1 Introduction

Explosions produce shock waves that propagate into the surrounding medium, imparting pressure and impulse throughout the blast field. The shock wave radius versus time for explosively driven shocks is one of the most fundamental measurements for explosive characterization. This profile yields not only time of arrival at individual locations, but can be differentiated to yield shock wave velocity and ultimately Mach number, which determines peak pressures throughout the blast field [1, 2].

Scaling is used to compare shock wave propagation between charges of different masses and between explosions of different energetic materials. The most fundamental scaling was developed by Hopkinson who demonstrated that the shock wave radius, R, scales with the cube root of the energy, E, or mass, m, of the explosive charge [3]. This scaling is typically used to identify an equivalent radius for explosions of two different masses of the same material:

$$\begin{aligned} \frac{R_1}{R_2}=\left( \frac{m_1}{m_2}\right) ^{1/3}=\left( \frac{E_1}{E_2}\right) ^{1/3} \end{aligned}$$
(1)

For explosions of the same mass of different energetic materials, the scaling can yield an equivalent energy release between the two materials. This is traditionally used to compare energy release to the energy release of TNT, resulting in a “TNT equivalence” [4]. Calculation of TNT equivalence is frequently reported as a single number [5, 6], or more recently a TNT equivalence as a function of position from a charge [1, 7, 8]. The idea of TNT equivalence as a function of radius means that the shock propagation from each explosive material is specific for that explosive, and detailed measurements of shock wave position versus time for each explosive material should be recorded.

Recent work by Wei and Hargather [9] presented a new scaling approach that demonstrated the ability to scale explosively driven shock waves in air and water. The work identified a single shock wave radius versus time relationship for a range of explosive formulations. Similarly, recent work by Diaz and Rigby developed a theoretical description of shock wave position and Mach number versus time starting from analysis of the hydrodynamic equations of motion [10]. Both of these new scaling approaches demonstrated good agreement with experimental data for explosive testing across orders of magnitude of scaling, and the identification of clear strong and weak shock limits to the curve fits. Both works have demonstrated that a singular shock wave radius versus time curve appears to exist that can characterize all explosive formulations, which appears to be a contradiction of the concept that each explosive has a unique shock wave radius versus time profile.

The goal of the present work is to compare the recent universal scaling approaches to each other and to traditional methods for representing shock wave radius versus time data. Insights into blast data and explosive energy release are obtained via scaling and comparison with the universal curve fits.

2 Explosive scaling approaches

Explosive scaling methods are generally similar in their approaches with differences primarily in the choice of the parameters used for non-dimensionalization, as reviewed by Wei and Hargather [9]. Scaling methods all assume that spherical shock wave expansion relates the volumetric expansion of the shock wave radius to an energy release from the explosion process, yielding the general form of the “cube-root” relationship (1). Sachs expanded on the work of Hopkinson by developing a method to scale the shock wave radius to account for different atmospheric temperature T and pressure P conditions [11]:

$$\begin{aligned} \frac{R_1}{R_2}=\left( \frac{m_1}{m_2}\right) ^{1/3}\left( \frac{P_{2}}{P_{1}}\right) ^{1/3} \end{aligned}$$
(2)

where time of arrival of the shock wave t is scaled according to:

$$\begin{aligned} \frac{t_1}{t_2}=\left( \frac{m_1}{m_2}\right) ^{1/3}\left( \frac{P_{2}}{P_{1}}\right) ^{1/3}\left( \frac{T_{2}}{T_{1}}\right) ^{1/2} \end{aligned}$$
(3)

Typically the data for a given explosive material are scaled to a 1-kg charge at normal temperature and pressure (NTP) conditions of 288.16 K and 101,325 Pa, respectively. The speed of sound a at NTP is denoted as \(a_0=340.29\) m/s. The scaling to NTP yields the scaling parameters S and c:

$$\begin{aligned} S=\left( \frac{m_1}{1\ \text {kg}}\right) ^{1/3}\left( \frac{101325\ \text {Pa}}{P_{1}}\right) ^{1/3} \end{aligned}$$
(4)

and

$$\begin{aligned} c=\frac{a}{a_0}=\left( \frac{T}{288.16\ \text {K}}\right) ^{1/2} \end{aligned}$$
(5)

which are used to define scaled radius \(R_\text {s}=R/S\) and scaled time \(t_\text {s}=tc/S\). Note that \(R_\text {s}\) and \(t_\text {s}\) are not non-dimensional, they have dimensions consistent with the dimensions of R and t used because the parameters S and c are dimensionless.

Note that the Sachs scaling does not use the energy of the explosive material as a parameter, but rather the mass of the explosive, so individual explosive materials should fall onto different scaled radius versus time curves. The Sachs scaling has been shown to be appropriate across many orders of magnitude from \(10^{-6}\) to \(10^{6}\) kg.

2.1 Wei–Hargather scaling

The scaling developed by Wei and Hargather [9] defines characteristic length and time scales, \(l_\text {c}\) and \(t_\text {c}\), respectively:

$$\begin{aligned}{} & {} l_\text {c}=\left( \frac{E}{\rho {a^2}}\right) ^{1/3} \end{aligned}$$
(6)
$$\begin{aligned}{} & {} t_\text {c} = \frac{l_\text {c}}{a} = \frac{1}{a}\left( \frac{E}{\rho {a^2}}\right) ^{1/3} \end{aligned}$$
(7)

These characteristic scales are used to create a non-dimensional shock wave radius \(R^*=R/l_\text {c}\) and time \(t^*=t/t_\text {c}\). Note that this scaling differs from Sachs’ scaling in the choice of atmospheric parameters, with atmospheric density \(\rho \) and sound speed a used here, whereas Sachs’ scaling uses temperature and pressure. This scaling also uses the explosive energy E as originally done by Hopkinson, whereas Sachs scaling uses explosive mass.

Wei and Hargather presented a universal relationship for the non-dimensional shock wave radius as a function of non-dimensional time for an explosion in a gas or liquid environment. The universal relationship was separated into strong shock (Mach number, \(M\gtrsim 5\), “Taylor regime”) and weak shock (\(M\lesssim 5\)) regimes, respectively [9]:

$$\begin{aligned}{} & {} R^* = (t^*)^{2/5} \quad {\textrm{for}}\quad M \gtrsim 5 \end{aligned}$$
(8)
$$\begin{aligned}{} & {} R^* = t^* + u \big [ {\textrm{ln}}(v+w(t^*)^p) \big ]^q \quad {\textrm{for}}\quad M \lesssim 5 \end{aligned}$$
(9)

where the coefficients \(u,w,p, \textrm{and}~q\) were determined by curve fitting to experimental data, with determined values of \(u = 0.45\), \(w=15\), \(p=1.4\), and \(q = 0.3\), and the coefficient v was set to 1 to match the origin (\(R^*=0\) at \(t^*=0\)). Equations (8) and (9) can be differentiated to yield the shock wave Mach number M as a function of time:

$$\begin{aligned} {\begin{matrix} M = \frac{2}{5} (t^*)^{-3/5} = \frac{2}{5} (R^*)^{-3/2} \quad {\textrm{for}}\quad M \gtrsim 5 \\ {\textrm{or}}\quad t^* \lesssim 10^{-2} \\ {\textrm{or}}\quad R^* \lesssim 0.16 \end{matrix}} \end{aligned}$$
(10)
$$\begin{aligned} {\begin{matrix} M = 1 + \frac{u \cdot w \cdot p \cdot q \cdot (t^*)^{p-1}}{(v+w(t^*)^p)\big [ \textrm{ln}(v+w(t^*)^p) \big ]^{1-q}} \\ \textrm{for }\quad M \lesssim 5 \end{matrix}} \end{aligned}$$
(11)

The Wei–Hargather non-dimensional scaling demonstrated a good fit of experimental data to these curves for a wide range of explosive masses, explosive materials, atmospheric conditions, and atmosphere phases for both liquid and gas environments [9]. The strong shock regime of the Wei–Hargather scaling matches the theoretical shock wave radius versus time decay to the 2/5 power as developed by Taylor [12], von Neumann [13], and Sedov [14], which is an important recovery of the theoretical shock wave propagation. Note that the coefficient symbols used in (9) have been changed here from the original publication [9] to prevent confusion with other variables and curve fits here.

2.2 Diaz–Rigby scaling

The recent work by Diaz and Rigby derived a relationship for the shock Mach number versus time starting from conservation of mass, momentum, and an equation of state in spherical coordinates [10]. The resulting equation for shock Mach number M is:

$$\begin{aligned} M=\frac{\text {d}z}{\text {d}\tau }=\left( 1+\frac{1}{K_0z^3}\right) ^{1/2} \end{aligned}$$
(12)

with dimensionless time

$$\begin{aligned} \tau =\frac{at}{R_0} \end{aligned}$$
(13)

and dimensionless distance \(z=R/R_0\), where the length scale

$$\begin{aligned} R_0=(E/P_0)^{1/3} \end{aligned}$$
(14)

is defined by the energy release of the explosive E, atmospheric pressure \(P_0\), and atmospheric sound speed a. The constant \(K_0\) is found to equal 7.86 from application of the limits for the shock behavior at zero and infinite times [10].

The Diaz and Rigby theoretical analysis showed good agreement to data from explosions of TNT, C4 (PE4), ammonium nitrate, and nuclear bombs, and recovers the theoretical strong shock relationship. The work also demonstrated the ability to estimate an explosive energy yield from the shock propagation data from nuclear blasts [10].

2.3 The Dewey equation

Shock wave radius versus time data are frequently fit to an empirical curve to simplify data handling, to improve differentiation of the data for determining shock wave velocities, and to simplify data archiving. The most common curve used in modern literature is an equation originally proposed by Dewey [6], and here is referred to as the “Dewey equation”:

$$\begin{aligned} R_\text {s}=A+Ba_0t_\text {s} + C \textrm{ln}\left( 1+a_0t_\text {s}\right) + D \sqrt{\textrm{ln}\left( 1+a_0t_\text {s}\right) } \end{aligned}$$
(15)

This curve fit provides a scaled radius \(R_\text {s}\) as a function of scaled time \(t_\text {s}\), where \(a_0\) is the sound speed in the scaled environment which generally is NTP. The four coefficients A, B, C, and D are unique to each explosive formulation, which are found by fitting (15) to experimental data. This curve fit has been used heavily in the literature, and the resulting coefficients A, B, C, and D are presented as the explosive characterization for a given material.

Equation (15) can be differentiated to yield a shock wave velocity as a function of time:

$$\begin{aligned} \frac{\text {d}R_\text {s}}{\text {d}t_\text {s}}=a_0\left( B+\frac{C}{1+a_0t_\text {s}} +\frac{D}{2\left( 1+a_0t_\text {s}\right) \sqrt{\textrm{ln}\left( 1+a_0t_\text {s}\right) }}\right) \nonumber \\ \end{aligned}$$
(16)

As proposed by Dewey [1, 6, 7], coefficient B is frequently set to a value of 1 so that the curve fit forces the shock wave to decay to a sound wave as \(t_\text {s} \rightarrow \infty \). For near-field data, the value of coefficient B is frequently found via the curve fit. Equation (15) is finite at \(t_\text {s}=0\), with the coefficient A representing the shock wave radius at \(t_\text {s}=0\). Values for A are reported in the literature as small or even negative numbers. This curve fit does not specifically return the strong shock behavior at early time.

The Dewey equation differs from the Wei–Hargather and Diaz–Rigby scalings in that there is a set of curve fit coefficients specific to each explosive formulation. Extraction of relative energy between explosive formulations is obtained by scaling the curves relative to one another, which yields a TNT equivalence if scaled to TNT [1, 8].

3 Non-dimensional Dewey equation

Inspection of the Dewey curve fit in (15) reveals that it is a dimensional equation. The variable \(R_\text {s}\) should have dimensions of length and \(t_\text {s}\) should have dimensions of time, presumably in seconds to match the time of \(a_0\). The coefficients A, B, C, and D each have dimensions, which are all different. In the case of coefficients C and D, the units are non-standard as a result of the natural log and square root of physical units in the operations. Published literature that uses this equation generally does not present the units for these coefficients. The choice of the units of \(R_\text {s}\) and \(t_\text {s}\), which are not always explicitly reported, determines the values of A, B, C, and D, which are also critical to document as part of the archiving of the A, B, C, and D coefficients.

A non-dimensional form of the Dewey equation is of interest for consistency. The development of a non-dimensional form is considered starting from writing the equation as radius R as a function of time t:

$$\begin{aligned} R=A+Ba_0t + C \textrm{ln}\left( 1+a_0t\right) + D \sqrt{\textrm{ln}\left( 1+a_0t\right) } \end{aligned}$$
(17)

This form is the same as (15), but the shock wave radius R and time t are considered instead of a scaled radius (\(R_\text {s}\)) and scaled time (\(t_\text {s}\)). The dimensional quantity inside the natural logarithms can be eliminated by dividing the term \(a_0t\) by a characteristic length \(L_\text {c}\), yielding a non-dimensional term \((1+a_0t/L_\text {c})\) inside of the natural logarithm. The coefficients C and D would thus be dimensionless with this adjustment. Examination of the B term shows that it should similarly be divided by \(L_\text {c}\) to allow B to be unitless. The A term stands alone and therefore has the same units as the rest. A dimensionally consistent and non-dimensional form of (15) is thus proposed as:

$$\begin{aligned} R^*=\frac{R}{L_\text {c}}=A+B\frac{a_0t}{L_\text {c}} + C \textrm{ln}\left( 1+\frac{a_0t}{L_\text {c}}\right) + D \sqrt{\textrm{ln}\left( 1+\frac{a_0t}{L_\text {c}}\right) } \end{aligned}$$
(18)

The term \(a_0t/L_\text {c}\) can be replaced with a non-dimensional time \(t^*\) using a characteristic time scale \(t_\text {c}=L_\text {c}/a_0\):

$$\begin{aligned} t^*=\frac{t}{t_\text {c}}=\frac{a_0t}{L_\text {c}} \end{aligned}$$
(19)

yielding

$$\begin{aligned} R^*=A+Bt^*+C\textrm{ln}(1+t^*)+D\sqrt{\textrm{ln}(1+t^*)} \end{aligned}$$
(20)

This equation is now entirely non-dimensional, and all coefficients are also dimensionless. This equation is of the same form that was proposed by Dewey in 1971 [15]; only a dimensional equation was proposed then. The dimensional form was modified at some point with the addition of \(a_0\) to the form that is used today and given in (15). The coefficient A represents the shock wave radius at \(t^*=0\). At this point, the choice of the characteristic length scale \(L_\text {c}\) has not been specified. It should be noted that the length scale choice should account for the charge mass and atmospheric conditions to maintain typical scaling achieved through Hopkinson and Sachs scalings. The length scale to be used here will be the Wei–Hargather scaling \(L_\text {c}=l_\text {c}=(E/\rho a^2)^{1/3}\) (6).

Differentiating (20) with respect to dimensionless time results in:

$$\begin{aligned} \frac{\text {d}R^*}{\text {d}t^*}=\frac{1}{a_0}\frac{\text {d}R_\text {s}}{\text {d}t_\text {s}} = M = B+\frac{C}{1+t^*}+\frac{D}{2\left( 1+t^*\right) \sqrt{\textrm{ln}\left( 1+t^*\right) }} \end{aligned}$$
(21)

where M is the Mach number of the shock in the atmosphere defined by \(a_0\). The coefficient B could still be set to 1 to force the shock wave propagation to decay to a sound wave as \(t^*\rightarrow \infty \).

Unfortunately, the new coefficients for the non-dimensional Dewey equation are not the same as those for the dimensional form, and there is no direct transformation between the coefficients because of the changes inside of the natural logarithm term. The coefficient values also will vary with the choice of \(L_\text {c}\).

4 Non-dimensional correlation comparisons

A range of published experimental data are used here for comparison with the Wei–Hargather, Diaz–Rigby, and non-dimensional Dewey empirical correlations. The experimental data are summarized in Table 1 including relevant material properties and notes about the experiments. The data span ideal and non-ideal energetic materials, with charge masses varying across several orders of magnitude. Material properties of each explosive and atmospheric conditions are extracted from the publications. The energy release E used here is the total energy of detonation for the explosive. The specific total energy of detonation e is calculated using CHEETAH, an equilibrium thermochemical calculation code developed by Lawrence Livermore National Laboratory[16, 17], and is related to the total energy via the charge mass \(E=me\).

Table 1 Experimental data sources and material properties
Full size table

The collected experimental data are plotted in Fig. 1 as (a) raw data, (b) scaled with traditional Sachs scaling, and (c) Wei–Hargather scaling. The unscaled data show that the experimental data span a wide range of length and time scales. Charges of similar masses are clustered together. Under traditional Sachs scaling (Fig. 1b), the data from individual explosive types collapse to individual radius versus time curves. This is the traditional point at which the Dewey equation is fit to each individual explosive material scaled shock wave radius versus scaled time data. This results in individual curve fits for each explosive.

Fig. 1
figure 1

Shock wave radius versus time presented as a raw data, b scaled with traditional Sachs scaling, and c Wei–Hargather scaling

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Under the Wei–Hargather scaling (Fig. 1c), the curves collapse to a single non-dimensional radius versus time curve. The theoretical relationship of (9) is plotted along with the data. Note that the strong shock region defined by (8) is only for \(t^*<10^{-2}\) and is not shown on these axes. This single curve was presented by Wei and Hargather [9] with similar data.

Fig. 2
figure 2

Scaled shock wave radius versus scaled time using the Wei–Hargather scaling presented on a linear and b log scales. The transition from the strong to weak shock regions is shown as the vertical dashed line on (b). Note that the Wei–Hargather strong shock almost perfectly overlaps the Diaz–Rigby curve in the strong shock regime, which is why the curve is barely observed in the plot

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The Diaz–Rigby scaled curve fit is also shown in Fig. 1c. In order to plot both the Wei–Hargather and the Diaz–Rigby curve fits together, the non-dimensionalization of the curves needed to be aligned. Examination of the scaling for Wei–Hargather versus the one for Diaz–Rigby shows that the primary difference is in their choice of characteristic length scale (6) versus (14), with Wei–Hargather using the term \(\rho a^2\) versus \(P_0\) for Diaz–Rigby. The time scaling in both approaches uses the ambient atmospheric sound speed and the characteristic length. The Diaz–Rigby non-dimensional time can be converted to the Wei–Hargather non-dimensional time via:

$$\begin{aligned} t^*=\frac{a}{l_\text {c}}\frac{R_0}{a}\tau = \left( \frac{\rho a^2}{P_0}\right) ^{1/3}\tau \end{aligned}$$
(22)

The non-dimensional radius can be converted via:

$$\begin{aligned} R^*=\frac{R_0}{l_\text {c}}z = \left( \frac{\rho a^2}{P_0}\right) ^{1/3}z \end{aligned}$$
(23)

The Mach number in both scalings is the same.

Both the Wei–Hargather and Diaz–Rigby fits are similar in their alignment with the data (Fig. 1c), with the Diaz–Rigby fit tapering to a lower shock velocity sooner than the Wei–Hargather fit. The difference in shock radius as a function of time is easily observed in linear axes but is not apparent when using a log–log plot as shown in Fig. 2. Examination of Fig. 4 in [10] shows that the Diaz–Rigby fit slightly under-predicts the radius as a function of time compared to ConWep, which appears to be similar to that observed here. The Diaz–Rigby theory includes the theoretical assumption that the pressure pulse is stretched from the shock front all the way to the center of the explosive charge, as initially proposed by Taylor [12], which could be why the shock appears to not propagate as fast in the intermediate region because its energy is distributed all the way back to the charge center.

Fig. 3
figure 3

A comparison of the four different non-dimensional Dewey equation fits showing shock wave radius versus time a with a log–log scale and shock Mach number versus radius with a b linear and c log–log scale. The Wei–Hargather strong and weak shock curve fits and the Diaz–Rigby curve fits are included. The vertical dashed line represents the transition from strong shock to weak shock for the Wei–Hargather scaling

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Careful examination of the data in Fig. 1c shows the C4 data from [2] significantly above the scaling fit. Revisiting the charge setup for that test series reveals that the charges were located above the ground by about 1 m, yielding a strong ground reflection. The data were extracted along the charge center, but a significant distance from the charge, to where the shock wave had become a roughly hemispherical blast, with no evidence of a Mach stem present. The data should thus be adjusted to account for the ground reflection. The blast literature generally applies a correction factor of between 1.7 and 2 to account for ground reflection [4, 20]. The value of 2 is used for an “unyielding surface,” and numbers less than 2 are used based on the energy absorbed by the ground into cratering. Because the Wei–Hargather scaling uses the explosive energy release E, the presence of a ground reflection in the data must be accounted for in the energy used for the scaling. The energy used to non-dimensionalize the C4 data extracted from [2] is therefore adjusted to be twice the value in Table 1. The data are observed to better collapse with all of the data in Fig. 2. This choice of energy is further discussed in Sect. 6.

The data from Rigby et al. [19] were obtained by tracking the expanding gases, or “fireball,” during early time expansion when the shock wave was assumed to still be attached to the expanding gases. The authors note that no apparent shock separation was observed, but there was variation in the fireball surface. Details on the fireball surface variation are reported in [19]. The mean fireball location was used here, and the variation observed is approximately captured by the data symbol sizes in the present plots. These data were excluded from the Dewey curve fitting here.

In Fig. 2b, the data from Rigby et al. [19] fall below the strong shock curve, which was also reported by Diaz and Rigby [10]. The data align with the TNT standard data [4] in this early time regime \(t^*<10^{-2}\). The PETN data from [18] do not appear to be along the same curve, but rather appear to stay along the strong shock curve. This indicates that there may be some differences in the strong shock regime between nuclear and chemical explosions, or even between different chemical compositions. These data are extremely difficult to measure in these early times, and thus, this region warrants continued investigation.

5 Non-dimensional Dewey equation curve fitting

The non-dimensional data can also be fit to the non-dimensional Dewey equation in (20). Here the data summarized in Table 1 are first non-dimensionalized with the Wei–Hargather scaling yielding a scaled radius versus scaled time (Fig. 2), which is then fit to (20). The choice of the Wei–Hargather scaling means that that \(L_\text {c}\) in (18) is equal to \(l_\text {c}\) in (6) and (19). As noted in Table 1, the PBXN-110 data from [18] are omitted from the curve fitting. These tests were omitted because their explosive energy release is not well known and will be estimated from the fit as discussed in Sect. 6. The C4 data from [2] included the energy release number multiplied by the factor of 2, as plotted in Fig. 2.

Four different curve fits are considered here: (1) fitting all four coefficients (A, B, C, and D), (2) a fit with \(B=1\), (3) a fit with \(A=0\) and \(B=1\), and (4) a fit with \(A=0.013\) and \(B=1\). The resulting curve fits are given in Table 2 and plotted in Fig. 3. A quality of the fit is presented as an R\(^2\) value from the nonlinear least-squares curve fit in Table 2, which shows essentially the same quality of fit for all coefficient sets.

Overall, the non-dimensional equations all fit the data extremely well, as expected. The four individual Dewey curve fits and the Wei–Hargather weak shock relationship are essentially indistinguishable on a linear axis for radius versus time, so that plot is not shown here. In the log–log plot of the radius versus time data in Fig. 3a, the differences in the curve fits are observable in early time expansion of the shock wave for \(t^*<10^{-1}\) and \(R^*<0.5\). Fits 1 and 2 asymptote toward a constant non-dimensional radius value. This is a result of the A coefficient, which acts as a non-dimensional shock wave radius at \(t^*=0\). The freely fit values of A result in larger radii at early time. It should be noted that the data sets do not have many points in this strong shock regime, which may be causing a poor fit in this region. The work by Wei and Hargather [9] identified that only nuclear blast data yielded data points in this strong shock regime. The addition of the Trinity test data [21] in the curve fitting does not significantly change any of the fits and the A coefficient remained approximately the same for Fits 1 and 2. The Trinity data were not included in the reported coefficients in Table 2.

The choice of the A coefficient was explored in Fits 3 and 4. In Fit 3, \(A=0\), assuming that the shock wave starts at a radius of 0 at a time of 0 (i.e., a true point-source). This resulted in prediction of the shock wave radius below the strong shock solution, which is considered not accurate.

For Fit 4, setting A equal to the charge radius was considered. This assumes that the detonation passes through the charge essentially instantaneously and thus the shock wave emerges from the charge surface at \(t=0\). The radius of each charge in the dataset was converted to a non-dimensional radius using the Wei–Hargather scaling (6). The charge radius was either directly reported in each test or an equivalent radius was calculated from the specified charge mass and explosive material density. The non-dimensional charge radius for all charges in the dataset ranged from 0.01414 to 0.01626 with a mean of 0.01542 and standard deviation of 0.00082. The A coefficient in Fit 4 was set to the mean value. The plot of Fit 4 in Fig. 3 shows the asymptote to the specified charge radius.

Table 2 Non-dimensional Dewey curve fit coefficients for (20)
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Fig. 4
figure 4

Comparison of the Dewey curve fits and theoretical profiles with experimental data in Mach number versus non-dimensional radius space plotted on a a linear scale, b log–log scale, and c zoomed-in log–log scale showing only the weak shock region

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All of the non-dimensional Dewey curve fits follow the same trends in the Mach number versus radius space as shown in Fig. 3b, c. For this plot, the A term plays a limited role, only affecting the horizontal axis because M is not a function of A. The plots of the Dewey curve fits all have similar shapes in the weak shock regime, but none of the Dewey fits agree well with the strong shock solution region. This may be a result of the limited number of points in the strong shock region used to generate the fits, but is likely more a result of the fit not representing the strong shock region well. The strong shock region, however, does not need to be represented by a curve fit as it could always use the simple 2/5 power law developed theoretically [12,13,14], which is also supported by the works of both Wei and Hargather [9] and Diaz and Rigby [10].

Figure 4 presents the fits in Mach number versus non-dimensional radius space along with the experimental data. The experimental data Mach number is calculated by performing a centered finite difference on the radius-time data. This leads to significant noise in the data, which is the spread of the data points. Examining Fig. 4, all of the fits pass through the experimental data and could all be reasonable approximations for the data in general. The Diaz–Rigby curve appears to represent a lower bound to the data and all curve fits. The Wei–Hargather fit appears to pass through the middle, or upper-middle of the experimental data in terms of Mach number at a given non-dimensional radius. It is interesting that the Dewey curve fits generally agree with the Wei–Hargather fit for non-dimensional radius \(R^*\gtrsim 0.6\) and transition toward the Diaz–Rigby fit at smaller radii. The Dewey curve fits all fail to accurately represent the strong shock regime. The Dewey curve fit should not be used in the strong shock regime because it does not accurately reproduce the 2/5 power-law profile. The data from [19] still fall below the strong shock region and continue to align with the TNT standard. In Fig. 4b, the data appear to follow reasonably well with Dewey fits 3 and 4, but examining the more zoomed-in plot in Fig. 4c the data are above these curve fit lines in this region.

Table 3 Experimental data sources and material properties
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Fig. 5
figure 5

Calculated explosive energy output as a function of scaled radius calculated by scaling individual data points to the universal shock radius profiles. a Scaling the PETN data and TNT data to the Wei–Hargather curve fit and the calculated singular least-square energy values which optimize the entire dataset to the theoretical profile. b The TNT data are scaled to the Wei–Hargather, non-dimensional Dewey Fit 1, and Diaz–Rigby profiles. All data sets show some variation in the calculated energy release as a function of radius

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The differences between the Dewey fits, the Wei–Hargather correlation, and the Diaz–Rigby theory are by no means large. The differences are subtle and still lie within the spread of the experimental data. In radius versus time space, the differences are barely perceptible, but can be observed in Mach number versus radius space, which is more useful for blast characterization. The consistency between the curves, however, does continue to support the idea that the non-dimensional shock wave propagation can be represented by a single curve for all explosive materials.

6 Energy calculation from measured shock data

The existence of a single curve, which represents shock propagation from all explosive materials, allows calculation of explosive energy release from measurements of shock wave propagation. This was demonstrated by Diaz and Rigby for nuclear blast energy release estimates [10]. Here the Wei–Hargather relationship is applied to calculate the energy release from a blast by scaling the recorded shock radius as a function of time to the generic profile. Each dataset considered is fit to the weak shock regime (9) using a nonlinear least-square calculation [22] to determine the value of E that minimizes the error from the Wei–Hargather weak shock profile. The value of the specific energy of detonation that is determined from the least squares fit (LS Fit) is shown in Table 3 compared to the calculated value from CHEETAH based on the explosive composition. The calculated energy is within about \(20\%\) of the CHEETAH calculation.

Note that many references exist showing similar computational predictions and experimental measurements for heats of detonation [23,24,25,26]. The CHEETAH values calculated here are similar to other computations. No consistent summary of experimentally measured heat of detonation for explosives exists, so no comparison to experimental measurements is presented here. The energy release amount for TNT is also regularly standardized as 4.184 MJ/kg, which is similar to the CHEETAH value here.

Fig. 6
figure 6

The first PETN data set is scaled with different choices of the energy E as obtained from CHEETAH, least-square fit to Wei–Hargather profile, and with the variable energy value shown in Fig. 5, plotted on various axes with the Wei–Hargather weak shock profile

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As discussed in Sect. 4, the C4 data from [2] when fit directly had a significant error compared to the CHEETAH calculations (values shown in parenthesis in Table 3). If the ground reflection is considered, then the calculated energy from the least-square fit should be divided by the factor of 2 for the actual energy release. This yields the detonation energy values shown of 6.245 and 4.975, with errors of \(-17.2\%\) and \(6.7\%\) for the 4.54-kg and 27.22-kg charges, respectively.

The energy calculation can also be applied at a single point instead of along an entire curve. The energy calculation from a single point could be useful for trials with limited experimental data or for analysis of unexpected explosions or industrial accidents like the Beiruit explosion in 2020 [27]. The energy release calculation requires knowledge of any combination of two parameters from: time of arrival, radius, or Mach number. For forensic analysis, time of arrival and radius would likely be known. The Mach number may be a useful parameter that could be derived from experimentally measured peak pressure [1, 8] at a known radius.

The calculation of energy from individual data points is performed on the PETN and TNT data here and plotted in Fig. 5a. The plot shows the calculated energy release as a function of scaled radius from the charge. The PETN and TNT data both show some variation of the energy value as a function of radius. Diaz and Rigby [10] discussed some pitfalls in calculating the energy by fitting data to the theoretical curve fit, in particular that the choice of radius range used is important and that large radii will cause a calculated energy amount to be non-physically high. The data in Fig. 5a exhibit the high energy calculation for the TNT data in particular, which has a large radii range used.

This calculation of the energy release at each radial position results in a curve and not a singular value, which indicates that there may be some variation in the shock propagation as a function of radius and that scaling with a single energy value is not appropriate. This is the traditional assessment that each explosive material has a unique shock wave position as a function of radius. The curve could also indicate that the universal scaling curve is not accurate and thus results in an apparent energy release variation.

The TNT data are used to examine the estimated energy release versus the chosen shock wave profile in Fig. 5b. The TNT data are related to the Wei–Hargather weak shock, Dewey Fit 1 (Table 2), and the Diaz–Rigby profiles to calculate energy release at each individual radial point. The Wei–Hargather data in Fig. 5a and b are the same. The TNT energy calculation has a different functional curve depending on the choice of universal shock wave function it is fit to. The Diaz–Rigby scaling results in a non-physically large energy number after a scaled radius of about 0.5, with the data significantly above the graph region shown. When scaled to the Dewey fit, the TNT data asymptotes to a constant value close to the traditional energy value 4.184 MJ/kg, which may be a result of the Dewey equation functional form being embedded in the TNT data.

The different energy values that can be obtained lead to questions of whether the idea of a universal shock profile is appropriate or whether individual explosives should have individual shock wave radius versus time profiles. To consider this, the first PETN data set from Table 1 is scaled with different energy values and plotted on different axes of interest in Fig. 6. The data are scaled using the values in Table 3 and also with the variable energy as plotted in Fig. 5. When plotted on scaled radius versus scaled time axes (Fig. 6a, b), the differences between the scaled values are almost indistinguishable and would certainly be within the experimental error variations. Plotted on Mach number versus scaled radius (Fig. 6c), the data points show some change in their relative radius position as a function of the choice of energy, but again the change is small. The differences are most observed at small scaled radii. Note that the Mach number is unaffected by the choice of energy scaling, only the scaled radius changes.

The plots show that the energy release of the explosive does not significantly change a data point’s position in the non-dimensional space when using the Wei–Hargather scaling. For a chemical explosive with mass up to several hundreds of kilograms, a detonation energy on the order of 5 MJ/kg, detonated in an air atmosphere, the characteristic length scale (6) is not significantly changed as the detonation energy varies by 30\(\%\). Most useful explosive formulations have detonation energies that fall within a \(30\%\) variation, so using the scaling fit to extract energy release is expected to have a large uncertainty. For nuclear explosions, the energy release is significantly larger than the non-dimensionalizing term \(\rho a^2\), which results in larger changes in \(l_\text {c}\) which will enable more unique determination of an energy release by scaling to the universal shock wave profile and especially when comparing to the strong shock region. These limitations are also expected for the Diaz–Rigby fit, which uses a similar non-dimensionalization, i.e., \(O(\rho a^2) \sim O(P)\).

7 Conclusions

Several representations of universal curves for explosively driven shock wave radius versus time have been considered and compared here. A new non-dimensional Dewey equation has been derived, and a universal Dewey fit to experimental data was developed for comparison with the Wei–Hargather and Diaz–Rigby equations. Both the Wei–Hargather and Dewey approaches are empirical correlations, with features that meet expected shock wave propagation limits at \(t=0\) and as \(t\rightarrow \infty \), whereas the Diaz–Rigby fit is derived analytically from conservation equations. The comparisons of the fits show that the curves are almost indistinguishable on scaled radius versus time profiles in linear space, and only limited variation can be observed on a log–log plot.

The differences between the fits mostly occur in a transition region \(0.15\lesssim R^*\lesssim 2\), which is a region that continues to merit study for measurement of shock propagation velocity (Mach number) versus radius. This region, which for a 1-kg TNT charge is approximately 0.45 m\(~\lesssim R\lesssim 6\) m, is the area in which the shock wave propagation may potentially be impacted by the fireball expansion process and additional “after burning” or continued acceleration of the expanding gases beyond that expected for a point energy release. The Diaz–Rigby fit may provide a lower-bound shock propagation profile for this region, because it assumes a point-source energy release, whereas the Wei–Hargather and Dewey fits have been made to experimental data, which would include any influence of fireball expansion processes. Detailed measurements of position and velocity should be pursued to examine these effects on shock propagation for different explosive formulations.

The region \(0.12\lesssim R^*\lesssim 0.7\) in particular has limited historical data and is a region in which additional measurements should be made. This region is approximately 0.35 \(\hbox {m}\lesssim R\lesssim 2.2\) m for a 1-kg TNT charge. Most of this experimental data gap is due to limited experimental capabilities to measure shock propagation in this early time and small radii. The Trinity data and nuclear testing data generally fall before this region, and most chemical explosive testing lies after this region. Trinity data fall before this due to the significant energy release and specialized cameras used for the early time measurements. The silver azide and likely most TNT data fall outside of this region because of the limited camera capabilities. The TNT data presented in Kinney and Graham [4] have data points that fall within \(R^*<0.1\), which is expected to be the strong shock regime. These data points fall below the strong shock regime in terms of Mach number. These data, however, in this region for TNT are suspect as it is unlikely that significant temporal resolution was available with cameras to document this region accurately. Likely the “data” in this region are a result of historical curve fitting and not actual historical data. The more recent data from Rigby et al. [19] filmed 0.100-kg charges at 160,000 frames per second and support the curve in this region, but the irregularities in the fireball surface add uncertainty and further support the importance of using modern instrumentation to continue studying early time shock and fireball expansion.

Accurate calculation of energy release from shock wave data for chemical explosives is possible, but the experimental errors need to be determined. In particular, detailed experiments with a consistent explosive charge material with detonation energy release and shock wave propagation measurements need to be developed. Energy calculation in the strong shock region, in comparison with the theoretical 2/5 power-law fit, yields reasonable results because the energy release is significant.

More experimental data should continue to be incorporated into fitting more accurate empirical coefficients. Analytical solutions for the shock wave propagation and evolution should also continue to be pursued. Non-dimensional scaling needs to continue to be considered with explosive data presentations. The strong shock regime for non-dimensional \(t^* \lesssim 10^{-2}\), \(R^* \lesssim 0.16\), or \(M \gtrsim 5\) should be treated separately and fit to the theoretical 2/5 power-law profile as theoretically derived [12,13,14]. The Dewey equation should not be used in this strong shock region as it does not accurately reproduce the 2/5 power-law profile.