Thermodynamics

where we must be careful to measure the angle relative to the new directionof flow downstream of the expansion, namely, parallel to the upper wall ofthe wedge in Fig. 17–45 if we neglect the influence of the boundary layeralong the wall. But how do we determine Ma 2 ? It turns out that the turningangle u across the expansion fan can be calculated by integration, makinguse of the isentropic flow relationships. For an ideal gas, the result is(Anderson, 2003),Turning angle across an expansion fan: u n 1Ma 2 2 n 1Ma 1 2 (17–48)where n(Ma) is an angle called the Prandtl–Meyer function (not to be confusedwith the kinematic viscosity),n 1Ma2 Bk 1k 1 tan1 c Bk 1k 1 1Ma2 12 d tan 1 a 2Ma 2 1 b(17–49)Note that n(Ma) is an angle, and can be calculated in either degrees or radians.Physically, n(Ma) is the angle through which the flow must expand,starting with n 0 at Ma 1, in order to reach a supersonic Mach number,Ma 1.To find Ma 2 for known values of Ma 1 , k, and u, we calculate n(Ma 1 ) fromEq. 17–49, n(Ma 2 ) from Eq. 17–48, and then Ma 2 from Eq. 17–49, notingthat the last step involves solving an implicit equation for Ma 2 . Since thereis no heat transfer or work, and the flow is isentropic through the expansion,T 0 and P 0 remain constant, and we use the isentropic flow relations derivedpreviously to calculate other flow properties downstream of the expansion,such as T 2 , r 2 , and P 2 .Prandtl–Meyer expansion fans also occur in axisymmetric supersonicflows, as in the corners and trailing edges of a cone-cylinder (Fig. 17–46).Some very complex and, to some of us, beautiful interactions involvingboth shock waves and expansion waves occur in the supersonic jet producedby an “overexpanded” nozzle, as in Fig. 17–47. Analysis of suchflows is beyond the scope of the present text; interested readers are referredto compressible flow textbooks such as Thompson (1972) and Anderson(2003).Ma 1 1Chapter 17 | 857ObliqueshockExpansionwavesm 1m 2Ma 2FIGURE 17–45An expansion fan in the upperportion of the flow formed by a twodimensionalwedge at the angle ofattack in a supersonic flow. The flowis turned by angle u, and the Machnumber increases across the expansionfan. Mach angles upstream anddownstream of the expansion fan areindicated. Only three expansion wavesare shown for simplicity, but in fact,there are an infinite number of them.(An oblique shock is present in thebottom portion of this flow.)duFIGURE 17–46A cone-cylinder of 12.5 half-angle ina Mach number 1.84 flow. Theboundary layer becomes turbulentshortly downstream of the nose,generating Mach waves that are visiblein this shadowgraph. Expansion wavesare seen at the corners and at thetrailing edge of the cone.Photo by A. C. Charters, Army Ballistic ResearchLaboratory.