Newtonian Theory

Return to Figure 14.1; note how close the shock wave lies to the body surface. This figure is redrawn in Figure 14.5 with the streamlines added to the sketch. When viewed from afar, the straight, horizontal streamlines in the freestream appear to almost impact the body, and then move tangentially along the body. Return to Figure 1.1, which illustrates Isaac Newton’s model for fluid flow, and compare it with the hypersonic flow field shown in Figure 14.5; they have certain distinct similarities. (Also, review the discussion surrounding Figure 1.1 before progressing further.) Indeed, the thin shock layers around hypersonic bodies are the closest example in fluid mechanics to Newton’s model. Therefore, we might expect that results based on Newton’s model would have some applicability in hypersonic flows. This is indeed the case; newtonian theory is used frequently to estimate the pressure distribution over the surface of a hypersonic body. The purpose of this section is to derive the famous newtonian sine – squared law first mentioned in Section 1.1 and to show how it is applied to hypersonic flows.

Consider a surface inclined at the angle в to the freestream, as sketched in Fig­ure 14.6. According to the newtonian model, the flow consists of a large number of individual particles which impact the surface and then move tangentially to the surface. During collision with the surface, the particles lose their component of mo­mentum normal to the surface, but the tangential component is preserved. The time rate of change of the normal component of momentum equals the force exerted on the surface by the particle impacts. To quantify this model, examine Figure 14.6. The component of the freestream velocity normal to the surface is Too sin в. If the area of the surface is A, the mass flow incident on the surface is p(A sin 9)V0O. Hence, the

Figure 14.5 Streamlines in a hypersonic flow.

time rate of change of momentum is

Mass flow x change in normal component of velocity

or (PccVccA sin0)(Voosin(9) = PocV^A sin2#

In turn, from Newton’s second law, the force on the surface is

IV = PocV^A sin2 в [14.1]

This force acts along the same line as the time rate of change of momentum (i. e., normal to the surface), as sketched in Figure 14.6. From Equation (14.1), the normal force per unit area is

= PccV^ sin20 [14.2]

Let us now interpret the physical meaning of the normal force per unit area in Equa­tion (14.2), N/A, in terms of our modem knowledge of aerodynamics. Newton’s model assumes a stream of individual particles all moving in straight, parallel paths toward the surface; that is, the particles have a completely directed, rectilinear motion. There is no random motion of the particles—it is simply a stream of particles such as pellets from a shotgun. In terms of our modern concepts, we know that a moving gas has molecular motion that is a composite of random motion of the molecules as well as a directed motion. Moreover, we know that the freestream static pressure рж is simply a measure of the purely random motion of the molecules. Therefore, when the purely directed motion of the particles in Newton’s model results in the normal force per unit area, N/A in Equation (14.2), this normal force per unit area must be construed as the pressure difference above px, namely, p — p^ on the surface. Hence, Equation (14.2) becomes

P – Poo = Poo sin2 в [ 14.3]

Equation (14.3) can be written in terms of the pressure coefficient Cp — (p — Poe)/ poeV£>’ as follows

= 2 sin2#

Equation (14.4) is Newton’s sine-squared law; it states that the pressure coefficient is proportional to the sine square of the angle between a tangent to the surface and the direction of the freestream. This angle в is illustrated in Figure 14.7. Frequently, the results of newtonian theory are expressed in terms of the angle between a normal to the surface and the freestream direction, denoted by ф as shown in Figure 14.7. In terms of ф, Equation (14.4) becomes

Cp — 2 cos2 ф

which is an equally valid expression of newtonian theory.

Consider the blunt body sketched in Figure 14.7. Clearly, the maximum pressure, hence the maximum value of Cp, occurs at the stagnation point, where в = тг/2 and ф — 0. Equation (14.4) predicts Cp = 2 at the stagnation point. Contrast this hypersonic result with the result obtained for incompressible flow theory in Chapter 3, where Cp = 1 at a stagnation point. Indeed, the stagnation pressure coefficient increases continuously from 1.0 at M= 0 to 1.28 at Мж = 1.0 to 1.86 for у = 1.4 as Mqo —► °o. (Prove this to yourself.)

The result that the maximum pressure coefficient approaches 2 at Moo —► oo can be obtained independently from the one-dimensional momentum equation, namely, Equation (8.6). Consider a normal shock wave at hypersonic speeds, as sketched in Figure 14.8. For this flow, Equation (8.6) gives

Poo + PooV^ = P2 + P2V2 [14.6]

Recall that across a normal shock wave the flow velocity decreases, V2 < Fool indeed, the flow behind the normal shock is subsonic. This change becomes more severe as Mco increases. Hence, at hypersonic speeds, we can assume that (pж V^) УР (p2

As stated above, the result that Cp = 2 at a stagnation point is a limiting value as Moo —*■ oo. For large but finite Mach numbers, the value of Cp at a stagnation point is less than 2. Return again to the blunt body shown in Figure 14.7. Considering the distribution of Cp as a function of distance 5 along the surface, the largest value of Cp will occur at the stagnation point. Denote the stagnation point value of Cp by Cp, max, as shown in Figure 14.7. Cp, max for a given Mx can be readily calculated from normal shock-wave theory. [If у = 1.4, then Cp, max can be obtained from Рол/Pi — Рол/Poo, tabulated in Appendix B. Recall from Equation (11.22) that Cp, max = (2/уМ^0)(рол/Poo ~ 1) ] Downstream of the stagnation point, Cp can be assumed to follow the sine-squared variation predicted by newtonian theory; that is,

Equation (14.7) is called the modified newtonian law. For the calculation of the Cp dis­tribution around blunt bodies, Equation (14.7) is more accurate than Equation (14.4).

Return to Figure 13.14, which gives the numerical results for the pressure distri­butions around a blunt, parabolic cylinder at = 4 and 8. The open symbols in this figure represent the results of modified newtonian theory, namely, Equation (14.7). For this two-dimensional body, modified newtonian theory is reasonably accurate only in the nose region, although the comparison improves at the higher Mach num­bers. It is generally true that newtonian theory is more accurate at larger values of both Moo and в. The case for an axisymmetric body, a paraboloid at Moo = 4, is given in Figure 14.9. Here, although Moo is relatively low, the agreement between the time- dependent numerical solution (see Chapter 13) and newtonian theory is much better.

paraboloid, = 4. Comparison of modified newtonian theory and time-dependent finite-difference calculations.

It is generally true that newtonian theory works better for three-dimensional bodies. In general, the modified newtonian law, Equation (14.7), is sufficiently accurate that it is used very frequently in the preliminary design of hypersonic vehicles. Indeed, extensive computer codes have been developed to apply Equation (14.7) to three­dimensional hypersonic bodies of general shape. Therefore, we can be thankful to Isaac Newton for supplying us with a law which holds reasonably well at hypersonic speeds, although such an application most likely never crossed his mind. Neverthe­less, it is fitting that three centuries later, Newton’s fluid mechanics has finally found a reasonable application.