Shock waves

A shock wave is characterized by a distinct wave front, traveling through a medium at supersonic speed compared to the undisturbed media. Shock waves in air are very thin transition layers of rapid changes of physical quantities such as pressure, density and temperature. The thickness of a strong shock wave is of the same order of magnitude as the mean-free pathⁱ of the molecules, i.e., about 10⁻⁷m (0.1µm) [88]. In the idealized case, the compressed gas reaches its equilibrium values of pressure, density and pressure in this distance.

The structure of the shock front and the processes through which the gas must pass to obtain its new equilibrium will depend upon the strength of the shock and the properties of the gas. Equilibrium in the state variables (T, p, ρandv) is established within the timescale of nanoseconds (10⁻⁹ seconds). It will be shown in the following that an idealized shock wave forms as a result of the steepening of the compressive part of a finite, continuous disturbance and that through the action of the pressure and inertia forces this gradient ultimately becomes nearly infinite. In a real gas this steepening is resisted by the diffusive effects of viscosity and heat conduction and the final form of the shock transition must involve a balance between these two events. It follows that in any real gas a shock wave will have a finite thickness and a definite structure. Viscous effects are important within the shock front since these effects cause the shock in the first place. The flow across a shock wave is considered adiabatic (i.e., no external heating) and the total enthalpy is constant across the wave. However, outside this layer, viscous effects are small on scales larger than the mean free path [201,204]. Thus, for most practical engineering purposes the shock thickness may be ignored, but in certain types of flow, at low gas densities for example, the shock structure becomes important. The details of the shock structure are beyond the scope of this thesis and the shock waves will in the following be considered as mathematical discontinuities. Thus, shock waves are considered as extremely thin regions and involve discontinuities which has to be carefully considered when solving the governing equations.

Shock waves are often classified based on the supersonic speed which is the rate of propagation of an object that exceeds the speed of sound (Mach 1). In fluid dynamics, the Mach numberM is a dimensionless quantity representing the ratio of the speed of an object moving through a fluid (gas or liquid) and the local speed of sound, given as

M =v

c (6.36)

iIn physics, the mean free path is the average distance traveled by a moving particle (such as an atom, a molecule, a photon) between successive impacts (collisions), which modify its direction or energy or other particle properties.

wherev is the velocity of the object relative to the medium andc is the speed of sound in the medium. Thus, any speed lower than the speed of sound in a sound-propagating medium is called subsonic (M <1), while any speed higher than the speed of sound in the medium is denoted supersonic (M >1). Speeds greater than five times the speed of sound (M >5) are often referred to as hypersonic. For objects traveling in dry air at a temperature of 20^◦ at sea level, M = 1 corresponds to a wave velocity ofv= 343.1 m/s.

In general, the principles are equally applicable for all types of matter (i.e., solids, liquids or gases) and it is important to distinguish between elastic and plastic waves. A starting point for understanding shock phenomena may therefore be obtained by considering the compression characteristics of most solid materials.

From 1D elastic stress-wave theory (see e.g. [203]), the longitudinal wave speed (or the speed of sound)cLin a solid is given as

cL= E

ρ (6.37)

where E is the elastic modulus (or elastic stiffness) andρ is (as before) the density of the medium. In the more general 3D case, the speed of sound in a solid becomes

cS =

K+ 3/4G

ρ (6.38)

whereK andGare the bulk and shear modulus, respectively. Introducing the possibility of non-linear material behaviour for the longitudinal wave speed in Eq. (6.37) may be carried out by replacing the elastic modulusE with the tangent stiffness∂σ/∂εof the material at a specific loading state (σ, ε). The 1D longitudinal waves speed then reads

cL= 1

ρ

∂σ

∂ε (6.39)

Similarly, in a perfect fluid or gas (3D medium withG= 0) the bulk speed of sound becomes

cB= K

ρ (6.40)

for hydrostatic pressure waves. Here,K is the bulk modulus that measures the substance’s resistance to uniform deformation under isentropic conditions (i.e., reversible and adiabatic conditions so that the entropy remains constant). It is defined as the derivative of pressurepwith respect to density (or volume),

6.3. Shock physics 133

i.e.,

K=ρ∂p

∂ρ (6.41)

The speed of sound may then be obtained by inserting Eq. (6.41) into Eq. (6.40) and reads which can be considered as a simple EOS that provides a mathematical rela-tionship between the change in pressure and density. Again, the subscripts denotes that the partial derivative is taken at constant entropy and it is seen that the speed of sound in the gas is related to the isentropic compressibility in Eq. (6.35).

Following Eqs. (6.37)-(6.42) it is observed that shock waves differ strongly from linear elastic waves regarding their expansion and propagation. High pressure magnitudes introduce the material to the non-linear and plastic region of the pressure-density relation. This is illustrated in Figure 6.3a where the gradient is the square of the speed of sound as given by Eq. (6.42). The speed of sound in the material is constant in the elastic region, which implies that the pressure and density are linearly related. Beyond the elastic region, the wave velocity increases with the pressure and density and the pressure-density relation becomes non-linear. That is, beyond the elastic region, the speed of sound increases with increasing pressure.

Density

Figure 6.3: Typical relation between pressure and density, and formation (shocking-up) of a shock wave [141]: (a) pressure versus density and (b) the steepening of the shock wave.

This is described by Cooper [141] and may be illustrated by studying the pressure front of the shock in Figure 6.3b. At pointAthe pressure is low and,

consequently, the speed of soundcAis low (see corresponding point in Figure 6.3a). Also the particle velocityvA, i.e., the speed to which the material locally has been accelerated, is relatively low. Thus, the velocity of the pressure wave cA+vA is quite low. At pointB there is an increase in pressure compared to pointA which implies that the speed of soundcB increases (since we are evaluating a strong shock beyond the elastic limit). The particle velocityvB is also higher, resulting in a pressure wave at pointB travelling faster than at pointA(cB+vB > cA+vA). Moreover, the same argument holds for point Cthat has a faster wave velocity than pointB. Thus, since the propagation velocity of the wave depends on the gradient of the pressure-density relation, the peak pressure (C in Figure 6.3b) of the shock wave propagates faster than its leading and trailing edges (A and B in Figure 6.3b), resulting in the formation of a steep shock front (right pressure profile in Figure 6.3b) characterized by a virtually discontinuity in pressure and density. Thus, the initially smooth pressure front disturbance has been "shocked-up" because the wave speed increases with increasing pressure.

When the pressure wave takes on this vertical front, it is, as pointed out earlier, called a shock wave. Thus, from a mathematical point of view, there is no smooth transition from the medium in front of the wave to the medium behind the wave. The material "jumps" from the non-shocked to the shocked state. It is important to keep track of the different velocities, i.e., the sound, particle and pressure wave, and to remember that the pressure wave velocity is the sum of the sound and particle velocity. It may be challenging to visualize how the pressure wave velocity can be faster than the particle velocity since the particles are also moving. However, one should be aware that the shock wave is caused by a sudden and violent disturbance of the material (e.g. a sudden release of high pressure or an explosive detonation). The shock wave then propagates through the undisturbed material, by accelerating, compressing and heating the material, inducing a mass motion with the particle velocity behind the shock wave. This will be shown mathematically using the method of characteristics in Section 6.4.2 (see Eq. (6.61) and Figure 6.8).

The pressure-density relation in Figure 6.3a is commonly found in the literature in terms of the pressure-specific volume relation (see Figure 6.4). This is called the Hugoniot curve and represents the locus of all the possible equilibrium states in which a particular material can exist [141]. It must be emphasized that this is not an EOS or a path along which shock waves arise. It should also be noted that the isentropic curve, i.e.,p^γ = constant and the path function that describes a continuity and not a jump, is different from the Hugoniot curve.

A rarefaction wave is an example of a continuous process and its path would be along the unloading isentrope. Remember that the rarefaction wave brings the pressure back down to the ambient pressure. Since the Hugoniot curve

6.3. Shock physics 135 represents the locus of all possible states (p2, ₂) behind the shock front reached from an initial state (p1, ₁) in front of the shock, the line joining the initial and final states on the Hugoniot curve represents the jump condition [141].

This line is called the Rayleigh line and is shown in Figure 6.4. That is, the shock is assumed to be acheived along a non-equilibrium path, assumed to be a straight line in the p−space. Above the value p₂ in Figure 6.4, strong shocks will occur.

Pressure

Specific volume,= 1/ρ

Plastic region (strong shocks)

Elastic - plastic region

Elastic region Hugoniot line

Rayleigh line Shock Rar

efacti on

Release isentrope, p^γ= constant

13

2

p1

p2

Figure 6.4: A typical representation of the pressure-specific volume (Hugoniot curve) [141]. Shock compression along Rayleigh line followed by an isentropic release resulting in dissipated energy. Shock waves produced at magnitudes beyondp2.

Figure 6.4 illustrates a shock compression from an ambient state (p1, 1) to a Hugoniot pressure p₂ followed by a subsequent isentropic release to the initial pressure p₁ at the end volume ₃. Whereas the loading path to the Hugoniot state is described by the Rayleigh line, i.e., a straight line of non-equilibrium states, the isentropic release follows a curved line in thep−plane.

Thus, the portion of dissipated energy in the irreversible process of a shock transition is the difference between the energy stored during the shock loading, i.e., the triangular area under the Rayleigh line, and the recovered energy during the isentropic release [203]. The dissipated energy is represented by the shaded area in Figure 6.4. It is emphasized that the shaded area is highly exaggerated. Moreover, the final state3is larger than the initial volume1and the released energy equals the difference between the two shaded areas indicated with vertical lines and in black, respectively. The irreversible thermodynamic process of an almost instantaneous jump from initial conditions at ambient or other equilibrium conditions to a Hugoniot state therefore enhances the entropy of the compressed gas. The released energy is dissipated as heat and results in a heating of the gas. The portion of dissipated energy can be calculated from the thermodynamics involved in the process [141].