Description of muzzle blast by modiﬁed ideal scaling models · the gun muzzle. From dimensional analysis, a scal-ing length, ˘, was obtained that is assumed to be pro-portional

1

Description of muzzle blast by modifiedideal scaling models

Kevin S. FanslerU.S. Army Research Laboratory, Aberdeen ProvingGround, MD 21005, USA

Gun blast data from a large variety of weapons are scaledand presented for both the instantaneous energy release andthe constant energy deposition rate models. For both idealexplosion models, similar amounts of data scatter occur forthe peak overpressure but the instantaneous energy releasemodel correlated the impulse data significantly better, par-ticularly for the region in front of the gun. Two parame-ters that characterize gun blast are used in conjunction withthe ideal scaling models to improve the data correlation.The gun-emptying parameter works particularly well withthe instantaneous energy release model to improve data cor-relation. In particular, the impulse, especially in the for-ward direction of the gun, is correlated significantly bet-ter using the instantaneous energy release model coupledwith the use of the gun-emptying parameter. The use ofthe Mach disc location parameter improves the correlationonly marginally. A predictive model is obtained from themodified instantaneous energy release correlation.

1. Introduction

Before guns are fired near personnel or fragileequipment that might be harmed, blast wave overpres-sure levels need to be accurately known over a largerange of distances from the gun muzzle. Personneland instruments may need to be located as close as 15to 20 calibers away from a gun. On the other hand,designing an enclosure for reducing impulsive noisemay require an estimate of the forces and impulses onits inside surfaces as far away as 100 calibers fromthe muzzle. Some of the quantities that can be usedto assess gun blast at a field point are the peak over-pressure, pp, the time of arrival for the front of theblast wave, ta, the impulse, I , and the positive phaseduration, τ . The positive phase duration is the timefor the positive phase of the wave to pass over thefield point. The time of arrival and positive phase du-

ration need to be known accurately when peak over-pressures and impulses at a field point result frommultiple reflections of the blast wave.

The impulse, I , is defined as the time integral ofthe positive phase for the overpressure:

I ≡∫ tc

ta

(p− p∞) dt, (1)

in which tc is the time for the overpressure to be-come zero in the blastwave. Here p is the pressureand p∞ is the atmospheric pressure, which near sealevel is approximately one bar. The impulse is a veryimportant quantity in determining if structures are re-inforced adequately to withstand the blast wave.

The first predictions of gun blast were done us-ing results obtained from instantaneous energy releaseblast from a point source. The Buckingham Pi the-orem can be applied to instantaneous energy releaseat a point source to obtain a fundamental length, λ(Baker [2]). The scaled peak overpressure, P , be-comes a function of the distance scaled by the fun-damental length (Hopkinson [11]; Baker [2]). Al-though explosive charges will not strictly generatepoint source blasts, they approximate blast wavesfrom these ideal point source blasts after a minimumscaled distance is reached (Brode [3]). Reynolds [14]applied this point source scaling theory developed byHopkinson [11] to gun blast problems. Westine [18]proposed a scaling prediction method based on high-explosive detonations in combination with the lengthof the gun barrel. He based his work primarily on20 mm data that included different projectile muz-zle velocities and developed contours for predictingpeak overpressure and time of arrival for a wide rangeof weapons, but these contours did not extend com-pletely around the weapons.

Smith [16] used a different approach in his studyof the blast wave produced by a 7.62 mm rifle. Hecombined scaled solutions to the problems of blastwaves generated by a constant energy efflux with blastwaves from asymmetrically initiated charges. Thetheory of asymmetrically initiated charges was applied

Shock and Vibration 5 (1998) 1–12ISSN 1070-9622 / $8.00 1998, IOS Press. All rights reserved

2 K.S. Fansler / Description of muzzle blast by modified ideal scaling models

to account for the peak overpressure values decreas-ing with increasing polar angle, with zero polar anglebeing along the boreline of the gun. The predictionmethod agreed well with the limited data.

Schmidt, Gion, and Fansler [15] used yet anotherapproach to predicting muzzle blast. They found thatthe peak overpressure correlated well with the locationof the Mach disc for an equivalent steady jet whoseexit conditions matched that for the pressure and thepropellant gas velocity. Later, Fansler and Schmidt [8]found that the Mach disc method of prediction wasdeficient in its ability to predict the peak overpressurechanges with the change in the propellant tempera-ture. However, Smith’s [16] prediction model pre-dicted the actual trends. Fansler and Schmidt [8] gen-eralized and extended Smith’s work [16] to developprediction methods for bare muzzle guns, based ondata collected in the range of 10 to 50 calibers fromthe gun muzzle. From dimensional analysis, a scal-ing length, ξ, was obtained that is assumed to be pro-portional to the square root of the peak energy ef-flux from the gun muzzle. This scaling length de-pends on such parameters as the exit muzzle pressure,exit temperature of the propellant, and the projectilevelocity at the gun muzzle. The distance from themuzzle to the field point divided by the scale length,r/ξ, yields the fundamental scaled distance. How-ever, the blast waves are highly directional, with theirpeak overpressures decreasing with increasing polarangle from the forward axis direction. The form forthe variation of peak overpressure with angle from theaxis is obtained from asymmetrically initiated charges,or equivalently, moving charge theory (Armendt andSperrazza [1]). This angle variation function, β, pos-sesses one free parameter that determines how rapidlythe peak overpressure falls off with increasing polarangle. The function, β(θ), is multiplied by the funda-mental scale length, ξ, to obtain the modified scaledlength, ξ′, for gun blast. The assumed formulation forthe scaled peak overpressure was

P = A

(ξ′

r

)a, (2)

in which r is the distance from the muzzle and the freeparameters, A and a together with the free parameterfor β, are determined by a least squares fit to peakoverpressure data. The other blast wave quantities ofinterest were also formulated and least square fittedto data.

The predicted muzzle blast quantities were imple-mented on a computer and can be applied for blast

waves incident on surfaces to obtain the reflected pres-sures (Heaps et al. [10]). Fansler [6], noting deficien-cies in the impulse model, examined additional im-pulse data and developed an expression that dependednot only on the scaling length but also on a dimen-sionless parameter that depended upon the time forthe gun barrel to empty. The computer-implementedtechnique (Heaps et al. [10]) was updated with theimprovement for treating impulse.

In the studies just cited, data were collected for dis-tances close to the muzzle. Other people have inves-tigated muzzle blast at greater distances. Soo Hooand Moore [17] primarily studied various naval gunswith data taken for distances between 20 and 110 cal-ibers but also obtained data for U.S. Army 20 mm M3and M197 cannon. Kietzman, Fansler, and Thomp-son [12] obtained overpressure data for a maximumof 400 calibers distance from a 105 mm tank cannonand noted that the angular distribution of the shockwave strength changed with distance. Pater [13] ob-tained additional data and combined his data with SooHoo and Moore’s data. He also noted that the peaksound pressure level (PSPS) decibel differences fromthe front of the gun to the rear of the gun decreasedwith distance.

Fansler et al. [9] obtained data from 7.62 mm ri-fles for a large range of distances (15 to 400 calibers)from the muzzle and for weapons shot at both highand low velocities with different lengths of barrels.The approach was similar to Fansler and Schmidt’s [8]approach but explored the use of gun blast parame-ters to modify the directional parameter, β. The datawere fitted using several trial parameters and func-tions to obtain a best representative function for thepeak overpressure. Time-of-arrival data were also ob-tained and a fit was obtained from a trial function.The impulse prediction function was obtained by as-suming a positive phase duration form and expressingthe impulse in terms of the peak overpressure and thepositive phase duration. The positive phase durationpossessed free parameters to be adjusted by a leastsquares fit to the impulse data.

Previous investigators usually developed a modelassuming only one approach. This report reconsidersboth the point source scaling approach and the energyefflux scaling approach as a basis for modification bythe parameters that characterize gun blast. The lengthscale for the chosen ideal explosion is modified byparameters for the gun blast, and coefficients for theseparameters are determined by least squares fitting tothe data. This study assumes that functions of the gunblast parameters will be good first order correctionsto the scale length for the ideal models.

K.S. Fansler / Description of muzzle blast by modified ideal scaling models 3

2. Data used in investigation

Most previous scaling investigations conductedused data with less variation in key parameters(Fansler and Schmidt [8]; Smith [16]; Reynolds [14])than are used in the current investigation. Thus, anearlier prediction model for gun blast might fit theearlier (limited) data but it might be a poor fit forother data used here. This investigation uses muchof the Fansler et al. [9] data, but in an attempt tomore accurately model the data nearer the gun, ig-nores the data taken at the longest distance, whichwas 400 calibers away from the gun muzzle. Thisinvestigation also uses some of the data of Fanslerand Schmidt [8] and Kietzman, Fansler, and Thomp-son [12] to provide a greater range in gun blast pa-rameters. In all these experiments, a gauge was po-sitioned near the muzzle to establish a zero referencetime for the experiment. The experiment to obtainthe small-caliber overpressure data (7.62 mm) wasconducted at ARL for distances 15 to 400 calibersfrom the muzzle (Fansler et al. [9]). For the cur-rent study, the data at 400 calibers will not be used.The weapons used in the test were a .300 Magnumbarrel, another .300 Magnum barrel that had a re-duced bore length, and a shortened carbine barrel. Aschematic of the gauge positions around the gun muz-zles is shown in Fig. 1 for the investigation by Fansleret al. [9].

In addition, 105 mm tank cannon data obtained inthe Kietzman, Fansler, and Thompson [12] investi-gation are included. Some 30 mm WECOM datafrom Fansler and Schmidt [8] are also included be-cause they can be used to extend the range of ap-plicability to more gun systems. The propellant forthe 30 mm WECOM cannon was specially selected toburn quickly in the barrel to maximize reproducibil-ity. The propellant speed of sound at the muzzle wasalso experimentally found.

Fig. 1. Gauge positions for the Fansler et al. study [9].

The loading, velocity of the projectile at the muz-zle, and muzzle pressure at projectile exit are shownin Table 1 for each configuration. The muzzle pres-sure in the last column is the peak value immediatelybefore the projectile exits the barrel. With the ex-ception of the last two rows, the first number in thedescription column of Table 1 refers to the chargemass in grains, while the second number refers to thecharge mass in grams. Further discussions about thefirings will use these descriptions for identification.The next to the last row refers to the parameters forthe 105 mm cannon shooting the M735 round (Kietz-man et al. [12]).

Table 1Loadings and characteristics

Barrel Description Propellant Charge Projectile Muzzletype mass velocity pressure

g (gr) m/s MPa

.300 Long Magnum 1220w70 4831 4.82 (70.0) 899 59.401125w75 4831 5.16 (75.0) 975 71.90

.300 Short Magnum s220w36 4227 2.48 (36.0) 594 74.90s125w42 4227 2.89 (42.0) 792 89.60

Carbine c13p6 2400 0.94 (13.6) 518 65.90105 mm 105mm M30 5966 (–) 1501 71.5030 mm m10f250 M10 16.2 (250) 572 8.73


3. Scaling approaches

Because the propellant empties from the barrel ina time that depends on parameters such as gun barrellength, and the propellant exits the barrel with con-siderable linear momentum, no simple ideal scalingtheory (e.g., neither instantaneous nor constant energyrelease models) can be successfully applied to gunblast phenomena without some modifications. Nev-ertheless, even though the blast wave from guns ismore complicated, the similarities to a particular idealexplosion may allow a related scaling technique to beused with modification by gun blast parameters to rep-resent gun blast waves. Both instantaneous energy re-lease explosions and constant energy efflux explosionsare treatable by scaling and have been used to help de-scribe gun blast (Reynolds [14]; Westine [18]; Fanslerand Schmidt [8]; Fansler et al. [9]). For blasts gen-erated by a constant energy efflux, dE/dt, the peakoverpressure, P ≡ pp−p∞, is expressed in functionalterms as

P = P (r, ρ∞, a∞, dE/dt), (3)

in which pp = peak value of the pressure at a givenfield point, r = distance from muzzle to field point,ρ∞ = ambient density, a∞ = ambient speed ofsound, and dE/dt = energy deposition rate into at-mosphere.

Using the Buckingham Pi theorem (Baker [2]), ascaling length for constant energy efflux explosion isobtained:

ξ ∼√

(dE/dt)/(p∞a∞) (4)

and

P ≡ P/p∞ = P (r/ξ). (5)

Dimensional analysis shows that the time of arrival,ta, the positive phase duration, τ , and the impulsive,I , also have their scaled equivalents,

ta ≡ taa∞/ξ = ta(r/ξ), (6)

τ ≡ τa∞/ξ = τ(r/ξ), (7)

and

I ≡ Ia∞/(ξp∞) = I(r/ξ). (8)

For blast waves assumed to be generated by instan-taneous energy deposition at a source point, dimen-sional analysis leads to scaling relationships that alsogenerate universal curves, but with the energy of theexplosive, E, considered a significant parameter in-

stead of dE/dt. Similarly, as for the constant energyefflux case, it is obtained that

P = P (r/λ), (9)

in which

λ =

(E

p∞

)1/3

. (10)

The point source expressions for the scaled time, pos-itive phase duration, and impulse are Eqs (6), (7),and (8) with ξ replaced by λ.

As related before, gun blast cannot be completelycharacterized by either of these ideal blast models.Nevertheless, the ideal models can be modified to de-scribe muzzle blast. The energy release rate of pro-pellant gas from the gun muzzle exit depends uponthe length of the barrel, the velocity of the projectile,and the sound speed of the propellant gas before pro-jectile exit. As a first step in characterizing muzzleblast in terms of the constant energy efflux model, theenergy deposition rate for gun blast is assumed to bethe energy efflux from the muzzle exit immediatelyafter the projectile back clears the muzzle exit. Theenergy deposition rate can then be written as

dEdt

=γepeue

γe − 1

[1 +

(γe − 1)

2M 2

e

]Ae, (11)

in which Ae is the area of the bore, Me is the exitMach number of the propellant flow immediately afterthe projectile exits the muzzle, pe is the peak muzzleoverpressure while the projectile exits the muzzle, ue

is the velocity of the propellant gas at the exit imme-diately after the projectile exits the muzzle, and γe isthe specific heat ratio for the exiting propellant.

To use the instantaneous energy release model as abasis for gun blast, a value for E must be assumed.The available energy, E, is assumed to be the totalenergy in the propellant minus the energy expendedin propelling the projectile and heating the gun tubeby friction and heat transfer. The kinetic energy ofthe propellant gas is part of the available energy forthe blast.

In addition to prescribing dE/dt and E (for usein the constant energy efflux and point source mod-els, respectively), other dimensionless parameters areneeded to modify the ideal models for accurately com-puting gun blast. For instance, if a gun barrel emptiesquickly, the peak overpressure will be higher than ifthe energy is released over a longer period of time.Neither the initial dE/dt nor E itself will account forthis emptying time effect. Hence, an additional pa-


Table 2Blast parameters for the point source and energy efflux models

Designation λ/D δλ ξ/D δξ xM/D

c13p6 39.8 0.437 9.44 1.84 15.80s220w36 51.4 0.712 9.45 3.87 17.60s125w42 55.6 0.462 10.40 2.47 22.301220w70 60.1 0.694 8.41 4.96 25.601125w75 63.8 0.598 9.80 3.89 26.60105mm 54.9 0.214 14.01 0.82 38.90

m10f250 22.2 0.955 3.08 6.88 6.58

rameter characterizing the blowdown process, whichcan be obtained with the Buckingham Pi theorem, isused (Fansler [6]; Baker [2]).

For the constant energy release model, the blow-down parameter is defined by the ratio of the scalefor the gun tube emptying time, L/Vp (Corner [4]), tothe time scale for muzzle blast, ξ/a∞.

δξ ≡La∞ξVp

, (12)

in which L is the effective length of the barrel and Vp

is the exit velocity of the projectile. The smaller thevalue of δξ, the more quickly the barrel is emptied.Its counterpart for the instantaneous energy releasemodel is

δλ ≡La∞λVp

. (13)

Again, when the blow-down parameter is small, thebarrel empties more quickly, which would result inthe energy from the propellant being used in a moreefficient manner by the gun blast wave. These expres-sions for the blow-down parameter are not unique butthey are simple and may be adequate representationsfor times to empty guns.

Another parameter that characterizes gun blast isthe axial location of the Mach disc or the recompres-sion shock that is centered on the gun bore axis (Er-dos and Del Guidace [5]; Schmidt et al. [15]). For thesteady jet, the axial position of the Mach disc relativeto the muzzle for the steady jet is given as

x′M/D = Me

√γepe/2, (14)

in which D is the diameter of the bore. The Machdisc location expression resembles the constant energyefflux expression but differs in the emphasis upon theMach number for the exit flow and the exit velocityfor the muzzle flow. The use of the Mach disc locationas a gun parameter allows other characteristics of thegun blast flow to be expressed. It has been used withsome success as a scaling factor (Schmidt et al. [15]).The scale lengths for the ideal types of explosions, the

blow-down parameters, and the Mach disc locationsare given in Table 2 for various gun test data presentedin this paper.

4. Results

A computer program was developed to extract thepeak overpressure, the impulse (numerically calcu-lated time integral of the positive phase for the over-pressure), and the time of arrival from the obtainedoverpressure waveforms. Then, both the scaled peakoverpressure data and the scaled impulse data is scaledwith the fundamental length for each model and pre-sented for the polar angles of 30◦, 90◦, and 150◦. Ifthe two models are comparable in their correlations,then modification of the scale length with the gunblast parameters are attempted by least square fittingto obtain improved correlations. Otherwise, if a cor-relation is superior for one of the ideal scaling mod-els, then only that model will be selected to improvecorrelation of the data with the use of the gun blastparameters.

4.1. Peak overpressure data presented with differentscaling approaches

The data can first be examined along selected rayswhen the distance from the gun tube muzzle is scaledby ξ, corresponding to an explosion generated by aconstant energy efflux. Figure 2 shows the peak over-

Fig. 2. Scaled peak overpressure data for θ = 30◦ (energy effluxassumption).


Fig. 3. Scaled peak overpressure data for θ = 30◦ (point sourceassumption) and calculation for TNT (Brode [3]).

pressure data along the ray directed 30◦ from the muz-zle axis.

When the data are scaled with the peak energy ef-flux approach, the slower emptying weapons (moreresembling constant energy efflux) should have highervalues of peak overpressure relative to the faster emp-tying weapons (less resembling constant energy ef-flux). Least square fitting, discussed later in this sec-tion, will show that the weapons with the largel muz-zle to Mach disc distances also tend to generate largerpeak overpressures, given equal values of δγ . TheMach disc location parameter is used because it wasfound to correlate data (Schmidt et al. [15]), but aphysical reason why this parameter could be used toimprove correlation for either the constant energy ef-flux model or the point blast model is not known.

The results for Fig. 2 indicate that the Mach discposition will correlate the data better than the blow-down parameter, when scaled for the constant energyefflux model. Nevertheless, the rapid blowdown forthe 105 mm cannon appears to bring the data pointsinto close company with other larger data values, eventhough the muzzle-to-Mach disc distance is larger thanfor any other firings. The c13p6 data have valuesfor their parameters that would predict that the peakoverpressure would be low compared to all the otherdata. Similar behavior occurs for data taken at otherangles and are discussed in detail by Fansler [7].

The peak overpressure data are next examined withthe distance scaled by λ, corresponding to the pointsource explosion. Figure 3 shows the peak overpres-

sure data along the 30◦ ray. Also shown is a calcu-lation for a spherical TNT charge in open air. TheTNT calculations were obtained with a finite differ-ence scheme developed by Brode [3]. The calcu-lated values for the TNT charge should better approxi-mate the ideal point source solution as the distance in-creases from the TNT charge. It is seen that the peakoverpressure for the TNT charge is initially higher butdecreases so rapidly that it is less than some of the gunblast data values at the longer scaled distances. Recallthat smaller values of δλ are associated with a quickeremptying time (more resembling a point source) andshould have higher peak overpressure values for agiven distance, which generally occurs. Some excep-tions result from data scatter, the other reasons arenot known. The trends are generally similar for dataat other polar angles (Fansler [7]). Both unmodifiedapproaches show deficiencies in correlating the data.It is not clear that either approach, when modified bythe characteristic gun parameters, would be notice-ably superior to the other. Accordingly, least squaresfitting are performed for both the point source modeland the energy efflux model.

4.2. Modeling approach for peak overpressure data

The ideal scaling relationships, which assumespherically symmetrical conditions, need to be modi-fied for gun blast, where the peak overpressure variesstrongly with the polar angle, θ, from the bore-line.Following Smith [16], a directional scaling length fac-tor is obtained, β, that depends on the polar angle,

β = µ cos θ +√

1− µ2 sin2 θ, (15)

in which µ is the momentum index, which determinesthe levels of peak overpressure (with distance heldconstant) as a function of the polar angle, θ. This ex-pression has been developed from moving blast the-ory. The momentum of the charge results in thestrength of the blast being increased in the direction ofthe momentum and decreased to the rear of the mov-ing charge. The directional length scaling factor, β, isreal and positive if µ < 1. The nearer µ is to zero, themore nearly spherical the blast. As µ approaches 1,the strength of the blast in front of the gun becomeslarge compared to the strength of the blast wave tothe rear.

To improve the prediction model, the scalinglengths, ξ and λ, need to be modified with the twogun blast parameters discussed earlier. Trial functionsof these parameters are assumed as first order terms


in a series expansion. Least square fitting showed thatthe most effective functions of these parameters in-volved the inverse of the blow-down parameter andthe first power of the Mach disc location. Along withβ to be determined by least squares fitting in termsof µ, two other constants that multiply these parame-ter functions need to be fitted to obtain the modifiedscaling lengths, ξ′ and λ′, defined by

λ′ ≡ λβ(

1 +C

δλ+H

xM

D

), (16)

ξ′ ≡ ξβ(

1 +C

δξ+H

xM

D

). (17)

With these modifications, the gun blast peak over-pressure is

P = P (r), (18)

in which r ≡ r/`, and ` assumes the value of ξ′ orλ′ depending on the scaling approach used.

Also, the time of arrival, ta, has its scaled equiva-lent,

ta(r/`) ≡ taa∞/`. (19)

The wave generated from gun blast decays to anacoustic wave (P ∝ 1/r) at large distances, but closerin, the peak overpressure versus distance relationshiphas a steeper slope (P ∝ 1/ra, a > 1), as also oc-curs with instantaneous explosions. It is attempted tomodel this behavior with the following expression forthe peak overpressure, P ,

P =A

r+B

r2 . (20)

Another more simple model will also be tried andcompared with the two-term model:

P =A

ra, (21)

in which a is assumed to be constant throughoutthe region of interest. Equation (20) or Eq. (21)is matched with pressure data in conjunction withEq. (15), which is the expression used to vary thestrength of the blast with the polar angle, θ. The val-ues of the significant parameters, λ/D, δλ, ξ/D, δξ,and XM/D, are given in Table 2. With these valuesand the peak overpressure data, fits are made to de-termine A, B, a, and µ for the one-term model, C,and H .

4.3. Peak overpressure fitting

Fits were made for both the point source modeland the energy efflux model with the use of the blow-down parameters and Mach disc position. For thepoint source model, Table 3 gives the parameter val-ues found for the predictive equations for both theone-term equation and the two-term equations.

The particular headings refer to the symbols occur-ing in Eqs (15), (16), (17), (20), and (21). The one-term models (1pa, 1pb, 1pc, and 1pd) are presentedfirst in Table 3, in which B = 0 and a 6= 1. When Cand H are assumed to be zero in the fitting procedure,the root mean square (RMS) error is 0.309. This valueis obtained from fitting the logarithm of the pressuredata with the logarithm of the fitting function. TheRMS value of 0.309 translates to an expectation thatthe data value will be 36% more or less than the curvefit value. Fitting with the inverse blow-down parame-ter (model 1pc) gives good improvement while fittingwith Mach disc position (model 1pb) yields signifi-cantly less improvement. Concurrent fitting with theblow-down and Mach disc position parameters givesonly marginal improvement over fitting with the blow-down parameter. The two-term models (2pa and 2pb)give comparable results. The physical intuition thatthe scaled (point mass scaling) peak overpressureswith longer blow-down times would be smaller rela-tive to the peak overpressures for smaller blow-downtimes is confirmed by the positive fitted values for C.

The results for the energy efflux model are pre-sented in Table 4. The important parameter to im-

Table 3Least squares fit results for peak overpressure data (point sourcemodel)

Model µ A B C H a RMS

1pa 0.78 0.270 – – – 1.10 0.3091pb 0.78 0.145 – – 0.0374 1.09 0.2471pc 0.76 0.117 – 0.562 – 1.14 0.1991pd 0.77 0.100 – 0.564 0.0159 1.13 0.1932pa 0.78 0.243 0.0220 – – – 0.3092pb 0.78 0.110 0.0061 0.592 – – 0.201

Table 4Least squares fit results for peak overpressure data (energy effluxmodel)

Model µ A B C H a RMS

1ea 0.77 1.89 – – – 1.13 0.2521eb 0.78 1.14 – – 0.0248 1.10 0.2081ec 0.77 1.77 – 0.107 – 1.12 0.2511ed 0.77 1.18 – −0.349 0.0299 1.12 0.2012ea 0.78 1.31 0.742 – – – 0.2522eb 0.78 0.89 0.247 – 0.0271 – 0.210


prove fitting for the energy efflux model is the Machdisc position instead of the inverse blow-down param-eter, 1/δξ, as it was for the point source model. Thefit marginally improved with the use of the inverseblow-down parameter. The fit performed to fit valuesfor both C and H (model led) gives a negative valuefor C and confirms the intuition that longer blow-down times result in higher scaled peak overpressureswith the constant energy efflux model. The positivevalue obtained for C when H is assumed to be zero(1ec) occurs because there is a correlation betweenthe blowdown and the Mach disc location parametervalues. For a given gun configuration, increasing thecharge mass results in a larger initial energy effluxand a smaller δξ. The trend is noted in Table 2. WhenH is assumed to be zero (model 1ec), the effect ofthe Mach disc location overshadows the blow-downparameter and produces a false illusion. For both thepoint source model and the energy efflux model, otherfits were attempted with the gun parameters raised toother powers in Eqs (16) and (17). These results arenot shown as these particular choices yielded notice-ably larger root mean square errors than those shown.

The use of the blow-down parameter with the pointsource model significantly improves the data correla-tion while the use of the Mach disc position parameterwith the constant energy efflux model markedly im-proves the data correlation. The point source modelwas selected as the basis for modification because,as will be seen in a later section, point source scal-ing correlates the impulse data better than the energyefflux scaling does. In terms of correlation, there islittle to choose between the two-term modified pointsource model and the one-term modified point sourcemodel. The two-term modified point source modelwas selected in preference to the one-term model be-cause the ratio of the peak overpressure to the frontof the gun over the peak overpressure to the rear ofthe gun becomes larger nearer the gun muzzle, as ex-periment shows. Also, the expression for the time ofarrival is simplified by the use of the two-term pointsource model with the blow-down parameter modifi-cation as compared with the use of the one-term ex-pression. The selected fit to calculate future predictedfits for the peak overpressure is model 2pb in Table 3,which gives

P = 0.11λ′

r+ 0.0061

(λ′

r

)2

, (22)

is which λ′ is given by Eq. (16), which varies with θthrough β, as given by Eq. (15). The scaled data with

Fig. 4. Peak overpressure versus r/λ′ with least squares fit.

the prediction curve is shown in Fig. 4. As expected,the carbine data (c13p6) show the most scatter, whichmay have to do with their gun blast parameter val-ues. The parameters for the carbine are an unusualcombination when compared with the other weapons.The carbine has the second smallest δλ and the sec-ond from the smallest muzzle-to-Mach disc distancein the group. If the other weapon’s value of δλ wereplaced in increasing order, the value of xM/D usuallyappears in declining order. Future studies should in-clude weapons with similar characteristics to the car-bine (c13p6).

Equation (22) gives good agreement over the pa-rameter ranges where data were obtained. One shouldemploy caution in extrapolating to parameter valuesoutside the range where data were taken.

4.4. Modeling and fitting time-of-arrival data

As in Fansler and Schmidt [8], the pressure-jumpMach relation,

P =2γγ + 1

(M 2

s − 1), (23)

can be equated to the predictive equation for peakoverpressure, Eq. (22). Here, Ms is the Mach num-ber for the shock moving through still air. Becausedta/dr = 1/Ms, the resulting expression can be in-tegrated to obtain a closed form expression for thetime of arrival, ta. The expression for the peak over-pressure, Eq. (20), when substituted into Eq. (23) and


integrated from the muzzle to the field point, r, be-comes

ta − t0 = X(r)− A′

2ln [2X(r) + 2r +A′], (24)

in which

X(r) =√r2 +A′r + B′,

A′ =2γAγ + 1

, B′ =2γBγ + 1

, (25)

and γ = 1.4 is the specific heat ratio for ambient air.The initial time value, t0, takes into consideration thenature of the formation of the blast wave and may varywith the angle. The zero value for time correspondsto the projectile exiting the muzzle.

The time-of-arrival expression, Eq. (24), can beused with the determined constants A, B, γ, and thetime-of-arrival data to obtain the value, t0, by a leastsquares fit

ta =X(r)

− 0.064 ln [2X(r) + 2r + 0.128]− 0.17, (26)

in which

X(r) =√r2 + 0.128r+ 0.0074.

Least squares fits were also tried to see if a directionaldependence existed for the scaled time of arrival. Thedirectional dependence was negligible.

Figure 5 shows the fitted curve for ta as a functionof the scaled length with the scaled time-of-arrivaldata. A logarithmic scale was used to emphasize com-parisons for the shorter scaled distances. The agree-ment of the data with the fitted time of arrival is sat-isfactory.

Fig. 5. Predicted ta compared with the observed ta.

4.5. Impulse data presented with different scalingapproaches

Similarly as for the peak overpressure data, the im-pulse data along selected direction rays (30◦, 90◦,and 150◦) are scaled with both the ideal point sourcescale length and the energy efflux scale length. Fig-ure 6 shows the impulse data obtained along a 30◦

ray when scaled using the constant energy efflux as-sumption. The data do not appear to be well corre-lated using energy efflux scaling. The carbine (c13p6)empties rapidly and has the lowest impulse valueswhile the long barrel Magnum with the largest pro-jectile (1220w70) empties its barrel more slowly andis among the data with the higher values for the im-pulse. The same trends occur for data taken at otherangles and with approximately the same spread in thedata values (Fansler [7]).

In general, constant energy efflux scaling for theimpulse data yields inferior correlations compared tothe corresponding correlations for the peak overpres-sure data. This result seems physically reasonable be-cause the peak value is more responsive to the earlypart of the energy efflux history while the shape of thewave would more strongly depend upon the completeenergy efflux history, which is partially described bythe blowdown parameter value.

Figure 7 shows the impulse data obtained along a30◦ ray when point source scaled. Nearer the gunmuzzle, the impulse decreases slowly at first and thendescends much more rapidly for the longer distances.

Fig. 6. Scaled impulse versus r/ξ for θ = 30◦ .


Fig. 7. Scaled impulse versus r/λ for θ = 30◦.

Apparently, energy flux flowing out of the gun con-tinues to add to the blast as the wave passes over thefirst gauges used in this investigation and as the en-ergy flux decreases, the curve steepens. The data arewell correlated by point source scaling. The correla-tion is improved for the larger distances, which wouldbe expected because almost all of the available energywould have been deposited by the time the front of thewave reaches the larger distances, while at the smallerdistances, the various guns would have deposited dif-ferent fractions of their total available energy. For thelarger scaled distances, the correlation for the impulseat 30◦ is superior to the correlation obtained from thescaling investigation of the peak overpressure. Theimpulse is the time integral of the positive phase ofthe overpressure and would be expected to be a betterindicator of the total energy deposited into generat-ing the blast wave as compared to the peak overpres-sure. Nevertheless, the data points are, on the whole,slightly lower for the more slowly emptying guns, asoccurs for the peak overpressure data.

The data values for other angles show less correla-tion and descend more slowly than observed for otherangles (Fansler [7]). The impulse data values for pointsource scaling are better correlated than the impulsedata values for constant energy efflux scaling. Ac-cordingly, only the data that are point source scaledwill be least squared fitted using the barrelemptyingparameter to improve the fit.

4.6. Modeling and fitting impulse data

With the observation noted previously, candidatefitting equations may be proposed. The best candidateequation with its fitting coefficients are

I = 0.0146

(λ′ir

)bi(θ)

, (27)

in which

bi(θ) = 0.954− 0.585 sinθ

2, (28)

and

λ′i = β

[1 + 0.444

sin(θ/2)

δλ

]λ. (29)

Note that the scaling length for the gun blast impulsediffers from the scaling length for the peak overpres-sure, with the emptying parameter characterizing thegun blast, δλ, being applied more strongly to the rearof the gun. The least square fit was obtained using thelogarithm of the impulse values as was done for thepeak overpressure data. The RMS value was equal to0.216, which is only a little larger than the RMS foundfor the peak overpressure data. Other more compli-cated fits were tried to account for the change in theslope with distance for the data forward of the gun,but only marginal improvements in the RMS valueswere found. The dimensionless form of the impulsefor gun blast is then

I ≡ Ia∞/(λ′ip∞

). (30)

Unlike the presentation of the peak overpressure,when scaled with the modified point source length,the impulse data scaled by its special modified pointsource length must be presented with the polar an-gle as a parameter because bi(θ) in Eq. (27) varieswith the polar angle. Figure 8 shows the impulse datascaled with the modified point source length for im-pulse together with the fitted curve for the 30◦ polarangle. The fitted curve is a compromise that, for smalldistances, is too high at first and then transitions to aregion where it is too low and then becomes too largeagain.

The flow processes are too complex to assume aform for the positive phase duration and to obtain anaccurate estimate of its value from some simple as-sumptions. Moreover, accurate values of the positive-phase-duration data are more difficult to obtain be-cause the overpressure varies slowly as the zero valueof overpressure is approached, with smaller superim-posed waves from turbulence in the flow resulting inlarge amounts of uncertainty for the time for the zerovalue of overpressure in the wave.


Fig. 8. Scaled impulse versus r/λ′i for θ = 30◦ with least squaresfit.

5. Conclusions and recommendations

Investigators have advocated particular approachesfor describing gun muzzle blast. This investigationexamines two principal approaches that use scalingideal blast waves as a basic starting point. The dataavailable consist of detailed overpressure records forweapons with a wide variety of locations, projectileweights, propellant weights, muzzle velocities, andbarrel lengths. From the data, the peak overpressure,the time of arrival, and the impulse obtained by in-tegrating the positive value of the overpressure wavewith time are obtained. The peak overpressure dataare scaled using both ideal scaling models: the instan-taneous energy release model and the constant energydeposition model. Neither of the scaling approachesgives a clearly superior correlation of the data.

Two parameters that characterize gun blast are in-troduced for modification of the basic scaling ap-proaches to improve data correlation. One is calledthe blow-down or emptying time parameter, while theother parameter gives the position of the Mach discin terms of calibers. For the peak overpressure datascaled with the energy efflux length, the Mach discposition parameter noticeably improves correlation ofthe data, while the blow-down parameter value im-proves the correlation negligibly. For the data scaledwith the point blast scaling length, the blow-down pa-rameter is important for improving correlation, andthe effect of the Mach disc location may be neglected.Slower emptying times result in an inefficient con-

version of energy into the muzzle blast wave that re-duces the peak overpressure. Both modified scalingapproaches give similar improvements in correlationof the peak overpressure data. The modified pointmass model was selected because the basic point massmodel yielded a superior correlation for the impulsedata. A two-term equation for predicting peak over-pressure was selected that permits the development ofa closed form expression describing the time of arrivaland also approximates the nonlinear wave behaviornear the muzzle.

The impulse data are also scaled with the two idealmodels. The instantaneous energy release model cor-relates the data well to the front of the gun withless correlation as the polar angle is increased. Theimpulse decreases more slowly with distance for in-creasing polar angle. The constant energy depositionmodel does not correlate the impulse data as well asthe point source model, particularly to the front of thegun. The point source model modified by the use ofthe gun-emptying parameter is used to develop a leastsquares fit function describing the data. The lengthscale developed for impulse data differs in form fromthe length scale for peak overpressure and dependsupon the polar angle, θ. The RMS error obtained withthe impulse data is only slightly larger than the errorobtained for the peak overpressure data.

A prediction method for the positive phase durationis left for further investigations. More needs to alsobe known about the detailed flow processes in gunblast. Computational fluid dynamics techniques couldbe used to investigate the flow of energy from the frontto the rear part of the wave and yield more insightinto gun blast phenomena.

Acknowledgements

I wish to thank William Thompson, John Carnahan,and Brendan Patton (all who then worked at ARL) fortheir assistance in conducting many of these experi-ments and obtaining the data.

References

[1] B.F. Armendt and J. Sperrazza, Air blast measurementsaround moving explosive charges, Part III, BRL-MR 1019,U.S. Army Ballistic Laboratory, Aberdeen Proving Ground,MD, 1956.

[2] W.E. Baker, Explosions in Air, University of Texas Press,Austin, TX, 1973, pp. 54–77.


[3] H.L. Brode, Numerical solutions of spherical blast waves,J. Appl. Phys. 26 (1955), 766–775.

[4] J. Corner, Theory of the Interior Ballistics of Guns, Wiley,New York, NY, 1950, pp. 339–383.

[5] J.I. Erdos and P. DelGuidice, Gas dynamics of muzzle blast,AIAA Journal 13 (1975), 1048–1055.

[6] K.S. Fansler, Dependence of free field impulse on the decaytime of energy efflux for a jet flow, Shock and VibrationBulletin, Bulletin 56, Part 1, The Shock and Vibration Center,Naval Research Laboratory, Washington, DC, 1986, pp. 203–212.

[7] K.S. Fansler, Description of muzzle blast by modified idealscaling models, ARL-TR 1434, U.S. Army Ballistic Re-search Laboratory, Aberdeen Proving Ground, MD 21005,1997.

[8] K.S. Fansler and E.M. Schmidt, The prediction of gunmuzzle blast properties utilizing scaling, ARL-TR 227,U.S. Army Ballistic Research Laboratory, Aberdeen ProvingGround, MD 21005, 1983.

[9] K.S. Fansler, W.P. Thompson, J.S. Carnahan and B.J. Patton,A parametric investigation of muzzle blast, ARL-TR 227,U.S. Army Ballistic Research Laboratory, Aberdeen ProvingGround, MD 21005, 1993.

[10] C.W. Heaps, K.S. Fansler and E.M. Schmidt, Computer im-plementation of a muzzle blast prediction technique, Shockand Vibration Bulletin, Bulletin 56, Part 1, The Shock andVibration Center, Naval Research Laboratory, 1986, pp. 213–230.

[11] B. Hopkinson, British Ordnance Minutes, No. 13565, 1915.[12] J. Kietzman, K.S. Fansler and W.G. Thompson, Muzzle blast

from 105 mm M735 round, ARLBR-MR 3957, U.S. ArmyBallistic Research Laboratory, Aberdeen Proving Ground,MD 21005, 1992.

[13] L.L. Pater, Gun blast far field peak overpressure contours,TR 79-442, Naval Surface Weapons Center, Dahlgren, VA,1981.

[14] B.T. Reynolds, Muzzle blast pressure measurements, ReportNo. PMR-21, Princeton University, 1944.

[15] E.M. Schmidt, E.J. Gion and K.S. Fansler, Analysis ofweapon parameters controlling the muzzle blast overpres-sure field, in: 5th Int. Symp. on Ballistics, Toulouse, France,1980.

[16] F. Smith, A theoretical model of the blast from stationaryand moving guns, in: First Int. Symp. on Ballistics, Orlando,FL, 1974.

[17] G. Soo Hoo and G.R. Moore, Scaling of naval gun blast peakoverpressures, TN-T7-72, Naval Surface Weapons Center,Dahlgren, VA, 1972.

[18] P. Westine, The blast field about the muzzle of guns, Shockand Vibration Bulletin 39 (1969), 139–149.

Received 6 May 1994; Revised 5 November 1997

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Description of muzzle blast by modiﬁed ideal scaling models · the gun muzzle. From dimensional analysis, a scal-ing length, ˘, was obtained that is assumed to be pro-portional