Fluid dynamics is one of the primary engineering sciences used to design a wide variety of vehicles, machines, and other devices. In the past, fluid-dynamic analysis was fairly simple, using empirical formulas from engineering handbooks or other simplified analysis techniques. However, as computer power becomes less expensive, fluid-dynamic analysis is shifting toward more-complex, powerful computational-fluid-dynamics (CFD) techniques.
A low-cost, rapid way to analyze performance and optimize designs comes from what is called “parametric CFD analysis.” It includes five steps: problem definition, dimensional reduction, experimental design, management of CFD simulations, and metadata analysis. The metadata analysis stage may also include surrogate modeling, visualization, optimization, and sensitivity analysis.\
Consider an aerodynamic design that entails placing vortex generators (VG) on the wing of a commercial airliner. VGs are typically added to correct undesirable flight attributes such as low-speed instability or too-little control. Even minor changes to an aerodynamic configuration such as this can significantly harm the aircraft’s handling and, thus, necessitate complex analysis. Engineers mastering the five stages of parametric CFD analysis better understand how the device behaves and, therefore, produce better designs. Here is an example of using the five steps of parametric CFD analysis.
Problem definition. For all designs, engineers must define requirements for the system, component, or process. For machines, requirements include the operating envelope of the equipment, which defines the range of conditions over which the device must operate safely.
In the case of commercial airliners, the operating envelope is the range of all parameters at which the aircraft may safely fly. Among the requirements for an airliner are that it be efficient and safe. Aircraft designers extensively test designs to meet these requirements using CFD simulations, wind-tunnel tests, and flight tests.
Undesirable flight attributes often arise because of a boundary-layer separation on the wing surfaces. This occurs when the boundary layer, a thin layer of slow-moving air caused by friction near the surface, separates from the surface of the wing. Due to the wing sweep, boundary-layer separation on the outer portion of the wing is a common problem on commercial jets. VGs circulate high-energy flow into the boundary layer, suppressing boundary-layer separation and mitigating the problems.
While common, vortex generators are actually an aerodynamic compromise. They delay the onset of boundary-layer separation, but also increase parasitic wing drag. Increasing drag reduces fuel efficiency, so aerodynamicists are careful about when and where they use VGs.
So where to place them? Obviously, aerodynamicists must test a wide variety of locations. Tests will likely focus on the landing configurations and high angle-of-attack regions where boundary-layer separation happens. However, other flight conditions and configurations can’t be ignored. For example, certain VG distributions may work to eliminate boundary-layer separation at landing configurations, but cause more drag at cruise configuration. Because the location of VGs alters flight attributes and aircraft performance, engineers must evaluate it throughout the flight envelope. The problem is a constrained-optimization problem, which in this case is best solved with a parametric analysis.
The placement of each VG is defined by three parameters: spanwise (wing root to wing tip) location, chordwise (leading edge to trailing edge) location, and angle relative to the aircraft plane of symmetry. In the example, the wing has 21 VGs; therefore, because each VG has three degrees-of-freedom (DOF), including all the DOF would add 63 parameters to the parametric analysis. When added to the five flight parameters and six (minimum) configuration parameters, the design space has 74 dimensions — quite large.
Dimensional reduction. Like most engineering-design problems, the full problem definition results in too many dimensions for a reasonable parametric analysis. Fortunately, it’s possible to reduce the number of dimensions by taking advantage of previous studies on similar devices. For example, VGs have been used since the 1940s and guidelines for their use are available in literature. These include suggested size, spacing, and angle of the vortex generators. The chordwise position is more problem-dependent and thus allowed to vary in the parametric study. This brings the number of dimensions down to 21 VGs, and 32 overall, still a large design space.
To further reduce the number of DOF, use analytical distributions for the chordwise position of the VGs. The outer five VGs are in a straight line at a constant chordwise position — one DOF instead of five. The chordwise positions of the inner 16 VGs could be approximated by a quadratic equation resulting in three DOF instead of 16. The upshot is a total of four DOF, which, when added to the 11 flight and configuration parameters, results in a total of 15 dimensions.
Experimental design. In this context, “experimental design” refers to the set of cases that the CFD code will run. The cases are selected to efficiently span the design space, with the goal of minimizing drag at cruise while retaining stability and control throughout the operating envelope. High-resolution CFD solutions of full aircraft configurations — especially landing configurations — are expensive, so the number of cases is limited. A space-filling design such as Latin hypercube (a statistical method), or one of its optimal variants, would probably be used to ensure coverage of the full range of each parameter.
For 15 dimensions, even a small parametric study would likely include hundreds of runs. Each run generates a large amount of 3D field data in addition to metadata results. Ideally, the 3D data is kept to verify solution quality and investigate the underlying cause of anomalies in the metadata
Management of CFD simulations. A key requirement for high-quality CFD solutions is that the grid-spacing be fine enough to resolve the dominant flow features. Ideally, a sufficiently fine grid would be used to resolve all of these flow features to a high degree, but there are seldom enough computing resources to do this. Also, certain subgrid quantities, such as turbulence, are modeled, so no model is ideal for all flows. For this reason, CFD experts must evaluate the results to ensure assumptions aren’t violated.
The analysis of VG placements strain the current capabilities of CFD simulations. When the simulation calculates the vertical flows, a fine grid is needed in the spanwise direction. Normally, the boundary layer is resolved using a fine wall-normal grid and the grid spacing in the spanwise direction. The streamwise directions are left coarse, reducing the number of grid points. Near the VG, the grid must also be refined in all three directions. When the velocity field in 21 streamwise vortices must be resolved, the number of grid points gets large, making the CFD simulations time consuming.
A CFD expert must verify that the fine grid is properly placed to compute the vortices. Finally, there have been efforts to reduce the number of grid points by modeling the vortex-generator geometry as a step-function in secondary velocities — a subgrid effect. This allows a comparatively coarse grid in the streamwise direction and reduces the computer resources required for the CFD simulations.
Ideally, the tools used to manage the parametric data make it easy to verify the quality of the CFD solutions. The metadata and links to 3D field data files get stored in a database. A graphical front-end to the database simplifies the access to the data, perhaps through clicking on scatter symbols representing individual cases or data in tables. This practice allows rapid evaluation of a flow field and grid to look for potential quality issues.
Metadata analysis. The effectiveness of the VGs is evaluated through metadata analysis. The first step in analyzing the metadata is to create a surrogate model. Like the CFD code itself, the surrogate model inputs the point in parametric space and returns a set of dependent variables. The surrogate model is an approximation of the values that are obtained by running the CFD code at that point in parametric space and runs much faster.
Surrogate models take a wide variety of forms. This aircraft wing example is a polynomial in 15 dimensions. If it were a quadratic response surface, a second-order polynomial with cross terms, it would have 136 terms. To use this model, the experimental design must contain at least 136 cases. Higher-order response-surface models, like fourth-order models, are also common but have far more terms and require more cases.
Surrogate models allow the visualization of metadata even when the experimental design has cases widely dispersed throughout the highly dimensional space. The surrogate model effectively fills the empty space between cases with an approximation of the true data. The surrogate model lets users create XY plots of the relationship between metadata variables along any line through parametric space. Likewise, it’s possible to create contour plots of any variable along any plane in the parametric space, or create an iso-surface plot of any variable in any 3D subspace of the parametric space. Using these visualizations, users can recognize relationships between variables and identify local minima and maxima in the subspaces.
A common concern in engineering design is how sensitive the device is to changes in the independent parameters. More specifically, it would be helpful to know the leverage of each independent variable on the changes in each dependent variable. Using a surrogate model, it is relatively straightforward to estimate the sensitivity or perform an analysis of variance (ANOVA).
Mastering CFD for engineering design. Parametric CFD analysis has been shown to play a crucial role in the engineering design of fluid-dynamic devices. Without it, engineering methods can be costly and time consuming. By the time a project is complete, there could already by a more-efficient way to utilize the device. An understanding of the five stages of parametric CFD analysis gives users a better understanding of how a device behaves over its entire operating envelope and quickly produces better designs.