Taylor–Maccoll Flow
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Taylor–Maccoll flow refers to the steady flow behind a conical
shock wave In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a me ...
that is attached to a solid cone. The flow is named after
G. I. Taylor Sir Geoffrey Ingram Taylor Order of Merit, OM Royal Society of London, FRS FRSE (7 March 1886 – 27 June 1975) was a British physicist and mathematician, who made contributions to fluid dynamics and wave theory. Early life and education Tayl ...
and J. W. Maccoll, whom described the flow in 1933, guided by an earlier work of
Theodore von Kármán Theodore von Kármán ( , May 11, 1881May 6, 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who worked in aeronautics and astronautics. He was responsible for crucial advances in aerodynamics characterizing ...
.Von Karman, T., & Moore, N. B. (1932). Resistance of slender bodies moving with supersonic velocities, with special reference to projectiles. Transactions of the American Society of Mechanical Engineers, 54(2), 303-310.Maccoll, J. W. (1937). The conical shock wave formed by a cone moving at a high speed. Proceedings of the Royal Society of London. Series A-Mathematical and Physical Sciences, 159(898), 459-472.


Mathematical description

Consider a steady
supersonic Supersonic speed is the speed of an object that exceeds the speed of sound (Mach 1). For objects traveling in dry air of a temperature of 20 °C (68 °F) at sea level, this speed is approximately . Speeds greater than five times ...
flow past a solid cone that has a semi-vertical angle \chi. 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 \chi upon crossing the shock wave so that streamlines behind the shock wave would be parallel to the wedge sides. 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.Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. section 123. pages 432-434. The direction and magnitude of the velocity immediately behind the oblique shock wave is given by weak branch of the shock polar. This particularly suggests that for each value of incoming Mach number M_1, there exists a maximum value of \chi_ 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 Mach reflection is a supersonic fluid dynamics effect, named for Ernst Mach, and is a shock wave reflection pattern involving three shocks. Introduction Mach reflection can exist in steady, pseudo-steady and unsteady flows. When a shock wave, ...
). These detached cases are not considered here. The flow immediately behind the oblique conical shock wave is typically supersonic, although however when \chi is close to \chi_, it can be subsonic. 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 Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
jump across the shock wave is also constant throughout. In this case, the flow behind the shock wave is a
potential flow In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
. Hence we can introduce the velocity potential \varphi such that \mathbf v = \nabla\varphi. Since the problem do not have any length scale and is clearly axisymmetric, the velocity field \mathbf v and the pressure field p will be turn out to functions of the polar angle \theta only (the origin of the spherical coordinates (r,\theta,\phi) is taken to be located at the vertex). This means that we have :\varphi=rf(\theta), \quad v_r = f(\theta), \quad v_\theta=f'(\theta), \quad v_\phi=0, \quad p = g(\theta). The steady
potential flow In fluid dynamics, potential flow or irrotational flow refers to a description of a fluid flow with no vorticity in it. Such a description typically arises in the limit of vanishing viscosity, i.e., for an inviscid fluid and with no vorticity pre ...
is governed by the equation :c^2\nabla\cdot\mathbf v - \mathbf v\cdot (\mathbf v \cdot \nabla)\mathbf v=0, where the
sound speed The speed of sound is the distance travelled per unit of time by a sound wave as it propagates through an elastic medium. More simply, the speed of sound is how fast vibrations travel. At , the speed of sound in air is about , or in or one m ...
c=c(v) is expressed as a function of the velocity magnitude v^2=(\nabla\phi)^2 only. Substituting the above assumed form for the velocity field, into the governing equation, we obtain the general Taylor–Maccoll equation :(c^2-f'^2) f'' + c^2 \cot\theta f' + (2c^2-f'^2) f = 0, \quad c = c(f^2+f'^2). The equation is simplified greatly for a polytropic gas for which c^2 = (\gamma-1)(h_0-v^2/2), i.e., :c^2 = (\gamma-1)h_0 \left(1-\frac\right), where \gamma is the
specific heat ratio In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...
and h_0 is the stagnation enthalpy. Introducing this formula into the general Taylor–Maccoll equation and introducing a non-dimensional function F(\theta) = f(\theta)/v_, where v_= \sqrt (the speed of the potential flow when it flows out into a vacuum), we obtain, for the polytropic gas, the Taylor–Maccoll equation, :\left fracF'^2-\frac(1-F^2)\right'' = (\gamma-1) (1-F^2) F + \frac\cot\theta(1-F^2)F' - \gamma F F'^2 - \frac\cot\theta F'^3. The equation must satisfy the condition that F'(\chi)=0 (no penetration on the solid surface) and also must correspond to conditions behind the shock wave at \chi=\psi, where \psi is the half-angle of shock cone, which must be determined as part of the solution for a given incoming flow Mach number M and \gamma. The Taylor–Maccoll equation has no known explicit solution and it is integrated numerically.


Kármán–Moore solution

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 Theodore von Kármán ( , May 11, 1881May 6, 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who worked in aeronautics and astronautics. He was responsible for crucial advances in aerodynamics characterizing ...
and Norton B. Moore in 1932. The solution is more apparent in the cylindrical coordinates (\rho,\varpi,z) (the \rho here is the radial distance from the z-axis, and not the density). If U is the speed of the incoming flow, then we write \varphi = Uz + \phi, where \phi is a small correction and satisfies :\frac\frac\left(\rho\frac\right) -\beta^2 \frac=0, \quad \beta^2 = M^2-1 where M=U/c_\infty is the Mach number of the incoming flow. We expect the velocity components to depend only on \theta, i.e., \rho/z=\tan\theta in cylindrical coordinates, which means that we must have \phi = zg(\xi), where \xi = \rho/z is a self-similar coordinate. The governing equation reduces to :\xi(1-\beta^2\xi^2) g'' + g'=0. On the surface of the cone \xi = \tan\chi \approx \chi, we must have v_\rho/v_z=(\partial\phi/\partial \rho)/(U+\partial\phi/\partial z)\approx (1/U)\partial\phi/\partial \rho=\chi and conesequently g'=U\chi. In the small-angle approximation, the weak shock cone is given by z=\beta \rho. The trivial solution for g 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 :g(\xi) = U \chi^2 \left(\sqrt-\cosh^\frac\right). We therefore have :\varphi = Uz +U \chi^2 \left(\sqrt-z\cosh^\frac\right) exhibiting a logarthmic singularity as \rho\to 0. The velocity components are given by :v_z = U - U\chi^2 \cosh^\frac, \quad v_\rho = \frac \sqrt. The pressure on the surface of the cone p_s is found to be p_s -p_\infty = \rho_\infty U^2\chi^2 ln (2/\beta\chi)-1/2/math> (in this formula, \rho_\infty is the density of the incoming gas).


See also

* Kármán–Moore theory


References

{{reflist, 30em * Fluid dynamics