Rankine–Hugoniot conditions

The Rankine–Hugoniot conditions, also referred to as Rankine–Hugoniot jump conditions or Rankine–Hugoniot relations, describe the relationship between the states on both sides of a shock wave in a one-dimensional flow in fluids or a one-dimensional deformation in solids. They are named in recognition of the work carried out by Scottish engineer and physicist William John Macquorn Rankine^[1] and French engineer Pierre Henri Hugoniot.^[2] See also Salas (2006)^[3] for some historical background.

In a coordinate system that is moving with the shock, the Rankine–Hugoniot conditions can be expressed as:

{\begin{aligned}&\rho _{1}\,u_{s}=\rho _{2}(u_{s}-u_{2})&\qquad {\text{Conservation of mass}}\\&p_{2}-p_{1}=\rho _{2}\,u_{2}\,(u_{s}-u_{2})=\rho _{1}\,u_{s}\,u_{2}&\qquad {\text{Conservation of momentum}}\\&p_{2}\,u_{2}=\rho _{1}\,u_{s}\,\left({\tfrac {1}{2}}\,u_{2}^{2}+e_{2}-e_{1}\right)&\qquad {\text{Conservation of energy}}\end{aligned}}

where u_s is the shock wave speed, ρ₁ and ρ₂ are the mass density of the fluid behind and inside the shock, u₂ is the particle velocity of the fluid inside the shock, p₁ and p₂ are the pressures in the two regions, and e₁ and e₂ are the specific (with the sense of per unit mass) internal energies in the two regions. These equations can be derived in a straightforward manner from equations (12), (13) and (14) below. Using the Rankine-Hugoniot equations for the conservation of mass and momentum to eliminate u_s and u₂, the equation for the conservation of energy can be expressed as the Hugoniot equation:

e_{2}-e_{1}={\tfrac {1}{2}}\,(p_{2}+p_{1})\,(v_{1}-v_{2})

where v₁ and v₂ are the uncompressed and compressed specific volumes per unit mass, respectively.

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:

(\;1)\quad \quad {\frac {\partial \rho }{\partial t}}\;\;=-{\frac {\partial }{\partial x}}\left(\rho u\right)

(\;2)\quad \quad {\frac {\partial }{\partial t}}(\rho u)\,=-{\frac {\partial }{\partial x}}\left(\rho u^{2}+p\right)

(\;3)\quad \quad {\frac {\partial }{\partial t}}(E^{t})=-{\frac {\partial }{\partial x}}\left[u\left(E^{t}+p\right)\right],

where

\rho =\,

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}=\rho e+\rho {\frac {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

(\;4)\quad \quad p=\left(\gamma -1\right)\rho e,

is valid, where $\gamma$ 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

(\;5)\quad \quad {\frac {p}{\rho ^{\gamma }}}={\text{constant}}.

For an extensive list of compressible flow equations, etc., refer to NACA Report 1135 (1953).^[4]

Note: For a calorically ideal gas $\gamma \,$ is a constant and for a thermally ideal gas $\gamma \,$ 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

(\;6)\quad \quad {\frac {d}{dt}}\int _{x_{1}}^{x_{2}}w\,dx=-\left.f\left(w\right)\right|_{x_{1}}^{x_{2}}

for any $x_{1}$ , $x_{2}$ , $x_{1}<x_{2}$ , and, therefore, by partial differential equation

(\;6')\quad \quad {\frac {\partial w}{\partial t}}+{\frac {\partial }{\partial x}}f\left(w\right)=0

for smooth solutions.^[5]

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

(\;7)\quad \quad {\frac {d}{dt}}\left[\left(\int _{x_{1}}^{x_{s}(t)}w\,dx+\int _{x_{s}(t)}^{x_{2}}w\,dx\right)\right]=-\int _{x_{1}}^{x_{2}}{\frac {\partial }{\partial x}}f\left(w\right)\,dx

(\;8)\quad \quad \therefore w_{1}{\frac {dx_{s}}{dt}}-w_{2}{\frac {dx_{s}}{dt}}+\int _{x_{1}}^{x_{s}(t)}w_{t}\,dx+\int _{x_{s}(t)}^{x_{2}}w_{t}\,dx=-\left.f\left(w\right)\right|_{x_{1}}^{x_{2}}

The subscripts 1 and 2 indicate conditions just upstream and just downstream of the jump respectively, i.e. $w_{1}=\lim _{\epsilon \rightarrow 0^{+}}w(x_{s}-\epsilon )$ and $w_{2}=\lim _{\epsilon \rightarrow 0^{+}}w(x_{s}+\epsilon )$ .

Note, to arrive at equation (8) we have used the fact that $dx_{1}/dt=0$ and $dx_{2}/dt=0$ .

Now, let $x_{1}\rightarrow x_{s}(t)-\epsilon$ and $x_{2}\rightarrow x_{s}(t)+\epsilon$ , when we have $\int _{x_{1}}^{x_{s}(t)-\epsilon }w_{t}\,dx\rightarrow 0$ and $\int _{x_{s}(t)+\epsilon }^{x_{2}}w_{t}\,dx\rightarrow 0$ , and in the limit

(\;9)\quad \quad u_{s}\left(w_{1}-w_{2}\right)=f\left(w_{1}\right)-f\left(w_{2}\right),

where we have defined $u_{s}=dx_{s}(t)/dt\,$ (the system characteristic or shock speed), which by simple division is given by

(10)\quad \quad u_{s}={\frac {f\left(w_{1}\right)-f\left(w_{2}\right)}{w_{1}-w_{2}}}.

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

(11)\quad \quad f^{\prime }\left(w_{1}\right)>u_{s}>f^{\prime }\left(w_{2}\right),

where $f^{\prime }\left(w_{1}\right)$ and $f^{\prime }\left(w_{2}\right)$ 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 $\left[\rho ,\rho u,E\right]^{T}$ and the jump conditions become

(12)\quad \quad \quad \quad \;u_{s}\left(\rho _{2}-\rho _{1}\right)=\rho _{2}u_{2}-\rho _{1}u_{1}

(13)\quad \quad \;u_{s}\left(\rho _{2}u_{2}-\rho _{1}u_{1}\right)=\left(\rho _{2}u_{2}^{2}+p_{2}\right)-\left(\rho _{1}u_{1}^{2}+p_{1}\right)

(14)\quad \quad u_{s}\left(E_{2}-E_{1}\right)=\left[\rho _{2}u_{2}\left(e_{2}+{\frac {1}{2}}u_{2}^{2}+p_{2}/\rho _{2}\right)\right]-\left[\rho _{1}u_{1}\left(e_{1}+{\frac {1}{2}}u_{1}^{2}+p_{1}/\rho _{1}\right)\right].

A schematic diagram of a shock wave situation with the density

\rho

, velocity

u

, and temperature

T

indicated for each region.

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

(15)\quad \quad u_{s}=u_{1}+c_{1}{\sqrt {1+{\frac {\gamma +1}{2\gamma }}\left({\frac {p_{2}}{p_{1}}}-1\right)}},

where $c_{1}={\sqrt {\gamma p_{1}/\rho _{1}}}$ is the speed of sound in the fluid at upstream conditions.

See Laney (1998),^[6] LeVeque (2002),^[7] Toro (1999),^[8] Wesseling (2001),^[9] and Whitham (1999)^[10] for further discussion.

Stationary shock

For a stationary shock $u_{s}=0$ , and for the 1D Euler equations we have

(16)\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \;\;\rho _{1}u_{1}=\rho _{2}u_{2}

(17)\quad \quad \quad \quad \quad \quad \quad \quad \;\;\rho _{1}u_{1}^{2}+p_{1}=\rho _{2}u_{2}^{2}+p_{2}

(18)\quad \quad \rho _{1}u_{1}\left(e_{1}+{\frac {1}{2}}u_{1}^{2}+p_{1}/\rho _{1}\right)=\rho _{2}u_{2}\left(e_{2}+{\frac {1}{2}}u_{2}^{2}+p_{2}/\rho _{2}\right).

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

(19)\quad \quad e_{1}+{\frac {1}{2}}u_{1}^{2}+p_{1}/\rho _{1}=e_{2}+{\frac {1}{2}}u_{2}^{2}+p_{2}/\rho _{2},

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:

(20)\quad \quad 2\left(h_{2}-h_{1}\right)=\left(p_{2}-p_{1}\right)\cdot \left({\frac {1}{\rho _{1}}}+{\frac {1}{\rho _{2}}}\right),

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

(21)\quad \quad {\frac {\rho _{2}}{\rho _{1}}}={\frac {{\frac {p_{2}}{p_{1}}}(\gamma +1)+(\gamma -1)}{(\gamma +1)+{\frac {p_{2}}{p_{1}}}(\gamma -1)}}={\frac {u_{1}}{u_{2}}}

(22)\quad \quad {\frac {p_{2}}{p_{1}}}={\frac {{\frac {\rho _{2}}{\rho _{1}}}(\gamma +1)-(\gamma -1)}{(\gamma +1)-{\frac {\rho _{2}}{\rho _{1}}}(\gamma -1)}}.

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 $\rho _{2}/\rho _{1}<(\gamma +1)/(\gamma -1)\,$ . As the strength of the shock increases, there is a corresponding increase in downstream gas temperature, but the density ratio $\rho _{2}/\rho _{1}$ approaches a finite limit: 4 for an ideal monatomic gas $(\gamma =5/3)$ and 6 for an ideal diatomic gas $(\gamma =1.4)$ . Note: air is comprised predominately of diatomic molecules and therefore at standard conditions $\gamma _{\mathrm {air} }\simeq 1.4$ .

Shock Hugoniot and Rayleigh line in solids

Shock Hugoniot and Rayleigh line in the p-v plane. The curve represents a plot of equation (24) with p₁, v₁, c₀, and s known. If p₁ = 0, the curve will intersect the specific volume axis at the point v₁.

Hugoniot elastic limit in the p-v plane for a shock in an elastic-plastic material.

For shocks in solids, a closed form expression such as equation (15) cannot be derived from first principles. Instead, experimental observations ^[11] indicate that a linear relation^[12] can be used instead (called the shock Hugoniot in the u_s-u_p plane) that has the form

(23)\qquad 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):^[13]

(24)\qquad p_{2}-p_{1}={\frac {c_{0}^{2}\,\rho _{1}\,\rho _{2}\,(\rho _{2}-\rho _{1})}{[\rho _{2}-s(\rho _{2}-\rho _{1})]^{2}}}={\frac {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)\qquad p_{2}-p_{1}=u_{s}^{2}\left(\rho _{1}-{\frac {\rho _{1}^{2}}{\rho _{2}}}\right)\,

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.

References

↑ Rankine, W. J. M. (1870). "On the thermodynamic theory of waves of finite longitudinal disturbances". Philosophical Transactions of the Royal Society of London. 160: 277–288. doi:10.1098/rstl.1870.0015.
↑ Hugoniot, H. (1887). "Mémoire sur la propagation des mouvements dans les corps et spécialement dans les gaz parfaits (première partie) [Memoir on the propagation of movements in bodies, especially perfect gases (first part)]". Journal de l'École Polytechnique (in French). 57: 3–97. See also: Hugoniot, H. (1889) "Mémoire sur la propagation des mouvements dans les corps et spécialement dans les gaz parfaits (deuxième partie)" [Memoir on the propagation of movements in bodies, especially perfect gases (second part)], Journal de l'École Polytechnique, vol. 58, pages 1-125.
↑ Salas, M. D. (2006). "The Curious Events Leading to the Theory of Shock Waves, Invited lecture, 17th Shock Interaction Symposium, Rome, 4–8 September." (PDF).
↑ Ames Research Staff (1953), "Equations, Tables and Charts for Compressible Flow" (PDF), Report 1135 of the National Advisory Committee for Aeronautics
↑ Note that the integral conservation law $\scriptstyle (6)$ could not, in general, be obtained from differential equation $\scriptstyle (6')$ by integraition over $\scriptstyle [x_{1};x_{2}]$ because $\scriptstyle (6')$ holds for smooth solutions only.
↑ Laney, Culbert B. (1998). Computational Gasdynamics. Cambridge University Press. ISBN 978-0-521-62558-6.
↑ LeVeque, Randall (2002). Finite Volume Methods for Hyperbolic Problems. Cambridge University Press. ISBN 978-0-521-00924-9.
↑ Toro, E. F. (1999). Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag. ISBN 978-3-540-65966-2.
↑ Wesseling, Pieter (2001). Principles of Computational Fluid Dynamics. Springer-Verlag. ISBN 978-3-540-67853-3.
↑ Whitham, G. B. (1999). Linear and Nonlinear Waves. Wiley. ISBN 978-0-471-94090-6.
↑ Ahrens, T.J. (1993), "Equation of state" (PDF), High Pressure Shock Compression of Solids, eds. J. R. Asay and M. Shahinpoor, Springer-Verlag, New York
↑ Though a linear relation is widely assumed to hold, experimental data suggest that almost 80% of tested materials do not satisfy this widely accepted linear behavior. See Kerley, G. I, 2006, "The Linear US-uP Relation in Shock-Wave Physics", http://arxiv.org/pdf/1306.6916.pdf; for details.
↑ Poirier, J-P. (2008) "Introduction to the Physics of the Earth's Interior", Cambridge University Press.

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