CC BY
24UDC 519.6 10.23947/2587-8999-2022-1-2-61-69
3D MODELING OF TURBULENT FLOWS USING LES AND RANS APPROACHES BASED ON FILTERED EXPEDITION DATA*
S.V. Protsenko
Taganrog Institute named after A.P. Chekhov (branch) of RSUE, Taganrog, Russia
The article describes the developed software that made it possible to process a large amount of data from observations of the movement and parameters of the aquatic environment in the Sea of Azov, which was obtained during expeditionary research using the ADCP hydrophysical probe, using the filtration procedure. The filtering procedure significantly reduces the spread of data and the amplitude of fluctuations, which, in turn, makes it possible to more adequately evaluate the information obtained during field experiments. For different filter widths, a box filter, a Gauss filter and a Fourier filter were used. In these calculations, the filter width was set based on the size of the hydrodynamics problem to be solved and the grid scale corresponding to this size. The obtained data are planned to be used for numerical simulation of three-dimensional turbulent flows using the LES approach and comparison with the results of averaging by RANS.
Keywords: modeling of hydrodynamic processes, three-dimensional turbulent flows, LES approach, RANS averaging.
Introduction. The forecast of development and ecological design of coastal and shallow water systems are currently necessary due to the negative impact on the environment and human health of chemical, biological and other hazardous substances, materials and waste, accidents and catastrophes of a natural and man-made nature. It is necessary to develop technologies for full-scale collection and accounting, multifactorial, multi-criteria analysis of the state of coastal and shallow-water systems using significant computing power in real time.
Vertical turbulent exchange plays a great role in coastal systems. In some cases, it determines the transport of nutrients, as well as the saturation of the aquatic environment with oxygen depends on it, as well as the occurrence of overseas phenomena in the absence of turbulent mixing in the water column. Setting the coefficient of vertical turbulent exchange in the form of a constant leads to a distorted picture of the distribution of velocities of the aquatic environment, as well as concentrations of nutrients and oxygen in the vertical direction and does not provide the required accuracy of calculating 3D flows, which is confirmed by comparing the results of numerical modeling and direct measurements of the 3D velocity vector of the aquatic environment using ADCP type equipment (Acoustic Doppler Current Profiler) [1].
The interest in turbulence in shallow waters, such as the Azov Sea, is caused by the fact that in the places of its existence there is an intensive transfer of the amount of motion and heat, the spread of passive impurities, the transfer of suspended particles. These processes significantly affect the formation and spatial structure of physical, chemical and biological fields of reservoirs and their
* The research was carried out at the expense of the grant of the Russian Science Foundation No. 22-71-00015, https://rscf.ru/project/22-71-00015/.
spatial and temporal changes [2]. The collected empirical material is currently being processed to study the internal structure of the recorded disturbances of small-scale turbulence, to determine the rate of dissipation of the energy of disturbances [3-4].
To assess turbulence characteristics using direct methods, there is a problem associated with the need to obtain large amounts of data, as well as long and expensive expeditionary measurements. It should be noted that, despite extensive experimental and theoretical studies, in the field of research of specific processes occurring in the coastal area of the reservoir, the effectiveness of the proposed approaches differs from the practically necessary one. The narrow scope of applicability of models built on the basis of laboratory research, the complexity and complexity of performing research in real sea conditions and, as a result, the limited data of field measurements cause the complexity of such studies.
2. Turbulence simulation based on field observations of water flow velocity profiles.
Turbulent structures exist against the background of the main motion, which can be distinguished by averaging, so it is usually called averaged motion. The definition of the averaged motion depends on the chosen averaging method: in time, in space, in ensemble, in phase. Thus, the turbulent flow can be divided into an averaged (deterministic) and pulsating component. Turbulent flows, in which the averaged component does not depend on time, are called stationary/
Subgrid direct methods are based on the definition of turbulent flows as the products of deviations of the components of the flow velocity and the transferred physical quantity averaged over space or time. For direct subgrid methods of estimating the characteristics of vertical turbulence, there is a problem consisting in the need for large amounts of information, long and expensive expeditionary measurements. In shallow water bodies, random velocity fluctuations in the vertical direction determine all hydrodynamic processes to an impressive extent, therefore in this paper we consider statistical models of parametrization of the vertical turbulent exchange coefficient.
The method of large eddies simulation (LES) is based on two assumptions. One of them is the possibility of dividing the flow field into the movement of large and small eddies. Large eddies under the direct influence of boundary conditions and carrying a maximum of Reynolds stresses are calculated. Small-scale turbulence is considered to be isotropic and having universal characteristics, and therefore less critical and more amenable to modeling. Another assumption is that it is possible to approximate nonlinear interactions between large and small eddies only by large eddies using subgrid models (SGS). In other words, the hypothesis of statistical independence of large and small eddies is accepted [5-6].
Small-scale motion is excluded from the Navier-Stokes equations by applying the filtering operation and is modeled using subgrid models. Among the most popular and frequently used filtering functions are the Gauss and Fourier filters, as well as the boxfilter. When performing calculations based on the control volume method, filtering is carried out as a result of integrating differential equations representing conservation laws over the control volumes of the difference grid. Classification of subgrid models is carried out according to the same criteria as in RANS (according to the number of relations introduced in addition to the system of filtered equations).
Large-scale motion is calculated by solving a filtered system of NavierStokes equations, which can be formally written in the same form as the system of Reynolds equations. The role of subgrid modeling increases with increasing Reynolds number [7].
The solution obtained using LES contains richer information compared to the solution based on the Reynolds equations, for example, not only the characteristics of the mean flow (velocity, concentration, temperature, pressure fields) and Reynolds stress distributions, but also spectral characteristics (velocity and pressure pulsation spectra), two-point moments (for example, spatial and spatio-temporal correlations of velocity and pressure pulsations), temporal and spatial scales of turbulence [8-9].
Pressure fluctuations in many cases are the cause of fatigue damage to structural elements. Based on LES, it is possible to calculate coherent vortex structures that control the dispersion of the impurity. The subgrid models used in LES usually have significant diffusion and dissipation, which makes it possible to overcome computational problems associated with the representation of very small eddies on the selected grid and to stabilize numerical calculations.
3. Data filtering. To obtain filtered Navier-Stokes equations, approaches with explicit and implicit introduction of the filtration operator are used.
We introduce a generalized filter that gives a formal definition of the averaging operation and allows us to exclude from consideration scales smaller than some predetermined value A, called the filter length. Vortices whose size is smaller than the filter width are not allowed. The generalized filter is defined as follows
f (x, t) = J f (f, t)g(x, f, A)df .
D
In the case when the function g(x, f) depends only on the difference x - £, the differentiation and filtering operations commute. Then the generalized filter is introduced as the convolution integral
f (x, t) = J f Of, t)g(x - f, A)df = f (x, t) • g(x, A).
D
It is assumed that the filtering function g(x), also called the filter kernel, it is even and infinitely differentiable in a bounded domain D, has a compact carrier and satisfies the normalization condition
g(x) = g(-x), J g(£, A)df = 1.
d
In the extreme case, there are relations
lim J f (f, t)g(x - f, A)df = f (x, t), lim g(x, A) = S(x).
D
Integration is carried out over the entire flow region D. The filtering function determines the structure and size of small-scale turbulent vortices resolved by a system of averaged equations.
According to Borel's convolution theorem, the Fourier transform of a convolution is equal to the product of the Fourier transforms
F [ f (x, t) • g( x)] - F [f (x, t)] F [ g( x)].
By the Fourier transform of the function f (x), absolute value \f (x)| which is integrable on
+ro
the interval -ro < x < + ro, is called the function c(k) = F[ f (x)], where c(k) = J f (x) exp(-ikx)dx.
-ro
The conversion formula has the form:
+ro
f (x) = J c(k )exp(ikx)dk.
-ro
There are various types of filters used in numerical calculations. Here are examples of some of them.
1. Box filter g(x - £) = ■
¡1/ A3, |x -£|<A*,. /2 , k -£.|>Ax /2
2. 3.
Gaussian Filter g(x - £) =
' 6 V/2
j
Fourier Filter g(x - £) = -1 n
exp -6|* - £| / A2 sin 2£ „ x -
With the exception of the sharp Fourier cutoff filter, filtering differs from the standard time averaging operation in that f ^ f.
To represent the smallest solvable scales, it is necessary that the filter width does not exceed the step of the difference grid. Usually, the difference between these two values is ignored, and the filter width is assumed to be equal
A = V1/3 = (AxAyAz )1/3,
where Vis the volume of the cell of the difference grid; Ax, Ay, Az are the grid steps in the coordinate directions x, y and z, respectively. Since the filter width depends on the difference grid, the filtering function is often called a grid filter.
To calculate the boundary layers, it is proposed to replace the grid pitch in the direction normal
to the wall Ay by the amount of Ay and find the filter width using the ratio A = (AxAyAz) .
With Ay = Ay near the wall and Ay = Ay near the wall and. For intermediate values y smooth transition between the specified limit values is used. The value Ay is the average value in the wall area Ay, and the value Ay is calculated by the formula
Ay =
Ay.
Ay.
-1/3
There are also other definitions of the filter width
A = ßN inAx
A
1/2
A = ß|ZAx2
j
1/2
. .=i
j
A = B min Ax ,
i=l,..., AT
A = B max Ax ,
7=1, ...,N
where N is the dimension of the problem, ( is the proportionality coefficient.
When filtering the Navier-Stokes equations, the filtering function is selected in such a way that the condition is met (g *, V)u = 0.
The form of writing the equations used in LES does not depend on one or another choice of the filtering function g(x). A specific type of filter plays a role only in statistical processing and comparison of numerical simulation results with experimental data or results obtained using DNS. At the same time, the results of numerical calculations, in particular, the dimensions of the solvable turbulence scales depend on the choice of the filter width. The acceptable filter width is chosen, as a rule, by trial and error. At A^-0 the LES method goes to DNS.
4. Filtering of data obtained using the ADCP probe during the expedition. The interest in turbulence in shallow waters, such as the Sea of Azov, is caused by the fact that in the places of its existence there is an intensive transfer of the amount of motion and heat, the spread of passive impurities, the transfer of suspended particles. These processes significantly affect the formation and spatial structure of physical, chemical and biological fields of reservoirs and their spatial and temporal changes. The collected empirical material is currently being processed to study the internal structure of the recorded disturbances of small-scale turbulence, to determine the rate of dissipation of the energy of disturbances.
To assess turbulence characteristics using direct methods, there is a problem associated with the need to obtain large amounts of data, as well as long and expensive expeditionary measurements.
Full-scale data were obtained during an expedition in the Central-Eastern part of the Azov Sea. The hydrophysical ADCP probe Workhorse Sentinel 600 was used to measure the three-dimensional velocity vector of the water medium. The research was carried out at 17 stations. Measurements of the water flow field in the Sea of Azov were carried out vertically, starting from the near sensitivity zone of the ADCP probe to the bottom. Measurements were recorded with an interval of 1 s every 10 cm at the measured depth. The speed was recorded in the corresponding file in mm/s. The columns show the time values on the device's clock and 128 measurements of the depth of one of the components of the velocity vector at the current time. In the described experiment, data was stored according to three components of the velocity vector of the water flow at the current time. Thus, with a vertical resolution of 10 cm, and a time step of 1 second for a time interval of 20-30 minutes, there are more than 3,000,000 initial measurements, at each point - more than 150,000.
Fig. 1. Application of the box filter: 1 - initial data, 2, 3, 4 - data obtained by filtering, with different filter
widths: A < A < A
Fig.2. Application of the Gauss filter: 1 - initial data, 2, 3, 4 - data obtained by filtering, with different
values of the filter width: A4 < A3 < A2
Fig. 3. Application of the Fourier filter: 1 - initial data, 2, 3, 4 - data obtained by filtering, with different
filter widths: A < A < A
The presence of errors in the measurements of the pulsations of the vertical velocity component is one of the intractable problems and is associated with many phenomena occurring at
the time of measurement, such as ship deviation, free surface fluctuations, changes in depth, stability, wind and waves. The degree of influence of pitching is taken into account using reduction coefficients. The reduction coefficients depend on the wavelength, and therefore on the frequency of the wave. They have the form of amplitude-frequency characteristics of linear low-frequency filters.
Consider the use of various filters for processing instantaneous water flow velocities obtained during measurements. We will use a box filter, a Gaussian filter and a Fourier filter with different filter widths. In these calculations, the filter width was set based on the dimension of the hydrodynamics problem to be solved and the grid scale corresponding to this dimension.
Figures 1-3 show an example of the operation of a program designed to eliminate the noise of the measured expedition data of the water flow velocity field. The filtering procedure significantly reduces the spread of data and the amplitude of fluctuations, which in turn allows for a more adequate assessment of the information obtained during field experiments.
Discussion and conclusions. The developed software made it possible to process a large volume of data from field observations of the movement and parameters of the aquatic environment in the water area of the Sea of Azov, which was obtained during expeditionary research using the ADCP hydrophysical probe, using the filtration procedure. The filtering procedure significantly reduces the spread of data and the amplitude of fluctuations, which in turn allows for a more adequate assessment of the information obtained during field experiments.A box filter, a Gauss filter and a Fourier filter were applied at different filter widths. In these calculations, the filter width was set based on the dimension of the hydrodynamics problem to be solved and the grid scale corresponding to this dimension. The obtained data are planned to be used for numerical simulation of three-dimensional turbulent flows using the LES approach and comparison with the results of averaging by RANS.
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Author:
Protsenko Sofya, Taganrog Institute of A.P. Chekhov (branch) RSUE (Initiative Street, Taganrog, Russian Federation), PhD of Science in Physics and Maths, [email protected]
УДК 519.6 10.23947/2587-8999-2022-1-2-61-69
3D-МОДЕЛИРОВАНИЕ ТУРБУЛЕНТНЫХ ПОТОКОВ С ИСПОЛЬЗОВАНИЕМ LES И RANS ПОДХОДОВ НА ОСНОВЕ ОТФИЛЬТРОВАННЫХ ЭКСПЕДИЦИОННЫХ ДАННЫХ*
С.В. Проценко
Таганрогский институт имени А.П. Чехова (филиал Ростовского государственного экономического университета), Таганрог, Российская Федерация
В статье описано разработанное программное обеспечение, которое позволило обработать большой объем данных наблюдений за движением и параметрами водной среды в акватории Азовского моря, который был получен в ходе экспедиционных исследований с использованием гидрофизического зонда ADCP, с использованием процедуры фильтрации. Процедура фильтрации значительно уменьшает разброс данных и амплитуду колебаний, что, в свою очередь, позволяет более адекватно оценивать информацию, полученную в ходе полевых экспериментов. При разной ширине фильтра применялись коробочный фильтр, фильтр Гаусса и фильтр Фурье. В этих расчетах ширина фильтра была установлена на основе размера решаемой задачи гидродинамики и масштаба сетки, соответствующего этому размеру. Полученные данные планируется использовать для численного моделирования трехмерных турбулентных течений с использованием подхода LES и сравнения с результатами осреднения по RANS.
Ключевые слова: моделирование гидродинамических процессов, трехмерные турбулентные течения, подход LES, осреднение по RANS.
Автор:
Проценко Софья Владимировна, Таганрогский институт им. А.П. Чехова (филиал) РГЭУ (РИНХ), кандидат физико-математических наук, доцент кафедры математики
* Исследование выполнено за счет гранта Российского научного фонда № 22-71-00015, https://rscf.ru/project/22-71-00015/.
