F18 – Lecture Notes

F18 – Lecture Notes 1. Prandtl-Meyer Waves 2. Shock-Expansion Theory (Supersonic Airfoils) Reading: Anderson 9.6, 9.7 Prandtl-Meyer Waves Wave flow relations An expansion fan, sometimes also called a Prandtl-Meyer expansion wave, can be considered as a continuous sequence of infinitesimal Mach expansion waves. To analyze this continuous change, we will now consider the flow angle � to be a flowfield variable, like M or V . Across each Mach wave of the fan, the flow direction changes by d�, while the speed changes by dV . Oblique-shock analysis dictates that only the normal velocity component u can change across any wave, so that dV must be entirely due to the normal-velocity change du. e av a M V M1 �1 V1 M2 �2 V2 w ch µ V d� V u µ dV tan µ � u du dV du dV From the u-V and du-dV velocity triangles, it is evident that d� and dV are related by d� = dV 1 tan µ V assuming d� is a small angle. With sin µ = 1/M , we have 1 cos µ = = tan µ sin µ � 1 − sin2 µ = sin µ � 1 − 1/M 2 1/M = � M2 − 1 and so the flow relation above becomes d� = � dV M2 − 1 V (1) This is a differential equation which relates a change d� in the flow angle to a change dV in the flow speed throughout the expansion fan. Prandtl-Meyer Function The differential equation (1) can be integrated if we first express V in terms of M . V = M a = M ao � 1+ 1 α −1 2 M 2 �−1/2 1 α −1 2 M ln V = ln M + ln ao − ln 1 + 2 2 � � α −1 2 −1 α −1 dV dM 1 M 1+ 2M dM = − 2 V M 2 2 dV 1 dM = �−1 2 V 1+ 2 M M � � Equation (1) then becomes d� = � M 2 − 1 dM 1 + �−1 M2 M 2 (2) which can now be integrated from any point 1 to any point 2 in the Prandtl-Meyer wave. � �2 M2 M 2 − 1 dM d� = �−1 �1 M1 1 + M2 M 2 �2 − �1 = �(M2 ) − �(M1 ) where �(M ) � � (3) � � α +1 α −1 arctan (M 2 − 1) − arctan M 2 − 1 α −1 α +1 (4) Here, �(M ) is called the Prandtl-Meyer function, and is shown plotted for α = 1.4 . 50 45 40 35 v(M) deg. 30 25 20 15 10 5 0 1.0 1.2 1.4 1.6 1.8 2.0 M 2.2 2.4 2.6 2.8 3.0 Equation (3) can be applied to any two points within an expansion fan, but the most common use is to relate the two flow conditions before and after the fan. Reverting back to our previous notation where � is the total turning of the corner, the relation between � and the upstream and downstream Mach number is � = �(M2 ) − �(M1 ) 2 (5) �(M ) � M1 M2 � M1 M M2 This can be considered an implicit definition of M2 (M1 , �), which can be evaluated graphi­ cally using the �(M ) function plot, as shown in the figure. Shock-Expansion Theory The combination of oblique-shock relations and Prandtl-Meyer wave relations constitutes Shock-Expansion Theory , which can be used to determine the flow properties and forces about simple 2-D shapes in supersonic flow. Flat-plate supersonic airfoil A flat plate is the simplest supersonic airfoil. When set at an angle of attack �, the leading edge point effectively is a convex corner to the upper surface flow, with turning angle � = �. The upper flow then passes through the resulting Prandtl-Meyer expansion, which increases its Mach number from M1 = M� , to a larger value M2 = Mu . The latter is implicitly determined via the Prandtl-Meyer relation (5). � = �(Mu ) − �(M� ) Mu (M� , �) � The corresponding upper-surface pressure is then given by the isentropic relation. pu = p� � 2 M� 1 + �−1 2 Mu2 1 + �−1 2 ��/(�−1) Conversely, the leading edge is a concave corner to the bottom surface flow, which then sees a pressure rise through the resulting oblique shock. The lower-surface Mach number M� and pressure p� are obtained from oblique-shock relations, with M1 = M� , � = � as the inputs. L’ M p pu pl R’ D’ � c The pressure difference produces a resultant force/span R� acting normal to the plate, which can be resolved into lift and drag components. R� = (p� − pu ) c L� = (p� − pu ) c cos � D � = (p� − pu ) c sin � 3 It’s worthwhile to note that this supersonic airfoil has a nonzero drag even though the flow is being assumed inviscid. Hence, d’Alembert’s Paradox definitely does not apply to supersonic 2-D flow. The drag D � computed here is associated with the viscous dissipation occurring in the oblique shock waves, and hence is called wave drag . This wave drag is an additional drag component in supersonic flow, and must be added to the usual viscous friction drag, and also the induced drag in 3-D cases. 4