета возрастёт примерно в 3 раза, в этом случае, часто считается, что результаты таких расчётов имеют физический смысл.…”

Section: гиперболичность и устойчивость модельной задачиunclassified

“…В настоящее время существуют два пути решения этой проблемы [4]. Первый -введение дополнительных, стабилизирующих слагаемых в уравнения импульса, подробная реализация этого подхода с обзором литературы дана в [5] и были установлены основные закономерности. В этой работе доказано, что недостаточно обеспечить гиперболичность системы, необходимо, чтобы стационарные решения были устойчивы; определены условия, когда неустойчивость решения приводит к невозможности получить достоверное решение, причем неустойчивость решения не является физической, а возникает из-за «дефекта» системы.…”

Section: Introductionunclassified

Flow stability and correctness of the Cauchy problem for a two-speed medium model with different phase pressures

Kroshilin¹,

Kroshilin

2021

MathMon

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At present, to describe the two-velocity flow of a dispersed mixture, as a rule, a two-fluid model is used with equal pressure of the phases of the medium and different velocities of the phases. The corresponding system of equations without special, postulated, stabilizing terms is non-hyperbolic. This can lead to difficulties in finding a solution. Recently, it has been proposed to use similar models more widely, but with different pressures of the phases of the medium. Such models allow one to take into account new physical effects associated with different phase pressures and often provide hyperbolicity of the corresponding system of equations. This article analyzes the influence of the difference in the pressure of the phases of the medium on the properties of the system: the importance of the corresponding new effects, the hyperbolicity of the system of equations, the stability of its stationary solutions, and the correctness of the corresponding Cauchy problem are investigated. Three systems are considered. The first, simplest model system is based on the well-known non-hyperbolic system, which has been modernized. It is shown that the Cauchy problem for the modified system is formally correct, but the practical possibility of using the calculation results obtained from the solution of this system should be investigated in each specific case, and depends on the calculated step and duration of the process under study. The techniques worked out to solve the first simplest system were used for other systems. As the second system, a model of the flow of a two-phase medium with different phase pressures and two momentum equations is considered. We will assume the phases are barotropic. Let us postulate an equation relating the pressure in the phases. It is proved that this system is always hyperbolic. The stability of its stationary solutions is investigated. Relationships are derived that make it possible to determine under what conditions, due to instability, the obtained solutions are unreliable. The properties of this system are compared with the system of two-speed flow of a dispersed mixture with equal pressure of the phases of the medium. As a third system, a two-pressure model describing bubble pulsations is considered. We will assume the phases are barotropic. Conditions are determined when the system is non-hyperbolic and the Cauchy problem is incorrect. It is investigated for what conditions the ill-posedness of the Cauchy problem leads to the unreliability of the solution, and under what conditions the ill-posedness of the Cauchy problem does not lead to the unreliability of the solution.

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Analytical Solutions of the Transport Equation for a Drift Model in an Unsteady Flow with Discontinuous Parameters

Kroshilin

2024

J. Engin. Thermophys.

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Correct numerical simulation of a two-phase coolant

Cited by 4 publications

References 2 publications

A drift model N of a velocity mixture and the evolution of a discontinuous solution of a two-velocity drift model

A drift model N of a velocity mixture and the evolution of a discontinuous solution of a two-velocity drift model

Flow stability and correctness of the Cauchy problem for a two-speed medium model with different phase pressures

Analytical Solutions of the Transport Equation for a Drift Model in an Unsteady Flow with Discontinuous Parameters

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