Wave motion over uneven surface Выпускная работа In work consider two problems about the impact of bottom shape on the profile of the free boundary. 1. Waves traveling along the surface of the liquid in the still water. 2. Fluid flow running on the underwater obstacle. Both problems can be solved using the method of numerical conformal mapping.

The mathematical formulation of the physical domain z=x+iy 1. Waves in still water (1) (2) (3) Рис.1 in approximate linear formulation speed potential φ found by solving the boundary value problem(1) If it is found in the form of a periodic function time (2), then the function Φ is obtained boundary value problem (3). In the simplest case, when the bottom is flat y = S (x) = 0, the solution problem (3) can be obtained in the form of a traveling wave (4) (4) (5) Where λ and ω - wavelength and the oscillation frequency, are connected by (5)

Two areas of solving the problem Boundary value problem (3) can be solved in the physical domain z = x + iy, and in the parameter region Ω = ξ + iη, which are shown in pictures 2 and 3, but the region Ω has a simple form, and so decision it turns out a lot easier. Suppose that conformal mapping is known (6). Then boundary value problems in domains Ω and z have the form (7) and (8). (6) (7) (8) We see that the problem (7) and (8) coincide with their equations and the condition at the bottom. The difference lies in the presence of varying factor f (ξ), which is calculated by the formula (9), and contains information about the entire shape of the bottom. (9) Рис.2 Рис. 3

Solution for the potential Φ in Ω Solution of the boundary problem (8) can be represented as a symbolic notation (10). In it function F (ξ) - the value of speed potential on the free boundary. (10) To find the function F (ξ), we must substitute (10) into the boundary condition at the upper boundary of the problem (8). Then, after appropriate simplifications, we obtain that the function F (ξ) must be a solution of the differential equation (11) (11) Equation (11) - a second order differential equation with variable coefficients. For its solution is often used in the physics of the asymptotic WKB method. With it, the solution is obtained in the form (12) (12) Thus, if the function f (ξ), obtained from a conformal mapping can be according to the formulas (12) and (10) to find the velocity potential at all points of the liquid. A surface wave (12) can be expressed by a coordinate x, using (13). (13)

The numerical realization of the conformal mapping (15) (16) (15) realize conformal mapping of the domain Ω = ξ + iη on the area z = x + iy. Here the choice of the function f0 (ξ) depends form the bottom. Equation (16) shows that the potential Φ (ξ, η) can be calculated inside the domain, if known waveform F (ξ), defined by the expression (12).

examples of calculation Рис. 4 for example take the function f0 (ξ) in the form (17). This is similar to square protrusion on the bottom. (17) Рис. 5 Then calculated using the formula (15) gives the contour lines η = const in the x, y, shown in pic 4. The same calculation with the function - f0 (ξ) of the opposite sign is shown in pic 5 where a representation of square basin. It turns out even a different size in the physical domain. As you can see, conformal mapping by the formulas (14) or (15) is acceptable to convert the area.

Wave profile of the velocity potential Φ (ξ, η), calculated according to the formula (16), of the projection and depression is shown in pic 6 and 7. The the potential oscillation amplitude decreases with depth. Рис. 6 Рис. 7 Рис. 8 Рис. 9 The result of the conversion of wave profiles in the physical domain is shown in Figures 8 and 9. The bandwidth of the fluid at infinity h: h = 2 for the projection, and h = 1 for depression. Wavelength λ in the program can also be set.

Рис. 10 Рис 11. change profile traveling waves on other forms bottom topography shown in pic 10 and 11. pic 10 -pologaya shelf with a steep drop. pic 11 - two-humped bottom surface.

2 The flow of heavy fluid over an uneven bottom In the final work is also considered another problem when the water flow depth h moves over uneven bottom with velocity c at infinity. Open surface such a flow is unknown curved line that you want to find. To find an approximate solution to this problem was the approximation of the linear theory waves and numerical conformal mapping, which has been proven by the example solutions previous wave problem in standing water. (18) (19) (20) The mathematical formulation in variables (x, y) is to find the velocity potential Φ, which satisfies the Laplace equation and the condition of nonflow at the bottom (18) It has been shown that at the free boundary must satisfy the condition (19), and and k is a real constant velocity flow c are related by (20).

Conditions (19) and (20) are derived from the assumption of boundedness of the solution. Therefore, the linear theory valid only for speeds less than the critical value (21). Critical rate √gh - is the speed with which Waves spreading of small amplitude on the water surface with the depth h. (21) The value of k can be regarded as well-known as it can always be found from the equation (20), if known speed c, satisfying (21). Solution of (18), (19) (22) To fulfill the condition uniform flow at infinity, we look for a solution in the form of a sum two potentials (22) Here the potential Φ0 known. It gives a uniform flow determined by a conformal mapping and obviously equal to (23) (23)

The second term φ (x, y) in the solution (22) determines the deviation free boundary from the horizontal level, and it must be to seek the solution of the boundary value problem (24). (24) To determine the form of the function F (x), which is included in (24), substitute it into the condition (18), and then obtain for function F (x) an ordinary differential equation (25) (25) Equation (25) can be easily solved numerically. it is the equation of a pendulum on which operates the driving force q (x), occurring due to irregularities in the bottom. Thus, the downstream surface flow should have an undulating appearance. Knowing the form of the function F (x), via the mapping to find the function F (ξ), and therefore a solution of (24) throughout the region Ω using Formula (26) (26)

Рис.12 Рис.13 Рис. 14 pic 12, 13 and 14 show examples calculation of the potential velocity at the top border and within liquid calculated formulas (25) and (26). elliptic hurdle form flows about flow of heavy liquids with different speed. according to Equation (20) wavelength increases when c → √gh

2. Движение потока тяжелой жидкости по неровному дну Рис. 15 Рис. 16 Рис. 17 pic and 17 shown wave flow in other forms the bottom.

Obtained solving two problems for waves over an uneven bottom, which have different mathematical formulation. 1 waves propagating in the still water. 2 perturbation flow irregularities bottom. Their example was tested numerical method for conformal mappings. Thus shown to be promising its applications in a variety of complex problems of hydrodynamics.