S. Mölder

Limiting Contractions for Starting Simple Ramp-Type Scramjet Intakes with Overboard Spillage

A method for obtaining the limiting contraction for supersonic intake-starting via overboard spillage is demonstrated for a simple ramp-type intake family. The strong-shock design principle is proposed on the basis of comparison of the limiting contraction line with the Kantrowitz (self-starting) lines of a few particular ramp intakes. Predicted starting characteristics compare favorably with two-dimensional inviscid numerical simulations.

Curved shock theory

Curved shock theory (CST) is introduced, developed and applied to relate pressure gradients, streamline curvatures, vorticity and shock curvatures in flows with planar or axial symmetry. Explicit expressions are given, in an influence coefficient format, that relate post-shock pressure gradient, streamline curvature and vorticity to pre-shock gradients and shock curvature in steady flow. The effect of pre-shock flow divergence/convergence, on vorticity generation, is related to the transverse shock curvature. A novel derivation for the post-shock vorticity is presented that includes the effects of pre-shock flow non-uniformities. CST applicability to unsteady flows is discussed.

Reflection of curved shock waves

Shock curvatures are related to pressure gradients, streamline curvatures and vorticity in flows with planar and axial symmetry. Explicit expressions, in an influence coefficient format, are used to relate post-shock pressure gradient, streamline curvature and vorticity to pre-shock gradients and shock curvature in steady flow. Using higher order, von Neumann-type, compatibility conditions, curved shock theory is applied to calculate the flow near singly and doubly curved shocks on curved surfaces, in regular shock reflection and in Mach reflection. Theoretical curved shock shapes are in good agreement with computational fluid dynamics calculations and experiment.

Flow behind concave shock waves

Curved shock theory is introduced and applied to calculate the flow behind concave shock waves. For sonic conditions, three characterizing types of flow are identified, based on the orientation of the sonic line, and it is shown that, depending on the ratio of shock curvatures, a continuously curving shock can exist with Type III flow, where the sonic line intercepts the reflected characteristics from the shock, thus preventing the formation of a reflected shock. The necessary shock curvature ratio for a Type III sonic point does not exist for a hyperbolic shock so that it will revert to Mach reflection for all Mach numbers. A demonstration is provided, by CFD calculations, at Mach 1.2 and 3.

Shock detachment from curved wedges

Curved shock theory is used to show that the flow behind attached shocks on doubly curved wedges can have either positive or negative post-shock pressure gradients depending on the freestream Mach number, the wedge angle and the two wedge curvatures. Given enough wedge length, the flow near the leading edge can choke to force the shock to detach from the wedge. This local choking can preempt both the maximum deflection and the sonic criteria for shock detachment. Analytical predictions for detachment by local choking are supported by CFD results.

On the Self-starting Constraints for Busemann Intakes with Overboard Spillage

The air intake is an important component of hypersonic airbreathing engines. It is essentially a converging duct decelerating and compressing airflow and supplying the compressed air to the engine’s combustor. Molder and Szpiro [1] proposed to use the Busemann flow [2] as the basis for hypersonic air intake design.

Startability analysis of Busemann intakes with overboard spillage

In this work quasi-steady starting of Busemann intakes with overboard mass spillage is considered analytically and simulated numerically using 2D and 3D unstructured adaptive Euler finite-volume flow solvers. Two designs of Busemann intakes are under consideration: the conventional one based on Busemann flow with a weak conical shock and a newly proposed one based on Busemann flow with a strong conical shock. For both designs, the overboard spillage is achieved by cutting out an angular section of the full Busemann intake and covering the cuts with flat plates allowing for overboard spillage and maintaining the Busemann flow in the started mode. The theory predicting the self-starting (spontaneous starting) area ratios for Busemann intakes with overboard spillage is outlined. Methodology of finding the self-starting area ratios for a given free-stream Mach number via numerical experiments is discussed and applied to Busemann intakes with different capture angles. The numerical findings confirm the theoretical predictions of self-starting boundaries. It is demonstrated that overboard spillage significantly improves the starting characteristics of Busemann intakes. Furthermore, the strong-shock design leads to markedly better starting characteristics.

Strength of Characteristics at a Curved Shock Wave

The strength of characteristic waves is related to the local gradient and streamline curvature. This relationship and the equations giving the pressure gradient and streamline curvature are used to determine the relative strengths (the reflection coefficient) of characteristics just downstream of a two-dimensional curved shock wave. It is shown that the characteristics’ strengths are a complex function of the specific heat ratio, the upstream Mach number and shock angle and vary directly with the shock curvature. The reflection coefficient, which is independent of shock curvature, is used to characterise four different types of shock wave whose existence depends on the specific heat ratio of the gas and the upstream flow Mach number. The nature of reflection at the shock’s downstream surface may change up to four times and this is posed as the explanation for the inflected shocks that have been observed both experimentally and computationally. It is concluded that such approximate analytical methods as the Tangent-Wedge should not be used when strong curved shocks are present and that the nature of wave reflections behind a weak shock in air is not properly simulated by tests with helium.