FUNDAMENTAL EQUATIONS OF AERODYNAMICS

In the following the tensor notation will be used. The components of a vector or a tensor are referred to a system of rectangular Cartesian coordinates (xv x2, xj which will be written as (x, y, z) if convenient. The components of a velocity vector will be denoted by (ult u2, us) or by (;и, v, w). Similarly the components of other quantities will be represented either by a subscript or by a self-explanatory triplet of letters, whichever be the more convenient in special instances. The Roman indices always range through 1, 2, 3, unless otherwise stated. The summation con­vention (p. 5) will be used: Any index repeated twice in the same term indicates a summation over the total range of that index.

In the tensor notation the fundamental equations of aerodynamics of a nonviscous fluid are

1. The Eulerian equation of motion (law of conservation of momentum)

2. The equation of continuity (law of conservation of mass)

where p is the fluid density, p the pressure, Fi the force per unit volume acting on the fluid (such as gravitation), щ the velocity components, and /the time. DuJDt denotes the acceleration of a particle of the fluid. To express DuJDt in terms of the space and time derivatives of the velocity field, note that, if the position of an element of the fluid is described by xjt), the velocity of the fluid element is

dx{

//, = — = щ(х1г x2, x3; t)

which is a function of space and time. By the usual rule of differentiation we obtain

Dui Э щ Э u{ dxs Эи і Э uf ~Dt = It + Э*, It = Э? + Щ Ц

The acceleration of a fluid particle is written as DuJDt to distinguish it from the partial derivative dujdt. Using Eq. 3, the equation of motion can be written as

The derivation of these equations can be found in any book on theoretical aerodynamics (e. g., Ref. 1.46). In the airfoil theory con­sidered below, the body force iq can be omitted, since the only significant body force, the gravitation, introduces only a field of hydrostatic pressure which does not concern us. The assumption that the fluid is nonviscous will be made throughout the following discussion, not because the effect of viscosity is unimportant, but because we shall consider only the flow over a thin airfoil at a small angle of attack without separation, in which case the boundary-layer theory shows that the fluid outside the boundary layer may be regarded as nonviscous and the effect of viscosity can be stated in a phenomenological rule that the velocity must remain finite and tangent to the airfoil at the sharp trailing edge. This assumption was put forward by M. Wilhelm Kutta (1867-1944) and Nikolai E. Joukowski (1847-1921) independently, and is called the Kutta-Joukowski condition.

For a compressible fluid, Eqs. 1 and 2 do not suffice in defining uniquely the flow. It is necessary to know also the thermal and caloric states of the fluid and the heat transfer. For an example of the ideal gas, the thermal equation of state is р/р = RT, and the caloric equation of state is given by the relationship between the internal energy and the tempera­ture. These, in addition to an equation expressing the balance of heat and mechanical energy (the first law of thermodynamics), define a flow uniquely for proper boundary conditions.

The analysis can be greatly simplified if it is possible to assume that the fluid is piezotropic, for which the density p is a unique function of pressure. For a piezotropic fluid the potential energy of the fluid can be defined by pressure alone and an integration of the equation of motion along a stream line defines the energy balance completely. Then Eqs. 1 and 2 are sufficient to determine the flow. Fortunately, this is the case in thin airfoil theory, which deals with airfoils of infinitesimal thickness at small angle of attack performing motions of infinitesimal amplitude. The disturbances caused by the airfoil in a flow is thus infinitesimal, and shock waves, if any, will be of infinitesimal strength. No external heat source will be considered. Under these circumstances it can be shown that the change of entropy in the entire field of flow is an infinitesimal quantity of higher order of smallness. Thus the flow may be correctly regarded as isentropic, and the relation

p/pY = const (5)

holds for the entire field, у being the ratio of the specific heats cjcv. c„ is the specific heat of the gas at constant pressure, and c„ is that at constant volume.