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Figure 5 Polynomial order and vorticity distributions obtained with simulation using adaptive polynomial order for the flow around a NACA 0012 profile with Re = 50,000 and a = 6°. Taken from [69].

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Figure 1: Coordinate systems and mappings between collapsed coordinates 7, reference coordinates € and Cartesian coordinates x for a high-order triangular element Q,.

1) A simple two-dimensional quadrilateral mesh consisting of a single (b) Directory-dataset structure that is used for storage of the topological lement. data in the left-hand figure.

Figure 3: Instantaneous acoustic source term as represented in CFD (proprietary finite volume flow solver with Ah < 1.4mm mesh) and CAA (Nektar++ AcousticSolver with Ah = 20mm mesh and fourth order expansion). Slice through a three-dimensional domain. universally used in Python scientific computing applications for data storage, and the Nektar++ storage Array<OneD, *> classes. This allows an ndarray to be passed into Nektar++ functions directly and vice versa. Moreover this interaction uses t t d he Boost.Python interface to NumPy to ensure that instead of copying data (which could be rather inefficient for large arrays), his interaction uses a shared memory space between the two ata structures. Reference counting is then used to ensure data persistence and memory deallocation, depending on whether memory was first allocated within the C++ environment or Py- t non. the Python C API also enables the use of higher-performance compiled code, making it suitable in instances where interpreted pure Python would be inefficient and impractical, as can be seen using packages such as NumPy and SciPy. These factors therefore make Python an ideal language through which to both introduce new users to a complex piece of software, interact with other software packages and, at the same time, retain a certain degree of performance that would not be possible from a purely interpreted perspective.

Figure 4: Numerical simulation of the MMF scheme in Nektar++ for several partial differential equations solved on the sphere.

Figure 6: Time-averaged streamwise reversing regions for incompressible flow over a wing with spanwise waviness with Re = 1,000 and a = 12°.

Figure 7: Example of split boundary layer mesh.

aerofoil generator. With just three inputs — a NACA code, an angle of attack and dimensions of the bounding box — a geometry is generated and passed to the meshing software. An example is shown in Figure 8 of a mesh generated around a NACA 0012 aerofoil at an angle of attack of a = 15°. Figure 8: Example of mesh generated around a NACA 0012 aerofoil.

Figure 9: Example of mesh generated around a t106c turbine blade. The other tool is based on the GEO geometry file format of the Gmsh [80] open source mesh generator. The GEO format is an intepreter for the procedural generation of geometries. NekMesh has been made capable to understand basic GEO commands. which gives the possibility to generate simple two-dimensional geometries. An example is shown in Figure 9 of a mesh gener. ated around a T106C turbine blade: the geometry was created using a GEO script of lines and splines.

(a) Normalized acoustic pressure distribution at t = 1s with the mesh shown in light gray and the sampling line in a black dashed line.

(b) Normalized acoustic pressure along sample line, obtained with the AcousticSolver and analytical solution [84].

Figure 11: Instantaneous vortex shedding visualized by the vortex-identification of Q-criterion (iso-surfaces of the Q-value= [—5, 5]) in body-fitted coordinates: (a) full domain view and zoom-in view of (b) first strip; (c) fourth strip and (d) 16th strip.

Figure 12: Instantaneous isosurfaces of Q-criterion (Q = 500) contoured by velocity magnitude, showing the vortical structures evolving on the suction surface and in the wake of a T106A cascade. The computational domain is replicated in the spanwise and pitchwise directions for visual clarity.

Figure 14: Skin friction coefficient for the SWBLI test case: blue line Nektar++ (60 x 40 quadrilateral elements, p = 3); triangles are from [96]; dotted line is empirical solution by [99]. Figure 13: Mach number field of SWBLI test case (6040 quadrilateral elements, p = 3); configuration based on [95].