0
Research Papers: Fundamental Issues and Canonical Flows

The Peak Overpressure Field Resulting From Shocks Emerging From Circular Shock Tubes

[+] Author and Article Information
A. J. Newman

Mechanical and Aerospace Engineering, SUNY, 302 Jarvis Hall, Buffalo, NY 14260-4400ajnewman@buffalo.edu

J. C. Mollendorf

Mechanical and Aerospace Engineering, SUNY, 335 Jarvis Hall, Buffalo, NY 14260-4400molendrf@buffalo.edu

J. Fluids Eng 132(8), 081204 (Aug 26, 2010) (7 pages) doi:10.1115/1.4002183 History: Received March 23, 2010; Revised July 13, 2010; Published August 26, 2010; Online August 26, 2010

A simple semi-empirical model for predicting the peak overpressure field that results when a shock emerges from a circular shock tube is presented and validated. By assuming that the shape of the expanding shock remains geometrically similar after an initial development period, an equation that describes the peak overpressure field in the horizontal plane containing the shock tube’s centerline was developed. The accuracy of this equation was evaluated experimentally by collecting peak overpressure field measurements along radials from the shock tube exit at 0 deg, 45 deg, and 90 deg over a range of shock Mach numbers from 1.15 to 1.45. It was found that the equation became more accurate at higher Mach numbers with percent differences between experimental measurements and theoretical predictions ranging from 1.1% to 3.6% over the range of Mach numbers considered. (1) Shocks do propagate in a geometrically similar manner after some initial development length over the range of Mach numbers considered here. (2) The model developed here gives reasonable predictions for the overpressure field from a shock emerging from a circular shock tube. (3) Shocks are expected to be completely symmetric with respect to the shock tube’s centerline, and hence, a three dimensional overpressure field may be predicted by the model developed here. (4) While there is a range of polar angle at which the shock shape may be described as being spherical with respect to the shock tube’s exit, this range does not encompass the entirety of the half space in front of the shock tube, and the model developed here is needed to accurately describe the entire peak overpressure field.

FIGURES IN THIS ARTICLE
<>
Copyright © 2010 by American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.

References

Figures

Grahic Jump Location
Figure 1

Physical situation considered here. Shown above is the emerging shock at three stages: developing, developed, and when the shock propagates in a geometrically similar manner. The polar coordinate system is shown in the plan view (b). Spatial points labeled in the elevation view (a) along the 90 deg radial are 0 at the opening of the shock tube and Rd, where the shock becomes fully developed.

Grahic Jump Location
Figure 2

Fully developed shock shape. Shown above is the typical shape of a fully developed shock emerging from the shock tube used in the current research as calculated by Whitham’s theory (5) (p. 161) for Ms=1.41. The view shown above is the plan view. The x axis indicates distance in meters from the shock tube exit along the 0 deg radial, and y indicates distance in meters along the 90 deg radial. Note, however, the shift in origin from the shock tube centerline to the edge. This is done to facilitate wave shape calculation.

Grahic Jump Location
Figure 3

Shown above is a comparison between the shape of the fully developed shock wave as calculated by the authors using Whitham’s (5) (p. 161) theory and the shape of the fully developed shock described using the equation of a cardioid

Grahic Jump Location
Figure 4

Experimental setup schematic detailing the major components of peak overpressure measurement experiments

Grahic Jump Location
Figure 5

Typical voltage/time signature. Clearly visible are the peak overpressure and the negative phase characteristic of blast waves. The above figures are for a fixed point in space. PabsP is the maximum pressure reached at the point when the blast wave passes determined by using an exponential least-squares fit over a portion of the positive phase extrapolated back to the shock arrival time. The negative phase represents the effects of the following rarefaction wave at the point.

Grahic Jump Location
Figure 6

(a) Comparison of experimental data (symbols) and theoretical predictions using Eq. 6 (lines) along the 0 deg, 45 deg, and 90 deg radials for Ms=1.15. (b) Comparison of experimental data (symbols) and theoretical predictions using Eq. 6 (lines) along the 0 deg, 45 deg, and 90 deg radials for Ms=1.20. (c) Comparison of experimental data (symbols) and theoretical predictions using Eq. 6 (lines) along the 0 deg, 45 deg, and 90 deg radials for Ms=1.29. (d) Comparison of experimental data (symbols) and theoretical predictions using Eq. 6 (lines) along the 0 deg, 45 deg, and 90 deg radials for Ms=1.35. (e) Comparison of experimental data (symbols) and theoretical predictions using Eq. 6 (lines) along the 0 deg, 45 deg, and 90 deg radials for Ms=1.41. (f) Comparison of experimental data (symbols) and theoretical predictions using Eq. 6 (lines) along the 0 deg, 45 deg, and 90 deg radials for Ms=1.45.

Grahic Jump Location
Figure 7

Average percent differences between experimental measurements and theoretical predictions as a function of Ms. A general trend of decreasing percent difference with increasing Ms is clearly visible.

Grahic Jump Location
Figure 8

Shown above is the effect that choice of reference point has on the cardioid approximation. One set of predictions is made using the first data point (lowest r/D value), one uses the fourth data point, and one uses the last data point (highest r/D value).

Grahic Jump Location
Figure 9

Based on the data, it is apparent that there is a certain range of polar angle, labeled β above, for which the wave shape may be accurately described as being spherical with respect to the shock tube exit. A detailed treatment of spherical shocks of this nature is given in Ref. 2.

Grahic Jump Location
Figure 10

Plot of isobars predicted by Eq. 6 and isobars given by assuming spherical shape both for Ms=1.41. At this value of shock Mach number, for the value for β shown above, the percent difference between isobars predicted by the cardioid approximation and isobars from assuming a spherical shape ranges between 0% (at 90 deg) and 4% (at 80 deg and 100 deg). These values are typical for the range of Ms considered here.

Grahic Jump Location
Figure 11

P versus max width/D for Ms=1.29. The difference between values calculated using the computational solution of Eq. 8 and the perturbation solution (Eq. 9) are virtually indistinguishable. Agreement between computational and perturbation solutions are equally as good for the range of Ms considered here.

Grahic Jump Location
Figure 12

(a) Comparison of experimental data points on the 0 deg radial (symbols labeled in lower case letters) and the isobars of identical peak overpressure predicted by the cardioid approximation (labeled in capital letters). Axes are scaled distances (r/D) at 0 deg and 90 deg; the center of the shock tube’s exit is located at (0,0). (b) Comparison of experimental data points (symbols labeled in lower case letters) and theoretical isobars (uppercase labels) on the 45 deg radial. (c) Comparison of experimental data points (symbols labeled in lowercase labels) and theoretical isobars (uppercase labels) on the 90 deg radial.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In