QUARTERLY PROGRESS REPORT NOS. 2 AND 3 CHARACTERISTIC SYSTEMS FOR ROTAT IONLAL FLOWS OF A COMPRESSIBLE GAS NATHANIEL COBURN Associate Professor of Mathematics Project 2201 ORDNANCE CORPS, U.S. ARMY, DETROIT ORDNANCE DII STRIT CONTRACT NO. DA-20-018-ORD-13282, DA PROJECT NO, 599-ci-O0C4 ORD PROJECT NO. TB2-001 (892) September, 1.954

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ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN CHARACTERISTIC SYSTEMS FOR ROTATIONAL FLOWS OF A COMPRESSIBLE GAS QUARTERLY PROGRESS REPORT NO. 2 Work Done by Mr. J. Klein from January 1, 1954 to April 1, 1954 The basic relations for straight line bicharacteristics in irrotational axial symmetric flows have been determined. These relations are of the form dJ H(a,,P,C,B,r), do = K(aP.,CBr), where B and r are linear functions of P and H and K are specified functions of their arguments. Thus, the necessary and sufficient condition for simple waves of this type is that dH dK _l = ~ ~ (2) If this last relation is solved for C, the sound speed, then Equation 1 furnishes two consistency relations. It is shown that C cannot be constant along any such simple wave. QUARTERLY PROGRESS REPORT NO. 3 Work Done by Mr. J. Klein from April 1, 1954 to July 1, 1954 Present computations indicate that no irrotational axial symmetric flows with straight line bicharacteristics exist. However, one class of

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN rotational axial symmetric flows with straight line bicharacteristics does exist. The properties of these flows will be studied in the future. Work Done by Mr. N. Coburn from June 15, 1954 to July 1, 1954 It is shown that the vortex flows previously described (see Quarterly Progress Report No. 1) are the only plane rotational flows with straight line bicharacteristics (simple waves). Future plans are to study those plane rotational flows for which the Mach number is constant along a bicharacteristic (simple waves of type two).

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