In this chapter, we review the governing equations of fluid mechanics that describe flow fields in mathematical terms. Modern computational aerodynamic methods are based on the particular governing equations that they solve. Since computational aerodynamics is now a necessity, not a luxury, for many aspects of configuration aerodynamics, it is crucially important for aerodynamicists to have a good understanding and appreciation of the underlying assumptions and approximations behind an equation before using its associated computational aerodynamic method. We will see that developments in computational methods have resulted in a slightly different approach to fundamental conservation statements compared with presentations before the digital computing age. This chapter also defines the nomenclature and terminology used in current aerodynamics practice. If you are fully conversant with the derivation and characteristics of the governing equations of fluid mechanics, you might consider skipping the remainder of this chapter and going straight to Chapter 3. 

The presentation of material in this chapter presumes that the reader has previously had a course in fluid mechanics or aerodynamics. Many excellent discussions of the foundations of fluid mechanics for aerodynamics application are available in books by Karamcheti,^[1] Bertin,^[2] Anderson,^[3] and Moran.^[4] The best formal derivation of the equations was assembled by Grossman.^[5] Indeed, an entire book has been written on the equations of fluid mechanics by Hoffman et al.^[6]

2.1 Governing Equations of Fluid Mechanics

Fluid is assumed to be a continuum. For virtually all aerodynamics work, this is a valid assumption. One case where this may not be true is when fluid density is so low that the actual molecular motion must be analyzed, which is the purview of the field of rarefied gas dynamics. This may be important for vehicles in hypersonic flight. Aeroassisted orbital transfer vehicles (AOTV’s) are examples of the vehicles requiring noncontinuum flow field analysis.

The fluid is defined by an equation of state and by the thermodynamic and transport properties, namely, the ratio of specific heats, γ, viscosity, μ, and the coefficient of heat conduction, k. Governing equations and boundary conditions control the motion of the fluid. The governing equations are given by conservation laws listed in table 2-1.

Table 2-1: Conservation laws for governing equations

Coordinate systems are also important in aerodynamics. The general equations of fluid motion are independent of the coordinate system. However, simplifying assumptions frequently introduce directional bias into approximate forms of the equations; they must be used with a specific coordinate system orientation relative to the flow field.

Traditionally, Cartesian coordinates were used to describe vehicle geometry. In this chapter, we will work entirely in the Cartesian coordinate system. It is frequently desirable to make calculations in non-Cartesian coordinate systems that are distorted to fit a particular shape. In general, these are non-orthogonal curvilinear coordinates. Even when using Cartesian coordinates, the x-, y-, and z-coordinates are oriented differently depending on whether the flow is two- or three-dimensional. Figure 2-1 shows the usual two-dimensional coordinate system.

The standard coordinate system used in three dimensions is illustrated in figure 2-2. Note the different orientation of the y coordinate in figures 2-1 and 2-2. In three dimensions, x is chordwise, much like it is in 2D, but y is spanwise instead of being normal to the chord. The right-hand rule determines the direction of the z coordinate. Figure 2-2 also shows the components of the freestream velocity vector, V_∞, in x, y, and z directions. The wing is flying at an angle of attack of α and a sideslip angle of β. The unit vectors in x, y, and z directions are denoted by i, j, and k, respectively.

In Cartesian coordinates, the independent variables are x, y, z, and t. Here t denotes time. We want to know the velocities (u, v, w) and the fluid properties (p, ρ, T). (Note that u, v, and w are the components of the flow velocity in the x, y, and z directions, respectively.) These six unknowns require six equations which are shown in table 2-2.

Table 2-2: Six equations of fluid mechanics

Assumptions frequently reduce the number of equations required; for example, incompressible, inviscid, and irrotational flow can be described by a single equation, as shown below in section 2.5. Prior to the 1980s, almost all aerodynamics work used a single partial differential equation, possibly coupled with another equation. An example of this approach is the calculation of potential flow for the inviscid portion of the flow field, which is used in combination with the boundary layer equations to compute the flow field where an estimate of the viscous effects is required.

2.2 Derivation of Governing Equations

Governing equations of fluid motion are mathematical models based on the conservation laws (for mass, momentum, and energy) and the fluid properties as listed in section 2.1. For most of the practical aerodynamic applications, the governing equations are numerically solved to determine the density, velocity, pressure, and temperature distributions. To develop the governing equations themselves, we can use two approaches, Lagrangian and Eulerian.

1. Lagrangian: In this approach, we observe the trajectory of an individual fluid particle as it moves in time through the space surrounding a body. The space itself is defined by a coordinate system fixed to the body, commonly referred as a body-fixed coordinate system. Figures 2-1 and 2-2 show body-fixed coordinates systems in two and three dimensions, respectively. Note that the motion of every fluid particle is described in terms of a “fixed reference configuration” which could be (but isn’t necessarily) the position in space occupied by each fluid particle at an initial time, t = 0. The Lagrangian approach may be considered as the conventional application of Newton’s Second Law to each fluid particle. Even in steady flow, the forces encountered by the particle are a function of the time history of its motion.
2. Eulerian: In contrast to the Lagrangian approach, the Eulerian approach considers the entire space around the body as a field, also called a flow field, and determines flow properties such as velocity and pressure at a given point in the field through which the fluid particles move. The space itself is defined by a body-fixed coordinate system. We essentially ignore the motion of individual fluid particles and instead consider the distribution of velocity and pressure throughout the space around a body. This approach imitates wind tunnels, in which fluid particles flow past a body and flow properties are measured at various points in the space surrounding the body.

It is worth noting that the Lagrangian and Eulerian approaches describe the same physical phenomena. Therefore, they are mathematically related. Karamcheti^[7] provides a particularly good explanation of the underlying ideas in his Chapters 4 through 7.

The Eulerian approach is almost universally used to develop the governing equations of fluid motion. The use of this approach requires careful attention in the application of the conservation concepts and of Newton’s Second Law in particular. Computational aerodynamic methods have been developed to numerically solve the resulting equations. Newton’s Law governs the motion of every fluid particle. To apply this law in the Eulerian approach, we define a control volume that maintains a fixed location relative to the coordinate system but through which fluid particles traverse. The connection between the rate of change of the properties of a fluid particle (e.g., velocity, density, pressure) and the rate of change of fluid properties flowing through the control volume* requires special consideration. The substantial derivative, discussed in section 2.2.2, is employed to define the rate of change of the properties of the fluid particle as it moves through the flow field relative to the fixed coordinate system. An integral approach to the description of the change of properties of a fluid particle relative to the fixed coordinate system is available through the use of the Reynolds transport theorem, which is described in Owczarek^[8] and section 1.2 of Grossman.^[9]

The conservation equations of fluid motion can be expressed in either a differential or integral viewpoint. The differential form of the equations is found in most aerodynamics books. However, the integral form is preferred for numerically solving the equations since integrals are more accurately computed than derivatives. The integral form is also better suited for simulating discontinuities (shocks) in the flow field than the differential form, which assumes fluid properties to be continuous. We will use aspects of each approach in this book.

2.2.1 Conservation of Mass: The Continuity Equation

In this section, we derive the continuity equation from a control volume viewpoint (in 2D), and then we look at the equivalent integral statement and the use of the Gauss divergence theorem to establish the connection. Other derivations are given by Moran (sections 2.2, 2.3, 2.4),^[10] Anderson (Chapters 2 and 6),^[11] and Bertin and Smith (Chapter 2).^[12]

From a control volume viewpoint, the statement of conservation of mass is in words simply:

Net outflow of mass through the surface surrounding the volume = decrease of mass within the volume.

To translate this statement into a mathematical form, consider the control volume given in figure 2-3. Here, u is the velocity in the x-direction, v is the velocity in the y-direction, and ρ is the density. The following represents the net mass flow rate, or flux,* out of the volume, as illustrated in figure 2-3:

$$[\mathrm{X}-\text{out}]-[\mathrm{X}-\mathrm{i}\mathrm{n}]+[\mathrm{Y}-\text{out}]-[\mathrm{Y}-\mathrm{i}\mathrm{n}]=\text{change of mass (decrease)}=-\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}\mathrm{\Delta }X\mathrm{\Delta }Y\text{(2-1)}$$

We use a Taylor series expansion of the mass fluxes into the volume around the origin of the volume. The flux per unit length through the surface is multiplied by the length of the surface.

$$\begin{array}{c}[\mathrm{X}-\text{out}]=[\rho u+\frac{\partial \rho u}{\partial x}\cdot \frac{\mathrm{\Delta }X}{2}]\mathrm{\Delta }Y\\ [\mathrm{X}-\mathrm{in}]=[\rho u-\frac{\partial \rho u}{\partial x}\cdot \frac{\mathrm{\Delta }X}{2}]\mathrm{\Delta }Y\\ [\mathrm{Y}-\mathrm{out}]=[\rho v+\frac{\partial \rho v}{\partial y}\cdot \frac{\mathrm{\Delta }Y}{2}]\mathrm{\Delta }X\\ [\mathrm{Y}-\mathrm{in}]=[\rho v-\frac{\partial \rho v}{\partial y}\cdot \frac{\mathrm{\Delta }Y}{2}]\mathrm{\Delta }X\end{array}\text{(2-2)}$$

Substituting these terms in equation (2-1), we arrive at equation (2-3) below.

$$\begin{array}{c}[\rho u+\frac{\partial \rho u}{\partial x}\cdot \frac{\mathrm{\Delta }X}{2}]\mathrm{\Delta }Y-[\rho u-\frac{\partial \rho u}{\partial x}\cdot \frac{\mathrm{\Delta }X}{2}]\mathrm{\Delta }Y\\ +[\rho v+\frac{\partial \rho v}{\partial y}\cdot \frac{\mathrm{\Delta }Y}{2}]\mathrm{\Delta }X-[\rho v-\frac{\partial \rho v}{\partial y}\cdot \frac{\mathrm{\Delta }Y}{2}]\mathrm{\Delta }X=-\frac{\partial \rho }{\partial t}\mathrm{\Delta }X\mathrm{\Delta }Y\end{array}\text{(2-3)}$$ $$$$ $$$$

Summing up and canceling ∆X∆Y, we obtain

$$\frac{\mathrm{\partial }\rho u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }\rho v}{\mathrm{\partial }y}=-\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}\text{(2-4)}$$

or, in three dimensions,

$$\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\frac{\mathrm{\partial }\rho u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }\rho v}{\mathrm{\partial }y}+\frac{\mathrm{\partial }\rho w}{\mathrm{\partial }z}=0.\text{(2-5)}$$

This is the differential form of the continuity equation. The more general vector form of the equation is

$$\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot (\rho \mathbf{V})=0\text{(2-6)}$$

where V = ui + vj + wk with i, j, and k denoting unit vectors in x-, y-, and z-coordinate directions, respectively.

Alternately, consider the arbitrary control volume shown in figure 2-4. The conservation of mass can then be written in an integral form quite simply. The surface integral of the flow out of the volume Ω with surface area S simply equals the change of mass in the volume as given in equation (2-7):

$$\text{∯}\rho \mathbf{V}\cdot \mathbf{n}dS=-\frac{\partial }{\partial t}\iiint \rho \mathrm{d\Omega }\text{(2-7)}$$

This is true without making any assumption requiring continuous variables and differentiability. It holds for all flows—viscous or inviscid, compressible or incompressible. To relate this expression to the differential form, we make use of the Gauss divergence theorem, which assumes continuous partial derivatives. It is given by

$$\text{∯}\mathbf{A}\cdot \mathbf{n}dS=\iiint \mathrm{\nabla }\cdot \mathbf{A}\mathrm{d\Omega .}\text{(2-8)}$$

The equivalent statement for a scalar is

$$\text{∯}\varphi \mathbf{n}dS=\iiint \mathrm{grad}\varphi \mathrm{d\Omega .}\text{(2-9)}$$

Using the divergence theorem, equation (2-8), the differential and integral forms can be shown to be the same. First, rewrite the surface integral in the conservation of mass, equation (2-7), as

$$\text{∯}\rho \mathbf{V}\cdot \mathbf{n}dS=\iiint \mathrm{\nabla }\cdot (\rho \mathbf{V})\mathrm{d\Omega .}\text{(2-10)}$$

Substituting it in equation (2-7), the continuity equation in integral form is written as equation (2-11).

$$\iiint \mathrm{\nabla }\cdot (\rho \mathbf{V})\mathrm{d\Omega }=-\frac{\mathrm{\partial }}{\mathrm{\partial }t}\iiint \rho \mathrm{d\Omega }\text{(2-11)}$$

Since Ω refers to a fixed volume, we can move ∂/∂t inside the integral and rewrite equation (2-11) as

$$\iiint [\mathrm{\nabla }\cdot (\rho \mathbf{V})+\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}]\mathrm{d\Omega }=0.\text{(2-12)}$$

For this to be true in general, the integrand must be zero, which is just the differential form! Further discussion and other derivations are available in Moran (sections 2.2, 2.3, and 2.4),^[13] Anderson (section 2.6),^[14] and Bertin and Smith (Chapter 2).^[15]

2.2.2 Conservation of Momentum and the Substantial Derivative

In this section, we derive the general equations for the conservation of momentum. This is a statement of Newton’s Second Law: The time rate of change of momentum of a body equals the net force exerted on it. For a fixed mass, this is the famous equation

$$\mathbf{F}=m\mathbf{a}=m\frac{D\mathbf{V}}{Dt}\text{.}\text{(2-13)}$$

Notice a new term, D/Dt, called the substantial derivative, on the right-hand side of equation (2-13). It is the consequence of the need to apply Newton’s Law to a moving fluid element (the “body” in the Second Law statement given above) in our fixed coordinate system, which introduces some extra complications. We discuss the substantial derivative next.

2.2.2.1 Substantial Derivative

Let us look at what D/Dt means in a body-fixed coordinate system. Figure 2-5 shows a moving fluid particle at two instances of time.^[16] In this figure, Q(r,t) represent any fluid property associated with the fluid particle at position r at time t; Q could be density or pressure or velocity. The change in the property Q between the position r at time t and r+∆s at t+∆t is

$$\mathrm{\Delta }Q=Q(\mathbf{r}+\mathrm{\Delta }s,t+\mathrm{\Delta }t)-Q(\mathbf{r},t).\text{(2-14)}$$

The space change ∆s is simply equal to V∆t. Thus we can write

$$\mathrm{\Delta }Q=Q(\mathbf{r}+\mathbf{V}\mathrm{\Delta }t,t+\mathrm{\Delta }t)-Q(\mathbf{r},t).\text{(2-15)}$$

Equation (2-15) is in a form that can be used to find the rate of change of Q:

$$\frac{DQ}{Dt}=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}=\underset{\mathrm{\Delta }t\to 0}{lim}\frac{Q(\mathbf{r}+\mathbf{V}\mathrm{\Delta }t,t+\mathrm{\Delta }t)-Q(\mathbf{r},t)}{\mathrm{\Delta }t}\text{(2-16)}$$

Note that the rate of change is in two parts, one for a change in time and the other for a change in position. Thus we write the change of Q as a function of both time and space using the Taylor series expansion as

$$Q(\mathbf{r}+\mathbf{V}\mathrm{\Delta }t,t+\mathrm{\Delta }t)=Q(\mathbf{r},t)+{\frac{\mathrm{\partial }Q}{\mathrm{\partial }t}|}_{\mathbf{r},t}\mathrm{\Delta }t+\dots +{\frac{\mathrm{\partial }Q}{\mathrm{\partial }s}|}_{\mathbf{r},t}V\mathrm{\Delta }t+\dots \text{(2-17)}$$

where the direction of s is understood from figure 2-5. Substituting into equation (2-16) and taking the limit, we obtain equation (2-18) below.

$$\underset{\mathrm{\Delta }t\to 0}{lim}\frac{\mathrm{\Delta }Q}{\mathrm{\Delta }t}=\frac{\mathrm{\partial }Q}{\mathrm{\partial }t}+\frac{\mathrm{\partial }Q}{\mathrm{\partial }s}V\text{(2-18)}$$

The noteworthy outcome of applying Newton’s Law for a moving particle in a stationary coordinate system is that the rate of change of Q at a point in space is the sum of two terms shown on the right-hand side of equation (2-18). The first term, local derivative, represents the temporal change or the instantaneous rate of change of Q at its location. The second term, convective derivative, represents the rate at which Q is convected to that location by the fluid motion. This term has the unknown velocity, V, multiplying a term containing the unknown Q. This is important as it introduces a fundamental nonlinearity into the system, as shown next.

We now put this result into a specific coordinate system:

$$\frac{\mathrm{\partial }Q}{\mathrm{\partial }s}={\mathbf{e}}_V\cdot \mathrm{\nabla }Q\text{(2-19)}$$

where e_V denotes the unit vector in the direction of V. Thus, V = Ve_V and

$$\frac{\mathrm{\partial }Q}{\mathrm{\partial }s}V=\mathbf{V}\cdot \mathrm{\nabla }\mathrm{Q.}\text{(2-20)}$$

Thus, we use equations (2-18) and (2-20) to write the substantial derivative, equation (2-16), as

$$\frac{D}{Dt}=\frac{\mathrm{\partial }}{\mathrm{\partial }t}+ \; (\mathbf{V}\cdot \mathrm{\nabla })\text{(2-21)}$$

which can be applied to either a scalar as

$$\frac{DQ}{Dt}=\frac{\mathrm{\partial }Q}{\mathrm{\partial }t}+(\mathbf{V}\cdot \mathrm{\nabla })Q\text{(2-22)}$$

or to a vector quantity as

$$\frac{D\mathbf{V}}{Dt}=\frac{\mathrm{\partial }\mathbf{V}}{\mathrm{\partial }t}+(\mathbf{V}\cdot \mathrm{\nabla })\mathbf{V.}\text{(2-23)}$$

In Cartesian coordinates, u, v, and w, are the components of V in the x, y, and z directions, respectively. The substantial derivative becomes equation (2-24).

$$\begin{array}{r}\frac{Du}{Dt}=\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\\ \frac{Dv}{Dt}=\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\\ \frac{Dw}{Dt}=\frac{\partial w}{\partial t}+u\frac{\partial w}{\partial x}+v\frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\end{array}\text{(2-24)}$$

To solve equations containing these nonlinear terms, we generally have to employ numerical methods. There are only a very few special cases where you can obtain analytic solutions to the nonlinear equations explicitly.

2.2.2.2 Forces

Next we need to find the net forces acting on the fluid element. The forces are:

• Body forces
• Pressure forces
• Shear forces

Each of these forces applies to the control volume. Surface forces are shown in figure 2-6, with τ as a commonly used symbol for stresses. [Note that the orientation of axes in figure 2-6 is different from that in figure 2-3. However, both systems follow the right-hand rule. Therefore, the conclusions are equally valid in both systems.] In the figure, the first subscript for τ indicates the direction normal to the surface, and the second subscript defines the direction in which the force acts. Fluids of interest in aerodynamics are isotropic (i.e., properties are uniform in all directions). Therefore, the following expressions hold to satisfy equilibrium of moments about each axis:

$$\begin{array}{rl}& {\tau }_{xy}={\tau }_{yx}\\ & {\tau }_{xz}={\tau }_{zx}\\ & {\tau }_{yz}={\tau }_{zy}\end{array}\text{(2-25)}$$

The connection between pressure and stress is defined more specifically when the properties of a fluid are prescribed. Figure 2-7 shows the details of the forces, with the surface forces expanded about the center of the control volume using a Taylor series and the body force per unit mass denoted by f acting at the center.

Considering the x-direction as an example and using the Taylor series expansion shown in figure 2-7, the net forces are found in a manner exactly analogous to the approach used in the derivation of the continuity equation. Thus, the net force in the x-direction is found to be

$$\rho \cdot \mathrm{\Delta }x\mathrm{\Delta }y{f}_x+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left({\tau }_{xx}\right)\mathrm{\Delta }x\mathrm{\Delta }y+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left({\tau }_{yx}\right)\mathrm{\Delta }y\mathrm{\Delta }\mathrm{x.}\text{(2-26)}$$

Now we combine the forces, including the z-direction terms, and use them to substitute for the force term F in the original statement, F = ma, as seen in equation (2-13). Using the substantial derivative and the definition of the mass, m = ρ∆x∆y∆z, the x-momentum equation is equation (2-13) written as ma=F, the usual fluid mechanics convention; considering the x component, ma_x = F_x,

$$\rho \mathrm{\Delta }x\mathrm{\Delta }y\mathrm{\Delta }z\frac{Du}{Dt}=\rho \mathrm{\Delta }x\mathrm{\Delta }y\mathrm{\Delta }z{f}_x+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left({\tau }_{xx}\right)\mathrm{\Delta }x\mathrm{\Delta }y\mathrm{\Delta }z+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left({\tau }_{yx}\right)\mathrm{\Delta }y\mathrm{\Delta }x\mathrm{\Delta }z+\frac{\mathrm{\partial }}{\mathrm{\partial }z}\left({\tau }_{zx}\right)\mathrm{\Delta }y\mathrm{\Delta }x\mathrm{\Delta }\mathrm{z.}\text{(2-27)}$$

Dividing both sides by ∆x∆y∆z, we obtain the final set of three equations by appending the y- and z-momentum equations that can be easily derived by applying the approach used for equation (2-27):

$$\begin{array}{rl}& \rho \frac{Du}{Dt}=\rho f_x+\frac{\mathrm{\partial }{\tau }_{xx}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{yx}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\tau }_{zx}}{\mathrm{\partial }z}\\ & \rho \frac{Dv}{Dt}=\rho f_y+\frac{\mathrm{\partial }{\tau }_{xy}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{yy}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\tau }_{zy}}{\mathrm{\partial }z}\\ & \rho \frac{Dw}{Dt}=\rho \begin{array}{r}{f}_z\end{array}+\frac{\mathrm{\partial }{\tau }_{xz}}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{\tau }_{yz}}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{\tau }_{zz}}{\mathrm{\partial }z}\end{array}\text{(2-28)}$$

These are general conservation of momentum relations, valid for anything!

To make equation (2-28) more specific for fluid flow problems, we need to relate the stresses to the motion of the fluid. For gases and water, stress is a linear function of the rate of strain. Such a fluid is called a Newtonian fluid, and equation (2-29) shows the expression for stress as

$$\tau =\mu \frac{\mathrm{\partial }u}{\mathrm{\partial }y}\text{(2-29)}$$

where μ is the coefficient of viscosity. In our work, we consider μ to be a function of temperature only. Note that in air, the viscosity coefficient increases with increasing temperature, and in water, the viscosity coefficient decreases with temperature increases.

To complete the specification of the relationship between stress and rate of strain, we need to define precisely the relation between the stresses and the motion of the fluid. This can become complicated. In general, the fluid description requires two coefficients of viscosity. The coefficient of viscosity arising from the shear stress is well defined. The second coefficient of viscosity is not. This coefficient depends on the normal stress and is only important in computing the detailed structure of shock waves. Various assumptions relating the coefficients of viscosity are made. The set of assumptions that leads to the equations known as the Navier–Stokes equations are:

• The stress-rate-of-strain relations must be independent of coordinate system.
• When the fluid is at rest and the velocity gradients are zero (the strain rates are zero), the stress reduces to the hydrostatic pressure.
• Stoke’s hypothesis is used to eliminate the issue of mean pressure versus thermodynamic pressure. This is the assumption between viscosity coefficients.

Details of the theory associated with these requirements can be found in Schlichting^[17] and Grossman.^[18] Using the set of assumptions listed above leads to the following relations:

$$\begin{array}{c}{\tau }_{xx}=-p-\frac{2}{3}\mu \mathrm{\nabla }\cdot V+2\mu \frac{\partial u}{\partial x}\\ {\tau }_{yy}=-p-\frac{2}{3}\mu \mathrm{\nabla }\cdot V+2\mu \frac{\partial v}{\partial y}\\ {\tau }_{zz}=-p-\frac{2}{3}\mu \mathrm{\nabla }\cdot V+2\mu \frac{\partial w}{\partial z}\end{array}\text{(2-30)}$$

and

$$\begin{array}{rl}& {\tau }_{xy}={\tau }_{yx}=\mu (\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})\\ & {\tau }_{xz}={\tau }_{zx}=\mu (\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x})\\ & {\tau }_{yz}={\tau }_{zy}=\mu (\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y})\end{array}\text{(2-31)}$$

If we combine equations (2-28), (2-30), and (2-31) and we neglect the body force term (which is very small compared to the other terms since the air is so light), we get the standard set of equations for aerodynamics:

$$\begin{array}{r}\rho \frac{Du}{Dt}=-\frac{\partial p}{\partial x}+\frac{\partial }{\partial x}(2\mu \frac{\partial u}{\partial x}-\frac{2}{3}\mu \mathrm{\nabla }\cdot V)+\frac{\partial }{\partial y}\left[\mu (\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})\right]+\frac{\partial }{\partial z}\left[\mu (\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z})\right]\\ \rho \frac{Dv}{Dt}=-\frac{\partial p}{\partial y}+\frac{\partial }{\partial x}\left[\mu (\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x})\right]+\frac{\partial }{\partial y}(2\mu \frac{\partial v}{\partial y}-\frac{2}{3}\mu \mathrm{\nabla }\cdot V)+\frac{\partial }{\partial z}\left[\mu (\frac{\partial w}{\partial y}+\frac{\partial v}{\partial z})\right]\\ \rho \frac{Dw}{Dt}=-\frac{\partial p}{\partial z}+\frac{\partial }{\partial x}\left[\mu (\frac{\partial w}{\partial x}+\frac{\partial u}{\partial z})\right]+\frac{\partial }{\partial y}\left[\mu (\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y})\right]+\frac{\partial }{\partial z}(2\mu \frac{\partial w}{\partial z}-\frac{2}{3}\mu \mathrm{\nabla }\cdot V)\end{array}\text{(2-32)}$$

These are the classic Navier–Stokes equations written in the standard aerodynamics form. Note that the equations are (i) non-linear (remember that superposition of solutions is not allowed—D/Dt!); (ii) second order; (iii) highly coupled, and (iv) long! As written above, it’s easy to identify F = ma, written in the fluid mechanics form ma = F.

There are also alternate integral formulations of the equations. Consider the momentum flux through an arbitrary control volume in a manner similar to the integral statement of the continuity equation pictured in figure 2-4 and given in equation (2-7). Here, the rate of momentum change is proportional to the force. The integral statement is

$$\text{∯}\rho \mathbf{V}(\mathbf{V}\cdot \mathbf{n})dS+\frac{\partial }{\partial t}\iiint \rho \mathbf{V}\mathrm{d\Omega }=\mathbf{F}={\mathbf{F}}_{\text{volume}}+{\mathbf{F}}_{\text{surface ,}}\text{(2-33)}$$ $$$$

and this statement can also be converted to the differential form using the Gauss divergence theorem. Note that we use the partial derivative notation ∂/∂t to denote the change in the fixed “porous” control volume with fluid moving across the boundaries. The derivation of the Navier–Stokes equations is for general unsteady fluid motion. Because of limitations in our computational capability (for some time to come), these equations are directly applicable to solve for laminar flows. When the flow is turbulent, the usual approach is to Reynolds-average the equations, with the result that additional Reynolds stresses appear in the equations. Clearly, the addition of new unknowns requires additional equations. This problem is treated through turbulence modeling and is discussed, for example, in Tannehill, et al.^[19]

2.2.3 The Energy Equation

The equation for the conservation of energy is required to complete the system of equations. This is a statement of the First Law of Thermodynamics: The sum of the work and heat added to a system will equal the increase of energy. Following the derivation given by White:^[20]

$$\underset{\begin{array}{c}\text{change of total energy}\\ \text{of the system}\end{array}}{\underbrace{d{E}_t}}=\underset{\text{change of heat added}}{\underbrace{\delta Q}}+\underset{\begin{array}{c}\text{change of work done}\\ \text{on the system}\end{array}}{\underbrace{\delta W}}\text{(2-34)}$$

For our control volume in a fixed coordinate system, the rate of change is

$$\frac{D{E}_t}{Dt}=\dot{Q}+\dot{W}\text{(2-35)}$$

where

$$E_t=\rho (e+\frac{1}{2}{V}^2-\mathbf{g}\cdot \mathbf{r})\text{(2-36)}$$

and e is the internal energy per unit mass. The last term is the potential energy (i.e., the body force). In aerodynamics, this term is neglected. E_t can also be written in terms of specific energy as

$$E_t=\rho e_0\text{(2-37)}$$

where

$$e_0=e+\frac{1}{2}{V}^2.\text{(2-38)}$$

To obtain the energy equation, we need to write the right-hand side (RHS) of equation (2-35) in terms of flow properties. Consider first the heat added to the system.* The heat flow into the control volume is found in the identical manner to the mass flow. Using figure 2-8 for reference, we obtain the expression for the net heat flow.

The heat fluxes are:

$$\begin{array}{c}{q}_{x_{\text{in}}}=(q_x-\frac{\partial q}{\partial x}\frac{\mathrm{\Delta }x}{2})\mathrm{\Delta }y\\ q_{x_{\text{out}}}=(q_x+\frac{\partial q}{\partial x}\frac{\mathrm{\Delta }x}{2})\mathrm{\Delta }y\end{array}\text{(2-39)}$$

and the net heat flow into the control volume in the x-direction is q_xin - q_xout, or

$$-\frac{\mathrm{\partial }q}{\mathrm{\partial }x}\mathrm{\Delta }x\mathrm{\Delta }\mathrm{y.}$$

Using the same analysis in the y- and z-directions, we obtain the net heat flux into the control volume (realizing that the ∆x∆y∆z terms will cancel):

$$\dot{Q}=-(\frac{\mathrm{\partial }{q}_x}{\mathrm{\partial }x}+\frac{\mathrm{\partial }{q}_y}{\mathrm{\partial }y}+\frac{\mathrm{\partial }{q}_z}{\mathrm{\partial }z})=-\mathrm{\nabla }\cdot \mathbf{q}\text{(2-40)}$$

Now relate the heat flow to the temperature field. Fourier’s law provides this connection:

$$\mathbf{q}=-k\mathrm{\nabla }T\text{(2-41)}$$

where k is the coefficient of thermal conductivity. Combining equation (2-41) with equation (2-40), we get the heat conduction in terms of the temperature gradient in the flow field as

$$\dot{Q}=-\mathrm{\nabla }\cdot \mathbf{q}=+\mathrm{\nabla }\cdot (k\mathrm{\nabla }T).\text{(2-42)}$$

Next we find the work done on the system. Using the definition of work = force × distance, the rate of work is

$$\dot{W}=\text{force}\times \text{velocity.}\text{(2-43)}$$

Using the control volume again, we find the work, which is equal to the velocity times the stress. The work associated with the x-face of the control volume (for two-dimensional flow) is

$$w_x=u{\tau }_{xx}+v{\tau }_{xy}\text{.}\text{(2-44)}$$

The complete description of the work on the control volume is shown in figure 2-9.

Using the x-component of net work as an example again, the work done on the system is w_xin or w_xout, or

$$(w_x-\frac{\mathrm{\partial }{w}_x}{\mathrm{\partial }x}\frac{\mathrm{\Delta }x}{2})\mathrm{\Delta }y-(w_x+\frac{\mathrm{\partial }{w}_x}{\mathrm{\partial }x}\frac{\mathrm{\Delta }x}{2})\mathrm{\Delta }y=-\frac{\mathrm{\partial }{w}_x}{\mathrm{\partial }x}\mathrm{\Delta }x\mathrm{\Delta }\mathrm{y.}\text{(2-45)}$$

Including the other two directions, namely y and z, and dropping the ∆x∆y∆z terms, which cancel out, we get

$$\dot{W}=-\mathrm{div}\mathbf{w}=\frac{\mathrm{\partial }}{\mathrm{\partial }x}(u{\tau }_{xx}+v{\tau }_{xy})+\frac{\mathrm{\partial }}{\mathrm{\partial }y}(u{\tau }_{yx}+v{\tau }_{yy}).*\text{(2-46)}$$

Substituting equations (2-37) and (2-38) into (2-35) for E_t, equation (2-42) for the heat, and equation (2-46) for the work, we obtain

$$\frac{D\rho (e+\frac{1}{2}{V}^2)}{Dt}=\mathrm{\nabla }\cdot (k\mathrm{\nabla }T)-\mathrm{div}\mathbf{w}\text{.}\text{(2-47)}$$

Many, many equivalent forms of the energy equation are found in the literature. Often the equation is thought of as an equation for the temperature. We now describe how to obtain one specific form. Substituting equation (2-46) into (2-47) and using the expressions for the τ’s in terms of μ and the velocity gradients in equations (2-29) and (2-30), we obtain the following lengthy expression (see Bertin and Smith,^[22] pp. 40–42). After making use of the momentum and continuity equations to simplify and finally introducing the definition of enthalpy, h = e + p/ρ, we obtain a frequently written form of the classical energy equation given as:

$$\rho \frac{Dh}{Dt}-\frac{Dp}{Dt}=\underset{\text{heat conduction}}{\underbrace{\mathrm{\nabla }\cdot (k\mathrm{\nabla }T)}}+\underset{\begin{array}{c}\text{viscous dissipation}\\ \text{(always positive)}\end{array}}{\underbrace{\mathrm{\Phi }}}\text{(2-48)}$$

where

$$\mathrm{\Phi }=\mu \left\{\begin{array}{c}2[{\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2+{\left(\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\right)}^2+{\left(\frac{\mathrm{\partial }w}{\mathrm{\partial }z}\right)}^2]+{(\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }y})}^2+{(\frac{\mathrm{\partial }w}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }z})}^2\\ +{(\frac{\mathrm{\partial }u}{\mathrm{\partial }z}+\frac{\mathrm{\partial }w}{\mathrm{\partial }x})}^2-\frac{2}{3}{(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }v}{\mathrm{\partial }y}+\frac{\mathrm{\partial }w}{\mathrm{\partial }z})}^2\end{array}\right\}\text{.}\text{(2-49)}$$

The energy equation can be written in numerous forms, and many different but entirely equivalent forms are available. In particular, the energy equation is frequently written in terms of the total enthalpy, H, to good advantage in inviscid and boundary layer flows. A good discussion of the energy equation is also given by White.^[23]

There is also an integral form of this equation:

$$\text{∯}\rho (e+V^2/2)(\mathbf{V}\cdot \mathbf{n})dS+\frac{\partial }{\partial t}\iiint \rho (e+V^2/2)d\text{Ω}=\dot{Q}+\dot{W}\text{(2-50)}$$

Here again note that we use the partial derivative notation ∂/∂t to denote the change in the fixed “porous” control volume with fluid moving across the boundaries.

2.3 Boundary Conditions

If all flow fields are governed by the same equations, what makes flow fields different? Boundary conditions are the means through which the solution of the governing equations produce different results for different shapes. For aerodynamic solutions, the specification of the arbitrarily shaped configuration geometry and of the boundary conditions constitutes the major part of any effort.

2.3.1 Boundary Conditions on the Surface

The boundary condition on the surface of the configuration is the most straightforward and most obvious physically. However, the statement of the boundary conditions is tightly connected to the flow field model in use and the type of surface (e.g., solid or porous).

For an inviscid steady flow over a solid surface, the statement of the boundary condition is

$${\mathbf{V}}_R\cdot \mathbf{n}=0.\text{(2-51)}$$

In this equation, V_R is the relative velocity defined as the difference between fluid (or freestream) velocity V and the surface velocity V_S, and n is the unit vector normal to the surface. Therefore, this boundary condition simply stipulates that the component of the relative velocity normal to the surface must be zero and is known as the no-normal-flow or nonpenetration condition. In other words, the flow must be parallel to the surface at every point on the surface. Equation (2-51) may be written as

$$(\mathbf{V}-{\mathbf{V}}_S)\cdot \mathbf{n}=0.\text{(2-52)}$$

If the surface is fixed, V_S = 0, and equation (2-52) becomes

$$\mathbf{V}\cdot \mathbf{n}=0.\text{(2-53)}$$

For a viscous flow over a solid surface, the statement becomes even simpler: V = 0, known as the no-slip condition. If the surface is porous and there is mass flow, the values of the surface velocity must be specified as part of the problem definition. Numerical solutions of the Euler and Navier–Stokes solutions require that other boundary conditions be specified. In particular, conditions on pressure and temperature are required.

Let us next review how to compute the unit normal vector, n. Recall that to obtain the unit normal, the body is defined (in 2D) in the form F(x,y) = 0, which is the traditional analytic geometry nomenclature.

In the usual two-dimensional notation, the body shape is given by y = f(x), which is written as

$$F(x,y)=0=y-f(x)\text{(2-54)}$$

and the unit normal is

$$\mathbf{n}=\frac{\mathrm{\nabla }F}{|\mathrm{\nabla }F|}\text{.}\text{(2-55)}$$

2.3.2 Boundary Conditions Away From the Surface

Conditions also must be specified away from the body. For most of the external flows, the most commonly used condition is that at large distances (approaching infinity) from the body, the flow field must approach the freestream conditions. In numerical computations, the question of the farfield boundary condition can become troublesome. How far away is infinity? Exactly how should you specify the farfield boundary condition numerically? How to best handle these issues is the basis for many papers in the literature, such as Thomas and Salas,^[24] Verhoff,^[25] and the survey by Colonius.^[26]

Another important use of boundary conditions arises as a means of modeling physics that would be neglected otherwise. When an approximate flow field model is used, the boundary conditions frequently provide a means of including key elements of the physics in the problem without having to include the physics explicitly. The most famous example of this is the Kutta condition, wherein the viscous effects at the trailing edge can be accounted for in an inviscid calculation without treating the trailing-edge problem explicitly. Karamcheti^[27] discusses boundary conditions in more detail.

2.4 Standard Forms and Terminology of Governing Equations

To understand the literature in computational aerodynamics, several other aspects of the terminology must be discussed. This section provides several of these considerations.

2.4.1 Nondimensionalization

The governing equations should be nondimensionalized. Considering fluid mechanics theory, nondimensionalization reveals important similarity parameters. In practice, many different nondimensionalizations are used, and for a particular computer program, care must be taken to understand exactly what nondimensionalization is used.

Sometimes the dimensional quantities are defined by *’s or ~’s. In other schemes, the nondimensionalized variables are designated by the special symbols. In the example given here, the nondimensionalized values are denoted by an asterisk (*). In this system, once the quantities are defined, the *’s are dropped and the nondimensionalization is understood.

Many different values can be used for nondimensionalization. We give an example here using the freestream velocity (V_∞), flow properties (ρ_∞, T_∞, etc.), and a reference length, L, as follows:

$$\begin{array}{llll}{x}^{\ast }=\frac{x}{L}& y^{\ast }=\frac{y}{L}& z^{\ast }=\frac{z}{L}& t^{\ast }=\frac{t{V}_{\mathrm{\infty }}}{L}\\ u^{\ast }=\frac{u}{V_{\mathrm{\infty }}}& v^{\ast }=\frac{v}{V_{\mathrm{\infty }}}& w^{\ast }=\frac{w}{V_{\mathrm{\infty }}}& p^{\ast }=\frac{p}{{\rho }_{\mathrm{\infty }}{V}_{\mathrm{\infty }}^2}\\ T^{\ast }=\frac{T}{T_{\mathrm{\infty }}}& {\rho }^{\ast }=\frac{\rho }{{\rho }_{\mathrm{\infty }}}& {\mu }^{\ast }=\frac{\mu }{{\mu }_{\mathrm{\infty }}}& e_o^{\ast }=\frac{e_o}{U_{\mathrm{\infty }}^2}\end{array}\text{(2-56)}$$

Each code will have a set of reference nondimensionalizations similar to these. A specific example is given in section 2.4.3. Frequently, the speed of sound is used as the reference velocity. Making sure that you understand the nondimensionalization used in a particular program is an important part of using computational aerodynamics programs to solve aerodynamic problems.

2.4.2 Use of Divergence Form of Governing Equations

The classical forms of the governing equations (shown in the preceding sections) that are normally given in textbooks are not used for computations. Instead, the divergence, or conservation, form* is preferred.

This form is found to be required for reliable numerical calculation. If discontinuities in the flow field exist, this form must be used to account for discontinuities correctly. It is a way to improve the capability of the differential form of the governing equations. For example, across a shock wave, the density and velocity both jump in value. However, the product of these quantities, the mass flow, is a constant. Thus we can easily see why it is better numerically to work with the product rather than the individual variables. In this section, we show how the divergence forms are obtained from the standard classical form. We use the 2D steady x-momentum equation as the example:

$$\rho u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+\rho v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{\mathrm{\partial }p}{\mathrm{\partial }x}\text{(2-57)}$$

This equation is written using the following identities:

$$\frac{\mathrm{\partial }\rho uu}{\mathrm{\partial }x}=\rho u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+u\frac{\mathrm{\partial }\rho u}{\mathrm{\partial }x}\text{(2-58)}$$

or

$$\rho u\frac{\mathrm{\partial }u}{\mathrm{\partial }x}=\frac{\mathrm{\partial }\left(\rho u^2\right)}{\mathrm{\partial }x}-u\frac{\mathrm{\partial }\rho u}{\mathrm{\partial }x}\text{(2-59)}$$

Similarly the second term on the left-hand side of equation (2-57) may be written as

$$\frac{\mathrm{\partial }\rho vu}{\mathrm{\partial }y}=\rho v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+u\frac{\mathrm{\partial }\rho v}{\mathrm{\partial }y}\text{(2-60)}$$

or

$$\rho v\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=\frac{\mathrm{\partial }\rho vu}{\mathrm{\partial }y}-u\frac{\mathrm{\partial }\rho v}{\mathrm{\partial }y}\text{.}\text{(2-61)}$$

Substituting (2-59) and (2-61) into (2-57), we get

$$\frac{\mathrm{\partial }\rho u^2}{\mathrm{\partial }x}-u\frac{\mathrm{\partial }\rho u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }\rho vu}{\mathrm{\partial }y}-u\frac{\mathrm{\partial }\rho v}{\mathrm{\partial }y}+\frac{\mathrm{\partial }p}{\mathrm{\partial }x}=0\text{(2-62)}$$

which can be written as

$$\frac{\mathrm{\partial }\rho u^2}{\mathrm{\partial }x}+\frac{\mathrm{\partial }\rho vu}{\mathrm{\partial }y}-u(\underset{=0\text{from continuity}}{\underbrace{\frac{\mathrm{\partial }\rho u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }\rho v}{\mathrm{\partial }x}}})+\frac{\mathrm{\partial }p}{\mathrm{\partial }x}=0.\text{(2-63)}$$

Finally, the x-momentum equation written in divergence form for 2D steady flow is given in equation (2-64).

$$\frac{\mathrm{\partial }(\rho u^2+p)}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(\rho vu)}{\mathrm{\partial }y}=0\text{(2-64)}$$

The equations must be written in divergence form to be valid when shock waves are present.

2.4.3 Standard Form of the Equations

Even after writing the governing equations in divergence form, the equations that you see in the literature won’t look like the ones we’ve been writing down. A standard form is used in the literature for numerical solutions of the Navier–Stokes equations. In this section, we provide one representative set used in the NASA Langley codes CFL3D and CFL3DE.^[28] Professors Walters and Grossman (of Virginia Tech) and their students have made contributions to these codes. The Navier–Stokes equations (and the other equations required in the system) are written in vector divergence form as follows:

$$\frac{\mathrm{\partial }\mathbf{Q}}{\mathrm{\partial }t}+\frac{\mathrm{\partial }(\mathbf{F}-{\mathbf{F}}_v)}{\mathrm{\partial }x}+\frac{\mathrm{\partial }(\mathbf{G}-{\mathbf{G}}_v)}{\mathrm{\partial }y}+\frac{\mathrm{\partial }(\mathbf{H}-{\mathbf{H}}_v)}{\mathrm{\partial }z}=0\text{(2-65)}$$

where the conserved variables are

$$\mathbf{Q}=\left\{\begin{array}{c}\rho \\ \rho u\\ \rho v\\ \rho w\\ E_t\end{array}\right\}=\left\{\begin{array}{c}\text{density}\\ \mathrm{x}\text{- momentum}\\ \mathrm{y}\text{- momentum}\\ \mathrm{z}\text{- momentum}\\ \text{total energy per unit volume}\end{array}\right\}\text{(2-66)}$$

There are two sets of flux vectors in equation (2-65), inviscid and viscous. The inviscid flux vectors are F, G, and H in x-, y-, and z-directions, respectively, and the viscous flux vectors are F_v, G_v, and H_v in x-, y-, and z-directions, respectively.

The flux vectors in the x-direction are described in (2-67) below.

$$\mathbf{F}=\left[\begin{array}{c}\rho u\\ \rho u^2+p\\ \rho uv\\ \rho uw\\ (E_t+p)u\end{array}\right]{\mathbf{F}}_v=\left[\begin{array}{c}0\\ {\tau }_{xx}\\ {\tau }_{xy}\\ {\tau }_{xz}\\ u{\tau }_{xx}+v{\tau }_{xy}+w{\tau }_{xz}-{\dot{q}}_x\end{array}\right]\text{(2-67)}$$

Similar expressions can be written down for the y- and z-direction fluxes, with the y-direction given as

$$\mathbf{G}=\left[\begin{array}{c}\rho v\\ \rho vu\\ \rho v^2+p\\ \rho vw\\ (E_t+p)v\end{array}\right]{\mathbf{G}}_v=\left[\begin{array}{c}0\\ {\tau }_{yx}\\ {\tau }_{yy}\\ {\tau }_{yz}\\ u{\tau }_{yx}+v{\tau }_{yy}+w{\tau }_{yz}-{\dot{q}}_y\end{array}\right]\text{(2-68)}$$

and with the z-direction given as

$$\mathbf{H}=\left[\begin{array}{c}\rho w\\ \rho wu\\ \rho wv\\ \rho w^2+p\\ (E_t+p)w\end{array}\right]{\mathbf{H}}_v=\left[\begin{array}{c}0\\ {\tau }_{zx}\\ {\tau }_{zy}\\ {\tau }_{zz}\\ u{\tau }_{zx}+v{\tau }_{zy}+w{\tau }_{zz}-{\dot{q}}_z\end{array}\right]\text{.}\text{(2-69)}$$

The equation of state (perfect gas) is written in this formulation as

$$p=(\gamma -1)[E_t-\rho (u^2+v^2+w^2)/2]\text{.}\text{(2-70)}$$

To complete the flow equations, we need to define (i) nondimensional variables using the freestream values of the flow properties like density, pressure, temperature, speed of sound, etc., and (ii) the shear stress and heat transfer nomenclature. In the following equations, the tilde (~) superscript denotes a dimensional quantity and the infinity (∞) subscript denotes evaluation at freestream conditions.

Equations (2-71) and (2-72) below show how nondimensional shear stress and heat transfer terms are written in indicial (or index*) notation.

$${\tau }_{x_i{x}_j}=\frac{M_{\mathrm{\infty }}}{{\mathrm{Re}}_L}[\mu (\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i})+\lambda \frac{\mathrm{\partial }{u}_k}{\mathrm{\partial }{x}_k}{\delta }_{ij}]\text{(2-71)}$$

and

$${\dot{q}}_{x_i}=-\left[\frac{M_{\mathrm{\infty }}\mu }{{\mathrm{Re}}_L\mathrm{Pr}(\gamma -1)}\right]\frac{\mathrm{\partial }\left(a^2\right)}{\mathrm{\partial }{x}_i}=-\left[\frac{M_{\mathrm{\infty }}\mu }{{\mathrm{Re}}_L\mathrm{Pr}(\gamma -1)}\right]\frac{\mathrm{\partial }T}{\mathrm{\partial }{x}_i}\text{(2-72)}$$

The nondimensional molecular viscosity is found using Sutherland’s law:

$$\mu =\tilde{\mu }/{\tilde{\mu }}_{\mathrm{\infty }}={\left(\frac{\tilde{T}}{{\tilde{T}}_{\mathrm{\infty }}}\right)}^{3/2}\left[\frac{{\tilde{T}}_{\mathrm{\infty }}+\tilde{c}}{\tilde{T}+\tilde{c}}\right]=(T{)}^{3/2}\left[\frac{(1+\tilde{c}/{\tilde{T}}_{\mathrm{\infty }})}{(T+\tilde{c}/{\tilde{T}}_{\mathrm{\infty }})}\right]\text{(2-73)}$$

where Sutherland’s constant (defined by the symbol c with a tilde (~) superscript) has a value of 198.6°R = 110.4°K.

Below are the definitions of the other quantities.

Reynolds number:

$${\mathrm{Re}}_L={\tilde{\rho }}_{\mathrm{\infty }}{\tilde{q}}_{\mathrm{\infty }}\tilde{L}/{\tilde{\mu }}_{\mathrm{\infty }}$$

Mach number:

$$M_{\mathrm{\infty }}={\tilde{q}}_{\mathrm{\infty }}/{\tilde{a}}_{\mathrm{\infty }}$$

Prandtl number:

$$\mathrm{Pr}=\tilde{\mu }{\tilde{c}}_p/\tilde{k}$$

Stoke’s hypothesis for bulk viscosity is used, meaning

$$\lambda +2\mu /3=0$$

and the freestream velocity magnitude is

$${\tilde{q}}_{\mathrm{\infty }}={[{\tilde{u}}_{\mathrm{\infty }}^2+{\tilde{v}}_{\mathrm{\infty }}^2+{\tilde{w}}_{\mathrm{\infty }}^2]}^{1/2}\text{.}$$

The nondimensional velocity components are given by

$$\begin{array}{ll}u=\tilde{u}/{\tilde{a}}_{\mathrm{\infty }}& u_{\mathrm{\infty }}=M_{\mathrm{\infty }}\cos \alpha \cos \beta \\ v=\tilde{v}/{\tilde{a}}_{\mathrm{\infty }}& v_{\mathrm{\infty }}=-M_{\mathrm{\infty }}\sin \beta \\ w=\tilde{w}/{\tilde{a}}_{\mathrm{\infty }}& w_{\mathrm{\infty }}=M_{\mathrm{\infty }}\sin \alpha \cos \beta \end{array}\text{(2-74)}$$

and the thermodynamic variables are given by

$$\begin{array}{cc}\rho =\tilde{\rho }/{\tilde{\rho }}_{\mathrm{\infty }},& {\rho }_{\mathrm{\infty }}=1\\ p=\tilde{p}/\tilde{\rho }{\tilde{a}}_{\mathrm{\infty }}^2,& p_{\mathrm{\infty }}=1/\gamma \\ T=\tilde{T}/{\tilde{T}}_{\mathrm{\infty }}=\gamma p/\rho =a^2& T_{\mathrm{\infty }}=1\end{array}\text{(2-75)}$$

$$E_t={\tilde{E}}_t/{\tilde{\rho }}_{\mathrm{\infty }}{\tilde{a}}_{\mathrm{\infty }}^2{E}_{t_{\mathrm{\infty }}}=1/[\gamma (\gamma -1)]+M_{\mathrm{\infty }}^2/2.\text{(2-76)}$$

This completes the nomenclature for one typical example of the application of the Navier–Stokes equations in an actual computer code, CFL3D. Note that these equations are for a Cartesian coordinate system. The general non-orthogonal curvilinear coordinate generalization is considerably more complicated.

2.5 The Gas Dynamics Equation and the Full-Potential Equation

For inviscid flow problems (and even some viscous flows), it is useful to combine the equations in a special form known as the gas dynamics equation. In particular, the gas dynamics form of the equation is used to obtain the complete or “full” nonlinear potential flow equation in section 2.5.3. Many valuable results can be obtained in computational aerodynamics using the potential flow approximation. When compressibility effects are important, a special form of the governing equation can be obtained based on the gas dynamics equation, which we derive here. The gas dynamics equation is valid for any flow assumed to be inviscid. The starting point for the derivation is the Euler equations, the continuity equation, and the equation of state.

2.5.1 The Gas Dynamics Equation

We demonstrate the derivation using two-dimensional steady flow. This is not required but is used for simplicity. Furthermore, the index notation x_i is used where the index i = 1, 2, 3, denote x-, y-, z-axes, respectively. To start, we make use of a thermodynamic definition to rewrite the pressure term in the momentum equation as equation (2-77).

$${\frac{\mathrm{\partial }p}{\mathrm{\partial }{x}_i}=\frac{\mathrm{\partial }p}{\mathrm{\partial }\rho })}_s\frac{\mathrm{\partial }\rho }{\mathrm{\partial }{x}_i}\text{(2-77)}$$

Using the definition of the speed of sound assuming isentropic or constant entropy flow,

$${a^2=\frac{\mathrm{\partial }p}{\mathrm{\partial }\rho })}_s\text{(2-78)}$$

allowing ∂p/∂x_i to be written as

$$\frac{\mathrm{\partial }p}{\mathrm{\partial }{x}_i}=a^2\frac{\mathrm{\partial }\rho }{\mathrm{\partial }{x}_i}\text{.}\text{(2-79)}$$

We next write the x-momentum equation multiplied by u- and y-momentum equation multiplied by v:

$$\begin{array}{rl}& u^2\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+uv\frac{\mathrm{\partial }u}{\mathrm{\partial }y}=-\frac{u}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }x}=-u\frac{a^2}{\rho }\frac{\mathrm{\partial }\rho }{\mathrm{\partial }x}\\ & vu\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v^2\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=-\frac{v}{\rho }\frac{\mathrm{\partial }p}{\mathrm{\partial }y}=-v\frac{a^2}{\rho }\frac{\mathrm{\partial }\rho }{\mathrm{\partial }y}\end{array}\text{(2-80)}$$

We use the continuity equation by expanding it from

$$\frac{\mathrm{\partial }\rho u}{\mathrm{\partial }x}+\frac{\mathrm{\partial }\rho v}{\mathrm{\partial }y}=0\text{(2-81)}$$

to

$$u\frac{\mathrm{\partial }\rho }{\mathrm{\partial }x}+\rho \frac{\mathrm{\partial }u}{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\rho }{\mathrm{\partial }y}+\rho \frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0\text{(2-82)}$$

or

$$u\frac{\mathrm{\partial }\rho }{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\rho }{\mathrm{\partial }y}=-\rho \frac{\mathrm{\partial }u}{\mathrm{\partial }x}-\rho \frac{\mathrm{\partial }v}{\mathrm{\partial }y}\text{.}\text{(2-83)}$$

We now add the modified x- and y-momentum equations, equation (2-80), which gives us equation (2-84) below.

$$\begin{array}{rl}{u}^2\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+uv\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+vu\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v^2\frac{\mathrm{\partial }v}{\mathrm{\partial }y}& =-u\frac{a^2}{\rho }\frac{\mathrm{\partial }\rho }{\mathrm{\partial }x}-v\frac{a^2}{\rho }\frac{\mathrm{\partial }\rho }{\mathrm{\partial }y}\\ & =-\frac{a^2}{\rho }(u\frac{\mathrm{\partial }\rho }{\mathrm{\partial }x}+v\frac{\mathrm{\partial }\rho }{\mathrm{\partial }y})\end{array}\text{(2-84)}$$

We substitute the rewritten continuity equation, equation (2-83), into equation (2-84), resulting in

$$\begin{array}{rl}{u}^2\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+uv\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+vu\frac{\mathrm{\partial }v}{\mathrm{\partial }x}+v^2\frac{\mathrm{\partial }v}{\mathrm{\partial }y}& =-\frac{a^2}{\rho }(-\rho \frac{\mathrm{\partial }u}{\mathrm{\partial }x}-\rho \frac{\mathrm{\partial }v}{\mathrm{\partial }y})\\ & =a^2\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+a^2\frac{\mathrm{\partial }v}{\mathrm{\partial }y}\end{array}\text{.}\text{(2-85)}$$

Finally, we rearrange the terms in equation (2-85) to obtain the following gas dynamic equation in two dimensions:

$$(u^2-a^2)\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+uv(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x})+(v^2-a^2)\frac{\mathrm{\partial }v}{\mathrm{\partial }y}=0\text{(2-86)}$$

Extension to three dimensions is straightforward, creating the corresponding gas dynamic equation:

$$(u^2-a^2)\frac{\mathrm{\partial }u}{\mathrm{\partial }x}+(w^2-a^2)\frac{\mathrm{\partial }w}{\mathrm{\partial }z}+(v^2-a^2)\frac{\mathrm{\partial }v}{\mathrm{\partial }y}+uv(\frac{\mathrm{\partial }u}{\mathrm{\partial }y}+\frac{\mathrm{\partial }v}{\mathrm{\partial }x})+vw(\frac{\mathrm{\partial }v}{\mathrm{\partial }z}+\frac{\mathrm{\partial }w}{\mathrm{\partial }y})+wu(\frac{\mathrm{\partial }w}{\mathrm{\partial }x}+\frac{\mathrm{\partial }u}{\mathrm{\partial }z})=0\text{(2-87)}$$

2.5.2 Derivation of the Classical Gas Dynamics-Related Energy Equation

The special form of the energy equation that is used to close the system (in two dimensions or 2D) is given below.

$$a^2=a_0^2-\left(\frac{\gamma -1}{2}\right)(u^2+v^2)\text{(2-88)}$$

We next show exactly how this relation is obtained. Let us start with the energy equation for inviscid, adiabatic flow expressed in the form of

$$\frac{DH}{Dt}=0\text{(2-89)}$$

where H is the total enthalpy, defined (in 2D) by

$$H=h+\frac{1}{2}(u^2+v^2).\text{(2-90)}$$

Equation (2-89) yields H = constant. Thus, we have a purely algebraic statement of the energy equation instead of a partial differential equation. This is an important reduction in complexity.

For a thermally and calorically perfect gas, h = c_pT and c_p = constant. Substituting for the enthalpy, we get equation (2-91).

$$c_p{T}_0=c_{p}T+\frac{1}{2}(u^2+v^2)\text{(2-91)}$$

Recalling that a²= γRT and R = c_p- c_v, with γ = c_p/c_v, we write

$$a^2=\frac{c_p}{c_v}(c_p-c_v)T=\left(\frac{c_p-c_v}{c_v}\right)c_{p}T\text{(2-92)}$$

or

$$c_{p}T=\left(\frac{c_v}{c_p-c_v}\right)a^2=\left(\frac{1}{\gamma -1}\right)a^2\text{.}\text{(2-93)}$$

Using equation (2-93), we can rewrite the total energy equation (H = constant), equation (2-91), as

$$\frac{a_0^2}{\gamma -1}=\frac{a^2}{\gamma -1}+\frac{1}{2}(u^2+v^2)\text{(2-94)}$$

or

$$a_0^2=a^2+\left(\frac{\gamma -1}{2}\right)(u^2+v^2).\text{(2-95)}$$

The two-dimensional form shown in equation (2-88) is readily obtained by solving equation (2-95) for a. Extension to three dimensions is straightforward, and the equation in 3D is

$$a^2=a_0^2-\left(\frac{\gamma -1}{2}\right)(u^2+v^2+w^2).\text{(2-96)}$$

2.5.3 Full Potential Equation

The gas dynamics equation is converted to the classical nonlinear potential equation when we make the irrotational flow assumption. The potential flow assumption requires that the flow be irrotational. This is valid for inviscid flow when the onset flow is uniform and there are no shock waves. However, we often continue to assume that the flow can be represented approximately by a scalar potential Φ when the Mach number normal to any shock wave is close to one (for example, M_n < 1.25). Recall that the irrotational flow assumption is stated mathematically as curl V= 0. When this is true, V can be defined as the gradient of a scalar potential, V = ∇Φ. Using the common subscript notation to represent partial derivatives, the velocity components are u = Φx ,v = Φy, and w = Φz. Using the gas dynamics equation, equation (2-87), the nonlinear or “full-potential” equation is written as

$$({\mathrm{\Phi }}_x^2-a^2){\mathrm{\Phi }}_{xx}+({\mathrm{\Phi }}_y^2-a^2){\mathrm{\Phi }}_{yy}+({\mathrm{\Phi }}_z^2-a^2){\mathrm{\Phi }}_{zz}+2{\mathrm{\Phi }}_x{\mathrm{\Phi }}_y{\mathrm{\Phi }}_{xy}+2{\mathrm{\Phi }}_y{\mathrm{\Phi }}_z{\mathrm{\Phi }}_{yz}+2{\mathrm{\Phi }}_z{\mathrm{\Phi }}_x{\mathrm{\Phi }}_{zx}=0.\text{(2-97)}$$

This is the classic form of the equation. This single nonlinear partial differential equation has been extensively used to obtain physical insights into a wide variety of flows. Note that, as written above, it is not in divergence (or conservation) form.

2.5.4 Equivalent Divergence Form and Energy Equation

The equivalent equation written in conservation form makes use of the continuity equation. This is the form that is used in most computational fluid dynamics codes. Equation (2-98) shows how it is written in two dimensions.

$$\frac{\mathrm{\partial }}{\mathrm{\partial }x}\left(\rho {\mathrm{\Phi }}_x\right)+\frac{\mathrm{\partial }}{\mathrm{\partial }y}\left(\rho {\mathrm{\Phi }}_y\right)=0\text{(2-98)}$$

The relation between ρ and the potential is given by

$$\rho ={[1-\left(\frac{\gamma -1}{\gamma +1}\right)({\mathrm{\Phi }}_x^2+{\mathrm{\Phi }}_y^2)]}^{\frac{1}{\gamma -1}}\text{(2-99)}$$

which is a statement of the energy equation (see section 2.5.5). Note that the full-potential equation is still nonlinear when the density varies, and ρ must be considered a dependent variable.

2.5.5 Derivation of Another Form of the Related Energy Equation

It is informative to demonstrate the derivation of the energy equation, equation (2-99), in section 2.5.4. To get this standard form, understand the specific nondimensionalization employed with this form:

$$\rho =\frac{\tilde{\rho }}{{\rho }_0},{\mathrm{\Phi }}_x=\frac{\tilde{u}}{a^{\ast }},{\mathrm{\Phi }}_y=\frac{\tilde{v}}{a^{\ast }},\text{(2-100)}$$

where a* denotes the sonic value. Start with the previous energy equation and work with dimensional variables for the moment:

$$a^2=a_0^2-\left(\frac{\gamma -1}{2}\right)(u^2+v^2)\text{(2-101)}$$

or

$$\frac{a^2}{a_0^2}=1-\left(\frac{\gamma -1}{2}\right)\left(\frac{u^2+v^2}{a_0^2}\right)\text{.}\text{(2-102)}$$

Now, let us get a relation for a_o in terms of the eventual nondimensionalizing velocity a*.

When the velocity is equal to the speed of sound, we can write

$$a_0^2=a^2+\left(\frac{\gamma -1}{2}\right)\underset{=a^2}{\underbrace{(u^2+v^2)}}\text{.}\text{(2-103)}$$

Since a = a* when the velocity is equal to the speed of sound, we obtain either

$$a_0^2=a^{\mathrm{\ast }2}+\left(\frac{\gamma -1}{2}\right)a^2=(1+\frac{\gamma -1}{2})a^{\mathrm{\ast }2}\text{(2-104)}$$

or

$$a_0^2=\left(\frac{\gamma +1}{2}\right)a^{\mathrm{\ast }2}\text{.}\text{(2-105)}$$

Using a² = γRT, the left-hand side of equation (2-102) becomes

$${\left(\frac{a}{a_0}\right)}^2=\frac{T}{T_0}\text{.}\text{(2-106)}$$

In addition, replacing a_o² with a*² in the denominator of the last term on the right-hand side of equation (2-102), we can rewrite the energy equation, equation (2-102), as

$$\frac{T}{T_0}=1-\frac{\gamma -1}{2}\frac{u^2+v^2}{\frac{\gamma +1}{2}a{\ast }^2}\text{(2-107)}$$

or

$$\frac{T}{T_0}=1-\left(\frac{\gamma -1}{\gamma +1}\right)[{\left(\frac{u}{a^{\ast }}\right)}^2+{\left(\frac{v}{a^{\ast }}\right)}^2]\text{.}\text{(2-108)}$$

Recall that entropy, s, is constant for isentropic flows (a consistent assumption if the use of Φ is valid):

$$s=\text{const}\text{(2-109)}$$

and

$$\frac{p}{{\rho }^{\gamma }}=\text{const}=\frac{p_0}{{\rho }_0^{\gamma }}\text{(2-110)}$$

Now, we introduce (~) to denote dimensional quantities and convert to the desired nondimensional form:

$$\left(\frac{\tilde{p}}{p_0}\right)={\left(\frac{\tilde{\rho }}{{\rho }_0}\right)}^{\gamma }={\left(\frac{\tilde{T}}{T_0}\right)}^{\frac{\gamma }{\gamma -1}}\text{(2-111)}$$

or

$$\frac{\tilde{T}}{T_0}={\left(\frac{\tilde{\rho }}{{\rho }_0}\right)}^{\gamma -1}\text{(2-112)}$$

Using equation (2-112) we write the energy equation, equation (2-108), as

$${\left(\frac{\tilde{\rho }}{{\rho }_0}\right)}^{\gamma -1}=1-\left(\frac{\gamma -1}{\gamma +1}\right)[{\left(\frac{\tilde{u}}{a^{\ast }}\right)}^2+{\left(\frac{\tilde{v}}{a^{\ast }}\right)}^2]\text{.}\text{(2-113)}$$

Using the nondimensionalizing definition given in section 2.4.3, we finally obtain equation (2-114) below.

$$\rho ={[1-\left(\frac{\gamma -1}{\gamma +1}\right)({\mathrm{\Phi }}_x^2+{\mathrm{\Phi }}_y^2)]}^{\frac{1}{\gamma -1}}\text{(2-114)}$$

This is an energy equation in ρ to use with the divergence form of the full-potential equation. It is also an example of how to get an energy equation in a typical nondimensional form used in the literature.

2.6 Special Cases

In this section, we present a number of special, simplified forms of the equations described above. These simplified equations are entirely adequate for many of the problems of computational aerodynamics, and until recently they were used nearly exclusively. The ability to obtain simpler relations that provide explicit physical insight into the flow field process has played an important role in the development of aerodynamic concepts. One key idea is the notion of small disturbance equations. The assumption is that the flow field is only slightly disturbed by the body. We expect this assumption to be valid for inviscid flows over streamlined shapes. These ideas are expressed mathematically by small perturbation or asymptotic expansion methods and are elegantly described in the book by Van Dyke.^[29] Figure 2-10 summarizes the theoretical path required to obtain these equations.

2.6.1 Small Disturbance Form of the Energy Equation

The expansion of the simple algebraic statement of the energy equation provides an example of a small disturbance analysis. In this case, the square of the speed of sound (or equivalently the temperature) is linearly related to the velocity field. Start with the energy equation:

$$a^2=a_0^2-\left(\frac{\gamma -1}{2}\right)(u^2+v^2)\text{(2-115)}$$

and

$$a_0^2=\text{const}=a^2+\left(\frac{\gamma -1}{2}\right)(u^2+v^2)=a_{\mathrm{\infty }}^2+\frac{\gamma -1}{2}{U}_{\mathrm{\infty }}^2\text{(2-116)}$$

Assuming the freestream velocity, U_∞, is aligned with the x-direction and letting the velocity components be u = U_∞ + u’ , v = v’ (where u’ and v’ are disturbance velocities in x- and y-directions, respectively), we can write equation (2-116) as 

$$a^2=a_{\mathrm{\infty }}^2+\frac{\gamma -1}{2}{U}_{\mathrm{\infty }}^2-\left(\frac{\gamma -1}{2}\right)[U_{\mathrm{\infty }}^2+2U_{\mathrm{\infty }}{u}^{\mathrm{\prime }}+u^{\mathrm{\prime }2}+v^{\mathrm{\prime }2}]\text{.}\text{(2-117)}$$

Combining terms, we obtain equation (2-118).

$$a^2=a_{\mathrm{\infty }}^2-\left(\frac{\gamma -1}{2}\right)[2U_{\mathrm{\infty }}{u}^{\mathrm{\prime }}+u^{\mathrm{\prime }2}+v^{\mathrm{\prime }2}]\text{(2-118)}$$

At this point, the relation is still exact, but now it is written in a form that can be easily simplified. The basic idea is to take advantage of the small perturbation assumption:

$$u^{\mathrm{\prime }}<U_{\mathrm{\infty }},v^{\mathrm{\prime }}<U_{\mathrm{\infty }}\text{(2-119)}$$

and thus,

$$\frac{u^{\mathrm{\prime }}}{U_{\mathrm{\infty }}}<1\Rightarrow {\left(\frac{u^{\mathrm{\prime }}}{U_{\mathrm{\infty }}}\right)}^2\approx 0.\text{(2-120)}$$

Equation (2-120) is applicable to v’ as well. The implications of the small disturbance assumption for equation (2-118) are

$$a^2=a_{\mathrm{\infty }}^2-\left(\frac{\gamma -1}{2}\right)[2U_{\mathrm{\infty }}{u}^{\mathrm{\prime }}+\underset{\begin{array}{c}\text{neglect as small}\\ \text{henceforth}\end{array}}{\underbrace{u^{\mathrm{\prime }2}+v^{\mathrm{\prime }2}}}].\text{(2-121)}$$

This is a linear relation between the disturbance velocity and the speed of sound. It is a heuristic example of the procedures used in a more formal approach known as perturbation theory.

2.6.2 Small Disturbance Expansion of the Full Potential Equation

We now use an approach similar to that in section 2.6.1 to show how to obtain a small disturbance version of the full-potential equation. Again consider the situation where we assume that the disturbance to the freestream is small. Now we examine the full-potential equation. First, we rewrite the full-potential equation, equation (2-97), in 2D for simplicity as

$$({\mathrm{\Phi }}_x^2-a^2){\mathrm{\Phi }}_{xx}+2{\mathrm{\Phi }}_x{\mathrm{\Phi }}_y{\mathrm{\Phi }}_{xy}+({\mathrm{\Phi }}_y^2-a^2){\mathrm{\Phi }}_{yy}=0.\text{(2-122)}$$

Now write the velocity as a difference from the freestream velocity. Introduce a disturbance potential, ϕ, defined by

$$\begin{array}{rl}& \mathrm{\Phi }=U_{\mathrm{\infty }}x+\varphi (x,y)\\ & {\mathrm{\Phi }}_x=u=U_{\mathrm{\infty }}+{\varphi }_x\\ & {\mathrm{\Phi }}_y=v={\varphi }_y\end{array}\text{(2-123)}$$

where we have introduced a directional bias. The x-direction is the direction of the freestream velocity. We next assume that ϕ_x and ϕ_y are small compared to U_∞. Using the idea of a small disturbance to the freestream, simplified (and even linear) forms of a small disturbance potential equation and an energy equation can be derived.

As an example of the expansion process, consider the first term in equation (2-122). Use the definition of the disturbance potential and the simplified energy equation as shown in equation (2-124) below.

$$\begin{array}{rl}({\mathrm{\Phi }}_x^2-a^2)& \cong {(U_{\mathrm{\infty }}+{\varphi }_x)}^2-\{a_{\mathrm{\infty }}^2-\left(\frac{\gamma -1}{2}\right)\left[2U_{\mathrm{\infty }}{u}^{\mathrm{\prime }}\right]\}\\ & \cong U_{\mathrm{\infty }}^2+2U_{\mathrm{\infty }}{\varphi }_x+{\varphi }_x^2-a_{\mathrm{\infty }}^2+\frac{\gamma -1}{2}2U_{\mathrm{\infty }}\underset{={\varphi }_x^{\mathrm{\prime }}}{\underbrace{u^{\mathrm{\prime }}}}\end{array}\text{(2-124)}$$

Regroup and drop the square of the disturbance velocity as small:

$$\begin{array}{rl}({\mathrm{\Phi }}_x^2-a^2)& \cong U_{\mathrm{\infty }}^2-a_{\mathrm{\infty }}^2+2U_{\mathrm{\infty }}{\varphi }_x+(\gamma -1)U_{\mathrm{\infty }}{\varphi }_x\\ & \cong U_{\mathrm{\infty }}^2-a_{\mathrm{\infty }}^2+\underset{\gamma +1}{\underbrace{[2+(\gamma -1)}}]U_{\mathrm{\infty }}{\varphi }_x\\ & \cong U_{\mathrm{\infty }}^2-a_{\mathrm{\infty }}^2+(\gamma +1)U_{\mathrm{\infty }}{\varphi }_x\end{array}\text{(2-125)}$$

Dividing by a_∞²,

$$\begin{array}{rl}(\frac{{\mathrm{\Phi }}_x^2}{a_{\mathrm{\infty }}^2}-\frac{a^2}{a_{\mathrm{\infty }}^2})& \cong \frac{U_{\mathrm{\infty }}^2}{a_{\mathrm{\infty }}^2}-1+(\gamma +1)\frac{U_{\mathrm{\infty }}}{a_{\mathrm{\infty }}}\frac{{\varphi }_x}{a_{\mathrm{\infty }}}\\ & \cong M_{\mathrm{\infty }}^2-1+(\gamma +1)M_{\mathrm{\infty }}\underset{\frac{U_{\mathrm{\infty }}}{U_{\mathrm{\infty }}}\frac{{\varphi }_x}{a_{\mathrm{\infty }}}}{\underbrace{\frac{U_{\mathrm{\infty }}}{a_{\mathrm{\infty }}}}}\frac{{\varphi }_x}{U_{\mathrm{\infty }}}\\ & \cong (M_{\mathrm{\infty }}^2-1)+(\gamma +1)M_{\mathrm{\infty }}^2\left(\frac{{\varphi }_x}{U_{\mathrm{\infty }}}\right)\end{array}\text{(2-126)}$$

Rewrite the potential equation, equation (2-122), by first dividing by a_∞² and then replacing the coefficient of the first term using equation (2-126):

$$\underset{[(M_{\mathrm{\infty }}^2-1)+(\gamma +1)M_{\mathrm{\infty }}^2\left(\frac{{\varphi }_x}{U_{\mathrm{\infty }}}\right)]}{\underbrace{(\frac{{\mathrm{\Phi }}_x^2}{a_{\mathrm{\infty }}^2}-\frac{a^2}{a_{\mathrm{\infty }}^2})}}{\mathrm{\Phi }}_{xx}+2\frac{{\mathrm{\Phi }}_x}{a_{\mathrm{\infty }}}\frac{{\mathrm{\Phi }}_y}{a_{\mathrm{\infty }}}{\mathrm{\Phi }}_{xy}+(\frac{{\mathrm{\Phi }}_y^2}{a_{\mathrm{\infty }}^2}-\frac{a^2}{a_{\mathrm{\infty }}^2}){\mathrm{\Phi }}_{yy}=0\text{(2-127)}$$