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If nothing happens, download Xcode and try again. If nothing happens, download the GitHub extension for Visual Studio and try again. It features second order accurate, unstructured, cell-centered, finite volume discretization of incompressible Navier-Stokes equations with heat transfer and species transport. Compilation of the main program and sub-programs depends on the following tools: ld, make, gfortran, git and mpi only for parallel execution.
The code is currently available under MIT license.
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To download the current version of the code use:. Code has four sub-programs at the moment and uses own proprietary format for computational grids. Xmgrace sources with referent DNS or experimental data are provided for the most of the test cases, as well as user functions for post-processing of data. The program produces files in vtu format that can be open by ParaView open-source software for postprocessing.
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Launching GitHub Desktop Go back. The function retained in the present work is a hat function Eqs.
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Figure 1 shows a two-dimensional scheme of an immersed boundary on a staggered finitevolume computational grid. It can be noted that the Lagrangian point - represented by the dots - are not coincident with the faces of the control volumes. The interpolation of the Eulerian velocities in this case is needed.
Using the hat function as the interpolation function, velocity values that lies within the blue circle shown in figure 1 will be interpolated to the Lagrangian point at the center of the circle. In a procedure similar to the interpolation of the "Eulerian" velocities to the Lagrangian points, once Eq. This is made with aid of Eq. Note that the interpolation procedure interpolates variables that can be positioned at faces, like velocities components, or at cell centers, like density for instance. Once the velocities are interpolated and the Lagrangian force is evaluated, such a force can also be distributed to faces or cell centers.
It depends on the arguments used in Eqs. In the present work, the Lagrangian forces are distributed to the cell faces. In Fig 2 , it is displayed the area doted circle in which the Lagrangian force evaluated at a point k is distributed over a Eulerian grid. It is noteworthy that a same Eulerian cell can receive contributions of more than one Lagrangian point, being these contributions summed. Concerning the parallelization of the immersed boundary method, two kinds of flows must be considered.
The flows under fluid-structure interaction, in which the immersed boundary moves within the computational domain, and therefore may cross sub-domains boundaries, and flows where a sub-domain boundary can cross an immersed object, but such a geometry is still. For the latter, the algorithm is quite simple. It is enough that in the beginning of the simulation each processor reads a file where the immersed geometry is described, and then, it is verified which Lagrangian points lies within each sub-domain.
Evidently, it may give a different number of Lagrangian points per processor, and static load balancing must be carefully evaluated. Considering that communication is required in each cycle of the multidirect forcing, such a communication must be as efficient as possible. If we consider figure 3 , it is possible to see that information must be interpolated from and distributed to overlapping cells. Since primitive variables, like velocity components, must be interpolated to a Lagrangian point k, from an overlapping cell, it causes no further difficulties, once at the moment of the interpolation, the velocity field is already updated in the overlapping cells, and therefore, any primitive variable can be interpolate to a Lagrangian point k regardless the subdivisions of the computational domain Vedovoto The procedure for distributing the Lagrangian force from the point k to a region of the Eulerian grid is a little more tricky however.
It can be noted in figure 3 , that a portion of the Lagrangian force once concentrated in a point k is distributed in overlapping cells in both processors IDs 1 and 2. One must remember that distributed Eulerian force is the result of a sum of the contributions of different Lagrangian points that lies within a given region see Eq. If there is fluid structure interaction, where the immersed object moves across sub-domains, it means that Lagrangian points must be transported across the sub-domains.
Since there is no need to retain a explicit connectivity between the triangles that compose the Lagrangian mesh, it is enough to transport information like area of the transported triangle, positions of the normal vector and centroids to another sub-domain. Since the details of the parallel implementation is not in the scope of the present work, the interested reader may gather more information on the subject in Vedovoto The verification of such implementation through the method of manufactured solutions was performed.
Moreover, for the sake of validation, two different test cases were simulated and the results will be shown in the next sections. The resort to the method of manufactured solutions is progressively being a well accepted methodology in the framework of numerical code verification Steinberg and Roache Such a method consists in developing a priori known analytical solutions of the retained system equations. These manufactured solutions modify the original equations by adding a 'source term', such as those presented in transport equations. There is an undeniable interest in the use of such a method to quantify accurately numerical capabilities before using computational programs to perform the simulation of more complicated physical systems.
This is the case when the immersed boundary is involved. The immersed boundary method creates a perturbation in the flow field in such a manner that it can cause a decay in the rate of convergence of the methods employed.
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The evaluation of the Lagrangian force necessary to stop a fluid particle placed at the proximity of an immersed object is dictated by the difference between the actual velocity at a Lagrangian point k and a velocity interpolated from the Eulerian domain at the same Lagrangian point.
It is natural, under the method of manufactured solutions, to consider the velocity of Eq. The subscript e stands for the manufactured solutions of the primary variables, i. It is observed that, if the divergence of Eqs. Such a set of parameter allow to apply the Method of Manufactured Solutions within a wide range of numerical verifications. The time step is controlled taken the C F L factor small than 0. The constant values of density and viscosity are set to unity and , respectively.
A sphere of radius 0. Figure 6 shows the immersed object and its location inside the computational domain. It is important to stress that this is not a physical problem. It is only a matter of verification of the numerical code implementation. The procedure used for evaluating the error rate of decay is the same presented in Vedovoto et al. From an Eulerian grid of , the Eulerian grid is halved three times, resulting 4 different grid sizes.
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The surface mesh of the immersed object follows the Eulerian mesh refinement. As the Eulerian grid becomes coarser or refined, the surface mesh also becomes coarser or refined. It is important to point-out that in the test presented here, the whole numerical scheme is tested again, i. The procedure for the evaluation of both the Eulerian and Lagrangian domains are very similar and were demonstrated in Vedovoto et al. The results gathered in the simulations are illustrated in Figure 7. The numerical velocity and pressure are compared with the analytical solutions given by equations The inhouse code developed has shown second order of convergence rate for velocity as well as for the immersed boundary.
It is an important result since it illustrates one of the main advantages of the direct forcing used in the present work. Since no ad-hoc constants are required in the modeling of the immersed boundary, the convergence rate is dictated by the Eulerian numerical scheme and by the interpolation and distribution functions. Such a characteristic is also observed in high order schemes such as pseudo-spectral based codes, Mariano et al. In this section it will be provided a series of applications of the methodologies depicted before.
In this section a synthetic inlet turbulence generator is presented to perform the numerical simulation of high velocity jets developing spatially. The predictive capability of this methodology is demonstrated via comparisons with experimental data associated to turbulent jet flow. Jet flows are very dependent on inlet boundary conditions such as mean velocity profile and turbulence intensity and length scale at the nozzle exit.
The mean velocity and the RMS profile used in the present work are given in the work of Chen et al. Only the inert configuration was used in order to perform comparisons. To correctly represent the anisotropic RMS velocity profile of the jet, the method of Smirnov et al.