Kármán–Moore theory

Kármán–Moore theory is a linearized theory for supersonic flows over a slender body, named after Theodore von Kármán and Norton B. Moore, who developed the theory in 1932.[1][2] The theory, in particular, provides an explicit formula for the wave drag, which converts the kinetic energy of the moving body into outgoing sound waves behind the body.[3]

Mathematical description

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Consider a slender body with pointed edges at the front and back. The supersonic flow past this body will be nearly parallel to the  -axis everywhere since the shock waves formed (one at the leading edge and one at the trailing edge) will be weak; as a consequence, the flow will be potential everywhere, which can be described using the velocity potential  , where   is the incoming uniform velocity and   characterising the small deviation from the uniform flow. In the linearized theory,   satisfies

 

where  ,   is the sound speed in the incoming flow and   is the Mach number of the incoming flow. This is just the two-dimensional wave equation and   is a disturbance propagated with an apparent time   and with an apparent velocity  .

Let the origin   be located at the leading end of the pointed body. Further, let   be the cross-sectional area (perpendicular to the  -axis) and   be the length of the slender body, so that   for   and for  . Of course, in supersonic flows, disturbances (i.e.,  ) can be propagated only into the region behind the Mach cone. The weak Mach cone for the leading-edge is given by  , whereas the weak Mach cone for the trailing edge is given by  , where   is the squared radial distance from the  -axis.

The disturbance far away from the body is just like a cylindrical wave propagation. In front of the cone  , the solution is simply given by  . Between the cones   and  , the solution is given by[3]

 

whereas the behind the cone  , the solution is given by

 

The solution described above is exact for all   when the slender body is a solid of revolution. If this is not the case, the solution is valid at large distances will have correction associated with the non-linear distortion of the shock profile, whose strength is proportional to   and a factor depending on the shape function  .[4]

The drag force   is just the  -component of the momentum per unit time. To calculate this, consider a cylindrical surface with a large radius and with an axis along the  -axis. The momentum flux density crossing through this surface is simply given by  . Integrating   over the cylindrical surface gives the drag force. Due to symmetry, the first term in   upon integration gives zero since the net mass flux   is zero on the cylindrical surface considered. The second term gives the non-zero contribution,

 

At large distances, the values   (the wave region) are the most important in the solution for  ; this is because, as mentioned earlier,   is a like disturbance propating with a speed   with an apparent time  . This means that we can approximate the expression in the denominator as   Then we can write, for example,

 

From this expression, we can calculate  , which is also equal to   since we are in the wave region. The factor   appearing in front of the integral need not to be differentiated since this gives rise to the small correction proportional to  . Effecting the differentiation and returning to the original variables, we find

 

Substituting this in the drag force formula gives us

 

This can be simplified by carrying out the integration over  . When the integration order is changed, the limit for   ranges from the   to  . Upon integration, we have

 

The integral containing the term   is zero because   (of course, in addition to  ).

The final formula for the wave drag force may be written as

 

or

 


The drag coefficient is then given by

 

Since   that follows from the formula derived above,  , indicating that the drag coefficient is proportional to the square of the cross-sectional area and inversely proportional to the fourth power of the body length.

The shape with smallest wave drag for a given volume   and length   can be obtained from the wave drag force formula. This shape is known as the Sears–Haack body.[5][6]

See also

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References

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  1. ^ 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.
  2. ^ Ward, G. N. (1949). Supersonic flow past slender pointed bodies. The Quarterly Journal of Mechanics and Applied Mathematics, 2(1), 75-97.
  3. ^ a b Landau, L. D., & Lifshitz, E. M. (2013). Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6 (Vol. 6). Elsevier. section 123. pages 123-124
  4. ^ Whitham, G. B. (2011). Linear and nonlinear waves. John Wiley & Sons. pages 335-336.
  5. ^ Haack, W. (1941). Geschossformen kleinsten wellenwiderstandes. Bericht der Lilienthal-Gesellschaft, 136(1), 14-28.
  6. ^ Sears, W. R. (1947). On projectiles of minimum wave drag. Quarterly of Applied Mathematics, 4(4), 361-366.