Chapter 2 provides a detailed account of the transport theories on continuum single-phase flows that include the laminar or turbulent flows of viscous fluids, the viscous fluid flows through porous media, and flows of granular particles. The most essential one is the continuum modeling of laminar flows of Newtonian fluids, which is the foundation of the Eulerian modeling approach of multiphase flows. The distinctly different transport mechanisms between a laminar flow and a turbulence flow as well as the flow regime transition conditions are presented. The essential aspects of the modeling of a turbulent flow of Newtonian fluid are also given. The basic theories of flows through porous media are represented by Darcy’s law, Ergun’s equation, and Brinkman's equation.The chapter also illustrates the continuum theory of granular flow, represented by the kinetic theory of granular flows whose transport mechanisms are dominated by inertia and interparticle collisions.

Chapter 4 discusses the mechanisms and formulation of various basic particle–particle interactions. The essential modes of these interactions include a pair of spheres interacting by head-on approaching or by wake attraction, flow through a uniformly suspended sphere, electrostatic field induced by the suspended charged particles, normal collision dynamics involving forces, deformation, contact area and duration for a pair of elastic spheres, van der Waals force, and capillary force due to liquid bridge between two particles. The chapter further discusses the nonidealized particle–particle interactions and associated formulation, including the radiation transport equation for thermal radiation within a particle cloud, collision dynamics with tangential friction and torsional traction of elastic spheres, inelastic collisions, and the concept of restitution coefficient, heat and charge transfer by particle collisions, and deformation, breakup, and coalescence of fluid particles. These particle–particle interactions are critical to the model formation of dense-phase multiphase flows.

In a continuum modeling approach of a multiphase flow, each transport phase is regarded as an individual pseudo-continuum fluid and all these “fluids” co-share the same space and time domains. Chapter 6 delineates the volume-averaging method to construct the pseudo-continuum fluids over which the volume-averaged Eulerian modeling approach is developed. The key concepts and formula of volume-averaged continuum modeling include the definitions of intrinsic and phase averages and their relationship; the volume-averaging theorems; the general form of volume-averaged transport equations and the individual equations of mass, momentum, and energy; the volume and mass balance conditions of all phases; the constitutive relations of the volume-averaged tensors by individual volume-averaged parameters; and the formulation of interfacial transport between phases. The effect of turbulence on phase transport is also handled via Reynolds decomposition and time-averaging over the volume-averaged equations. The detailed formulation of various turbulent transport coefficients and the salient behavior of the turbulence modulation from various interactions among phases are presented.

The knowledge gained in the previous two chapters leads to procedures for computing solutions to the Navier–Stokes equations in 2D and 3D. Chapter 6 explains the major components and functions of a typical Reynolds-averaged Navier–Stokes (RANS) code, including the modeling of turbulence in steady or unsteady flows. Convergence acceleration devices, including multigrid techniques, are explained. Finite-volume formulation and standard physical modeling for turbulence yields the RANS equations used in most computational fluid dynamics (CFD) codes directed toward compressible-flow aeronautical applications. By taking the reader through a RANS application step by step, this chapter illustrates the process that an informed CFD user needs to know for applying a typical code of this genus to aerodynamic design. Two practical cases of transonic flow over an airfoil – one in steady flow and the other in unsteady buffeting flow – demonstrate execution of the workflow. Computing a Mach sweep across the entire transonic regime, the steady-flow example exhibits the nonlinear phenomenon of shock stall. Mastering this chapter makes the student a reasonably well-informed CFD user who understands how to carry out a sensitivity analysis to demonstrate CFD due diligence.

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.

Our treatment of aerodynamic performance (i.e. the mapping from shape to lift and drag for clean wings) idealized the plane as a mass point with lift and drag forces. The variation of the aerodynamic forces on the aircraft along the flight path determines its stability and the need for control with sustained authority. Addressing this issue requires an airplane model responding to gravity, thrust, and realistic aerodynamic forces and moments. A six-degree-of-freedom Newtonian rigid body model is compiled from the mass and balance properties of the airframe. Computational fluid dynamics (CFD) is used to predict the aerodynamic forces and moments, expressed in look-up tables of coefficients, and a major part of the text explains how such tables can be populated efficiently. The stability properties describe how well the aircraft recovers from external disturbances and how it reacts to commanded changes in flight attitude. The response in steady flight to small disturbances can be represented as a superposition of a small number of natural flight modes, the quantitative properties of which provide the quantified flight-handling qualities. A number of examples are given, from redesign of the Transonic Cruiser configuration for better pitch stability to CFD investigation of vortex interference on control surfaces on an unmanned aerial vehicle.​

Wings for three speed regimes with potential for efficient flight are investigated. We compute and analyse our own data for the designs in search of a coherent explanation for why these aircraft are the shapes they are for the tasks they have to perform. The subcritical speed case is a straight, high-aspect-ratio wing designed to maintain attached flow to the trailing edge. Two swept wings for supercritical flow are studied: the first one is from the late 1940s, when transonic problems were not understood. The second is a modern transport wing (Common Research Model (CRM)) showing what was learned in 70 years of transonic wing design. Attached flow is harder to sustain since shock waves interacting with the boundary layer may cause premature separation and drag increase. The slender Mach 2 Concorde-like example is marked by its low-aspect-ratio delta-like wing. This class breaks the paradigm of attached flow. Instead, the design creates a lift-enhancing controlled vortex separating from the leading edge, as seen also on modern fighters. Most of the work analyzes a given shape for aerodynamic performance. In our discussion of the CRM wing, we examine the minimum wave drag shape produced by mathematical optimization to learn how the optimizer changed the geometry.

Computational fluid dynamics (CFD) requires a computational mesh: a subdivision of the flow region into millions of computational “cells” as a basis for making a computationally feasible discrete mathematical representations of the governing partial differential equation (PDE). Tools are needed to make a computerized geometric model of the aircraft skin, possibly extended by details of propulsion – propeller disks, jet engine intakes, and exhausts. Once the airframe geometry is defined, its exterior volume must be subdivided into small cells – the computational grid or mesh – for the numerical solution of the PDE. This was a trivial task for the 1D nozzle problem, but grid generation for detailed configurations is very demanding of the engineer's time. This chapter presents relevant details of computational geometry applied to the representation and manipulation of aircraft surfaces. The tutorial Surface Modeler indicates how this is done with open-source software. Intended to be a fail-safe unsupervisedtool in algorithmic shape optimization, automated grid generation is currently becoming a reality. This chapter uses the Sumo and TetGen tools, which take a geometry description format, specialized for aircraft, into a high-quality grid for Euler CFD, demonstrated in hands-on tutorials with many examples of generated grids.

Aero-elasticity is concerned with the interaction of air flow with flexible structures. The load-carrying structures are designed to sustain the loads encountered during operation, but the wings will bend and twist under the airloads – the pressure forces acting over the surface of the vehicle. Depending on structural and flow characteristics, this leads to static (wing divergence, control reversal) and dynamic (flutter) effects that limit the airspeed. This chapter shows how the wing shape can be determined for a given flight condition. Computational fluid dynamics (CFD) delivers the airloads and a structural model provides the resulting deformations in the iterative process of the aero-elastic loop to find the deformation to the flight shape. The structural model is a finite-element discretization of a simple beam approximation of the real structure, and the transfer of forces from CFD and of deformations from the finite-element model is explained. The software is a loosely coupled, modular framework that illustrates the required data exchange. The loop is exemplified by low-speed static aero-elastic analyses of wing deformation, divergence, and control reversal in two case studies: a wing model in a wind tunnel undergoing divergence and control reversal; and the determination of the flight shape of a unmanned aerial vehicle in pull-up and push-down maneuvers.​

The aerodynamics of airplanes designed before and during World War II springs from linear potential theory (discussed in Chapter 3) together with empirical data and lessons learned from previous airplane designs. Jet engines and rocket propulsion enabled vehicles to fly much faster. This uncovered high-speed aerodynamic phenomena that must be understood for the successful design of airplanes capable of trans- and supersonic flight and of space vehicles. This chapter presents the numerical methods employed in computational fluid dynamics (CFD) to treat shock waves. A complete recipe for inviscid nozzle flow is given, with accompanying tutorial software. Mature tools are now standard in the form of industrial-strength CFD codes. A perusal of the user manual for any one of them shows many options and functions. A basic understanding of the theory is needed for the user to set up the code properly for the intended case. While not focusing on constructing such a CFD code, this chapter lays the foundations for training the reader to become an "informed user" of these codes by learning CFD "due diligence." It spells out CFD fundamentals such as constructing the numerical flux, artificial dissipation, approximate Riemann, high-resolution schemes. explicit and implicit time integration, and convergence to steady state.