Comparison of Solutions to a Hydrodynamic Problem in a Rectangular Cavity Using Initial Velocity Field Damping and Acceleration Methods

Introduction. This study investigates the numerical solution of a two-dimensional hydrodynamic problem in a rectangular cavity using the method of initial velocity field damping and the method of accelerating the initial conditions in terms of stream function and vorticity variables. The damping method was applied at Reynolds numbers Re ≤ 3000, and the acceleration method was used for Re = 8000.
Materials and Methods. To speed up the numerical solution of the problem using an explicit finite-difference scheme for the vorticity dynamics equation, the method of initial condition damping and the method of n-fold splitting of the explicit difference scheme (with n = 100) were used. Compared to the traditional method of accelerating from stationary fluid, the initial velocity field damping method reduced the computation time by a factor of 57. The splitting method used a maximum time step proportional to the square of the spatial step, while maintaining spectral stability of the explicit scheme in the vorticity equation. The majority of computation time was spent solving the Poisson equation in the “stream function — vorticity” variables. By freezing the velocity field and solving only the vorticity dynamics equation, computation time was further reduced in the splitting method. The inverse matrix for solving the Poisson equation using a finite number of elementary operations were computed using the Msimsl library.
Results. Numerical solutions demonstrated the equivalence of the damping and acceleration methods for the initial velocity field at low Reynolds numbers (up to 3000). The equivalence of solutions obtained using the “stream function — vorticity” algorithm and the implicit iterated polyneutic recurrent method for accelerated initial conditions was numerically confirmed. For the first time, an initial horizontal velocity field was proposed, smooth at internal points and composed of two sine waves, with a stationary center of mass for the fluid in the rectangular cavity.
Discussion and Conclusion. An algorithm for numerically solving a two-dimensional hydrodynamic problem in a rectangular cavity using “stream function — vorticity” variables is proposed. The approximation of the equations in system (1) has sixth-order accuracy at internal grid points and fourth-order accuracy at boundary points. A novel damping method is introduced using an initial horizontal velocity field formed by smoothly connecting two sine waves. The proposed algorithms enhance the efficiency of solving hydrodynamic problems using an explicit finite-difference scheme for the vorticity equation

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