David Nixon

Nonisentropic Potential Formulation for Transonic Flows

A potential equation for nonisentropic transonic flows is formulated. This procedure captures shock waves with Rankine-Hugoniot strengths, but retains the simplicity of the traditional potential equation. Numerical computations are presented that verify the efficiency of the present procedure. It is also shown that Croccos theorem is not applicable to potential flows. A new Croccos theorem valid for potential flows is derived and it is shown that transonic flows with nonconstant shock strengths can never be irrotational, irrespective of whether the flow is isentropic or not.

Calculation of Unsteady Transonic Flows Using the Integral Equation Method

The basic integral equations for a harmonically oscillating airfoil in a transonic flow with shock waves are derived; the reduced frequency is assumed to be small. The problems associated with shock wave motion are treated using a strained coordinate system. The integral equation is linear and consists of both line integrals and surface integrals over the flow field which are evaluated by quadrature. This leads to a set of linear algebraic equations that can be solved directly. The shock motion is obtained explicitly by enforcing the condition that the flow is continuous except at a shock wave. Results obtained for both lifting and nonlifting oscillatory flows agree satisfactorily with other accurate results.

Transonic small disturbance theory with strong shock waves

Introduction T most common methods of predicting aerodynamic characteristics at transonic speeds are either the transonic small disturbance (TSD) theory or the full potential equation (FPE) theory. The more accurate Euler equation solutions are expensive to obtain, although for flows with strong shock waves such solutions are essential. The FPE theory requires that the flow is irrotational and treats the wing boundary conditions exactly (numerically). The TSD theory is an approximation to the FPE theory. One advantage of the TSD theory is the flexibility in deriving the approximate equation. This flexibility is generally utilized by a choice of a transonic scale parameter. The basic assumption of irrotationality in both these theories is only valid when the flow is shock free or contains only weak shocks. Both TSD and FPE solutions are in satisfactory agreement with realistic Euler equation solutions, provided that the basic restriction to weak shock waves is not violated. The thin wing boundary conditions can also introduce errors into the TSD solutions. If the flow has strong waves, however, then there is considerable disagreement among all three theories. Generally the predicted shock locations for the potential theories are much further aft than for the Euler equations. The problem addressed in this paper is to examine the error in the shock location in the TSD theory in two-dimensional flow and to derive a correction procedure within the confines of small disturbance theory. The basic hypothesis of the present theory is that the error in shock location is primarily due to the stronger shock strength predicted by TSD theory compared to that of the Euler equations. The technique uses two TSD solutions with different scaling parameters and an interpolation scheme derived for discontinuous transonic flows to give a corrected shock strength.

Rapid computation of unsteady transonic cascade flows

Results The typical case considered is for a freestream Mach number of 2.0, a Reynolds number of 0.296X 10 based on the distance XSHK from the leading edge to the shock impingement point, and an incident shock angle of 32.585 deg. For this set of data, the shock is strong enough (pressure ratio =1.4) to trigger separation. The computation was done for five cases: 1) no suction along the wall, 2) normal suction, 0 = 90 deg at the location from X/XSHK = 0.7817 to 1.1569,3) vectored suction, 0 = 45 deg at the locations as in case 2, 4) normal suction from X/XSHK = Q.1\92 to 0.8442, and 5) vectored suction at the same locations as in case 4. The computed surface pressure distribution in the interaction region is presented in Fig. 2. For the vectored upstream suction case 5, the pressure jump is closes to the inviscid flow conditions case and reaches its postshock value quite smoothly. It is interesting to note that the pressure rise is steeper than that corresponding to normal suction. Though the pressure plateau indicating separation has vanished for all the examples in which suction is considered, the upstream vectored suction has a minimum effect in the downstream direction (Fig. 3). This study indicates that the upstream vectored suction not only eliminates separation but that its influence in the neighborhood is limited. If the prior knowledge of separation bubble location is not available or some minor changes in the input data are required, a judicious choice of upstream vectored suction can control the flow effectively. However, a detailed assessment regarding the locations, rates, and angles, for such suction should be carried out to optimize its fruitful usage.