Taylor–Maccoll flow
Taylor–Maccoll flow refers to the steady flow behind a conical shock wave that is attached to a solid cone.
The flow is named after G. I. Taylor and J. W. Maccoll, whom described the flow in 1933, guided by an earlier work of Theodore von Kármán.
[1][2][3] Consider a steady supersonic flow past a solid cone that has a semi-vertical angle
A conical shock wave can form in this situation, with the vertex of the shock wave lying at the vertex of the solid cone.
If it were a two-dimensional problem, i.e., for a supersonic flow past a wedge, then the incoming stream would have deflected through an angle
Such a simple turnover of streamlines is not possible for three-dimensional case.
After passing through the shock wave, the streamlines are curved and only asymptotically they approach the generators of the cone.
The curving of streamlines is accompanied by a gradual increase in density and decrease in velocity, in addition to those increments/decrements effected at the shock wave.
[4] The direction and magnitude of the velocity immediately behind the oblique shock wave is given by weak branch of the shock polar.
beyond which shock polar do not provide solution under in which case the conical shock wave will have detached from the solid surface (see Mach reflection).
The flow immediately behind the oblique conical shock wave is typically supersonic, although however when
The supersonic flow behind the shock wave will become subsonic as it evolves downstream.
Since all incident streamlines intersect the conical shock wave at the same angle, the intensity of the shock wave is constant.
This particularly means that entropy jump across the shock wave is also constant throughout.
Since the problem do not have any length scale and is clearly axisymmetric, the velocity field
This means that we have The steady potential flow is governed by the equation[4] where the sound speed
is expressed as a function of the velocity magnitude
Substituting the above assumed form for the velocity field, into the governing equation, we obtain the general Taylor–Maccoll equation
The equation is simplified greatly for a polytropic gas for which
(the speed of the potential flow when it flows out into a vacuum), we obtain, for the polytropic gas, the Taylor–Maccoll equation, The equation must satisfy the condition that
(no penetration on the solid surface) and also must correspond to conditions behind the shock wave at
is the half-angle of shock cone, which must be determined as part of the solution for a given incoming flow Mach number
The Taylor–Maccoll equation has no known explicit solution and it is integrated numerically.
When the cone angle is very small, the flow is nearly parallel everywhere in which case, an exact solution can be found, as shown by Theodore von Kármán and Norton B. Moore in 1932.
[2] The solution is more apparent in the cylindrical coordinates
We expect the velocity components to depend only on
The governing equation reduces to On the surface of the cone
In the small-angle approximation, the weak shock cone is given by
describes the uniform flow upstream of the shock cone, whereas the non-trivial solution satisfying the boundary condition on the solid surface behind the shock wave is given by We therefore have[4] exhibiting a logarthmic singularity as
The velocity components are given by The pressure on the surface of the cone
