Numerical modelling of long waves in wide rectangular channels in lagrangian coordinates

. The paper discusses the possibility of using Lagrangian coordinates to predict the transformation of long waves of different origins near the coasts, necessary in the design, construction, and operation of protective structures, as well as a wide range of environmental problems related, among others, to the spread of pollutants. Results of numerical investigations, compared with analytical solutions, showed satisfactory coincidence.


Introduction
Engineering problems associated with the propagation of long waves and their interaction with hydraulic structures require hydrodynamic calculations of currents in areas with variable boundaries where it is necessary to predict: x transformation of surge, tidal, and tsunami waves coastal security; x breakwater and flood propagation parameters x water movement during furrow irrigation; or carry out x ecological assessments of shallow waters and coastal marshes x predicting HPP-PSPP operation regimes.
Three approaches for implementing pass-through methods are described in (Fedotova Z.I., 2002): x on land, a thin film of water held on the surface by frictional force is assumed, x a body of water of simple geometry is embedded into the area occupied by land, In the non-flooded area, the depth and flow velocity are set to zero.In pass-through, the calculations are carried out in Eulerian coordinates.In the case of boundary selection, coordinates that are not purely Euler coordinates but some coordinate's line that moves with the boundary of the computational domain(Belikov V.V., Norin S.V., Shkolnikov S.Ya., 2014).The motivation for this research was to overcome the problem of separating the physical and schematic diffusion of a fluid using Eulerian coordinates in numerical calculations of wave processes and also the fact of wide application of computational methods using Lagrangian coordinates in the solution of gas dynamics problems in regions with variable boundaries (Samarskiy A.A., Popov Yu.P., 1992).
The following is an explicit finite-difference method for calculating wave currents off the coast, using Lagrangian coordinates under the assumption of planar symmetry.

Saint-Venin equations in Lagrangian coordinates
Currently, the most widely used mathematical model describing currents in open unstratified bodies of water is the Saint-Venin system of equations.This system is based on the hypothesis of hydrostatic pressure distribution over depth.As a rule (not always), this hypothesis is supplemented by the hypothesis that the velocity epure is close to a rectangular one and that the friction laws used in steady-state currents apply to unsteady currents.
The Saint-Venin equations in Lagrange coordinates assuming planar symmetry can be represented in integral form (Krutov, A.N. Shkolnikov, S.Ya., 2022): is the spatial Eulerian coordinate of the station moving with the fluid flow, linked with the Lagrangian coordinate of the station by the relation: is velocity of liquid particles, q is specific water flow, h is depth, g=9,8 m/s is acceleration of gravity, 2 2 gh U is hydrostatic pressure divided by width unit, ρ is liquid density, ρR is the projection onto the X axis of the pressure acting from the bottom on the fluid, numerically equal to the projection onto the same axis of the fluid pressure on the bottom of the stream, opposite in direction to it, O is coefficient of hydraulic friction using Manning's formula (Manning, R., 1891, Gauckler, Ph. 1867): n is bottom roughness coefficient, the value of which depends on the underlying surface (Kiselev P.G. (ed.), 1974), ρ, ) is other forces affecting the flow: wind stress on the free water surface, projection of the tidal force on the flow axis, etc.In this paper, it is assumed that the influence of these forces is negligible and ) is not taken into account further in the calculations.
The first equation of the system (1) expresses the law of conservation of mass of an incompressible fluid; the second equation is the law of conservation of momentum for this fluid.
Note that, for flows in areas with variable boundaries, drying of zones within an initially single-bonded area is possible.Such phenomena are not considered in this paper.

Numerical implementation of the Saint-Venin equations in Lagrangian coordinates without bottom topography and friction
In (Krutov, A.N. Shkolnikov, S.Ya., 2022), an explicit finite-difference scheme for a channel without gradient and friction and without considering the bottom relief is proposed, implementing (1.1), in which the value R=0 has the following form: °°°°® ' ' here: subscript k: 1≤k≤K is cell number of the finite-difference scheme, fractional lower index (subscript) k+1/2 indicates that the corresponding value is assigned to the boundary between k-th and k+1-th cells, τ is time step, ' : is specific (by flow width) fluid volume in k-th cell of the finite-difference scheme, located between moving sections and ' does not change with time and can be calculated once, at the beginning of calculations; hereinafter the values with upper index "1" refer to a new time step, without indices -to the old one, scheme parameters of flow are marked the same way as their analogs in the description of the mathematical model in section 1.The proposed explicit finite-difference scheme (6) has a time step limit Courant condition (R. Courant, K. Friedrichs, H. Lewyt ,1928.): As a test case, the solution to a classical hydrodynamic problem -the problem of dam failure with an outflow into a filled or dry downstream -was considered (Stoker J.J., 1959, Kochin N.E., Kibel I.A., Rose N.V., 1963).It is assumed that a finite-difference grid that has a constant length step at the initial moment of time X k+1/2 -X k-1/2 =const k , on the left boundary of the calculation domain, at X=0, an impermeable wall is given and the dam site at the initial time t=0 coincides with the initial left-hand boundary of the calculation area, i.e., the solution of the problem corresponds to the analytical one until the downgoing wave reaches the right boundary of the calculation area 0 X .Here it should be noted that the system of equations ( 5) does not require a boundary condition on this (right) boundary in the case of dam failure with a spill into a dry channel or at shallow downstream, as the flow in its vicinity will be turbulent ( . However, the layout view for the right boundary cell is not obvious and will be discussed below.
Next, consider a more general problem in which it is assumed that there is a device on the right-hand boundary of the calculation area that ensures that a constant depth is maintained on it: Note that the solution to the long-wave fluid flow problem under this boundary condition is contained in the solution to the "dam break" problem (Fig. 1) at the initial depth in the tailwater h ∞ ≠0.III-IV is "flat" (plateau); V-VI is undisturbed downstream; VII-VIII is unflooded part of the downstream.
X is length, t is time, h is depth, V is velocity, g is acceleration of gravity, H is reservoir depth, h ∞ is downstream water depth, V R and h R are speed and water depth on the flat Dboron propagation speed.Indeed, the line with the self-similar coordinate gH V R separates the liquid which has escaped from the reservoir and the downstream liquid which has been perturbed by the breakthrough wave, and if condition ( 9) is fulfilled, the analytical solution coincides with the solution of the dam failure problem, where the boundary of the calculation area moves with the velocity of water on the "flat" R V .In this case In this case, the length of the computational domain at time t becomes equal to it does not contradict the conditions of the task since there is such a filling of the downstream with which the depth at the "flat" is equal to R h .
Option 1: One of the boundary conditions was that the depth in the last cell of the area at the initial moment of time is h R, and it did not change with time, nor does the length of the last cell > @ Numerical experiments have shown that at 9 / 4 / H h s there are strong circuit pulsations in the zone of the boundary K-sell (Fig. 2 and 3).Legend: 1 is initial depth profile with a discontinuity at Х=0; 2 is wave profiles: analytical and calculated according to (6); 3 is velocity graphs analytical and calculated according to (6).H=10 m, h R =1 m.

H h s /
decreases, and at small values of H h s / , they are so strong that calculations become impossible.Note that the solution becomes unstable only in the nearest vicinity of the K-cell, which is explained by the fact that the turbulent flow "blows away" the arising oscillations and does not let them penetrate deep into the computational area.
Option 2: The velocity of the right-hand boundary of the area was calculated using the finite-difference formula: which formally coincides with the third formula (2.1), that is, under the assumption that there is a "borderline" K+1-cell with specific volume and constant depth h R .Numerical experiments have shown that wave flow calculations using formula (16) give satisfactory results for all values of h R <H, including h R =0.It can be seen that using formula (16), the results of the numerical experiment do not practically differ from the analytical solution.15) is applied.

3 Numerical implementation of the Saint-Venin equations taking into account bottom topography and friction
In the same way, as for the horizontal bottom, a spaced finite-difference grid is used (Fig. 5).To account for the bottom topography, we assume that the bottom shape is set independently of the finite-difference grid by the function x Z b , which can be found for any x (for example, set at some node points, and for points located between them is found by interpolation).Z is water surface mark (consider it within the interval > @ ' is specific (relative to flow width unit) water volume at interval > @ The following values have been determined in the half-integer number sections: The bottom is set independent of the moving Lagrangian finite-difference grid, e.g., on a separate time-invariant grid, so that the bottom mark of the k+1/2-cell line here: is water depths at the center of mass of the fluid compartments in the intervals > @ respectively (here indices "k" and "k+1" and the positions of the centers of mass of the corresponding compartments 1 сk X and W Construction of a finite-difference scheme for the K-th cell, similar to (16): The constant water level hypothesis is unsuitable for the 'overseas' liquid compartment, as plausible results are also obtained at h R =0.This paper accepts the hypothesis that the ' , given by the initial conditions.In contrast to the horizontal-bottom situation, in which coordinate 1 R X was not involved in any way, when calculating the flow in an area with complex bottom topography, it is necessary to know 1 R X , as the pressure on the bottom side must be determined.
There are also 2 possible variants of the scheme on the boundary, which are analogous to schemes (18) and (19).
In the conducted numerical experiments, the difference in results when using variants of the finite-difference scheme ( 18) and ( 19) was insignificant.However, in the case of velocity setting at K+1/2-th point, when using (20) to solve the problem of coastal slope runup, the scheme loses its stability.When using (21), the results obtained are satisfactory.Apparently, in case of coastal escarpment problem, for the boundary velocity better to use (18), because application of ( 19) results in instability.

Results and Discussion
As an example of a numerical experiment to calculate wave run-up on a slope, the problem of wave flow during an initial water level jump in an area bounded on the left by a vertical wall and on the right by a slope was considered.In this numerical experiment, friction was not considered.Figure 7 shows the wave profiles for several consecutive points in time.At the time t = 100 s, a boron is formed at the wavefront that has not collapsed even in the relatively shallow coastal area.The boron cannot move along the dry land (Belikov V.V., Norin S.V., Shkolnikov S.Ya., 2014); therefore, after the onset of coastal flooding, the boron collapses and flattens out (t = 125 c).After the maximum inundation (t=235 c), the outflow begins, with a turbulent flow regime in the upper zone of the flooded area, coupled with a calm flow at a sufficiently deep depth through the boron (t=375 c).The next period of bank inundation then begins (t=430 c).
When a rollback occurs in the vicinity of a zero-depth boundary point, the flow is formally turbulent (Fr>1), and two boundary conditions are required to solve the flow problem.One boundary condition is that h R =0; the second condition seems to be insignificant in rollback problems since, at small values of h, there is no momentum entering the calculation domain (the same statement can be formulated differently: the role of the second boundary condition is played by the equality to zero of the water flow coming through the boundary q R =0).

Conclusion
1.The proposed calculation method has no schematic mass diffusion, which is essential when modeling water quality.2. Testing of the proposed finite-difference schemes showed good agreement with analytical solutions.3. The proposed method allows modeling the transformation of the surge and tidal waves near coasts and the interaction of waves, including tsunamis, with coastal defenses, forecasting the operation of tidal HPPs, HPP-GHPPs, and the spread of pollutants.
4. Long-wave transformation studies in areas with water exchange are envisaged as a next step.

Zρ 4
is stream bottom mark, not changing in time in Eulerian coordinates, is hydraulic friction stress at the bottom of the stream, using the Darcy- Weisbach formula (McKeon, B. J.; Zagarola, M. V; Smits, A. J. 2005:

Fig. 1 .
Fig. 1.Diagram of the problem solution for dam failure with a spill into the filled downstream reservoir (solid line) and onto a dry downstream (dashed line).At the initial point in time at X=0, there is a depth discontinuity in the upstream and downstream reaches -the "dam".a) -wave profile, b) -velocity graph along the flow.I-II is undisturbed part of the upstream; II-III and II-III-VII are simple depression wave for filled and dry downstream, respectively;III-IV is "flat" (plateau); V-VI is undisturbed downstream; VII-VIII is unflooded part of the downstream.X is length, t is time, h is depth, V is velocity, g is acceleration of gravity, H is reservoir depth, h ∞ is downstream water depth, V R and h R are speed and water depth on the flat Dboron propagation speed.Indeed, the line with the self-similar coordinate

Fig. 2 .
Fig. 2. Wave profile and longitudinal velocity plot when formula (2.8) is applied at the boundary point.

. 2022 Fig. 3 .
Fig. 3.The graph of the calculated depth change over time at the design points closest to the righthand boundary.The symbols are the same as in Fig. 2.

Fig. 5 .
Fig. 5. Spread out finite-difference grid for non-horizontal bottom.The following values have been determined in the whole-numbered sections:

'
does not change over time).
Conferences 365, 03013 (2023) https://doi.org/10.1051/e3sconf/202336503013CONMECHYDRO -2022 calculated from the Eulerian coordinate of The finite-difference scheme(17) generalizes the above-constructed scheme (2.1) for the Saint-Venin equations in Lagrangian coordinates with a horizontal bottom to the case of an arbitrarily shaped bottom (under flat symmetry and without considering hydraulic friction): of the pressure and friction forces, referred to the unit of flow width, acting on the fluid compartment in mind that it is an arbitrary value:

E3SFig. 6 .
Fig. 6.Diagram of the vicinity of the boundary of the computational area with adjacent to a border compartment Ω K and transborder compartment Ω R. (h R =Z fsR -Z bR ).The task is to determine the coordinate

Fig. 7 .
Fig. 7. Wave profiles in the area bounded on the left by the vertical wall and on the right by the shore slope at different points in time.(1 -"outboard" liquid compartment).