The article has been devoted to the problem of improvement real numerical modeling accuracy for the viscous uid ow between two coaxial half-cylinders on rectangular grids taking into account the lling of cells are used to solve this problem. Approximation of the problem with respect to time is performed on the basis of splitting schemes for physical processes. The simulation was performed on a sequence of condensing computed grids of sizes 11 21, 21 41, 41 81, and 81 161 nodes for the areas of smooth and piecewise rectangular boundaries. The grids taking into account the lling of cells are used to improve the smoothness of the solution. In the case of piecewise rectangular approximation the numerical solution error reaches 70%. The grids taking into account the lling of cells reduce the numerical solution error to 6% for the test problem. The test problem shows that using the grid condenced in each spatial direction by 8 times does not lead to increasing the accuracy solutions whereas the solutions accuracy obtained on the basis proposed approach has signi cant advantage in accuracy.
grids that approximate the boundary ear used as an alternative to rectangular grids, which have low accuracy in the case of direct piecewise rectangular approximation of the boundary [Mur16, Lan86].
When solving the problems of hydrodynamics, inaccuracies arise associated with a stepwise representation of the interface between two media, which can reach 70% of the solution. This error has a high-frequency character. The smoothness of the solution indicates the absence of these assimilations. This scheme has found its application in solving the problems of hydrodynamics of shallow water bodies. The proposed original method is described.
The constructing problem of optimal three-dimensional computational grids remains open nowadays for the 3D regions of common con gurations in computational uid dynamics [Hir81]. Di erence schemes accuracy comparison has been discussed in this paper in cases of direct rectangular grids usage and additional involvement of the cell lling function for the Taylor-Couette ow numerical modeling. The proposed method is likely close to Volume Of Fluid (VOF) method [Bun06, Bel75]. 2
In this work we study the viscous incompressible uid motion in two-dimensional region between two in nitely long coaxial circular cylinders. We introduce the Cartesian coordinate system xOy perpendicular to the axis of the cylinders. The coordinate system origin coincides with the cylinders' axis. In the section of the cylinder by the plane x = 0 de nes the eld of velocity. It is required to determine the liquid motion. The initial equations for the mathematical description of the uid dynamics problem are [Bel87, Suk13']: { Navier-Stokes equation: u0t + uu0x + vu0y = vt0 + uvx0 + vvy0 =
P 0
x + ( u0x)0x + ( u0y)0y; P 0
y + ( vx0)0x + ( vy0)0y; { the continuity equation for incompressible uid:
u0x + vy0 = 0: Equations (1)-(3) are considered under the following boundary conditions: { the ows are de ned on the input and output boundaries:
u(x; y; t) = U (x; y); v(x; y; t) = V (x; y); Pn0 (x; y; t) = 0; { the frictionless and slip conditions are set on the lateral surfaces (in the case j j = 0, that is, without friction):
Pn0 (x; y; t) = 0; un(x; y; t) = 0; where u = fu; vg is the water medium velocity vector; (x, y ) is Cartesian coordinates, t is time, P is pressure; is the turbulent exchange coe cient; is the liquid density; n is the normal vector; x, y are the tangential stress components at the bottom of the liquid.
The wind stress according to the Van Dorn law, is calculated by the formulas [Mon73]: f x; yg =
Cp (juj) u juj : 3
The computational domain inscribed in a rectangle. For numerical realization of the discrete mathematical model of the formulated wave hydrodynamics problem, uniform grid is introduced: wh = tn = n ; xi = ihx; yj = jhy; n = 0; ::: ; Nt; i = 0; ::: ; Nx; j = 0; ::: ; Ny;
Nt = T; Nxhx = lx; Nyhy = lyg ; where is the time step, hx, hy are steps in space, Nt is the step number on the time coordinate, Nx, Ny are spacing steps on the spatial coordinates x and y, respectively.
We use the splitting schemes for physical processes [Sam89, Sam95]. In this case, the solution of the problem (1)-(3) reduces to solving the following system of equations: 0 y ; un+1 un+ =
Px0 vn+1 ; vn+ =
P 0 y :
The calculated cells are rectangles, which may be lled, partially lled, or empty. The cell centers and nodes
are separated apart hx/2and hy/2 on the coordinates x and y, respectively. Fig. 1 (a) shows that the velocity
eld and pressure are calculated at the tops of the cells. The cells' vertices (i; j) are nodes (i; j), (i 1; j),
(i; j 1), (i 1; j 1).
(8)
(9)
(10)
(11)
(
Let us introduce grid value oi; j for the notation of the cell lling. The lling of cells means the value of cell part volume (area) which has been lled with a liquid medium. Fig. 1 (b) shows that in the neighborhood of the node are cells (i; j), (i + 1; j), (i; j + 1), (i + 1; j + 1).
We introduce the coe cients k0, k1, k2, k3, k4, describing the lling of regions located in the neighborhood of the cell. The value k0 characterizes the lling of the region 0: x 2 (xi 1; xi+1), y 2 (yj 1; yj+1), k1 { 1: x 2 (xi; xi+1), y 2 (yj 1; yj+1), k2 { 2: x 2 (xi 1; xi), y 2 (yj 1; yj+1), k3 { 3: x 2 (xi 1; xi+1), y 2 (yj ; yj+1), k4 { 4: x 2 (xi 1; xi+1), y 2 (yj 1; yj ). The lled parts of the regions m is called Dm, where m = 0; :::; 4. The coe cients km can be calculated from the formulas: (km)i; j = SSDmm ; (k0)i; j = oi; j + oi+1; j + o4i+1; j+1 + oi; j+1 ; (k1)i; j = oi+1; j +2oi+1; j+1 ;
(k2)i; j = oi; j +2oi; j+1 ; (k3)i; j = oi+1; j+12+ oi; j+1 ; (k4)i; j = oi; j +2oi+1; j : The boundary conditions for the rst subproblem of wave hydrodynamics (8), (9) take form: u0x(x; y; t) = u; xu + u; x; vx0(x; y; t) =
v; xv + v; x; After calculating separately each of the integrals we obtain:
D0 Calculating the integrals over the regions 1 and 2, we obtain
In the last equality, let us assume that SD1 > SD2 , where we select from the region D1 fragment D1; 2, adjacent to the regionD2, and SD2 = SD1; 2 (Fig. 3). 1 D2 As a result, we get:
ZZ
D0 ( u0x)0xdxdy =
ZZ uin; j hxhy+ (k1)i; j ui+1=2; jhy (ui+1; j ui; j) + (k2)i; j ui 1=2; jhy (ui; j ui 1; j) .2+ + (k3)i; j vi; j+1=2hx (ui; j+1 ui; j) + (k4)i; j vi; j 1=2hx (ui; j ui; j 1) .2 = ui; j
ui; j 1 hy If we divide the obtained expression by the area of the cell hxhywe are coming to: uin;+j uin; j + (k1)i; j ui+1=2; j ui+1; j 2hx ui; j + (k2)i; j ui 1=2; j ui; j
ui 1; j + 2hx + (k3)i; j vi; j+1=2 ui; j+1 2hy ui; j + (k4)i; j vi; j 1=2 ui; j
ui; j 1 = (k1)i; j i+1=2; j 2hy ui+1; j h2 x
ui; j ui; j
i; j ui; j
ui; j 1 h2 y u; xui; j + u; x + hx 2hy (k4)i; j i;j 1=2 (k3)i; j 2hx = + (k3)i; j i; j+1=2 under sticking condition:
n + (k3)i; j vi; j+1=2 vin; +j+=12 (k0)i; j vin; j + (k1;2)i; j ui+1=2; j n
vin; +j =2 2hx + (k1;2)i; j uin 1=2; j 2hx vn+ =2 i 1; j + 2hy h2
x + (k3)i; j i; j+1=2 under sticking condition: uin;+j+=12
uin;+j =2 h2 y 2hy h2 x h2
y 2hy h2
x for the velocity vector vi; j component under slip conditions: vin; +j vin; j + (k1)i; j ui+1=2; j n { equations to re ne the velocity eld by pressure:
vin; +j =2 2hy h2 x (k0)i; j (k2)i; j
Pi; j (k1)i; j uin++1=2; j
Pi 1; j + (k3)i; j Pi; j+1 h2x h2y (k1)i; j
Suppose, on the internal side, the rotation speed is jujjr=r1 = u1, on the external side, the rotation speed is jujjr=r2 = u2. The polar coordinate system was introduced to solve the problem (x = r cos ; y = r sin ) The analytical solution of this system of equations is: u (r) = A1r + A2=r; P (r) = P (r1) +
u2=r dr: = 0: Z r r1 To compare the results of numerical calculations with the analytical solution, we take r1 = 5 m, r2 = 10 m, u1 = 1 m/s, u2 = 0:5 m/s.
In this case, the analytical solution takes the form u (r) = 5=r; P (r) = P (r1) The analytical solution in the Cartesian coordinate system takes the form u (x; y) =
5y 5x x2 + y2 ; v (x; y) = x2 + y2 ; P (x; y) = P (r1) 12:5 x2 + y2 + =2: 5
The problem of nding the numerical ow of a viscous uid between two coaxial cylinders (x 0) is considered. The inside cylinder radius is r1 = 5 m. The outside cylinder radius is r2 = 10 . The calculated domain is inscribed in a rectangle with dimensions 10 20 m (0 x 10; 10 y 10). In the section of the cylinder by the plane x = 0 sets the velocity eld u (0; y) = 5=y m/s, v (0; y) = 0 m/s. In all other grid nodes, the velocity eld is calculated. On the inside and outside walls of the cylinder, the conditions for slip and non- ow are speci ed.
Defects of numerical solutions are most clearly visible on coarse grids. We describe the parameters of a coarse grid. The steps in the spatial directions are 1 m, the time step is 0.1 s, the mesh size is 21 11 knots, the length of the counting interval is 10 s, the density is = 1000 kg/m3, the turbulent exchange coe cient is = 1 m2/s. Fig. 2 shows the contents of an array describing the degree of lling of cells in the case of using the grid of 21 11 nodes.
Fig. 6 shows the error values of the numerical solution of the uid ow problem, depending on the radius (circles indicate the error in case of a smooth boundary, circles indicate the error in the case of a step boundary).
Fig. 4, 6 show that the increase in the size of the calculated grids for the problem of ow of the aqueous medium does not lead to an increase in the accuracy in case of piecewise rectangular approximation of the boundary, but to a decrease in the linear dimensions of the border region where the solutions of the solution associated with rough approximation of the boundary are manifested. It should also be noted that when using grids taking into account the lling of cells, the error in the numerical solution of model hydrodynamic problems caused by the approximation of the boundary does not exceed 6% of the solution of the problem.
Table 1 presents the error values of the numerical solution of the uid ow problem between two coaxial cylinders obtained from a sequence of condensing computed grids 11 21, 21 41, 41 81, and 81 161 nodes in case of a smooth and stepped boundary.
The analysis of the error calculating results of the numerical solution of the problem of uid ow between two cylinders on the sequence of condensing grids presented in Table 1 allows us to conclude that the use of di erence schemes taking into account the lling of cells is e ective. The grid splitting by 8 times in each of the spatial directions does not lead to an increase in the accuracy that solutions obtained on grids taking into account the lling of the cells possess.
The paper considers the problem of searching the numerical ow of a viscous uid between two coaxial halfcylinders. Analytic solution describing the Taylor-Couette ow is used as a standard to evaluate the accuracy of the numerical solution of hydrodynamic problems. The simulation was performed on a sequence of condensing computed grids of sizes 11 21, 21 41, 41 81, and 81 161 nodes in cases of smooth and piecewise rectangular boundaries. To improve the solution smoothness the we used grids taking into account the lling of the cells. When solving the hydrodynamics of shallow reservoirs, rectangular meshes are mainly used. This is due to the large di erence in step lengths in horizontal and vertical directions.
In the case of piecewise rectangular approximation the error of numerical solution reaches 70%. The grids taking into account the lling of cells reduce the numerical solution error to 6%. It is shown that crushing the grid by 8 times in each spatial direction does not lead to increasing the accuracy solutions whereas the solutions accuracy obtained on grids taking into account the lling of cells signi cantly increases. 6.0.1
Acknowledgements This work was supported by the Russian Foundation for Basic Research (project code 19-07-00623). [Suk12] A. I. Sukhinov, A. E. Chistyakov, E. F. Timofeeva, A. V. Shishenya. Mathematical model for calculating coastal wave processes // Mathematical Models and Computer Simulations, 2012. V. 24, pp. 32-44. [Suk13] A. I. Sukhinov, A. E. Chistyakov, E. V. Alekseenko Numerical realization of the three-dimensional model of hydrodynamics for shallow water basins on a high-performance system// Mathematical Models and Computer Simulations, 2011. V. 23, pp.3-21.