Theory of shock waves and introduction to gas dynamics/Chapter 10

by Yakov Zeldovich

Chapter 10

The History of the Shock Wave Problem

The equation of the connection between pressure and density in a substance subjected to the action of a shock wave, which was derived from elementary considerations and from the study of the conservations laws, led to an unexpected result, namely, the increase in entropy with compression of the ideal gas in a shock wave. Entropy increase follows directly from juxtaposing the initial and final state of the substance, which are associated with one another by the conservation equations. We did not investigate the processes that occurred between the control surfaces A and B (Fig. 23b) which led to entropy increase. Formally, as already mentioned, only the conservation equations are symmetrical with respect to $\rho _{1}$, $p_{1}$ and $\rho _{2}$, $p_{2}$. We could also satisfy the conservation equations by investigating the inverse motion, viz., a expansion wave in which expansion occurs within a small interval AB (which we shall not investigate closer) in accordance with the Hugoniot equation. In actual fact, however, such a motion is impossible since entropy would drop in it (this is the so-called Zemplen theorem [99] mentioned earlier). This particular feature of the result of Chapter 9 where, without considering dissipation processes, we came to a change in entropy, creates specific difficulties in the understanding of the theory of shock waves which can be overcome only if we observe the processes inside the region of the change of state proper (between the control surfaces A and B, (Fig. 23b). This has held up considerably the evolution of the theory of shock waves.

It is remarkable that the first three most important works on the theory of shock waves were produced at different time periods but, apparently, completely independently from one another. We shall therefore investigate them not in their chronological order.

Riemann [81] set up for the first time two equations, one for the conservation of matter and one for the conservation of momentum. As a third equation he took Poisson's equation, i.e., he preassigns the conservation of entropy in a shock wave, similarly to the conservation of entropy in non-shock waves in which the effect of dissipation forces, viscosity and thermal conduction, is not considered. The relation between pressure and density obtained by him is pretty close to the real one, and so is the general picture of motion which he discovered. However, Riemann's equations do not fully satisfy the law of energy conservation. Hence we have to regard them as erroneous.

It is interesting that in the 1925 edition of the well-known book "Partial Differential Equations in Mathematical Physics", compiled by Weber on the basis of Riemann's lectures [97], even after the problem had been entirely clarified, he (Weber) expresses peculiar doubts as to whether or not Riemann's equations may still hold when considering disturbance.

The conclusion by Hugoniot [56], with whose name Eq. (VIII-7) is usually associated, has been dealt with in the preceding Chapter.

We shall now take a look at Rankine's book [78], which is most interesting from the viewpoint of physical gas dynamics because the author has a deep understanding of the phenomena occurring in a shock wave.

Rankine examines a motion which could propagate ad infinitum without changing its form, i.e., he studies a disturbance that propagates steadily in a gas. He establishes two control planes (like we did when deriving Hugoniot's adiabatic curve) and sets up the law of conservation of matter and the law of conservation of momentum. Rankine studies a substance which has thermal conductivity but no viscosity. He formulates principles of self-modelling which are of the utmost importance for shock waves. Specifically, he emphasizes that numerically the coefficient of thermal conductivity of a substance may be infinitesimal, but we may not neglect it in a shock wave because the width of a shock wave as well as the magnitude of the gradients are not pre-assigned. The smaller the coefficient of thermal conductivity, the greater we may expect the gradients to be in a shock wave, so that the product of the temperature gradient times the coefficient of thermal conductivity (equal to the amount of heat transferred by thermal conductivity in a unit of time) can remain finite as the coefficient itself approaches zero. This makes us thoroughly understand when we can ignore dissipation forces, in particular thermal conductivity, which is when the magnitude of the gradients is preassigned by the equations of motion without thermal conductivity. It also makes us thoroughly understand why we cannot ignore thermal conductivity when the magnitude of the gradient is not pre-assigned or predetermined. An example of the first case is a expansion wave for which we have plotted a solution assuming the absence of thermal conductivity. We found that the width of a expansion wave is of the same order as the distance covered by disturbance. The width of a expansion wave increases linearly in time, and in order of magnitude is equal to

$\Delta x={\frac {\Delta p}{p}}ct.$

It we take this to be the first approximation since in the plotting of the expansion wave thermal conductivity and viscosity were not considered, and if we want to consider in the following approximation the effect of thermal conductivity and viscosity on the temperature and velocity fields found in the first approximation, then we will see that all the gradients will rapidly grow so small that thermal conductivity and viscosity will have virtually no effect on the result. This, however, is not the case in a shock wave. Should we take as a first approximation an infinitely steep discontinuity, obtained when thermal conductivity and viscosity are equal to zero, then in the next approximation, introducing thermal conductivity and viscosity, we obtain infinite heat flow and an infinitely great increase in entropy. In the case of a shock wave where the equations of motion without thermal conductivity and viscosity do not give any specific value for wave width, the gradients and the wave width connected with them can only be obtained from the consideration of dissipative forces. The width turns out to be precisely such that it gives the increase in entropy required by the conservation equations. Conversely, if in a expansion wave with a finite width commensurable with the dimensions of the system we could disregard the effect of dissipative forces, then in a shock wave, in order that dissipative forces could give a finite increase in entropy, it is necessary that the width of the shock wave should be very small as compared with the dimensions of the system. Owing to this we can disregard dissipative forces everywhere except on the surface of shock waves. These relations have been well explained by Rankine qualitatively for the particular case when the only dissipative factor is the thermal conductivity of the substance.

Rankine's further explanations suffer from excessive complexity. He does set up the energy equation quite correctly, but in the general case of an arbitrary substance he does not express intrinsic energy in an explicit form as a function of pressure and density. Instead he uses general thermodynamic formulas which include entropy.

On the processes of heat transfer within the discontinuity, he imposes a condition, $\int TdS=0$, the physical significance of this condition is that in a shock wave there occurs only an exchange of heat between neighboring layers, so that the amount of heat removed from one layer is equal to the amount of heat received by the other one, which means that there are no exterior heat sources.

It takes Rankine some effort to derive a system of equations equivalent to that in Chapter 8 from the combination with the general thermodynamic formulas, and he then writes the equations for an ideal gas. Thus, Hugoniot's adiabatic equation in its customary form (Eq. (VIII-10)), could be derived from the formulas contained in Rankine's work by means of elementary algebraic transformations. Let us remind the reader, however, that Rankine preceded Hugoniot's work by some fifteen years.

Rayleigh summarized in 1910 the evolution of the history of shock waves [79]. He particularly emphasizes the unfairness involved in the term "Hugoniot's adiabatic curve".

Among the occasional papers it is interesting to note that as early as 1858 the English priest Earnshaw [49] came quite close to creating a theory of shock waves. Like Riemann he proceeded from the investigation of a compression wave of finite width in which (see Chapter 2) the wave crest overtakes the region of low pressure thus resulting in a discontinuity. However, the Reverend Earnshaw all of a sudden makes the surprising inference that nature does not suffer discontinuities or jumps. He makes some obscure statements on reflections, and implies that nature will somehow manage to prevent the formation of a shock wave or of a discontinuity. This is an educational example of the bad influence exerted by an erroneous philosophy on scientific research.

In a latter time, already after the discoveries of Riemann, Rankine and Hugoniot, the French scientist Pierre Duhem (one of the leaders of the "energetics" movement fashionable at the beginning of the twentieth century) denied the existence of shock waves on the assumption that in equations of gas dynamics involving viscosity and thermal conductivity there can be no strict discontinuity [46,47]. Emile Jouguet, a pupil of Duhem, followed Rankine and pointed out that dissipation forces result in an exceedingly small width. If one disregards it, then one can speak of a discontinuity or a shock wave. Not only did Jouguet clarify Duhem's error, but he greatly contributed to an advance in the theory of shock waves and detonation waves [58,59,60]. Yet, to this day French authors, probably on account of Duhem's remarks, frequently speak of "quasi-waves", with a view on the finite width of the front.

Essentially we are dealing here with the general problem of the value and significance of approximate methods or approximate solutions in physics (see the remarkable paper by V. A. Foch [29]). This involves also the question as to when as approximate realization of some formulas or relations justifies the creation of new qualitative concepts.

Rankine also touches upon the problem of expansion waves, and refers to an oral communication by Thomson according to which an expansion wave must be mechanically unsteady. In point of fact, however, Rankine already implies the impossibility of a expansion wave (and not its unsteadiness or instability). In fact, if we study the processes of thermal conductivity inside the wave then, besides the conservation equation written by Rankine, $\int TdS=0$, which states that in a process of thermal conductivity the amount of heat received by one layer's equal to the amount of heat released by other layers, we must take account, at least qualitatively, of the elementary fact according to which in the process of thermal conductivity heat always passes from a hotter body to a cooler one. Hence, of course, we get that in a shock wave entropy can only increase. Thus, were we to type to plot a expansion wave by inverting in a shock wave all the velocities, then inside the shock wave front, inside the "discontinuity" we would also run into the necessity of inverting the heat flow and achieve a transfer of heat from cooler gas layers to hotter ones -- which is impossible. We cannot but regret that these elementary considerations are sometimes ignored even in the contemporary literature (see Chapter 1 of Vlasov's book [3], which is otherwise quite valuable).

---