Vortex Trapping Cavity on Airfoil: High-Order Penalized Vortex Method Numerical Simulation and Water Tunnel Experimental Investigation

This paper presents a two-dimensional implementation of the high-order penalized vortex in cell method applied to solve the flow past an airfoil with a vortex trapping cavity operating under moderate Reynolds number. The purpose of this article is to investigate the fundamentals of the vortex trapping cavity. The first part of the paper treats with the numerical implementation of the method and high-order schemes incorporated into the algorithm. Poisson, stream-velocity, advection, and diffusion equations were solved. The derivation, finite difference formulation, Lagrangian particle remeshing procedure, and accuracy tests were shown. Flow past complex geometries was possible through the penalization method. A procedure description for preparing geometry data was included. The entire methodology was tested with flow past impulsively started cylinder for three Reynolds numbers: 550, 3000, 9500. Drag coefficient, streamlines, and vorticity contours were checked against results obtained by other authors. Afterwards, simulations and experimental results are presented for a standard airfoil and those equipped with a trapping vortex cavity. Airfoil with an optimized cavity shape was tested under three angles of attack: 3°, 6°, 9°. The Reynolds number is equal to Re = 2 × 104. Apart from performing flow analysis, drag and lift coefficients for different shapes were measured to assess the effect of vortex trapping cavity on aerodynamic performance. Flow patterns were compared against ultraviolet dye visualizations obtained from the water tunnel experiment.

An improved third-order HWCNS for compressible flow simulation on curvilinear grids

Due to the very high requirements on the quality of computational grids, stability property and computational efficiency, the application of high-order schemes to complex flow simulation is greatly constrained. In order to solve these problems, the third-order hybrid cell-edge and cell-node weighted compact nonlinear scheme (HWCNS3) is improved by introducing a new nonlinear weighting mechanism. The new scheme uses only the central stencil to reconstruct the cell boundary value, which makes the convergence of the scheme more stable. The application of the scheme to Euler equations on curvilinear grids is also discussed. Numerical results show that the new HWCNS3 achieves the expected order in smooth regions, captures discontinuities sharply without obvious oscillation, has higher resolution than the original one and preserves freestream and vortex on curvilinear grids.

Alternating Polynomial Reconstruction Method for Hyperbolic Conservation Laws

We propose a new multi-moment numerical solver for hyperbolic conservation laws by using the alternating polynomial reconstruction approach. Unlike existing multi-moment schemes, our approach updates model variables by implementing two polynomial reconstructions alternately. First, Hermite interpolation reconstructs the solution within the cell by matching the point-based variables containing both physical values and their spatial derivatives. Then the reconstructed solution is updated by the Euler method. Second, we solve a constrained least-squares problem to correct the updated solution to preserve the conservation laws. Our method enjoys the advantages of a compact numerical stencil and high-order accuracy. Fourier analysis also indicates that our method allows a larger CFL number compared with many other high-order schemes. By adding a proper amount of artificial viscosity, shock waves and other discontinuities can also be computed accurately and sharply without solving an approximated Riemann problem.

An Improved Third-order HWCNS for Compressible Flow Simulation on Curvilinear Grids

Due to the very high requirements on the quality of computational grids, stability property and computational efficiency, the application of high-order schemes to complex flow simulation is greatly constrained. In order to solve these problems, the third-order hybrid cell-edge and cell-node weighted compact nonlinear scheme(HWCNS3) is improved by introducing a new nonlinear weighting mechanism. The new scheme uses only the central stencil to reconstruct the cell boundary value, which makes the convergence of the scheme more stable. The application of the scheme to Euler equation on curvilinear grids is also discussed. Numerical results show that the new HWCNS3 achieves the expected order in smooth region, captures discontinuities sharply without obvious oscillation, has higher resolution than the original one and preserves freestream and vortex on curvilinear grids.

Positivity preserving high order schemes for angiogenesis models

Hypoxy induced angiogenesis processes can be described by coupling an integrodifferential kinetic equation of Fokker–Planck type with a diffusion equation for the angiogenic factor. We propose high order positivity preserving schemes to approximate the marginal tip density by combining an asymptotic reduction with weighted essentially non oscillatory and strong stability preserving time discretization. We capture soliton-like solutions representing blood vessel formation and spread towards hypoxic regions.