Galilean-Invariant Characteristic-Based Volume Penalization Method for Supersonic Flows with Moving Boundaries

This work extends the characteristic-based volume penalization method, originally developed and demonstrated for compressible subsonic viscous flows in (J. Comput. Phys. 262, 2014), to a hyperbolic system of partial differential equations involving complex domains with moving boundaries. The proposed methodology is shown to be Galilean-invariant and can be used to impose either homogeneous or inhomogeneous Dirichlet, Neumann, and Robin type boundary conditions on immersed boundaries. Both integrated and non-integrated variables can be treated in a systematic manner that parallels the prescription of exact boundary conditions with the approximation error rigorously controlled through an a priori penalization parameter. The proposed approach is well suited for use with adaptive mesh refinement, which allows adequate resolution of the geometry without over-resolving flow structures and minimizing the number of grid points inside the solid obstacle. The extended Galilean-invariant characteristic-based volume penalization method, while being generally applicable to both compressible Navier–Stokes and Euler equations across all speed regimes, is demonstrated for a number of supersonic benchmark flows around both stationary and moving obstacles of arbitrary shape.

Numerical Simulation of Wall-Bounded Flows using a Spectral Method with Volume Penalization

The volume penalization method, which allows to impose no-slip boundary conditions, is assessed for wall-bounded flows. For the numerical solution of the penalized equations a spectral method is used. Considering a two-dimensional Poiseuille flow, the solution of the Navier-Stokes penalized equation is computed analytically and the convergence of the numerical solution is studied. To illustrate the properties of the approach we compute a three-dimensional turbulent channel flow imposing a constant flow rate. The obtained results are compared with reference data of Kim et al. [10].

A Coupled Lattice Boltzmann-Volume Penalization for Flows Past Fixed Solid Obstacles with Local Mesh Refinement

A coupled Lattice Boltzmann-Volume Penalization (LBM-VP) with local mesh refinement is presented to simulate flows past obstacles in this article. Based on the finite-difference LBM, the local mesh refinement is incorporated into the LBM to improve computing efficiency. The volume penalization method is introduced into the LBM by an external forcing term. In the LBM-VP method, the processes of interpolating velocities on the boundaries points and distributing the force density to the Eulerian points near the boundaries are unnecessary. Performing the LBM-VP on a certain point, only the variables of this point are needed, which means the whole procedure can be conducted parallelly. As a consequence, the whole computing efficiency can be improved. To verify the presented method, flows past a single circular cylinder, a pair of cylinders in tandem arrangement, and a NACA-0012 are investigated. A good agreement between the present results and the data in the previous literatures is achieved, which demonstrates the accuracy and effectiveness of the present method to solve the flows past obstacle problems.

Mechanism of aeroacoustic sound generation and reduction in a flow past oscillating and fixed cylinders

The aeroacoustic sound generated in a flow past two cylinders, one of which is oscillating and the other is fixed, is studied by direct numerical simulation. This problem involves key ingredients of the aeroacoustic noise generated from wind turbines, helicopters, axial flow fans and other turbomachinery: flow, a moving body and a fixed body. The corrected volume penalization method is successfully applied to resolve the sound pressure of aeroacoustic waves as a solution of the compressible Navier–Stokes equations. The sound pressure was shown to be in good agreement with the prediction by the Ffowcs Williams–Hawkings aeroacoustic analogy, which takes account of the cylinder motion, confirming the accuracy of the corrected volume penalization method. Prior to the case of two cylinders, sound generation in flow past a single oscillating cylinder is considered. The fluid motion can be either periodic or non-periodic depending on the frequency and the amplitude of cylinder oscillation. The acoustic power is significantly reduced when the fluid motion locks in to a frequency lower than the natural frequency of vortex shedding from a fixed cylinder. When a fixed cylinder is added, the acoustic power depends strongly on the distance between the cylinders, since that determines whether synchronization occurs and the phase difference between the three forces: the lift forces exerted on the two cylinders and the inertial force due to volume displacement effect of the oscillating cylinder. In particular, significant sound reduction is observed when the fixed cylinder is placed upstream and the frequency of the cylinder oscillation is set to the frequency for which the acoustic power is minimized in the single-cylinder case.