Coefficient of Pressure

Once source strength distributions Xi are obtained, the velocity tangential to the surface at each control point can be calculated as follows.

Let s be the distance along the body surface, as shown in Figure 7.3, measured positive (+ve) from front to rear. The component of freestream velocity tangential to the surface is:

Vc»,s V(X> sin Pi.

The tangential velocity Vs at the control point of the ith panel induced by all the panels is obtained by differentiating Equation (7.1) with respect to s. That is:

Note that the tangential velocity Vs on a flat source panel induced by the panel itself is zero; hence in Equation (7.6), the term corresponding to j = i is zero. This is easily seen by intuition, because the panel can emit volume flow only in a direction perpendicular to its surface and not in the direction tangential to its surface.

The surface velocity Vi at the control point of the ith panel is the sum of the contribution V, x,s from the freestream and Vs given by Equation (7.6).

The pressure coefficient Cp at the ith control point is:

(7.8)

Note: It is important to note that the pressure coefficient given by Equation (7.8) is valid only for incompressible flows with freestream Mach number less than 0.3. For compressible flows the pressure coefficient becomes:

where pi is the static pressure at the ith panel and px, px and Vx, respectively, are the pressure, density and velocity of the freestream flow. The dynamic pressure can be expressed as:

– p V[8] [9] = 2 Px x 2 yRTx

Dividing the numerator and denominator of the right-hand side by y, we have the dynamic pressure as:

yp x Vx

2 aL, ’

since aI = yRTx. This simplifies to:

2 PxV2 = Y2x mi.

Thus the pressure coefficient for compressible flows becomes:

7.2.1.1 Test on Accuracy

Let Sj be the length of the /h panel and Xj be the source strength of the /h panel per unit length. Hence, the strength of the jth panel is XjSj. For a closed body, the sum of the strengths of all the sources and sinks must of zero, or else the body itself would be adding or absorbing mass from the flow. Hence, the values of the Xjs obtained above should obey the relation:

This equation provides an independent check on the accuracy of the numerical results.