The normal reflection of a blast wave

The normal reflection of a blast wave  (1943)
by Subrahmanyan Chandrasekhar

From: Ballistic Research Laboratories, 20 December 1943

Ballistic Research
Laboratory Report No. 439

Chandrasekhar/emh
Aberdeen Proving Ground, Md.
20 December 1943

THE NORMAL REFLECTION OF A BLAST WAVE

Abstract

In this paper a numerical method is outlined for determining the impulse ${\displaystyle I_{w}}$ imparted to a rigid wall by the normal reflection of a blast wave. The method is illustrated by a numerical example and the corresponding pressure-time curve on the wall obtained.

1. Introduction. The study of the reflection of blast waves from rigid surfaces was initiated by G. I. Taylor (R.C., 118) who considered the case of normal reflection. In Taylor's investigation (and also in those which have followed his) attention has been directed almost exclusively on the instantaneous increase in the pressure on the wall at the moment of reflection. Thus, it can be readily shown that if the peak pressure in the incident blast wave is ${\displaystyle y}$ (in units of the pressure in the undisturbed region in front of the blast wave before reflection) then at the moment of reflection of the shock front, the pressure on the wall jumps instantaneously to the value ${\displaystyle \alpha }$ (in the same units as ${\displaystyle y}$) where

(1)${\displaystyle \alpha =y{\frac {(3\gamma -1)y-(\gamma -1)}{\gamma +1+(\gamma -1)y}}.}$

However, the determination of this instantaneous increase of pressure on the wall provides only a partial answer to the problem of reflection of a blast wave which requires for its complete solution the determination of the entire pressure-time curve on the wall from the moment of reflection of the shock front to the time the original conditions on the wall are restored. In this paper we shall outline a simple numerical method by which this pressure-time curve on the wall can be derived. For the sonic case the corresponding problem has been treated in considerable detail by Taylor (R.C., 235) and others.

For later reference we may include a table of ${\displaystyle \alpha }$ for the case ${\displaystyle \gamma =1.4}$ (air).

Table

 y ${\displaystyle \alpha }$ ${\displaystyle y}$ ${\displaystyle \alpha }$ ${\displaystyle y}$ ${\displaystyle \alpha }$ ${\displaystyle y}$ ${\displaystyle \alpha }$ 1.0 1.000 2.4 5.200 3.8 11.400 6.0 23.500 1.2 1.433 2.6 5.986 4.0 12.400 7.0 29.615 1.4 1.930 2.8 6.809 4.2 13.424 8.0 36.000 1.6 2.484 3.0 7.667 4.4 15.536 9.0 42.600 1.8 3.092 3.2 8.557 4.6 15.536 10.0 49.375 2.0 3.750 3.4 9.477 4.8 16.622 2.2 4.454 3.6 10.425 5.0 17.727

2. Outline of the Numerical Method. The method we are about to describe is similar to that adopted by Penney in his investigations on the decay of shock waves (R.C., 260 and 301). For the sake of simplicity we shall restrict ourselves to the case of the normal reflection of plane blast waves of moderate intensities (Mach numbers less than 1.5). Then under these circumstances the changes in entropy as we cross the shock front can be ignored and to a sufficient approximation we can write (cf. Penney, loc. cit., or Chandrasekhar, B.R.L. Report No. 423).

 ${\displaystyle \left.{\begin{array}{lll}dP&={\frac {\partial P}{\partial x}}[dx-(u+c)dt]\\\\dQ&={\frac {\partial Q}{\partial x}}[dx-(u-c)dt]\end{array}}\right.}$ (2)

where

 ${\displaystyle P={\frac {2}{\gamma -1}}c+u;\quad Q={\frac {2}{\gamma -1}}c-u,}$ (3)

${\displaystyle c}$ and ${\displaystyle u}$ denoting the local sound and mass velocities respectively. According to equations (2)

 ${\displaystyle dP=0\quad {\text{for}}\quad dx=(u+c)dt,}$ (4)

and

 ${\displaystyle dQ=0\quad {\text{for}}\quad dx=(u-c)dt.}$ (4)

Thus if the ${\displaystyle (P,x)}$ and the ${\displaystyle (Q,x)}$ curves are known at any instant ${\displaystyle t}$, those for ${\displaystyle t+\Delta t}$ can be obtained by a simple deformation process. The boundary conditions to be used are (1) that ${\displaystyle u=0}$ at the reflecting surface at all times (i.e., ${\displaystyle P=Q}$ on the wall) and (2) that the Rankine-Hugoniot equations are satisfied at the front. More particularly the details of the computations are as follows:

We start with a shock pulse (at the instant immediately before the reflection from the wall) in which we assign the distribution of pressure ${\displaystyle p}$ and the mass velocity ${\displaystyle u}$. This initial distribution of ${\displaystyle p}$ and ${\displaystyle u}$ can be assigned arbitrarily except for the condition that at the shock front the Rankine-Hugoniot equations are satisfied. On reflection the pressure at the wall becomes ${\displaystyle \alpha p}$ and a shock moves forward into the initial pulse. Suppose that the ${\displaystyle (P,x)}$ and the ${\displaystyle (Q,x)}$ curves are known at the time ${\displaystyle t}$. Then these curves can be deformed according to equations (4) and (5) to obtain the curves for the slightly later time ${\displaystyle t+\Delta t}$. However, the shock front moves to a new position obtained by displacing the position at ${\displaystyle t}$ by the amount ${\displaystyle U\Delta t}$ where ${\displaystyle U}$ is the shock velocity. The result of the deformation process will therefore be that the new ${\displaystyle P}$-curve crosses beyond the new shock front while a gap in the ${\displaystyle Q}$-curve has arisen just behind the shock front.[1] We erase the part of the ${\displaystyle P}$-curve which is beyond the new shock front and determine the new value of ${\displaystyle Q}$ at the front as follows: The deformation of the original ${\displaystyle P}$ and ${\displaystyle Q}$ curves yields at the same time the new values for ${\displaystyle P}$ and ${\displaystyle Q}$ in front of the shock. Thus at the new position of the shock, the value of ${\displaystyle P}$ immediately behind the shock, and the values of ${\displaystyle P}$ and ${\displaystyle Q}$ in front of the shock are all known. These together with the Rankine-Hugoniot equations are sufficient to determine the physical conditions behind the shock front[2] thus determining the value of ${\displaystyle Q}$ immediately behind the shock front and also the new shock velocity. The ${\displaystyle P}$ and ${\displaystyle Q}$ curves for time ${\displaystyle t+\Delta t}$ are known and we are ready to take the next step. In this manner the reflection of a given initial shock pulse can be followed in its entirety.

3. A Numerical Example. The Impulse Imparted to the Wall. As an illustrative example, an initial shock pulse was considered in which at the instant before reflection the peak pressure was ${\displaystyle 1.5}$ (in units of the pressure in the undisturbed regions). Further the pressure was assumed to decrease linearly to the value ${\displaystyle 1}$. At the shock front the material velocity (as determined by the Rankine-Hugoniot equation for ${\displaystyle \gamma =1.4}$) is ${\displaystyle 0.3}$ in units of the velocity of sound in the undisturbed regions. The material velocity was further assumed to decrease linearly to the value zero where ${\displaystyle p=1}$. The derived distributions of pressure and velocity at various instants during the reflection of such a pulse are illustrated in Figures 1 and 2. The corresponding pressure-time curve on the wall is shown in Figure 3. And integrating the pressure-time curve of Figure 3 we can obtain the impulse ${\displaystyle I_{w}}$ imparted to the wall. It was thus found that

 ${\displaystyle I_{w}=\int (p-1)dt=5.22.}$ (6)

The definition of impulse in an incident shock pulse is arbitrary to some extent though the definition most commonly used is

 ${\displaystyle {\frac {1}{U}}\int (p-1)dx}$ (7)
Fig. 1
Fig. 2
Fig. 3
where ${\displaystyle U}$ is the shock velocity. For the numerical example considered the impulse as defined by (7) is ${\displaystyle 2.09}$. However, Dr. Lewy has suggested that it would perhaps be more consistent to define the impulse in a shock pulse at a given instant by the formula
 ${\displaystyle I_{p}=\int pudx,}$ (8)

where the integration on the right hand side is extended over the pulse. For a pulse in which ${\displaystyle p}$ and ${\displaystyle u}$ decrease linearly it can be shown that

 ${\displaystyle I_{p}={\frac {\gamma ^{2}x_{o}u_{o}}{(\Delta p)^{2}(\gamma +1)(\gamma +2)}}\left[(1+\Delta p)^{(\gamma +1)/\gamma }(2\Delta p-\gamma )+\gamma \right]}$ (9)

where ${\displaystyle \Delta p(=y-1)}$ denotes the over pressure at the shock front (in units of the pressure in the undisturbed regions), ${\displaystyle u_{o}}$ the maximum material velocity (in units of the velocity of sound in the undisturbed regions) and ${\displaystyle x_{o}}$ the length of the pulse. For the particular case considered the foregoing formula gives

 ${\displaystyle I_{p}=2.26.}$ (10)

Accordingly,

 ${\displaystyle {\frac {I_{w}}{2I_{p}}}=1.15,}$ (11)

i.e., a departure from the sonic case to the extent of ${\displaystyle 15\%}$.

It is possible that for small over pressures the factor corresponding to (11) can be found from an interpolation formula of the type

 ${\displaystyle I_{w}=2I_{p}(1+a\Delta p)}$ (12)

where ${\displaystyle a}$ is some appropriately chosen constant. Equation (11) would suggest that ${\displaystyle a\sim 0.3}$ since for the case we have considered ${\displaystyle \Delta p=0.5}$.

S. Chandrasekhar

APPENDIX

It is thus evident that the normal reflection of a shock from a rigid surface is equivalent to the head on collision of two shocks of equal intensity. Thus reflection is a special case of the collision of two shocks of unequal intensities. In the latter case it is known that a contact discontinuity will in general be formed which will move with a certain constant velocity. If ${\displaystyle p_{m}}$ and ${\displaystyle p_{o}}$ denote the pressure in the central zone before and after the two shocks have collided, it can shown that the ratio ${\displaystyle \alpha =p_{m}/p_{o}}$ is the positive root of equation

 ${\displaystyle {\frac {\alpha -y_{1}}{\sqrt {(\gamma -1)y_{1}+(\gamma +1)\alpha }}}{\sqrt {\frac {\gamma +1+(\gamma -1)y_{1}}{\gamma -1+(\gamma +1)y_{1}}}}+{\frac {\alpha -y_{2}}{\sqrt {(\gamma -1)y_{2}+(\gamma +1)\alpha }}}{\sqrt {\frac {\gamma +1+(\gamma -1)y_{2}}{\gamma -1+(\gamma +1)y_{2}}}}={\frac {y_{1}-1}{\sqrt {\gamma -1+(\gamma +1)y_{1}}}}+{\frac {y_{2}-1}{\sqrt {\gamma -1+(\gamma +1)y_{2}}}}}$ (13)

where

 ${\displaystyle p_{1}=y_{1}p_{o}\quad {\text{and}}\quad p_{2}=y_{2}p_{o},}$ (14)

${\displaystyle p_{1}}$ and ${\displaystyle p_{2}}$ denoting the pressures behind the two shocks before collision. For ${\displaystyle \gamma =1.4}$ equation (13) reduces to

 ${\displaystyle {\frac {\alpha -y_{1}}{\sqrt {y_{1}+6\alpha }}}\left[{\frac {6+y_{1}}{1+6y_{1}}}\right]^{1/2}+{\frac {\alpha -y_{2}}{\sqrt {y_{2}+6\alpha }}}\left[{\frac {6+y_{2}}{1+6y_{2}}}\right]^{1/2}={\frac {y_{1}-1}{\sqrt {1+6y_{1}}}}+{\frac {y_{2}-1}{\sqrt {1+6y_{2}}}}.}$ (15)

In the following table ${\displaystyle \alpha }$ is given for various pairs of values for ${\displaystyle y_{1}}$ and ${\displaystyle y_{2}}$. Page:Chandrareflection1943.pdf/11 Page:Chandrareflection1943.pdf/12 Page:Chandrareflection1943.pdf/13

1. A similar gap which arises in the ${\displaystyle P}$-curve near the wall can be filled at once since ${\displaystyle P=Q}$ on the wall.
2. This part of the calculation is best carried out by trial and error.