3D turbulent currents simulation in the coastal zone using the LES approach based on filtered ADCP data

. 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 spatial and temporal changes. Depending on the smoothing scale of the series of initial readings of the flow meters, the pulsation characteristics of the flow velocity were determined, which were processed in order to obtain data on turbulence and its scales. The research results provided empirical data on the conditions of generation and existence of small-scale turbulence. The collected empirical material is currently being processed to study the internal structure of the recorded disturbances of small-scale turbulence. In this paper, small-scale motion is excluded from the Navier-Stokes equations by applying the filtration operation and is modeled using subgrid models. To do this, in two-dimensional and three-dimensional cases, various types of filters are used: a box filter, a Gaussian filter and a Fourier filter, with a gradual decrease in the filter width, which allows you to reproduce a wider frequency range of fluctuations of the solution. 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. The article considers the possibilities of using various types of approximations for parametrization of vertical turbulent exchange. Algebraic models for calculating the coefficient of vertical turbulent exchange and semi-empirical turbulence models are compared.


Introduction
Vertical turbulent exchange plays a great role in coastal systems.In a number of cases, it determines the transport of biogenic substances, as well as the saturation of the aquatic environment with oxygen, 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 as 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).
The three-dimensional nature of the flow, the stochastic nature and the wide space-time spectrum of turbulence make predictive modeling of the characteristics of complex turbulent flows a difficult and time-consuming task.The initial premise of mathematical modeling of turbulent flows within the framework of the phenomenological approach consists in the assumption of the acceptability of the Navier-Stokes equations for the interpretation of turbulent flows and the prediction of their instantaneous characteristics.
There is a need to calculate the vertical structure of the current to solve a number of applied problems, primarily anthropogenic pollution of water areas, as well as to assess the reliability of hydraulic structuresprotective structures, oil platforms, wave converters and other devices installed in the areas of the shelf zone and shallow seas, with the effects of high tide and storm surge [1].
Despite conducting a wide range of studies focused on the problem under consideration, they did not fully reflect the totality of various factors and processes affecting the structure and parameters of vertical turbulent mixing [2][3][4].This indicates the need for a systematic analysis of the problem and the construction of an interconnected set of models, highperformance algorithms and programs.A relatively small number of publications devoted to modeling of multi-scale vertical turbulent mixing and the complexity of obtaining fullscale data in the real field indicate the need to attract 3D models of hydrodynamics developed in the author's team, which take into account the specifics of coastal systems, and have, in comparison with known models, better accuracy and an increased margin of stability, with depth differences of 15-20 times.The review of the currently existing mathematical models of hydrodynamics has shown that a hydrostatic approximation is used to calculate the hydrodynamic component, such an approach does not allow taking into account the acceleration of the movement of the water flow along the vertical component.When parameterizing models of turbulent mixing, simplified functional dependencies are widely used that are not related to realistic models of hydrophysics, which leads to models that do not have the proper predictive value.Few of the papers are devoted to the parallel numerical implementation of problems of this class.Despite the large number of existing software systems that allow modeling hydrodynamic processes: POM (Princeton Ocean Model), EFDC (The Environmental Fluid Dynamics Code), DELFT, Mars3D, CARDINAL (Coastal Area Dynamics Investigation Algorithm, wind-wave models of the third generation WAM, SWAN (Simulation Waves Nearshore), WaveWatch, these developments have a number of significant drawbacks [5][6][7][8].The analysis of these and other models shows that the vast majority of existing approaches to constructing models of hydrodynamic processes in the coastal zone of shallow reservoirs are based on the shallow water approximation, or at best on the hydrostatic approximation, which does not take into account turbulent heat and mass transfer in the vertical direction.On the other hand, the available experimental data indicate a significant influence of these processes on the hydrophysics of the coastal zone, the dislocation of salts and the gas regime.Therefore, the construction, study and application of spatial-three-dimensional models of hydrodynamics involving mechanisms of turbulent exchange along a vertical coordinate seems to be an urgent scientific and important applied problem.

Large eddy simulation (LES)
Among the main methods of numerical modeling of three-dimensional turbulent flows, it is necessary to distinguish direct numerical Simulation (DNS), large eddy simulation (LES) and solving Reynolds Averaged Navier-Stokes equations (RANS).There are also various intermediate approaches combining certain features of RANS, LES and DNS, for example, the method of modeling Detached Eddy Simulation (DES) [9].
Direct numerical simulation (DNS) involves the numerical solution of complete nonstationary three-dimensional Navier-Stokes equations.With this approach, all the scales of turbulent motion are resolved.To use DNS, powerful computational resources are required, and the possibilities of its application are limited to calculations of flows with simple geometry and small Reynolds numbers.
The use of Reynolds averaged Navier-Stokes equations (RANS) requires much less computational resources.This approach is successfully applied in practical calculations.However, turbulence models used to close Reynolds equations do not have acceptable universality, and therefore cannot be used to solve a wide range of applied problems.
The method of modeling large vortices (LES) is a compromise option between DNS and the RANS solution.This approach is limited to the study of flows only at scales exceeding a certain set value.The LES method solves the space-filtered Navier-Stokes equations, and only large vortices are allowed to move.Small vortices have a more universal structure and are modeled using Subgrid Scale Models (SGS) based on the concept of vortex viscosity or other rational approximations of transport processes.
The method of modeling large vortices (LES) is based on two assumptions.One of them is the possibility of dividing the flow field into the movement of large and small vortices.Large vortices 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 vortices only by large vortices using subgrid models (SGS).Otherwise, the hypothesis of statistical independence of large and small vortices is accepted.
While DNS displays the entire range of vortex sizes, the LES method considers large vortices corresponding to small wave numbers to be the most important.At the same time, subgrid models do not have a critical impact on the results as a whole.The statistics of large vortices are usually not sensitive to subgrid modeling.Large-scale motion is calculated by solving a filtered system of Navier-Stokes equations, which can be formally written in the same form as the Reynolds system of equations.The role of subgrid modeling increases with an increase in the Reynolds number.
Small-scale motion is excluded from the Navier-Stokes equations by applying a 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, filtration 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) [10][11][12].
The calculation results obtained using LES depend on the filter width Δ, which is included in the filtration operator and is usually associated with the step size of the difference grid.Reducing the filter width allows you to reproduce a wider frequency range of fluctuations of the solution.Increasing Δ contributes to smoothing the solution, and at Δ→0, the LES method switches to DNS.Nevertheless, the LES method is a promising direction in the development of methods for calculating turbulent flows and seems to be a significant alternative to DNS and RANS.

Filter operator selection
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 Δ, called filter length.Vortices whose size is smaller than the filter width are not allowed.The generalized filter is defined as follows In the case when the function ā(ý, ) depends only on the difference ý 2 , differentiation and filtering operations commute.Then the generalized filter is introduced as a convolution integral 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 In the extreme case, there are relations 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 By the Fourier transform of the function Ā(ý), the absolute value of which The conversion formula has the form Ā(x) = ∫ ý(k) ÿý ( +∞ 2∞ kx)þk.
There are various types of filters used in numerical calculations.Here are examples of some of them.With the exception of the sharp Fourier cutoff filter, filtering differs from the standard time averaging operation in that о Ā ̄b Ā ̄.
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 Δ = 1/3 = (ΔýΔþΔÿ) 1/3 , where V is the volume of the difference grid cell; Δý, Δþ, Δÿ are grid steps in x, y and z coordinate directions, 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 þ by the amount of ̂þ and find the filter width using the ratio Δ = (ΔýΔ ̂þΔÿ) 1/3 .
And ̂þ = þ near the wall and Δ ̂þ = Δ ̄þ away from the wall.For intermediate values, a smooth transition between the specified limit values is used.The value is the average value þ in the wall area, and the value ̂þ is calculated by the formula There are also other definitions of the filter width where N is the dimension of the problem, β is the proportionality coefficient.
When filtering the Navier-Stokes equations, the filtering function is selected in such a way that the condition is met (ā * , )þ = 0.
The form of writing the equations used in LES does not depend on one or another choice of the filtering function ā(ý).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 Δ→0, the LES method goes to DNS.

Filtering of data obtained by the ADCP probe during the expedition
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.
The field data were obtained during the expedition in the Central-Eastern part of the Azov Sea and in the Taganrog Bay.To measure the three-dimensional velocity vector of the water medium, a hydrophysical ADCP probe Workhorse Sentinel 600 was used.The research was carried out at 17 stations.The measurements of the water flow field in the Azov Sea 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 s for a time interval of 20-30 minutes, there are more than 3,000,000 initial measurements, at each point (at each station, of which -17)more than 150,000.The expeditions were conducted in order to obtain data to accumulate information about the state and changes in hydrophysical and hydrochemical parameters.During the flights, the following measured parameters were studied: pulsations of the velocity components; measurement error; depth of the reservoir at the measuring point; wind value.There is no density stratification in shallow water bodies, the processes in the vertical direction are essentially nonlinear, so we have to resort to a non-standard third method of measuring and estimating the coefficient of vertical turbulent viscosity, based on calculating the gradient of the average flow velocity.
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 being solved and the grid scale corresponding to this dimension.Fig. 4-6 demonstrate the result of the software designed to eliminate the noise of expedition measurements, using the example of one of the components of the velocity vector of the water flow, in the two-dimensional case.The color indicates the velocity of the water flow in mm/s in accordance with the given color scale.

Parameterization of the vertical turbulent exchange process using the LES approach
Parametrization of the turbulent exchange process in discrete models is primarily performed for the vertical coordinate direction taking into account the density gradient of the aquatic environment on the basis of modern subgrid models.
To parametrize the coefficient of vertical turbulent exchange, algebraic subgrid models 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 are considered.Experiments were performed on the basis of several approaches to calculate the coefficient of turbulent exchange according to the vertical: parametrization of Belotserkovsky, Boussinesq, Smagorinsky (Fig. 7).Filtered data on the distribution of the three-dimensional vector of the instantaneous velocity of the water flow for depths from 0.8 to 20 m and more, obtained using the ADCP WHS 600 WHS 1200 Sentinel, were used.
All methods of parametrization of the vertical turbulent coefficient allow in most cases to obtain similar distributions of the vertical turbulent exchange coefficient in order of magnitude and localization of maxima-minima.
The phenomenon of sharp jumps in the coefficient on all graphs is associated with errors in the measurements of pulsations of the vertical velocity component, which is included in the calculation formula of the method.The presence of errors in the measurements of the pulsations of the vertical velocity component is associated with many phenomena occurring at the time of measurement, such as the deviation of the vessel, fluctuations of the free surface, changes in depth, stability, wind and waves.The profiles of the coefficient of vertical turbulent exchange at the moment of time show that the parametrization of Belotserkovsky and Boussinesq most adequately reflects the processes of turbulent exchange for shallow water bodies, but to assess the quality of parametrization, a more in-depth analysis using methods of mathematical statistics is needed.

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 Azov Sea, which was obtained during expedition studies using the ADCP hydrophysical probe, using the filtration procedure.The filtering procedure significantly reduces the spread of data and the amplitude of oscillations, 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.
The article considers the possibilities of using various types of approximations for parametrization of vertical turbulent exchange.Using ADCP data on velocity pulsations for several stations measuring hydrological characteristics, the results of parameterization of the vertical turbulent exchange coefficient were analyzed.All the considered methods of parametrization of the vertical turbulent exchange coefficient allow in most cases to obtain similar distributions of the vertical turbulent exchange coefficient in order of magnitude and localization of maxima-minima.

Fig. 1 -
3 demonstrate an example of a program designed to eliminate the noise level 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.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.

Fig. 7 .
Fig. 7. Vertical turbulent exchange coefficient calculated on the basis of various types of approximations for parameterization of vertical turbulent mixing (horizontal values in m2/s).