15.4. Normal Shock Waves

A shock wave is similar to a step-change compression acoustic wave except that it has finite strength. The thickness of such waves is typically of the order of micrometers, so that fluid properties vary almost discontinuously across a shock wave. The high gradients of velocity and temperature result in entropy production within the wave so isentropic relations cannot be used across a shock. This section presents the relationships between properties of the flow upstream and downstream of a normal shock, where the shock is perpendicular to the direction of flow. Here, the shock wave is treated as a discontinuity and the actual process by which entropy is generated is not addressed. However, the entropy rise across the shock predicted by this analysis is correct. The internal structure of a shock, as predicted by the Navier-Stokes equations under certain simplifying assumptions, is given at the end of this section.

Stationary Normal Shock Wave in a Moving Medium

To get started, consider the thin control volume shown in Figure 15.7 that encloses a stationary shock wave. The control surface locations 1 and 2, shown as dashed lines in the figure, can be taken close to each other because of the discontinuous nature of the shock wave. In this case, the area change and the wall-surface friction between the upstream and the downstream control volume surfaces can be neglected. Furthermore, external heat addition is not of interest here so the basic equations are (15.18) and (15.24) with F = 0, both simplified for constant area, and (15.22) with Q = 0:

ρ1u1=ρ2u2,p1−p2=−ρ1u12+ρ2u22,andh1+12u12=h2+12u22.

(15.35, 15.36, 15.37)

Figure 15.7 A normal shock wave trapped in a steady nozzle flow. Here, a control volume is shown that has control surfaces immediately upstream (1) and downstream (2) of the shock wave. Shock waves are very thin in most gases, so the area change and wall friction of the duct need not be considered as the flow traverses the shock wave.

The Bernoulli equation cannot be used here because the process inside the shock wave is dissipative. The equations (15.35) through (15.37) contain four unknowns (h₂, u₂, p₂, ρ₂). The necessary additional relationship comes from the thermodynamics of a perfect gas (15.1):

h=cpT=γRγ−1pρR=γp(γ−1)ρ,

so that (15.37) becomes:

γ(γ−1)p1ρ1+u122=γ(γ−1)p2ρ2+u222.

(15.38)

There are now three unknowns (u₂, p₂, ρ₂) and three equations: (15.35), (15.36), and (15.38), so the remainder of the effort to link the conditions upstream and downstream of a shock is primarily algebraic. Elimination of ρ₂ and u₂ from these equations leads to:

p2p1=1+2γγ+1[ρ1u12γp1−1]=1+2γγ+1[M12−1],

(15.39)

where the second equality follows because ρu²/γp = u²/γRT = M².

With this relationship, an equation can be derived for M₂ in terms of M₁. Because ρu² = ρc²M² = ρ(γp/ρ)M² = γpM², so the momentum equation (15.36) can be written:

p1+γp1M12=p2+γp2M22.

Using (15.39), this gives:

M22=(γ−1)M12+22γM12+1−γ,

(15.40)

which is plotted in Figure 15.8. Because M₂ = M₁ (state 2 = state 1) is a solution of (15.35), (15.36), and (15.38), that is shown as well, indicating two possible solutions for M₂ for all M₁ > [(γ − 1)/2γ]^1/2. As is shown below, M₁ must be greater than unity to avoid violation of the second law of thermodynamics, so the two possibilities for the downstream state are: 1) no change from upstream, and 2) a sudden transition from supersonic to subsonic flow with consequent increases in pressure, density, and temperature. The density, velocity, and temperature ratios can be similarly obtained from the equations provided so far. They are:

ρ2ρ1=u1u2=(γ+1)M12(γ−1)M12+2,andT2T1=1+2(γ−1)(γ+1)2γM12+1M12(M12−1).

(15.41, 15.42)

The normal shock relations (15.39) through (15.43) were worked out independently by the British engineer W. J. M. Rankine (1820–1872) and the French ballistician Pierre Henry Hugoniot (1851–1887). These equations are sometimes known as the Rankine-Hugoniot relations. The results of (15.39), (15.40) and (15.42) are tabulated for γ = 1.4 in Table 15.2.

In terms of stagnation properties, T₀ and h₀ are constant across the shock because of the adiabatic nature of the process. In contrast, the stagnation properties p₀ and ρ₀ decrease across the shock due to the dissipative processes inside the shock zone that increase the entropy (see Exercise 15.13). Using (15.1) and (15.2), the entropy change is:

Figure 15.8 Normal shock wave solution for M₂ as function of M₁ for γ = 1.4. The trivial (no change) solution is also shown as the straight line with unity slope. Asymptotes are [(γ − 1)/2γ]^1/2 = 0.378 for M₁ or M₂ → ∞. The second law of thermodynamics limits valid shock wave solutions to those having M₁ > 1.

s2−s1cv=ln{p2p1(ρ1ρ2)γ}=ln{[1+2γγ+1(M12−1)]((γ−1)M12+2(γ+1)M12)γ},

(15.43)

which is plotted in Figure 15.9. This figure shows that the entropy would decrease across an expansion shock in a perfect gas, which is impermissible. However, expansion shocks may be possible when the gas follows a different equation of state (Fergason et al., 2001). When the upstream Mach number is close to unity, Figure 15.9 shows that the entropy change may be very small. The dependence of s₂ – s₁ on M₁ in the neighborhood of M₁ = 1 can be ascertained by treating M12−1 as a small quantity and expanding (15.43) in terms of it (see Exercise 15.8) to find:

s2−s1cv≅2γ(γ−1)3(γ+1)2(M12−1)3.

(15.44a)

Table 15.2

One-Dimensional Normal-Shock Relations (γ = 1.4)

M1	M2	p2/p1	T2/T1	(p0)2/(p0)1	M1	M2	p2/p1	T2/T1	(p0)2/(p0)1
1	1	1	1	1	1.62	0.663	2.895	1.402	0.888
1.02	0.98	1.047	1.013	1	1.64	0.657	2.971	1.416	0.88
1.04	0.962	1.095	1.026	1	1.66	0.651	3.048	1.43	0.872
1.06	0.944	1.144	1.039	1	1.68	0.646	3.126	1.444	0.864
1.08	0.928	1.194	1.052	0.999	1.7	0.641	3.205	1.458	0.856
1.1	0.912	1.245	1.065	0.999	1.72	0.635	3.285	1.473	0.847
1.12	0.896	1.297	1.078	0.998	1.74	0.631	3.366	1.487	0.839
1.14	0.882	1.35	1.09	0.997	1.76	0.626	3.447	1.502	0.83
1.16	0.868	1.403	1.103	0.996	1.78	0.621	3.53	1.517	0.821
1.18	0.855	1.458	1.115	0.995	1.8	0.617	3.613	1.532	0.813
1.2	0.842	1.513	1.128	0.993	1.82	0.612	3.698	1.547	0.804
1.22	0.83	1.57	1.14	0.991	1.84	0.608	3.783	1.562	0.795
1.24	0.818	1.627	1.153	0.988	1.86	0.604	3.869	1.577	0.786
1.26	0.807	1.686	1.166	0.986	1.88	0.6	3.957	1.592	0.777
1.28	0.796	1.745	1.178	0.983	1.9	0.596	4.045	1.608	0.767
1.3	0.786	1.805	1.191	0.979	1.92	0.592	4.134	1.624	0.758
1.32	0.776	1.866	1.204	0.976	1.94	0.588	4.224	1.639	0.749
1.34	0.766	1.928	1.216	0.972	1.96	0.584	4.315	1.655	0.74
1.36	0.757	1.991	1.229	0.968	1.98	0.581	4.407	1.671	0.73
1.38	0.748	2.055	1.242	0.963	2	0.577	4.5	1.688	0.721
1.4	0.74	2.12	1.255	0.958	2.02	0.574	4.594	1.704	0.711
1.42	0.731	2.186	1.268	0.953	2.04	0.571	4.689	1.72	0.702
1.44	0.723	2.253	1.281	0.948	2.06	0.567	4.784	1.737	0.693
1.46	0.716	2.32	1.294	0.942	2.08	0.564	4.881	1.754	0.683
1.48	0.708	2.389	1.307	0.936	2.1	0.561	4.978	1.77	0.674
1.5	0.701	2.458	1.32	0.93	2.12	0.558	5.077	1.787	0.665
1.52	0.694	2.529	1.334	0.923	2.14	0.555	5.176	1.805	0.656
1.54	0.687	2.6	1.347	0.917	2.16	0.553	5.277	1.822	0.646
1.56	0.681	2.673	1.361	0.91	2.18	0.55	5.378	1.837	0.637
1.58	0.675	2.746	1.374	0.903	2.2	0.547	5.48	1.857	0.628
1.6	0.668	2.82	1.388	0.895	2.22	0.544	5.583	1.875	0.619
Table Continued

M1	M2	p2/p1	T2/T1	(p0)2/(p0)1	M1	M2	p2/p1	T2/T1	(p0)2/(p0)1
2.24	0.542	5.687	1.892	0.61	2.64	0.5	7.965	2.28	0.445
2.26	0.539	5.792	1.91	0.601	2.66	0.499	8.088	2.301	0.438
2.28	0.537	5.898	1.929	0.592	2.68	0.497	8.213	2.322	0.431
2.3	0.534	6.005	1.947	0.583	2.7	0.496	8.338	2.343	0.424
2.32	0.532	6.113	1.965	0.575	2.72	0.494	8.465	2.364	0.417
2.34	0.53	6.222	1.984	0.566	2.74	0.493	8.592	2.386	0.41
2.36	0.527	6.331	2.003	0.557	2.76	0.491	8.721	2.407	0.403
2.38	0.525	6.442	2.021	0.549	2.78	0.49	8.85	2.429	0.396
2.4	0.523	6.553	2.04	0.54	2.8	0.488	8.98	2.451	0.389
2.42	0.521	6.666	2.06	0.532	2.82	0.487	9.111	2.473	0.383
2.44	0.519	6.779	2.079	0.523	2.84	0.485	9.243	2.496	0.376
2.46	0.517	6.894	2.098	0.515	2.86	0.484	9.376	2.518	0.37
2.48	0.515	7.009	2.118	0.507	2.88	0.483	9.51	2.541	0.364
2.5	0.513	7.125	2.138	0.499	2.9	0.481	9.645	2.563	0.358
2.52	0.511	7.242	2.157	0.491	2.92	0.48	9.781	2.586	0.352
2.54	0.509	7.36	2.177	0.483	2.94	0.479	9.918	2.609	0.346
2.56	0.507	7.479	2.198	0.475	2.96	0.478	10.055	2.632	0.34
2.58	0.506	7.599	2.218	0.468	2.98	0.476	10.194	2.656	0.334
2.6	0.504	7.72	2.238	0.46	3	0.475	10.333	2.679	0.328
2.62	0.502	7.842	2.26	0.453

This equation explicitly shows that s₂ – s₁ will only be positive for a perfect gas when M₁ > 1. Thus, stationary shock waves do not occur when M₁ < 1 because of the second law of thermodynamics. However, when M₁ > 1, then (15.40) requires that M₂ < 1. Thus, the Mach number changes from supersonic to subsonic values across a normal shock, and this is the only possibility. A shock wave is therefore analogous to a hydraulic jump (see Section 8.6) in a gravity current, in which the Froude number jumps from supercritical to subcritical values; see Figure 8.21. Equations (15.39), (15.41), and (15.42) then show that the jumps in p, ρ, and T are also from lower to higher values, so that a shock wave leads to compression and increased fluid temperature at the expense of stream-wise velocity.

Interestingly, terms involving (M12−1) and (M12−1)2 do not appear in (15.44a). Using the pressure ratio from (15.39), (15.44a) can be rewritten:

s2−s1cv≅γ2−112γ2(p2−p1p1)3.

(15.44b)

This shows that as the wave amplitude Δp = p₂ – p₁ decreases the entropy jump goes to zero like (Δp)³. Thus, weak shock waves are nearly isentropic and this is the primary reason that loud acoustic disturbances are successfully treated as isentropic.

Figure 15.9 Entropy change (s₂ − s₁)/c_v as a function of M₁ for γ = 1.4. Note higher-order contact at M = 1 to the horizontal line corresponding to zero entropy change as M₁ → 1 from above. Negative entropy changes (a violation of the second law of thermodynamics) are predicted for M₁ < 1, so shock waves do not occur unless the upstream speed is supersonic, M₁ > 1.

Moving Normal Shock Wave in a Stationary Medium

Frequently, one needs to calculate the properties of flow due to the propagation of a shock wave through a still medium, for example, that caused by an explosion. The Galilean transformation necessary to analyze this problem is indicated in Figure 15.10. The left panel shows a stationary shock, with incoming and outgoing velocities u₁ and u₂, respectively. To this flow we add a velocity u₁ directed to the left, so that the fluid ahead of the shock is stationary, and the fluid behind the shock is moving to the left at a speed u₁ − u₂, as shown in the right panel of the figure. This is consistent with acoustic results in Section 15.2 where it was found that the fluid within a compression wave moves in the direction of the wave propagation. The shock speed is therefore u₁, with a supersonic Mach number M₁ = u₁/c₁ > 1. It follows that a finite pressure disturbance propagates through a still fluid at supersonic speed, in contrast to infinitesimal waves that propagate at the sonic speed. The expressions for all the thermodynamic properties of the flow, such as (15.39) through (15.44), are still applicable since their values are frame-independent.

Figure 15.10 Stationary and moving shocks. The stationary shock shown in the left panel corresponds to a situation like that depicted in Figure 15.7 where the incoming flow moves toward the shock. The moving shock situation shown on the right corresponds to blast wave that propagates away from an explosion into still air.

Normal-Shock Structure

We conclude this section on normal shock waves with a look into the structure of a shock wave. The viscous and heat conductive processes within the shock wave result in an entropy increase across the wave. However, the magnitudes of the viscosity μ and thermal conductivity k only determine the thickness of the shock wave and not the magnitude of the entropy increase. The entropy increase is determined solely by the upstream Mach number as shown by (15.43). We shall also see later that the wave drag experienced by a body due to the appearance of a shock wave is independent of viscosity or thermal conductivity. (The situation here is analogous to the viscous energy dissipation in fully turbulent flows, Section 12.7, in which the average kinetic-energy dissipation rate ε¯ is determined by the velocity and length scales of a large-scale turbulence field (12.49) and not by the magnitude of the viscosity; a change in viscosity merely changes the length scale at which the dissipation takes place, namely, the Kolmogorov microscale.)

A shock wave can be considered a very thin boundary layer involving a large stream-wise velocity gradient du/dx, in contrast to the cross-stream (or wall-normal) velocity gradient involved in a viscous boundary layer near a solid surface. Analysis shows that the thickness δ of a shock wave is given by:

(u1−u2)δ/ν∼1,

where the left side is a Reynolds number based on the velocity change across the shock, its thickness, and the average kinematic viscosity. Taking a typical value for air of ν ∼ 10^–5 m²/s, and a velocity jump of Δu ∼ 100 m/s, we obtain a shock thickness of 10^–7 m. This is not much larger than the mean-free path (average distance traveled by a molecule between collisions), which suggests that the continuum hypothesis and the assumption of local thermodynamic equilibrium are both of questionable validity in analyzing shock structure.

With these limitations noted, some insight into the structure of shock waves may be gained by considering the one-dimensional steady Navier-Stokes equations, including heat conduction and Newtonian viscous stresses, in a shock-fixed coordinate system. The solution we obtain provides a smooth transition between upstream and downstream states, looks reasonable, and agrees with experiments and kinetic theory models for upstream Mach numbers less than about 2. The equations for conservation of mass, momentum, and energy, respectively, are the steady one-dimensional versions of (4.7), (4.38) without a body force, and (4.60) written in terms of enthalpy h:

d(ρu)dx=0,ρududx+dpdx=ddx((43μ+μυ)dudx),andρudhdx−udpdx=(43μ+μυ)(dudx)2+ddx(kdTdx).

By adding the product of u and the momentum equation to the energy equation, these can be integrated once to find:

ρu=m,mu+p=μ″dudx+mV,andm(h+12u2)=μ″ududx+kdTdx+mI,

where m, V, and I are the constants of integration and μ″=43μ+μυ. When these are evaluated upstream (state 1) and downstream (state 2) of the shock where gradients vanish, they yield the Rankine-Hugoniot relations derived earlier. We also need the equations of state for a perfect gas with constant specific heats to solve for the shock structure: h = c_pT, and p = ρRT. Multiplying the energy equation by c_p/k we obtain the form:

mcpk(cpT+12u2)−μ″cp2kdu2dx−cpdTdx=mcpkI,

This equation has an exact integral in the special case Pr″ ≡ μ″c_p/k = 1 that was found by Becker in 1922. For most simple gases, Pr″ is likely to be near unity so it is reasonable to proceed assuming Pr″ = 1. The Becker integral is c_pT + u²/2 = I. Eliminating all variables but u from the momentum equation, using the equations of state, mass conservation, and the energy integral, we reach:

mu+(m/u)(R/cp)(I−u2/2)−μ″(du/dx)=mV.

With c_p/R = γ/(γ − 1), multiplying by u/m, leads to:

−[2γγ+1]μ″mududx=−u2+[2γγ+1]uV−2Iγ−1γ+1≡(U1−U)(U−U2).

Divide by V² and let u/V = U. The equation for the structure becomes:

−U(U1−U)−1(U−U2)−1dU=[(γ+1)/2γ](m/μ″)dx,

where the roots of the quadratic are:

U1,2=[γγ+1]{1±[1−2(γ2−1)I/(γ2V2)]1/2},

the dimensionless speeds far up- and downstream of the shock. The left-hand side of the equation for the structure is rewritten in terms of partial fractions and then integrated to obtain:

[U1ln(U1−U)−U2ln(U−U2)]/(U1−U2)=[(γ+1)/2γ]m∫dx/μ″≡[(γ+1)/2γ]η.

The resulting shock structure is shown in Figure 15.11 in terms of the stretched coordinate η = ∫(m/μ″)dx where μ″ is often a strong function of temperature and thus of x. A similar structure is obtained for all except quite small values of Pr″. In the limit Pr″ → 0, Hayes (1958) points out that there must be a “shock within a shock” because heat conduction alone cannot provide the entire structure. In fact, Becker (1922, footnote, p. 341) credits Prandtl for originating this idea. Cohen and Moraff (1971) provided the structure of both the outer (heat conducting) and inner (isothermal viscous) shocks. Here, the variable η is a dimensionless length scale measured very roughly in units of mean-free paths. We see that a measure of shock thickness is of the order of 5 mean-free paths from this analysis.

Figure 15.11 Shock structure velocity profile for the case U₁ = 0.848485, U₂ = 0.31818, corresponding to M₁ = 2.187. The units of the horizontal coordinate may be approximately interpreted as mean-free paths. Thus, a shock wave is typically a small countable number of mean-free-paths thick.

Example 15.4

A normal shock wave forms just ahead of a bullet as it travels at 750 m/s through still air at 100 kPa and 295 K. What are the pressure, temperature, and density of the air immediately behind the shock wave?

Solution

Ahead of the bullet, the density is:

ρ1=p1/RT1=(100kPa)/[(287m2s−2K−1)(295K)]=1.181kgm−3,

 

and the sound speed is:

c1=γRT1=1.4(287m2s−2K−1)295K=344m/s

 

Thus, the shock Mach number is M₁ = 750/344 = 2.18. So, from Table 15.2, or equations (15.39) to (15.42), M₂ = 0.550 and the ratios across the shock are: p₂/p₁ = 5.378, T₂/T₁ = 1.837, and ρ₂/ρ₁ = 2.924. Hence, p₂ = 538 kPa, T₂ = 542 K, and ρ₂ = 3.453 kgm^–3.