DERIVATION OF THE
SMALL DISTURBANCE POTENTIAL FLOW EQUATION

2.1 Preliminary Remarks

	In this chapter, and the next chapter, we develop approximate forms of the governing equations for subsonic and supersonic  flow over thin airfoils, wings and bodies of revolution. Our ultimate interest is in techniques for predicting the pressure distribution Cp and the loads (lift Cl, drag Cd and the pitching moment about the quarter chord cm) over these geometries. For simplicity, we extensively discuss only 2-D planar flows.  We recall that the exact governing equations are:

(2.1)
and

(2.2)

	It is possible to eliminate the explicit appearance of density r from equations (3.1) and (3.2), and arrive at the following quasi-linear form of the full potential equation:

(2.3)

	Equation 2.3 is called quasi-linear because it is linear in its highest derivatives fxx, fyy and fxy. It is, of course, nonlinear in f.  It is this nonlinearity that allows us to model shock waves, a very non-linear phenomenon.  From a mathematical theory called the method of characteristics, one can show that equation (3.3) is elliptic if
 (u2+v2)/a2 < 1

and hyperbolic if 
(u2+v2)/a2 > 1

and parabolic if
(u2+v2)/a2 = 1

2.2 Loads over the Airfoil

We define the surface pressure coefficient Cp as,

(2.4)

	The airloads over the airfoil may be found once Cp is known as follows:

(2.5)

	where Y(x) is the airfoil shape and a is the angle of attack.

	We are interested in the solution of the governing equations, in the entire region between the airfoil and infinity. Because of the nonlinearity of the exact form of the governing equations, we can not analytically solve the governing equations except in supersonic flows, using the method of characteristics. Thus, further approximations to the governing equations are necessary.

2.3 Small Disturbance Assumptions

	The assumptions we make are:

(i)	The body is thin, has a small angle of attack, and has only a mild camber. As a result, the body slope dY/dx, in a coordinate system attached to the freestream (known as the wind tunnel coordinate system) is small.

(ii)	As a consequence, we assume that the local flow velocity components u and v are not significantly different from their freestream values. 

Disturbance Potential j:

	We introduce a disturbance potential, j, related to the full potential f as follows:

f = V∞ x +  j
(2.6)

	It must be remembered that in this chapter we are using the wind tunnel coordinate system, and the freestream velocity is parallel to the x- axis. From equation (3.6), we obtain the flow velocity components terms of j as follows:

(2.7)
	From the small disturbance assumptions, it then follows that

(2.8)
3.4 Derivation of the Small Disturbance Equation

	Our starting point is the quasi-linear form of the full potential equation 2.3.  From equation 2.6 we note that

(2.9)

	Next, consider the second term in equation (2.3),  -2uvfxy . This term may be viewed as

         		 small
(2.10)
	Next, consider the coefficient a2-u2 , in front of the first term in (2.3).  Noting the fact that the speed of sound is related to the flow speed by the energy equation:

(2.11)
We can approximate this coefficient as follows:

(2.12)
	Examine the above approximation carefully. Note that we have neglected second powers of the "disturbance velocities" as small. We have, however, kept the first power of the term  jx/V∞ . This is because the term (1-M∞2) itself may be small in transonic flows. Thus, in transonic flows only the last term in the above approximation for (a2-u2) may be comparable in magnitude to (1-M∞2) , and can not be neglected.

	In a very similar manner, we can show that

(2.13)
	With these approximations, the quasi-linear form of the full potential equation takes on the following simpler form for transonic flows:

(2.14)

	In subsonic and supersonic flows, the small disturbance equation becomes:

(2.15)
Exercise 2.1

	Show that the axi-symmetric form of the transonic small disturbance equation is given by

2.5 Mathematical Characteristics of the Small Disturbance Equation:

(a)	The transonic small disturbance equation is still nonlinear, in spite of the approximations that were made to arrive at this equation.  In subsonic and supersonic flows, for thin airfoils, wings and bodies, the governing equation (2.15) is linear.

(b)	Equation (2.15) may be elliptic, hyperbolic or parabolic.  This was given as a homework problem earlier.

2.6  Small Disturbance Approximation for Surface pressure Coefficient., Cp:

	We next develop a small disturbance approximation for the surface pressure. Our starting point is equations (2.2) and (2.4), which may be combined to yield:

	In arriving at the above form, we have neglected second powers of terms such as jx/V¥ and jy/V¥ .

	Next, we use the binomial expansion

(2.16)
	where e is any small quantity e << 1 and n is any real number. Then,

Simplifying, the following small disturbance approximation to surface pressure coefficient results:

(2.17)

2.7 Boundary Conditions

	Before the transonic small disturbance equation (2.15) may be solved, we need to specify the boundary conditions. Of course, the boundary conditions must take into account the physics of the problem, and the mathematical characteristics of the equation. 

Boundary Conditions at the Solid Boundary:

y = Y(x)

 

	At nay point on the body surface, the flow must be tangential to the body. In other words, the slope of the velocity vector  must equal the body slope. 

	Neglecting the disturbance velocity contribution jx/V¥  in comparison to unity, we get the following boundary condition at the airfoil surface.

(2.18)

	Where should this boundary condition be applied?  We have two choices. This boundary condition may be applied either at the actual airfoil surface, or on a slit along the chord line, located on the x- axis.  The latter choice makes the solution procedure simpler because we can use a Cartesian coordinate system, rather than a curvilinear coordinate system that is wrapped around the body. Within the assumptions built into the small disturbance theory, these two approaches may be shown to be equivalent.  Note that this slit is a discontinuity, across which both the disturbance velocity potential j and its y- derivative are discontinuous.