# 50 Introduction to Advanced Computational Methods

# Introduction

Advanced computational methods continue to be developed to analyze challenging problems in aerospace engineering. Such methods include aerodynamics, structural analysis, and the coupling of disciplines such as with aeroelasticity. In particular, computational fluid dynamics, usually called “CFD,” and the finite element method, or “FEM,” have been revolutionary computational tools for predicting aerodynamics and structural loads, respectively. The intention of this chapter is not to review the details of such techniques and their associated numerical algorithms *per se,* but to give an *introduction and general overview* of the foundational principles and how they are used, along with a summary of their capabilities and some of their limitations. Nor is it meant to be an all-encompassing treatment.

Traditionally, the aircraft industry has relied on quicker and more economical computational methods for designing aircraft and predicting their flight performance, and successfully too. One does not need CFD or the FEM to design and make a flight vehicle work in the way it was intended; history speaks for itself. However, both tools can help significantly refine the aircraft design to extract its maximum performance at the lowest cost, improving safety and reliability. Knowing why it works (or doesn’t) is one key to commercial success and continued advancements. Wind tunnel testing has always been a critical part of this design process, and will continue to be essential for CFD validation. Advances in computer technology, such as code execution speed, memory and data storage access speed, and the availability of inexpensive mass data storage systems, continue to encourage the development of more ambitious numerical methods that can be used during design.

Much new fundamental research work in modeling and algorithmic development has occurred over the last two decades. Aerodynamics and structural problems, once relegated to the fastest available supercomputers, can now be solved using local computational resources. For example, most universities today have powerful computer clusters, allowing for research in a broad range of technical fields. As shown in the image below, the ability to predict flows about complete aircraft can open up opportunities in new design directions, such as the reconsideration of supersonic transports (SSTs). It has also inspired broader innovations in flight vehicle designs, allowing concepts outside the “domain of conventional wisdom” to be investigated confidently. Thinking “out of the box” can now be quantified for the right reasons!

Continued advances in computer capabilities, which continue to accelerate,^{[1]} has also played a large part in motivating and accelerating the development of more* integrated computational techniques*, encompassing several traditionally separate engineering disciplines. For example, the governing equations and solution methods used in one field are rarely the same as those used in another, so some form of coupling of methods is required. Increasingly tighter mathematical, algorithmic, and numerical coupling have allowed, for example, between aerodynamics and structural dynamics or between aerodynamics and flight mechanics, to be solved simultaneously. If fully realized, this integration strategy will eventually predict the behavior of an entire flight vehicle in an arbitrary flight condition, and so allow more capable and efficient aircraft to be designed. However, significant integration problems and more ambitious numerical techniques are needed to push forward this technology before they can be used to fully predict aircraft characteristics before their first flights.

Learning Objectives

- Understand the foundational elements of modern computational aerodynamic methods starting from the governing Navier-Stokes equations.
- Appreciate the hierarchy of such aerodynamic methods in terms of their relative predictive advantages and disadvantages.
- Know about surface singularity methods and their use in aerodynamic prediction.
- Understand why boundary layers significantly affect aerodynamic behavior and why their modeling is essential.
- Appreciate the fundamentals of the structural finite element method and its uses.
- Learn about the field of aeroelasticity and why it is essential in the design of flight vehicles.

# Aerodynamics

Over the past three decades, advances have been made toward understanding challenging problems in aerodynamics by using CFD. Numerical methods, such as finite-difference methods (FDM) or finite-volume methods (FVM), applied to the governing flow equations (e.g., Navier-Stokes), are used to model the flow field about an aircraft. The choice of which governing equations to use affects the level of physics that can be approximated by the CFD scheme, as well as the computational effort and time taken to solve it. Not all numerical methods are created equally, and the choice of method affects the solution’s accuracy, stability, and cost. The definition of “cost” comes down to how much time^{[2]} and effort goes into obtaining a solution, and most CFD methods are considered costly by any design standard.

Developing comprehensive CFD methods is usually considered the ultimate in aerodynamic modeling. So far, a wide range of numerical techniques have been used to solve a hierarchy of equations governing the flow, ranging from irrotational and inviscid potential flow approximations to the Euler and Navier-Stokes equations; the latter set of equations can resolve the effects of compressibility, thermodynamics, and turbulence. Each level in the hierarchy of governing equations increases physical modeling capability, but also requires more computational effort and cost for the solution.

A key issue in using these numerical methods is the definition and creation of appropriate computational grids over which to solve the governing equations, requiring considerable skill and effort. Other problems include the difficulties in modeling flow turbulence, for which no single method exists. Furthermore, CFD methods must be validated against measurements to realize the high predictive confidence levels needed for aircraft design work. Commercially available tools such as ANSYS Meshing and Pointwise can aid in grid generation, while CFD solvers like ANSYS Fluent and OpenFOAM can run the actual CFD simulations. However, there are many other options for grid generation and CFD solvers.

## History

The earliest numerical solutions to viscous flow problems about bodies go back almost a century. Alexander Thom, a Carnegie Fellow at the University of Glasgow, published a paper in the Proceedings of the Royal Society in 1933 titled: “The Flow Past Circular Cylinders at Low Speeds.”^{[3]} In this work, the viscous flow was solved over a numerically defined grid by “arithmetical” means. Back then, there were no digital computers, so the calculations were performed “by hand,” i.e., using pencil, paper, slide rule, and perhaps a mechanical calculator. Thom remarked in his paper that the “whole process is long,” but his computed results compared favorably with flow experiments.

The advent of digital computers in the 1950s provided the computational power to begin to tackle numerical solutions to the governing equations of fluid dynamics. Early work by pioneers such as John von Neumann and his collaborators, laid the groundwork for numerical methods to solve partial differential equations. Von Neumann is often called the “Father of Modern Computer Architectures.” During the 1960s, CFD began to evolve more rapidly in parallel with advances in computer technologies. Developing algorithms based on finite difference, finite element, and finite volume methods enabled more accurate and efficient solutions of the various forms of the governing equations of fluid dynamics. Finite-difference techniques were already well established by then, starting from Brook Taylor^{[4]}in the early 1700s, with other contributions by Leonhard Euler and Carl Runge. There was also much new work in numerical methods, and the accuracy and stability of such schemes were increasingly important for solving aerospace problems.

The publication of many technical papers and essential textbooks during the 1960s, helped to disseminate these numerical methods and apply them to fluid flow problems. Textbooks by Brian Spalding^{[5]} and Suhas V. Patankar^{[6]} were used to teach CFD to students as part of courses in fluids. NASA and other aerospace laboratories worldwide played a crucial role in advancing CFD for practical applications in aerospace engineering. In 1987, NASA founded the Advanced Supercomputing (NAS) Division at NASA Ames. In addition, the U.S. Army Aeronautical Research Laboratory played an essential role in advancing the CFD field. The first commercial CFD software packages also emerged during this period, making the technology accessible to more engineers and researchers.

The 1980s and 1990s saw substantial improvements in numerical techniques, computational power, digital memory, and storage devices. Desktop computers were still somewhat limited in their capabilities, and most serious computational work was done on “mainframe” computers. Remotely accessible supercomputers, such as the CRAY, became available, and desktop computers also became increasingly capable, allowing for more complex CFD simulations at a local level. Advances in turbulence models, grid generation techniques, and numerical methods continued to contribute to more accurate and robust CFD solutions. During this period, CFD also started to find applications beyond the aerospace field, including automotive, chemical, and civil engineering.

CFD has continued to advance from the early 2000s until today, with the development of more sophisticated algorithms, increased computational power, and the advent of parallel computing. High-performance computing (HPC) and graphics processing units (GPUs) have become commonplace, and have enabled the simulation of more complex and larger-scale aerospace problems. Additionally, CFD has benefited from advancements in data visualization techniques, making it easier for engineers to interpret and analyze large quantities of data and understand complex flows. Today, CFD is an essential tool in engineering analysis, with applications ranging from designing flight vehicles to weather prediction, biomedical engineering, and environmental studies. The continuous development of software, increased accessibility of computational resources, and ongoing research mean that CFD will remain at the forefront of future scientific and engineering innovation.

## Navier-Stokes Equations

Most aerodynamic problems with flight vehicles involve viscous effects and turbulence. These effects are significant in the turbulent boundary layers found on the wings and airframe, as well as with a host of component interaction phenomena, such as wing/fuselage interference, powerplant/wing interference, etc. Modeling flow separation is also essential, as the onset of which can set boundaries for an aircraft’s operational flight envelope. The Navier-Stokes equations are the fundamental equations that govern fluid dynamic behavior. They encompass the principles of conserving mass, momentum, and energy and their interchange within a fluid.

The Navier-Stokes equations are a set of coupled, highly nonlinear partial differential equations. This means they can only be solved by making simplifying assumptions, which in some cases may be rather sweeping. The most common practical simplification is to avoid treating turbulence and the associated viscous shearing effects by assuming a laminar flow. While this approach simplifies the Navier-Stokes equations, it should be noted that the flows will be turbulent even at the lowest Reynolds numbers found on aircraft.

## Tensor Form

The tensor form is often used to express the Navier-Stokes equations. The conservation of mass (continuity equation) is given by

(1)

(2)

where is the flow density and , , is the velocity vector. For an incompressible flow, where = 0, then

(3)

Conservation of momentum (momentum equation) gives

(4)

and in tensor form by

(5)

where is the pressure, is the *stress tensor*, and is a body force per unit mass, e.g., acceleration under gravity.

**Modeling Viscous Effects**

The Navier-Stokes equations are general and apply to all types of flows, including flows with turbulence. The flow about an aircraft, at least at lower Mach numbers, can be readily approximated as an isotropic Newtonian fluid, for which the relationship between stress and strain in the Navier-Stokes equations is given by

(6)

where is the Kronecker delta function, i.e., for , and otherwise, and is the coefficient of dynamic viscosity. The parameter is called the second coefficient of viscosity or *bulk viscosity*, usually given as , which is referred to as Stokes’s hypothesis.^{[7]} The bulk viscosity, which can be measured, as can , is essential in describing the compression and energy dissipation in fluids, both gasses and liquids, such as in sound wave propagation, shock waves, and other phenomena.

Notice from either Eq. 1 2, for an incompressible flow where , then the term involving vanishes, so the stress tensor depends only on the dynamic viscosity, . For a compressible flow, however, the term represents the rate at which the fluid’s density changes under compression or expansion. Therefore, the term in the stress tensor accounts for the viscous stresses arising from these volume changes.

Rounding out the conservation laws is the conservation of energy (energy equation), which can be expressed as

(7)

and in tensor notation

(8)

where is the internal energy per unit mass, is temperature, is the bulk thermal conductivity, and is the external heat per unit mass. Finally, the ideal gas equation (equation of state) can be used to relate the thermodynamic quantities, i.e.,

(9)

where is the gas constant. Recall that the equation of state is one of the few truly “handy” equations in aerodynamics.

## Conservation Form

For practical use, the Navier-Stokes equations are usually rewritten more compactly in the so-called *conservation form*, compared to the tensor differential form given previously, i.e.,

(10)

where is the conserved variable vector and , , and are the flux vectors. The flux vectors express the rate at which mass, momentum, and energy are being transported at any point in the flow. This latter equation is often expanded into a form with the viscous terms on the right-hand side to give

(11)

where , , and are the fluxes resulting from the viscosity of the flow.

This form of the governing equations allows a mathematical reduction to a simpler form called the t*hin-layer Navier-Stokes equations*, where the viscous derivatives with respect to the and directions are neglected to give

(12)

for a suitably defined viscous flux, , that accounts for the neglected derivatives.

The thin-layer Navier-Stokes equations are particularly useful in scenarios where the flow can be approximated as two-dimensional or quasi-two-dimensional, simplifying computational efforts and modeling complexities. Engineers often apply these simplified equations in boundary layer analyses, fluid film lubrication, and other situations where the flow dynamics are primarily in the and directions.

## Reynolds-Averaged Navier-Stokes (RANS) Form

The Reynolds-Averaged Navier-Stokes (RANS) equations are a form of the Navier-Stokes equations to describe the flow of incompressible fluids. These equations are particularly useful in the study of turbulent flows. The key idea behind the RANS equations is to decompose the instantaneous quantities (such as velocity and pressure) into their mean (time-averaged) and fluctuating components.

The Navier-Stokes equations for an incompressible fluid can be written as

(13)

and

(14)

where represents the external body forces per unit mass.

In the Reynolds decomposition, the instantaneous velocity and pressure are decomposed into mean and fluctuating components, e.g., for the component of velocity, then

(15)

and for the static pressure

(16)

Here, and are the mean (time-averaged) components, while and are the fluctuating components, as illustrated in the figure below.

When the Navier-Stokes equations are averaged over time, the nonlinear term introduces additional terms that account for the Reynolds stresses, which are the stresses created by the turbulent fluctuations. The averaged form of the continuity and momentum equations then become

(17)

and

(18)

The term represents the divergence of the Reynolds stress tensor, which arises from the turbulent fluctuations. This term is often modeled using various turbulence models, which will be discussed later, because it cannot be directly calculated from the mean flow variables.

The RANS equations provide a CFD method that balances computational cost and accuracy well, making them a popular choice for many practical fluid dynamics problems. RANS methods are widely used in engineering and scientific applications to predict and analyze turbulent flows in aircraft and automobile design, ship hull design, water flow in pipes, weather forecasting, and environmental engineering, including pollutant dispersion. Several CFD software packages are available to solve the RANS equations, including ANSYS Fluent, OpenFOAM, COMSOL Multiphysics, and STAR-CCM+.

## Griding of the Computational Domain

There have been many algorithmic developments in the numerical solution of the Navier-Stokes equations. A fundamental issue in any flow problem about a body, is the generation of the grid structure representing the body’s geometry on which to solve the governing equations. This means that the size of the computational domain should be as small as possible for economy, yet large enough to ensure that any nonphysical issues within the computational domain are avoided, making computer memory requirements very large.

Several types of grid systems are appropriate for analyzing flow problems; a summary is shown in the figure below. Grids are usually categorized according to their geometric shape, e.g., O-grids, C-grids, and H-type grids. The O-grids and C-grids are often used to analyze the flows about wings as they can conform to the curved airfoil shape of the surface, which are called *body-fitted grids*, as shown in the figure below. H-type grids are also used; they allow a compromise between the advantages of a Cartesian grid geometry and the ability to refine the grid near a surface to resolve the boundary layer. Nodes are the intersection points on the grid, and the nodes make up a cell. The numerical solution to the flow properties is then based on finite-difference or finite-volume solutions applied over the nodes and cells.

- Moore's Law, observed by Gordon E. Moore in 1965, states that the number of transistors on a microchip doubles approximately every two years, leading to exponential increases in computing power and cost reductions. ↵
- Time = money. In this regard, wall clock time = money plus compute time = more money. ↵
- Published September 1933 ↵
- Of "Taylor series" fame. ↵
- "Numerical Solution of Convection-Diffusion Problems" by D. Brian Spalding (1972) ↵
- "Numerical Heat Transfer and Fluid Flow" by Suhas V. Patankar (1980) ↵
- Stokes assumed that , although later research suggests it is not always accurate. ↵
- By definition, this is where the local velocity approaches 99% of the external flow (or edge) velocity . ↵
- Jameson, A., "Transonic Potential Flow Calculation using Surface Transpiration Boundary Conditions," AIAA Journal, 13(7), 1975, pp. 826–833. ↵
- Theodore Theodorsen, "General Theory of Aerodynamic Instability and the Mechanism of Flutter," NACA-TR-496 (1949). ↵
- Meaning the the aerodynamics adjust instantaneously to changes in pitch angle with no time lag. ↵