The (non-dimensional) point vortex model describes the dynamics of N point-like vortices in a two dimensional...

 

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The (non-dimensional) point vortex model describes the dynamics of N point-like vortices in a two dimensional incompressible fluid having position r;= (zi, yi) and circulation Fi, with i = 1,..., N The equation of motion of the ith vortex results in dy dx; dt 2T Yi-Yj r-r and 2 T r - r ji Design your own Runge-Kutta 4th order method (also called RK45) method to numerically study the dynamics of the point vortex model and address the following points. 1/2 (a) Consider the case N = 2 with I = 1, F2 = -1 and initial conditions r = (0, -1), r2 = (0, 1) at t = 0. Evolve numerically the initial condition until t = 10 using a time-step of At = 0.01, and show that the two vortices move along a straight line with a constant speed to be determined. (b) Consider the case N = 2 with I = I2 = 1 and initial conditions r = (1.0), r = (-1,0) at t = 0. Evolve numerically the initial condition until t = 50 using a time-step of At = 0.1, and show that the two vortices move around the circumference of a circle. (c) Consider the case N = 3 with F = F = 1, F3 = -1 and initial conditions r = (1,0), r = (-1,0), r3 = (0,3) at t = 0. Evolve numerically the initial condition until t = 50 using a time-step of At = 0.1, and plot the trajectories r(t) in the (x,y) plane. By interpolating the trajectory of the point vortex #1 by with the Splines technique, plot the extrapolated vortex acceleration components in time. The (non-dimensional) point vortex model describes the dynamics of N point-like vortices in a two dimensional incompressible fluid having position r;= (zi, yi) and circulation Fi, with i = 1,..., N The equation of motion of the ith vortex results in dy dx; dt 2T Yi-Yj r-r and 2 T r - r ji Design your own Runge-Kutta 4th order method (also called RK45) method to numerically study the dynamics of the point vortex model and address the following points. 1/2 (a) Consider the case N = 2 with I = 1, F2 = -1 and initial conditions r = (0, -1), r2 = (0, 1) at t = 0. Evolve numerically the initial condition until t = 10 using a time-step of At = 0.01, and show that the two vortices move along a straight line with a constant speed to be determined. (b) Consider the case N = 2 with I = I2 = 1 and initial conditions r = (1.0), r = (-1,0) at t = 0. Evolve numerically the initial condition until t = 50 using a time-step of At = 0.1, and show that the two vortices move around the circumference of a circle. (c) Consider the case N = 3 with F = F = 1, F3 = -1 and initial conditions r = (1,0), r = (-1,0), r3 = (0,3) at t = 0. Evolve numerically the initial condition until t = 50 using a time-step of At = 0.1, and plot the trajectories r(t) in the (x,y) plane. By interpolating the trajectory of the point vortex #1 by with the Splines technique, plot the extrapolated vortex acceleration components in time.

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Designing a RungeKutta 4th order RK45 method for the point vortex model involves solving a system of ordinary differential equations ODEs that describe the motion of the vortices The ODEs are given by ... View the full answer