An overview of the three modern categories of methods for numerical prediction of turbulent flows is provided: direct numerical simulation (DNS), solution of the Reynolds-averaged Navier-Stokes (RANS) equations, and large-eddy simulation (LES). We describe zero-equation, one-equation, two-equation, and Reynolds stress transport models for the RANS equations. RANS computations require significantly fewer grid points and lower computational cost since the solutions are smooth and turbulent structures are not captured, but there is a need to tune model parameters for different flows to match experimental data. In LES, only the large-scale motions are resolved, whereas unresolved small scales are modeled. We introduce the notion of filtering, subgrid-scale parameterization, as well as the seminal dynamic Smagorinsky subgrid-scale model. Wall-resolved and wall-modeled LES are briefly discussed. With ever increasing computer power, as well as advances in numerical methods and subgrid-scale models, LES is rapidly becoming a viable tool for practical computations. In selecting a method, one should consider quantities to be predicted, accuracy of the predictions, and the computational cost.

A vehicle in an airstream sets up a pressure field on its surface, resulting in forces acting on it. Thus, the aerodynamic design task becomes: determine the shape that produces a surface pressure distribution yielding optimal flight performance. Based on the principles of flow physics, computational fluid dynamics (CFD) maps out how an aircraft's shape affects the flow patterns around it. Combined with mathematical techniques for shape optimization, CFD offers a powerful tool for sophisticated aerodynamic design. The goal is to achieve those vital features stemming from the concept of a "healthy flow," namely that these specific flow patterns and associated surface pressures are efficient means of generating aerodynamic lift with acceptable drag and are capable of persisting in a steady and stable form over ranges of Mach numbers, Reynolds numbers, angles of incidence, and sideslip embracing the flight envelope of the aircraft. In the parlance of multidisciplinary design and optimization, this chapter talks about the level of fidelity of the models and solutions. L0 methods are based on empiricisms and statistics. L1–L3 are physics-based models. The governing equations in L1 are linear potential flow, in L2 are inviscid compressible flow, and in L3 are nonlinear viscous turbulent flow.