CGE: Cartesian Grid Euler Solver

CGE is an advanced adaptive Cartesian grid-based (or octree-based) CFD solver that is ideally suited to steady and unsteady flow prediction over complicated real-life configurations. The flow solver is fast (comparable to fast panel methods) and can automatically generate grids around imperfect surface geometries - making it ideal for design and routine engineering applications.

Key Features

  • Automated "push" button grid generation
    • Solution-based adaptive mesh refinement
    • Support for imperfect geometries (non-watertight, overlapping surface elements)
  • Steady and unsteady compressible flow
    • Stable at low Mach numbers
  • Engine and rotor models
    • Actuator disk (rotating and stationary stages)
    • Heat addition

    A perennial difficulty with grid-based CFD solvers is the reliable generation of a good quality mesh about complex geometries. The three most common grid topologies are: structured grids, unstructured grids and Cartesian meshes, although others such as hex and dragon meshes have also been used. Advanced grid concepts such as hybrid meshes and Chimera/Overset grids essentially derive from one or more of these basic mesh types. Structured grids employ a logically rectangular indexing structure, which allows for trivial identification of neighboring elements/faces and simple interpolation rules, and are thus highly efficient in terms of both storage and CPU. Unfortunately, structured grids are extremely difficult to generate about complex configurations and require considerable user intervention to ensure good quality (i.e., adequate resolution and minimal skewing) and proper contiguity between blocks (for multiple block structures). Unstructured meshes on the other hand, readily accommodate complicated surfaces, facilitate solution adaptation and can be generated in a nearly autonomous fashion. However, mesh generation can be expensive and care must be taken to ensure mesh quality particularly in re-entrant corners. Moreover, the resolution of the volume mesh is tightly wedded to that of the surface. The third basic mesh type is the Cartesian grid which employs the data structure known as an octree to decompose the flow domain into a collection of hierarchically nested cubes. Unlike the previous two grid constructs, the Cartesian mesh is not inherently boundary conforming. Instead, surface and volume clipping procedures are employed to explicitly deduce the necessary connectivity relations, volumes and surface areas for grid cells that intersect the body surface. The main advantage of a properly formulated Cartesian grid generation algorithm is that a mesh can be generated for virtually any closed surface or collection of surfaces in a virtually autonomous manner. The list of parameters controlling the mesh generation process is very short (and simple) and consists of specifying the range of mesh spacings desired at the surface resolutions and the rate at which cell size is allowed to grow away from the surface.

    To relieve the time consuming and laborious duties of mesh quality improvement for both structured and unstructured meshes, CDI developed an unsteady 3D Cartesian Grid based Euler solver (CGE) for performing routine design and analysis work. This solver exploits CDI's significant experience in octree generation and handling, polygon clipping algorithms and unstructured grid-based flow solvers. CGE incorporates state-of-the art flux splitting routines, implicit time marching algorithms, higher order interpolation methods and multigrid-based acceleration schemes together with flow-based adaptive mesh routines (see Figure 1). It has been extensively validated and demonstrated for complicated geometries such as the ONERA M6 wing and the space shuttle booster combination, as well as engine (rotor/stator) configurations (Figure 1), commercial aircraft (Figure 2), missiles (Figure 1), rotorcraft, automotive configurations (Figure 2) and ships.

    Figure 1: Slices through the adaptive grid, colored by Mach number, of a CGE prediction of a missile travelling at M=1.25 (left). Slice through the center of the grid and surface contours on the duct and center-body of a CGE predicted flow through an engine model with rotor and stator (right)

    Figure 2: CGE predicted flow around a generic civil-transport aircraft at high angle of attack (left) and around a Ferrari 333 (right)

    Application to Imperfect Geometries

    To support general design work, the grid generation routines have been extended to support imperfect (i.e. non-singly defined and non-watertight) surface definitions that are often encountered in "real-life" configurations. This addition has significantly expanded CGE's ability to solve flows about realistic geometries, a capability not offered by competing CFD solvers since they require watertight surface representations - even if the actual vehicle is not watertight.

    An example for a commercially sourced fighter configuration at high angle of attack is of particular relevance to this application since while the 3D surface geometry (see Figure 3) was acceptable for 3D rendering and visualizations, it was far from ideal for contemporary CFD. The surface mesh is somewhat representative of the types of geometry generated during design, or provided by customers/commercial vendors and contains a coarse representation of the wings, with holes, pitot-probes, antennae and even meshing of the inside of the cockpit. Given the Cartesian grid nature of CGE and the ability to handle non-watertight geometries, CGE was able to automatically generate a mesh without any user-intervention or mesh generation problems in only a few seconds (Figure 4).

    Figure 3: F-18 surface grid. Top view (left), close-up of the nose showing pitot-probes (center) and close-up of the cockpit showing internal meshing of the instrument panel etc. (right)

    The calculation consisted of the F-18 at a Mach number of 0.3 at an angle of attack of 60°, and results are presented in Figure 5. CGE is capable of predicting the classical delta-wing vortex flow around this configuration, and predictions are consistent with the flight test and full-scale wind tunnel tests performed by NASA in the 1990's (see Figure 6 and NASA-TM-112360). As the flow wraps around the nose of the aircraft and the cockpit area, a strong vortex is formed on either side of the aircraft. These vortices are relatively strong and coherent and remain close to the fuselage until about mid chord along the wing. At that point, the vortices, as shown in the right plot in Figure 5, burst and entrain into the remaining flow. Such structures are very important on predicting the performance, stability and handling qualities of aircraft, since they are critical to phenomena such as wing rock.

    Figure 4: Close-up of perpendicular slices through the F-18 grid

    Figure 5: Predicted flow around an F-18 at high angle of attack with streamtraces colored by vorticity magnitude. Overall view (left) and top view of vortex from the port side (right)

    Figure 6: Experimental visualization of high-angle of attack flow over an F-18 configuration; water tunnel (left) and flight test (right)

    Computational Performance in the Design Environment

    Panel methods, such as CDI's FPV (click here) offer a low cost alternative for many routine analysis tasks. They provide efficient 3D modeling capability especially when combined with Fast Multipole Methods to reduce the cost from O(N2) to O(NlogN). Panel methods can support surface definitions consisting of quadrilaterals and triangles and tolerate small gaps and overlapping surfaces; however accuracy is sensitive to panel alignment and shape. These methods are inherently inviscid and incompressible, and while the definition of the wake lines from sharp edges can be automatically determined, lift carryover at a wing-body junctures must be defined by the user, as must separation locations for bluff edges. Extensions to include non-linear compressibility effects are generally expensive and provide little additional fidelity. Currently, panel methods maintain a significant advantage over conventional CFD when simulating low Mach number flows about surfaces executing large amplitude relative motions since they eliminate the need for 3D volumetric remeshing and generally allow larger time steps to be taken. For flows about fixed geometries however, CGE offers advantages in terms of computational speed, capturing of shocks and nonlinear compressibility, and automated locating of wake separation.

    In work for NASA developing aerodynamic design tools for integration into optimization environments, a direct comparison was undertaken between the linear and non-linear (i.e. compressible) variants of FPV and CGE for a realistic geometry. A surface geometry for a simplified space shuttle booster configuration was extracted from a multiblock (OVERFLOW) CFD mesh consisting of 35,022 panels (mostly quads). This mesh, with wake lines assigned to the wings of the shuttle, was used for linear and non-linear FPV calculations, and was converted into 69,120 triangles for use in CGE. Predictions were performed at M=0.5 with an angle of attack of 10°.

    Plots of the surface pressure and flow vectors in Figure 7 indicate that FPV and CGE predict similar loading over much of the configuration, especially from the middle of the fuselage forward. The methods predict distinctly different loadings around the wings and the aft of the shuttle and fuel tank. The wings have relatively blunt trailing edges and thus the assumption of attached flow imposed by assigning wake lines in FPV results in a more idealized loading. The flow around the aft of the shuttle and fuel tank is highly separated with discrete vortices that directly impact loading. The fast panel analyses assume that the flow remains attached, whereas CGE is able to better represent the flow separation (see Figure 7 right) and base drag. Predictions with the non-linear panel method are similar to the linear variant (see Figure 8), though the pressure peaks around the windshield and the trailing edges are slightly reduced (though not to the level predicted by CGE).

    Figure 7: Predictions of surface pressure and velocity vectors for a simplified space-shuttle booster configuration. Linear panel method (left) CGE (center) and close-up of the CGE prediction of the aft section of the fuel tank (right)

    A direct comparison of computational cost is presented in Table 1 for the three methods where the times have been normalized by the linear fast panel analysis. Using the FMM methods, a factor of 80 reduction in cost was obtained over conventional direct methods (i.e. 15.2M influence coefficients and wake entries in contrast to 1.23B entries for the direct calculation), however, even with these acceleration techniques, the non-linear panel method is very expensive (>24 times the linear version) for this realistic configuration, with fidelity only slightly improved over the linear method. The Cartesian grid solver on the other hand converged in 91% of the time of the linear method and provided a more realistic and accurate representation of the flow.

    Figure 8: Comparison of linear and non-linear panel method predictions of surface pressure along the centerline

    Table 1: Run metrics for space-shuttle booster calculations (time normalized by linear FPV calculation time)
    Non-Dimensional Calculation time Surface Grid (number of elements) Volumetric Grid
    Linear Panel Method 1.00 35,022 na
    Non-Linear Panel Method 24.44 35,022 na
    Cartesian Grid Solver 0.91 69,120 130,929

    Application to Vorticity Dominated and Bluff Body-Type Flows

    Though CGE has its roots in aerodynamic design and analysis, it has been aggressively transitioned to complex flow environments in recent years. CGE has been validated for a variety of configurations ranging from isolated wings to complete aircraft and even ships. Despite its compressible nature, CGE is somewhat unique amongst Euler solvers in that it remains stable even at low Mach numbers without preconditioning. As a further demonstration of this capability is shown in Figure 9, where predictions of flow separation on the deck of the "simple frigate shape" ship geometry are compared to experiments and reveal accurate prediction of separation line. An important practical advantage of this CFD approach, over traditional RANS solvers, particularly for building/urban environments, ship and military vehicle flows (see Figure 10) that are dominated by geometries with sharp edges, is that numerical viscosity of the scheme ensures that Kutta conditions are implicitly enforced at sharp corners and edges (i.e. the flow separates appropriately without the need to model surface friction and turbulence as would be the case for a smooth surface such as a sphere or cylinder).

    It is well known that CFD methods are subject to numerical diffusion effects that smear and diffuse vortex structures, slip lines, wakes, etc. To counter diffusion, it is normally necessary to reduce mesh resolution to sufficiently fine levels, resulting in large increases in mesh node counts and thus storage and CPU requirements. For many applications, such as vehicular load and stability estimation, wake diffusion can be tolerated. However, when long time wake effects are important - rotorcraft flows, wake interaction between aircraft, dynamic interface (aircraft operating in ship air wakes), agricultural aircraft product deposition and drift, helicopter operation in urban environments - diffusion of vorticity must be explicitly addressed. CDI has developed a family of novel vorticity-velocity flow analyses (VorTran-M/ VorTran-M 2) that provides a low cost means of accurately evolving vortex structures over long distances and times. VorTran-M/VorTran-M2 requires a means of initializing/specifying vorticity which can be provided by an alternate CFD code such as CGE. Thus, to facilitate efficient predictions of long-time vorticity dominated flows, CGE has been coupled to CDI's VorTran-M solver (which itself employs a modified adaptive flow tracking Cartesian grid system, click here for more information).

    The resulting coupled CGE/VorTran-M code has been used to develop the first commercial ship awake database for a real-time tactical flight simulator (currently in use in the U.S. Navy's SH-60B/MH-60R TOFT) consisting of over 192 ship/flow condition combinations. These ships contain numerous gaps (e.g., the flight deck to hull connections), overlapping elements (e.g., numbers painted on the deck that are retained for visualization purposes) and protrusions such as surface details (turrets, superstructure) that penetrate into the deck. In addition, the geometries contain numerous small details such as antennas that are aerodynamically insignificant to the scales of interest. Manually eliminating unnecessary components and repairing surface inconsistencies is labor-intensive. With CGE no manual intervention is necessary to obtain high quality CFD solutions, and software routines have been generated to preprocess several standard geometry formats to the appropriate format for CGE. Sample CGE/VorTran-M ship airwake predictions are shown in Figure 11. CGE/VorTran-M has also been used to investigate the flow through cities (downtown Chicago, IL and Philadelphia, PA) - see Figure 12.

    Figure 9: Simple Frigate Shape (SFS) geometry (left), measured surface velocity (Oil Flow) profile for SFS at zero yaw angle (center) and CGE predicted surface velocity vectors (right)

    Figure 10: CGE/VorTran-M predictions of HMMWV aerodynamics at 130kts (left) and 200kts (right) to simulate a slung-load. Pressure contours plotted on the surface of the vehicle and streamtraces and an iso-contour of vorticity magnitude plotted to illustrate the near wake

    Figure 11: CGE/VorTran-M predictions of the flow across the deck of a LHA (left) and the entire wake behind a LPD-13 plotted as an iso-surface of vorticity (right)

    Figure 12: CGE/VorTran-M predictions of the flow around downtown Chicago, IL (left) and downtown Philadelphia, PA (right)


    For a trial of CGE, to discuss licensing or to simply request more information, please contact Dr. Alexander Boschitsch or Dr. Glen Whitehouse.


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