This study is focused on the development of longitudinal aerodynamic models for steady flight conditions. While several commercial solvers are available for this type of work, we seek to evaluate the accuracy of an open source software. This study aims to verify and demonstrate the accuracy of the OpenFoam solver when it is used on basic computers (32–64GB of RAM and eight cores). A new methodology was developed to show how an aerodynamic model of an aircraft could be designed using OpenFoam software. The mesh and the simulations were designed only using OpenFoam utilities, such as blockMesh, snappyHexMesh, simpleFoam and rhoSimpleFoam. For the methodology illustration, the process was applied to the Bombardier CRJ700 aircraft and simulations were performed for its flight envelope, up to M0.79. Forces and moments obtained with the OpenFoam model were compared with an accurate flight data source (level D flight simulator). Excellent results in data agreement were obtained with a maximum absolute error of 0.0026 for the drag coefficient, thus validating a high-fidelity aerodynamic model for the Bombardier CRJ-700 aircraft.

Direct numerical simulations of the transition process from steady laminar to chaotic flow are considered in this study with the relatively new incompressible lattice Boltzmann equation. Numerically, a multiple relaxation time fully incompressible lattice Boltzmann equation is implemented in a 2D driven cavity. Spatial discretization is 2nd-order accurate, and the Kolmogorov length scale estimation based on Reynolds number (Re) dictates grid resolution. Initial simulations show the method to be accurate for steady laminar flows, while higher Re simulations reveal periodic flow behavior consistent with an initial Hopf bifurcation at Re 7,988. Non-repeating flow behavior is observed in the phase space trajectories above Re 13,063, and is evidence of the transition to a chaotic flow regime. Finally, flows at Reynolds numbers above the chaotic transition point are simulated and found with statistical properties in good agreement with literature.

An immersed boundary method is proposed for the simulation of the interaction of an incompressible flow with rigid bodies. The method is based on a new interpretation of velocity-vorticity formulation and no longer includes the force term which is an essential issue of common immersed boundary methods. The system is considered in an Eulerian frame and retrieving the vorticity in this formulation enforces continuity at the fluid-solid interface and rigid motion of the solid. The method focuses on the mutual kinematic relations between the velocity and vorticity fields and with retrieving the vorticity field and recalculating the velocities yields the solenoidal velocity field. The method is applied to the two dimensional problems and the results show that the solenoidality is satisfied acceptably. The comparisons with 2D test cases are provided to illustrate the capabilities of the proposed method.

In this paper, we consider a two-scale stabilized finite volume method for the two-dimensional stationary incompressible flow approximated by the lowest equal-order element pair P1 — P1 which do not satisfy the inf-sup condition. The two-scale method consist of solving a small non-linear system on the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimal order in the H1-norm for velocity and the L2-norm for pressure are obtained. The error analysis shows there is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation h = 𝾪 (H2). Numerical experiments completely confirm theoretic results. Therefore, this method presented in this paper is of practical importance in scientific computation.

A lattice Boltzmann flux solver (LBFS) is presented in this work for simulation of incompressible viscous and inviscid flows. The new solver is based on Chapman-Enskog expansion analysis, which is the bridge to link Navier-Stokes (N-S) equations and lattice Boltzmann equation (LBE). The macroscopic differential equations are discretized by the finite volume method, where the flux at the cell interface is evaluated by local reconstruction of lattice Boltzmann solution from macroscopic flow variables at cell centers. The new solver removes the drawbacks of conventional lattice Boltzmann method such as limitation to uniform mesh, tie-up of mesh spacing and time interval, limitation to viscous flows. LBFS is validated by its application to simulate the viscous decaying vortex flow, the driven cavity flow, the viscous flow past a circular cylinder, and the inviscid flow past a circular cylinder. The obtained numerical results compare very well with available data in the literature, which show that LBFS has the second order of accuracy in space, and can be well applied to viscous and inviscid flow problems with non-uniform mesh and curved boundary.

We have introduced a fully second order IMplicit/EXplicit (IMEX) time in-tegration technique for solving the compressible Euler equations plus nonlinear heat conduction problems (also known as the radiation hydrodynamics problems) in Kadioglu et al., J. Comp. Physics [22,24]. In this paper, we study the implications when this method is applied to the incompressible Navier-Stokes (N-S) equations. The IMEX method is applied to the incompressible flow equations in the following manner. The hyperbolic terms of the flow equations are solved explicitly exploiting the well understood explicit schemes. On the other hand, an implicit strategy is employed for the non-hyperbolic terms. The explicit part is embedded in the implicit step in such a way that it is solved as part of the non-linear function evaluation within the framework of the Jacobian-Free Newton Krylov (JFNK) method [8,29,31]. This is done to obtain a self-consistent implementation of the IMEX method that eliminates the potential order reduction in time accuracy due to the specific operator separation. We employ a simple yet quite effective fractional step projection methodology (similar to those in [11,19,21,30]) as our preconditioner inside the JFNK solver. We present results from several test calculations. For each test, we show second order time convergence. Finally, we present a study for the algorithm performance of the JFNK solver with the new projection method based preconditioner.

We consider a non-standard mixed method for the Stokes problem in ℝn, n Є {2,3}, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor σ and the velocity vector u become the only unknowns. Then, we apply the Babuška-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k≥0 for σ and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order k≥0 for σ and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided.

A local domain-free discretization-immersed boundary method (DFD-IBM) is presented in this paper to solve incompressible Navier-Stokes equations in the primitive variable form. Like the conventional immersed boundary method (IBM), the local DFD-IBM solves the governing equations in the whole domain including exterior and interior of the immersed object. The effect of immersed boundary to the surrounding fluids is through the evaluation of velocity at interior and exterior dependent points. To be specific, the velocity at interior dependent points is computed by approximate forms of solution and the velocity at exterior dependent points is set to the wall velocity. As compared to the conventional IBM, the present approach accurately implements the non-slip boundary condition. As a result, there is no flow penetration, which is often appeared in the conventional IBM results. The present approach is validated by its application to simulate incompressible viscous flows around a circular cylinder. The obtained numerical results agree very well with the data in the literature.

In this paper, new finite difference methods based on the augmented immersed interface method (IIM) are proposed for simulating an inextensible moving interface in an incompressible two-dimensional flow. The mathematical models arise from studying the deformation of red blood cells in mathematical biology. The governing equations are incompressible Stokes or Navier-Stokes equations with an unknown surface tension, which should be determined in such a way that the surface divergence of the velocity is zero along the interface. Thus, the area enclosed by the interface and the total length of the interface should be conserved during the evolution process. Because of the nonlinear and coupling nature of the problem, direct discretization by applying the immersed boundary or immersed interface method yields complex nonlinear systems to be solved. In our new methods, we treat the unknown surface tension as an augmented variable so that the augmented IIM can be applied. Since finding the unknown surface tension is essentially an inverse problem that is sensitive to perturbations, our regularization strategy is to introduce a controlled tangential force along the interface, which leads to a least squares problem. For Stokes equations, the forward solver at one time level involves solving three Poisson equations with an interface. For Navier-Stokes equations, we propose a modified projection method that can enforce the pressure jump condition corresponding directly to the unknown surface tension. Several numerical experiments show good agreement with other results in the literature and reveal some interesting phenomena.

It is well known that the classical local projection method as well as residual-based stabilization techniques, as for instancestreamline upwind Petrov-Galerkin (SUPG), are optimal on isotropicmeshes. Here we extend the local projection stabilization for the Navier-Stokes system to anisotropic quadrilateral meshes in two spatial dimensions. Wedescribe the new methodand prove an a priori error estimate. This method leads on anisotropic meshes to qualitatively better convergence behavior than other isotropicstabilization methods.The capability of the methodis illustrated by means of two numerical test problems.

We study a two-grid scheme fully discrete in time andspace for solving the Navier-Stokes system. In the first step, thefully non-linear problem is discretized in space on a coarse gridwith mesh-size H and time step k. In the second step, theproblem is discretized in space on a fine grid with mesh-size hand the same time step, and linearized around the velocity u H computed in the first step. The two-grid strategy is motivated bythe fact that under suitable assumptions, the contribution ofu H to the error in the non-linear term, is measured in theL 2 norm in space and time, and thus has a higher-order than ifit were measured in the H 1 norm in space. We present thefollowing results: if h = H2 = k, then the global error ofthe two-grid algorithm is of the order of h, the same as wouldhave been obtained if the non-linear problem had been solveddirectly on thefine grid.