Rankine–Hugoniot conditions

Basics: Euler equations in one dimension

Consider gas in a one-dimensional container (e.g., a long thin tube). Assume that the fluid is inviscid (i.e., it shows no viscosity effects as for example friction with the tube walls). Furthermore, assume that there is no heat transfer by conduction or radiation and that gravitational acceleration can be neglected. Such a system can be described by the following system of conservation laws, known as the 1D Euler equations, that in conservation form is:

where

ρ = fluid mass density, [kg/m³] u = fluid velocity, [m/s] e = specific internal energy of the fluid, [J/kg] p = fluid pressure, [Pa] t = time, [s] x = distance, [m], and E t = ρ e + ρ 1 2 u 2 , is the total energy density of the fluid, [J/m³], while e is its specific internal energy [J/kg]

Assume further that the gas is calorically ideal and that therefore a polytropic equation-of-state of the simple form

is valid, where γ is the constant ratio of specific heats c p / c v . This quantity also appears as the polytropic exponent of the polytropic process described by

For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953).

Note: For a calorically ideal gas γ is a constant and for a thermally ideal gas γ is a function of temperature. In the latter case, the dependence of pressure on mass density and internal energy might differ from that given by equation (4).

The jump condition

Before proceeding further it is necessary to introduce the concept of a jump condition – a condition that holds at a discontinuity or abrupt change.

Consider a 1D situation where there is a jump in the scalar conserved physical quantity w , which is governed by integral conservation law

for any x 1 , x 2 , x 1 < x 2 , and, therefore, by partial differential equation

for smooth solutions.

Let the solution exhibit a jump (or shock) at x = x s ( t ) , where x 1 < x s ( t ) and x s ( t ) < x 2 , then

The subscripts 1 and 2 indicate conditions just upstream and just downstream of the jump respectively, i.e. w 1 = lim ϵ → 0 + w ( x s − ϵ ) and w 2 = lim ϵ → 0 + w ( x s + ϵ ) .

Note, to arrive at equation (8) we have used the fact that d x 1 / d t = 0 and d x 2 / d t = 0 .

Now, let x 1 → x s ( t ) − ϵ and x 2 → x s ( t ) + ϵ , when we have ∫ x 1 x s ( t ) − ϵ w t d x → 0 and ∫ x s ( t ) + ϵ x 2 w t d x → 0 , and in the limit

where we have defined u s = d x s ( t ) / d t (the system characteristic or shock speed), which by simple division is given by

Equation (9) represents the jump condition for conservation law (6). A shock situation arises in a system where its characteristics intersect, and under these conditions a requirement for a unique single-valued solution is that the solution should satisfy the admissibility condition or entropy condition. For physically real applications this means that the solution should satisfy the Lax entropy condition

where f ′ ( w 1 ) and f ′ ( w 2 ) represent characteristic speeds at upstream and downstream conditions respectively.

Euler equations shock condition

In the case of the hyperbolic conservation law (6), we have seen that the shock speed can be obtained by simple division. However, for the 1D Euler equations ( 1), ( 2) and ( 3), we have the vector state variable [ ρ , ρ u , E ] T and the jump conditions become

Equations (12), (13) and (14) are known as the Rankine–Hugoniot conditions for the Euler equations and are derived by enforcing the conservation laws in integral form over a control volume that includes the shock. For this situation u s cannot be obtained by simple division. However, it can be shown by transforming the problem to a moving co-ordinate system (setting u s ′ := u s − u 1 , u 1 ′ := 0 , u 2 ′ := u 2 − u 1 to remove u 1 ) and some algebraic manipulation (involving the elimination of u 2 ′ from the transformed equation (13) using the transformed equation (12)), that the shock speed is given by

where c 1 = γ p 1 / ρ 1 is the speed of sound in the fluid at upstream conditions.

See Laney (1998), LeVeque (2002), Toro (1999), Wesseling (2001), and Whitham (1999) for further discussion.

Stationary shock

For a stationary shock u s = 0 , and for the 1D Euler equations we have

In view of equation (16) we can simplify equation (18) to

which is a statement of Bernoulli's principle, under conditions where gravitational effects can be neglected.

Substituting u 1 and u 2 from equations (16) and (17) into equation (19) yields the following relationship:

where h = p ρ + e represents specific enthalpy of the fluid. Eliminating internal energy e in equation (19) by use of the equation-of-state, equation ( 4), yields

From physical considerations it is clear that both the upstream and downstream pressures must be positive, and this imposes an upper limit on the density ratio in equations (21) and (22) such that ρ 2 / ρ 1 < ( γ + 1 ) / ( γ − 1 ) . As the strength of the shock increases, there is a corresponding increase in downstream gas temperature, but the density ratio ρ 2 / ρ 1 approaches a finite limit: 4 for an ideal monatomic gas ( γ = 5 / 3 ) and 6 for an ideal diatomic gas ( γ = 1.4 ) . Note: air is comprised predominately of diatomic molecules and therefore at standard conditions γ a i r ≃ 1.4 .

Shock Hugoniot and Rayleigh line in solids

For shocks in solids, a closed form expression such as equation (15) cannot be derived from first principles. Instead, experimental observations indicate that a linear relation can be used instead (called the shock Hugoniot in the u_s-u_p plane) that has the form

( 23 ) u s = c 0 + s u p = c 0 + s u 2

where c₀ is the bulk speed of sound in the material (in uniaxial compression), s is a parameter (the slope of the shock Hugoniot) obtained from fits to experimental data, and u_p=u₂ is the particle velocity inside the compressed region behind the shock front.

The above relation, when combined with the Hugoniot equations for the conservation of mass and momentum, can be used to determine the shock Hugoniot in the p-v plane, where v is the specific volume (per unit mass):

( 24 ) p 2 − p 1 = c 0 2 ρ 1 ρ 2 ( ρ 2 − ρ 1 ) [ ρ 2 − s ( ρ 2 − ρ 1 ) ] 2 = c 0 2 ( v 1 − v 2 ) [ v 1 − s ( v 1 − v 2 ) ] 2 .

Alternative equations of state, such as the Mie–Gruneisen equation of state may also be used instead of the above equation.

The shock Hugoniot describes the locus of all possible thermodynamic states a material can exist in behind a shock, projected onto a two dimensional state-state plane. It is therefore a set of equilibrium states and does not specifically represent the path through which a material undergoes transformation.

Weak shocks are isentropic and that the isentrope represents the path through which the material is loaded from the initial to final states by a compression wave with converging characteristics. In the case of weak shocks, the Hugoniot will therefore fall directly on the isentrope and can be used directly as the equivalent path. In the case of a strong shock we can no longer make that simplification directly. However, for engineering calculations, it is deemed that the isentrope is close enough to the Hugoniot that the same assumption can be made.

If the Hugoniot is approximately the loading path between states for an "equivalent" compression wave, then the jump conditions for the shock loading path can be determined by drawing a straight line between the initial and final states. This line is called the Rayleigh line and has the following equation:

( 25 ) p 2 − p 1 = u s 2 ( ρ 1 − ρ 1 2 ρ 2 )

Hugoniot elastic limit

Most solid materials undergo plastic deformations when subjected to strong shocks. The point on the shock Hugoniot at which a material transitions from a purely elastic state to an elastic-plastic state is called the Hugoniot elastic limit (HEL) and the pressure at which this transition takes place is denoted p_HEL. Values of p_HEL can range from 0.2 GPa to 20 GPa. Above the HEL, the material loses much of its shear strength and starts behaving like a fluid.