A Note for Finding Complex Roots of a Special Kind of Quartic Equation

Abstract

The author's2 numerical method of solution of blunt-bod}^ flow fields was modified to allow for this heat loss by the addition of the appropriate term to the one-dimensional energy equation which is applied along each stream tube and the calculation of the progressive change in entropy along each stream tube. The method was then used to calculate the details of the flow behind a paraboloidal shock wave of nose radius 5 ft. traveling at a Mach number of 20 in air at 220°K. and 0.01 atm. Fig. 1 shows the path on a thermodynamic chart of an element of gas ver}^ close to the stagnation streamline (minimum velocity 25 ft./sec.) when thermal radiation is taken into account and also when it is neglected; and it is seen that the parameter <£ is here equal to 2/3. The greater part of the heat loss and consequent change in entropy occurs in the stagnation region where the flow velocity is small. However, there was no detectable change in the geometry of the flow or in the pressure distribution on the body, and the only significant change was in the temperature of the gas layers near the surface. The near constancy of the flow geometry and pressure distribu­ tion can be explained by the following argument. The flow is constructed as a series of one-dimension al stream tubes and the geometry of the flow remains unchanged unless there is a change in the flow velocity or density along them. The radiant heat loss causes a significant drop in temperature onl}^ in the stream tubes which pass near the stagnation point (such as that in Fig. 1). The small change in area of these stream tubes due to the re­ sultant density change does not produce any significant change in the overall flow pattern. Therefore, as long as there is no change in the flow velocity, the pressure gradients and pressures will re­ main almost unchanged; and it can be easily shown3 that, ir­ respective of the amount of heat lost, there is no change in veloc­ ity unless there is a change in pressure. This study indicates that, for practical hypersonic blunt-body flows, the velocity gradient outside the boundary layer will not be significantly altered by thermal radiation effects, and the effects on the temperature can be calculated by a step-by-step construction along the stagnation streamline using the pressure distribution given by theories which neglect radiation. These conclusions will only be affected if the parameter $ is increased by at least an order of magnitude above its value in the present example.