TY - JOUR

T1 - Direct calculation of the average local Mach number in converging-diverging nozzles

AU - Majdalani, Joseph

AU - Maicke, Brian A.

N1 - Funding Information:
This work was sponsored partly by the National Science Foundation , and partly by the University of Tennessee Space Institute .

PY - 2013

Y1 - 2013

N2 - Foremost amongst rocket nozzle relations is the area-Mach number expression linking the local velocity normalized by the speed of sound to the area ratio At/A, and the ratio of specific heats. Known as Stodolas equation, the attendant expression is transcendental and requires iteration or numerical root finding in extracting the solution under subsonic or supersonic nozzle operation. In this work, a novel analytical inversion of the problem is pursued to the extent of providing the local Mach number directly at any given cross-section. The inversion process is carried out using two unique approaches. In the first, Bürmanns theorem is employed to undertake a functional reversion from which the subsonic solution may be retrieved. In the second, the Successive Approximation Approach is repeatedly applied to arrive at a closed-form representation of the supersonic root. Both methods give rise to unique recursive approximations that permit the selective extraction of the desired solution to an arbitrary level of accuracy. Results are verified numerically and the precision associated with the supersonic solution is shown to improve with successive increases in the ratio of specific heats.

AB - Foremost amongst rocket nozzle relations is the area-Mach number expression linking the local velocity normalized by the speed of sound to the area ratio At/A, and the ratio of specific heats. Known as Stodolas equation, the attendant expression is transcendental and requires iteration or numerical root finding in extracting the solution under subsonic or supersonic nozzle operation. In this work, a novel analytical inversion of the problem is pursued to the extent of providing the local Mach number directly at any given cross-section. The inversion process is carried out using two unique approaches. In the first, Bürmanns theorem is employed to undertake a functional reversion from which the subsonic solution may be retrieved. In the second, the Successive Approximation Approach is repeatedly applied to arrive at a closed-form representation of the supersonic root. Both methods give rise to unique recursive approximations that permit the selective extraction of the desired solution to an arbitrary level of accuracy. Results are verified numerically and the precision associated with the supersonic solution is shown to improve with successive increases in the ratio of specific heats.

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U2 - 10.1016/j.ast.2011.10.009

DO - 10.1016/j.ast.2011.10.009

M3 - Article

AN - SCOPUS:84875528811

SN - 1270-9638

VL - 24

SP - 111

EP - 115

JO - Aerospace Science and Technology

JF - Aerospace Science and Technology

IS - 1

ER -