This section presents a brief introduction to the elements of a CFD analysis. We introduce the governing fluid flow equations and explain the hierarchy of simpli- fied forms of the governing equations which are commonly used in aerodynamic design. The CFD process is then illustrated with an example two-dimensional simulation of the flow past an airfoil at transonic flow conditions.
9.3.1 Governing Equations
The fluid flow past an aerodynamic surface is governed by the Navier-Stokes equ- ations. For a Newtonian fluid, compressible viscous flows in two dimensions, without body forces, mass diffusion, finite-rate chemical reactions, heat conduc- tion, or external heat addition, the Navier-Stokes equations, can be written in Cartesian coordinates as [24]
=0
∂ +∂
∂ +∂
∂
∂
x x t
F E U
(9.4)
where U, E, and F are vectors given by
⎥⎥
⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎣
⎡
= Et
v u ρ ρ ρ
U
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢
⎣
⎡
−
− +
−
−
= +
xy xx t
xy xx
v u u p E
uv p u
u
τ τ τ ρ
τ ρ
ρ
) (
2
E
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢
⎢⎢
⎢
⎣
⎡
−
− +
− +
= −
yy xy t
yy xy
v u v p E
p v
uv v
τ τ
τ ρ
τ ρ
ρ
) ( F 2
(9.5)
Here, ρ is the fluid density, u and v are the x and y velocity components, respec- tively, p is the static pressure, Et = ρ (e+V2/2) is the total energy per unit volume, e is the internal energy per unit mass, V2/2 is the kinetic energy, and τ is the viscous shear stress tensor given by [24]
⎟⎟⎠⎞
⎜⎜⎝⎛
∂
−∂
∂
= ∂
y v x u
xx 2
3 2à τ
⎟⎟⎠⎞
⎜⎜⎝⎛
∂
−∂
∂
= ∂
x u y v
yy 2
3 2à τ
⎟⎟⎠⎞
⎜⎜⎝⎛
∂ +∂
∂
= ∂
x v y u
xy à
τ
(9.6) where à is the dynamic viscosity of the fluid.
The first row of Eq. (9.6) corresponds to the continuity equation, the second and third rows are the momentum equations, and the fourth row is the energy equation. These four scalar equations contain five unknowns, namely (ρ, p, e, u, v). An equation of state is needed to close the system of equations. For most prob- lems in gas dynamics, it is possible to assume a perfect gas, which is defined as a gas whose intermolecular forces are negligible. A perfect gas obeys the perfect gas equation of state [24]
RT p=ρ
(9.7) where R is the gas constant.
The governing equations are a set of coupled, highly nonlinear partial differen- tial equations. The numerical solution of these equations is quite challenging.
What complicates things even further is that all flows will become turbulent above a critical value of the Reynolds number Re = VL/υ, where V and L are representa- tive values of velocity and length scales and υ is the kinematical viscosity. Turbu- lent flows are characterized by the appearance of statistical fluctuations of all the variables (ρ, p, e, u, v) around mean values.
By making appropriate assumptions about the fluid flow, the governing equa- tions can be simplified and their numerical solution becomes computationally less expensive. In general, there are two approaches that differ in either neglecting the effects of viscosity or including them into the analysis. The hierarchy of the gov- erning flow equations depending on the assumptions made about the fluid flow situation is shown in Fig. 9.4.
Navier-Stoke Equations
Newtonian fluid, compressible, viscous, unsteady, heat-conducting
Direct Numerical Simulation (DNS) Euler Equations
Potential or Full Potential Equation
Laplace’s Equation
Prandtl-Glauert Equation Transonic Small-Disturbance Equation
Large-Eddy Simulation (LES)
Reynolds Equations (RANS)
Thin Layer N-S Equations
Boundary Layer Equations Inviscid flow assumption
Irrotational flow Weak shocks
Incompressible flow
Small disturbance approximation
DNS of large scale fluctuations Model small scales
Treat turbulence via Reynolds averaging and use a turbulence model
Restrict viscous effects to gradients normal to bodies (directional bias)
Prandtl boundary layer assumption (pressure constant across layer and
leading viscous term only)
Fig. 9.4 A hierarchy of the governing fluid flow equations with the associated assumptions and approximations
Direct Numerical Simulation (DNS) has as objective to simulate the whole range of the turbulent statistical fluctuations at all relevant physical scales. This is a for- midable challenge, which grows with increasing Reynolds number as the total computational effort for DNS simulations is proportional to Re3 for homogeneous turbulence [25]. Due to limitations of computational capabilities, DNS is not avail- able for typical engineering flows such as those encountered in airfoil design for typical aircraft and turbomachinery, i.e., with Reynolds numbers from 105 to 107.
Large-Eddy Simulation (LES) is of the same category as DNS, in that it com- putes directly the turbulent fluctuations in space and time, but only above a certain length scale. Below that scale, the turbulence is modeled by semi-empirical laws.
The total computational effort for LES simulations is proportional to Re9/4, which is significantly lower than for DNS [25]. However, it is still excessively high for large Reynolds number applications.
The Reynolds equations (also called the Reynolds-averaged Navier-Stokes eq- uations (RANS)) are obtained by time-averaging of a turbulent quantity into their
mean and fluctuating components. This means that turbulence is treated through turbulence models. As a result, a loss in accuracy is introduced since the available turbulence models are not universal. A widely used turbulence model for simula- tion of the flow past airfoils and wings is the Spalart-Allmaras one-equation turbu- lence model [43]. The model was developed for aerospace applications and is con- sidered to be accurate for attached wall-bounded flows and flows with mild separation and recirculation. However, the RANS approach retains the viscous ef- fects in the fluid flow, and, at the same time, significantly reduces the computa- tional effort since there is no need to resolve all the turbulent scales (as it is done in DNS and partially in LES). This approach is currently the most widely applied approximation in the CFD practice and can be applied to both low-speed, such as take-off and landing conditions of an aircraft, and high-speed design [25].
The inviscid flow assumption will lead to the Euler equations. These equations hold, in the absence of separation and other strong viscous effects, for any shape of the body, thick or thin, and at any angle of attack [44]. Shock waves appear in transonic flow where the flow goes from being supersonic to subsonic. Across the shock, there is almost a discontinuous increase in pressure, temperature, density, and entropy, but a decrease in Mach number (from supersonic to subsonic). The shock is termed weak if the change in pressure is small, and strong if the change in pressure is large. The entropy change is of third order in terms of shock strength.
If the shocks are weak, the entropy change across shocks is small, and the flow can be assumed to be isentropic. This, in turn, allows for the assumption of irrota- tional flow. Then, the Euler equations cascade to a single nonlinear partial differ- ential equation, called the full potential equation (FPE). In the case of a slender body at a small angle of attack, we can make the assumption of a small distur- bance. Then, the FPE becomes the transonic small-disturbance equation (TSDE).
These three different sets of equations, i.e., the Euler equations, FPE, and TSDE, represent a hierarchy of models for the analysis of inviscid, transonic flow past airfoils [44]. The Euler equations are exact, while FPE is an approximation (weak shocks) to those equations, and TSDE is a further approximation (thin airfoils at small angle of attack). These approaches can be applied effectively for high-speed design, such as the cruise design of transport aircraft wings [13, 14] and the design of turbomachinery blades [2].
There are numerous airfoil and wing models that are not typical CFD models, but they are nevertheless widely used in aerodynamic design. Examples of such methods include thin airfoil theory, lifting line theory (unswept wings), vortex lat- tice methods (wings), and panel methods (airfoils and wings). These methods are out of the scope of this chapter, but the interested reader is directed to [45] and [46] for the details. In the following section, we describe the elements of a typical CFD simulation of the RANS or Euler equations.
9.3.2 Numerical Modeling
In general, a single CFD simulation is composed of four steps, as shown in Fig. 9.5: the geometry generation, meshing of the solution domain, numerical so- lution of the governing fluid flow equations, and post-processing of the flow
results, which involves, in the case of numerical optimization, calculating the objectives and constraints. We discuss each step of the CFD process and illustrate it by giving an example two-dimensional simulation of the flow past the NACA 2412 airfoil at transonic flow conditions.
9.3.2.1 Geometry
Several methods are available for describing the airfoil shape numerically, each with its own benefits and drawbacks. In general, these methods are based on two different approaches, either the airfoil shape itself is parameterized, or, given an initial airfoil shape, the shape deformation is parameterized.
Evaluate objective(s) and constraints
Flow solution Generate grid Generate geometry
Fig. 9.5 Elements of a single CFD simulation in numerical airfoil shape optimization The shape deformation approach is usually performed in two steps. First, the surface of the airfoil is deformed by adding values computed from certain functions to the upper and lower sides of the surfaces. Several different types of functions can be considered, such as the Hicks-Henne bump functions [1], or the transformed cosine functions [47]. After deforming the airfoil surface, the compu- tational grid needs to be regenerated. Either the whole grid is regenerated based on the airfoil shape deformation, or the grid is deformed locally, accounting for the airfoil shape deformation. The latter is computationally more efficient. An exam- ple grid deformation method is the volume spline method [47]. In some cases, the first step described here above is skipped, and the grid points on the airfoil surface are used directly for the shape deformation [14].
Numerous airfoil shape parameterization methods have been developed. The earliest development of parameterized airfoil sections was performed by the Na- tional Advisory Committee for Aeronautics (NACA) in the 1930’s [48]. Their de- velopment was derived from wind tunnel experiments, and, therefore, the shapes generated by this method are limited to those investigations. However, only three parameters are required to describe their shape. Nowadays, the most widely used airfoil shape parameterization methods are the Non-Uniform Rational B-Spline