talk.org (5976B)
1 * Slide 1 2 3 My next talk is about ocean wave reflection from the ship hull. In this talk we 4 again apply the law of reflection to simulate diffraction of the waves near the 5 ship hull. 6 7 * Slide 2 8 9 The initial goal of this work was to replace empirical methods (namely, the 10 method of added masses) currently used in ship motions simulators to model ship 11 resistance to flow with diffraction and radiation. But in the course of the 12 experiments we changed the goal to just simulation of diffraction: it is too 13 early to talk about replacement of the method of added masses. We need more time 14 to understand and simulate every physical phenomena that contributes to ship 15 resistance to flow. In this talk I will present only wave /diffraction/ 16 17 * Slide 3 18 19 Wave diffraction is the change of wave direction due to obstacles: when a wave 20 meets obstacle (like an island or a ship) it slowly changes its direction to be 21 tangent to the boundary of the obstacle. There are three regions associated with 22 the diffraction: 23 - in the first region the wave slows down before approaching the 24 obstacle, 25 - in the second region the wave goes around the obstacle and its direction becomes 26 tangent to the boundary, and 27 - in the third region waves that went around different sides of the obstacle interfere 28 with each other. 29 The solution used in this work simulates only the first and the second region. 30 The third region is simulated the same way as the first one, but the wave 31 increases its speed instead of slowing down. 32 33 To simulate the change of wave direction due to diffraction we use the law of reflection. 34 Informally, the law of reflection states that 35 - the incident ray, the reflected ray and the surface normal lie in the same plane and 36 - the angle of incidence equals the angle of reflection. 37 We use it for ocean waves instead of light rays. This approach can also be found in the 38 literature. 39 40 We describe incident wave direction as the vector of wave numbers third 41 component of which is nought (because we consider surface waves, not spherical 42 ones). Reflected wave direction vector is derived using basic geometric 43 principles (you can see the formula on the slide). On order to write potentials 44 using vector notation we introduce pseudo-vector \(\vec{d}_k\) that contains 45 decay coefficient. The potential has the usual form of complex exponent, in 46 order to get the real potential we need to take double real part of the 47 exponent. Then the total potential is written as weighted sum of the potentials 48 for incident and reflected wave and it is the form of the solution that we seek 49 in this problem. 50 51 * Slide 4 52 53 The governing system of equations for potential flow includes 54 - equation of continuity (that describes conservation of mass), 55 - equation of motion (that describes conservation of momentum) and 56 - boundary condition on the ship hull (that nullifies water velocity on the 57 boundary). 58 The ship hull is defined by the collection of triangular panels each of which 59 has the normal \(\vec{n}\) and the point \(\vec\zeta_0\) that lies in the plane 60 of the panel. 61 62 The solution /on/ the boundary, where neighbouring panels do not affect each 63 other, has simple form with \(C_1=1\) and \(C_2\) being equal to some exponent. 64 The solution /near/ the boundary, where all panels contribute to the velocity 65 potential is constructed using smoothing kernel (similar to air flow from the 66 previous talk). We compute weighted sum of reflected wave potentials for all 67 panels, the potential decays quadratically with the distance to the panel. So, 68 the solution on the boundary is precise, and the solution near the boundary is 69 just a weighted sum that happens to satisfy all equations in the system. For 70 the purpose of simulating ship motions we need only the solution on the 71 boundary. 72 73 * Slide 5 74 75 We simulated diffraction around Aurora's ship hull. In the picture we can see 76 that waves near the hull have more pronounced crests and troughs (i.e. slightly 77 larger amplitude). We can clearly see the first region where the wave slows 78 down, and the second region where the wave direction follows ship waterline (it 79 is more visible with huge cylinder: the ship is too small compared to the island 80 to substantially change the direction of the wave). So, our solution correctly 81 simulates wave diffraction. 82 83 * Slide 6 84 85 Here is the same picture in three dimensions and without the ship. In greyscale 86 the change in amplitude is more visible. 87 88 * Slide 7 89 90 Finally, we benchmarked our solution on CPU and GPU using three ships with 91 different number of panels and three computers with different performance 92 characteristics. In this work we used local memory to optimised handling of 93 large number of panels in GPU kernel. As you can see from the table, this 94 optimisation allowed us to get much larger speedups than in air flow simulation. 95 In the best case GPU performance is three orders of magnitude higher than CPU 96 performance. 97 98 * Slide 8 99 100 Although, our solution for wave diffraction near ship hull works, it can not 101 replace the method of added masses. We can not use it compute ship resistance to 102 flow. The future work is to simulate wave radiation and check if it can be used 103 to compute ship resistance to flow. And if it is not we have to resort to 104 viscous flows near the ship hull, which is a different and much more difficult 105 problem compared to diffraction and radiation. 106 107 To summarise the two talks, the law of reflection can be successfully applied 108 not only to light ray reflection but also for ocean wave reflection and air 109 particle reflection from the boundary. In case of air flow the solution you 110 obtain using law of reflection is equivalent to the known solution for potential 111 flow around a cylinder, in case of ocean waves it is not equivalent but close to 112 the solution from linear wave theory (I did not show this in the paper). The 113 advantage of the law of reflection-based solutions is that they can be easily 114 generalised to the object of any form using smoothing kernel with quadratic 115 decay.