Supersonic Flow Over a Cone (Taylor-Maccoll)

Overview

A sharp cone at zero incidence in a uniform supersonic stream produces an attached conical shock and a flow field that is self-similar along rays from the tip: every flow quantity is constant along a cone of constant half-angle, depending only on the conical angle. This structure admits an exact reduction of the steady Euler equations to a single ordinary differential equation , the Taylor-Maccoll equation, whose solution gives the cone-surface pressure, Mach number and temperature with no empirical input. That makes the case a verification benchmark: the reference is an exact solution of the governing equations, not measured data.

We solve the ${10}^{\circ }$ half-angle cone at free-stream Mach $M_{\infty }=2.35$ (the NPARC Alliance verification geometry) and compare Luminary's cone-surface state to the Taylor-Maccoll solution. Because the body is axisymmetric and Luminary has no dedicated axisymmetric mode, the cone is modelled as a thin angular wedge sector of the meridional half-plane revolved a few degrees about the axis, with both angular faces treated as symmetry planes, a way to recover an axisymmetric flow on a three-dimensional unstructured solver.

Problem Setup

The geometry is a cone of half-angle ${10}^{\circ }$ aligned with the free stream. A meridional slice (axial coordinate $x$, radial coordinate $r$) is revolved through a small wedge angle to build a one-cell-thick sector mesh; the two angular faces are symmetry planes, the cone surface is an inviscid (slip) wall, and the outer and upstream boundaries are supersonic free-stream. The sharp cone apex sits exactly on the rotation axis, where the sector cells reduce to standard prism, pyramid and tetrahedral elements.

The flow is inviscid and compressible (ideal gas, $\gamma =1.4$), solved with a density-based formulation using a second-order FDS scheme with the Venkatakrishnan limiter for shock-stable capture of the conical shock. The free stream is $M_{\infty }=2.35$. Because the field is fully supersonic, information only propagates downstream and the conical solution establishes almost immediately.

The reference is generated by integrating the Taylor-Maccoll equation for these exact conditions ($M_{\infty }=2.35$, ${10}^{\circ }$ cone, $\gamma =1.4$), giving the shock angle and the uniform cone-surface state.

Quantities of Interest

The verification quantity is the cone-surface state, the pressure ratio $p/p_{\infty }$, the surface Mach number $M_c$, and the temperature ratio $T/T_{\infty }$, averaged over the conical region of the surface (away from the apex transient and the downstream boundary). These are compared directly to the Taylor-Maccoll values.

Since the reference is an exact analytic solution rather than a published code-on-grids dataset, the grid is anchored to NASA's own published meridional grid for this geometry and uniformly refined once coarser and once finer. We track the mean cone-surface pressure ratio as a function of grid spacing $h=(1/N{)}^{1/2}$; convergence toward the Taylor-Maccoll value confirms the agreement is grid-converged rather than an artifact of a single mesh.