Hugoniot Condition

The Hugoniot conditions are the conditions that must be satisfied on the discontinuity lines of the solutions of the equations of gas dynamics , as a consequence of the integral conservation laws .

Let be $x=x(t)-$ ${\ displaystyle x = x (t) -}$ ${\ displaystyle x = x (t) -}$ the equation of one of the discontinuity lines of hydrodynamic quantities, which we will assume for the considered segment $t_{1}\leqslant t\leqslant t_{2}$ ${\ displaystyle t_ {1} \ leqslant t \ leqslant t_ {2}}$ ${\ displaystyle t_ {1} \ leqslant t \ leqslant t_ {2}}$ continuous tangent

Let be $f(x,t)$ ${\ displaystyle f (x, t)}$ ${\ displaystyle f (x, t)}$ suffers a break on the line $x=x(t)$ ${\ displaystyle x = x (t)}$ $x = x (t)$ .
Denote:
$f_{1}(t)=f(x(t)-0,t);$ ${\ displaystyle f_ {1} (t) = f (x (t) -0, t);}$ ${\ displaystyle f_ {1} (t) = f (x (t) -0, t);}$
$f_{2}(t)=f(x(t)+0,t);$ ${\ displaystyle f_ {2} (t) = f (x (t) + 0, t);}$ ${\ displaystyle f_ {2} (t) = f (x (t) + 0, t);}$
$[f]=f_{2}(t)-f_{1}(t)$ ${\ displaystyle [f] = f_ {2} (t) -f_ {1} (t)}$ ${\ displaystyle [f] = f_ {2} (t) -f_ {1} (t)}$

The integral conservation laws in Euler coordinates have the form
$(1){\begin{cases}\oint \limits _{C}\rho x^{\nu }\cdot dx-\rho ux^{\nu }\cdot dt=0,\\\oint \limits _{C}\rho ux^{\nu }\cdot dx-(p+\rho u^{2})x^{\nu }\cdot dt=-\oint \limits _{G}\oint \limits _{C}\nu px^{\nu -1}\cdot dxdt,\\\oint \limits _{C}\rho (\varepsilon +{\frac {u^{2}}{2}})x^{\nu }\cdot dx-\rho u(\varepsilon +{\frac {p}{\rho }}+{\frac {u^{2}}{2}})x^{\nu }\cdot dt=0.\end{cases}}$ ${\ displaystyle (1) {\ begin {cases} \ oint \ limits _ {C} \ rho x ^ {\ nu} \ cdot dx- \ rho ux ^ {\ nu} \ cdot dt = 0, \\\ oint \ limits _ {C} \ rho ux ^ {\ nu} \ cdot dx- (p + \ rho u ^ {2}) x ^ {\ nu} \ cdot dt = - \ oint \ limits _ {G} \ oint \ limits _ {C} \ nu px ^ {\ nu -1} \ cdot dxdt, \\\ oint \ limits _ {C} \ rho (\ varepsilon + {\ frac {u ^ {2}} {2}}) x ^ {\ nu} \ cdot dx- \ rho u (\ varepsilon + {\ frac {p} {\ rho}} + {\ frac {u ^ {2}} {2}}) x ^ {\ nu} \ cdot dt = 0. \ end {cases}}}$ ${\ displaystyle (1) {\ begin {cases} \ oint \ limits _ {C} \ rho x ^ {\ nu} \ cdot dx- \ rho ux ^ {\ nu} \ cdot dt = 0, \\\ oint \ limits _ {C} \ rho ux ^ {\ nu} \ cdot dx- (p + \ rho u ^ {2}) x ^ {\ nu} \ cdot dt = - \ oint \ limits _ {G} \ oint \ limits _ {C} \ nu px ^ {\ nu -1} \ cdot dxdt, \\\ oint \ limits _ {C} \ rho (\ varepsilon + {\ frac {u ^ {2}} {2}}) x ^ {\ nu} \ cdot dx- \ rho u (\ varepsilon + {\ frac {p} {\ rho}} + {\ frac {u ^ {2}} {2}}) x ^ {\ nu} \ cdot dt = 0. \ end {cases}}}$

We write the conservation laws (2) for the circuit AA 'BB' , assuming that the lines A'B and B'A of circuit C , as well as the double integral $\oint \limits _{G}\oint \limits _{C}\nu px^{\nu -1}\cdot dxdt$ ${\ displaystyle \ oint \ limits _ {G} \ oint \ limits _ {C} \ nu px ^ {\ nu -1} \ cdot dxdt}$ ${\ displaystyle \ oint \ limits _ {G} \ oint \ limits _ {C} \ nu px ^ {\ nu -1} \ cdot dxdt}$ . Along the line $x=x(t)$ ${\ displaystyle x = x (t)}$ $x = x (t)$ we have $dx=Ddt$ ${\ displaystyle dx = Ddt}$ ${\ displaystyle dx = Ddt}$ where $D=D(t)=x^{\prime }(t)$ ${\ displaystyle D = D (t) = x ^ {\ prime} (t)}$ ${\ displaystyle D = D (t) = x ^ {\ prime} (t)}$ .
Therefore, for example, from the first equation of (1) , we obtain
$\int \limits _{t_{1}}^{t_{2}}x^{\nu }\left\{(\rho _{2}(t)-\rho _{1}(t))D(t)-(\rho _{2}(t)u_{2}(t)-\rho _{1}(t)u_{1}(t))\right\}\cdot dx=0$ ${\ displaystyle \ int \ limits _ {t_ {1}} ^ {t_ {2}} x ^ {\ nu} \ left \ {(\ rho _ {2} (t) - \ rho _ {1} (t )) D (t) - (\ rho _ {2} (t) u_ {2} (t) - \ rho _ {1} (t) u_ {1} (t)) \ right \} \ cdot dx = 0}$ ${\ displaystyle \ int \ limits _ {t_ {1}} ^ {t_ {2}} x ^ {\ nu} \ left \ {(\ rho _ {2} (t) - \ rho _ {1} (t )) D (t) - (\ rho _ {2} (t) u_ {2} (t) - \ rho _ {1} (t) u_ {1} (t)) \ right \} \ cdot dx = 0}$ $,(2)$ ${\ displaystyle, (2)}$ ${\ displaystyle, (2)}$
Due to the arbitrariness of the integration limits in (2) , the integrand must be equal to zero i.e. $x^{\nu }(t)\left\{D(t)[\rho ]-[\rho u]\right\}=0$ ${\ displaystyle x ^ {\ nu} (t) \ left \ {D (t) [\ rho] - [\ rho u] \ right \} = 0}$ ${\ displaystyle x ^ {\ nu} (t) \ left \ {D (t) [\ rho] - [\ rho u] \ right \} = 0}$ .

Reducing Equality by $x^{\nu }$ ${\ displaystyle x ^ {\ nu}}$ ${\ displaystyle x ^ {\ nu}}$ , we see that the conditions on the discontinuity line are the same for the three cases of symmetry $\nu =0,1,2$ ${\ displaystyle \ nu = 0,1,2}$ ${\ displaystyle \ nu = 0,1,2}$ .
Acting in a similar way with all conservation laws (1) , we obtain conditions on the discontinuity line $x=x(t)$ ${\ displaystyle x = x (t)}$ $x = x (t)$

$D[\rho ]=[\rho u],$ {\ displaystyle D [\ rho] = [\ rho u],} ${\ displaystyle D [\ rho] = [\ rho u],}$

$D[\rho u]=[p+\rho u^{2}],$ ${\ displaystyle D [\ rho u] = [p + \ rho u ^ {2}],}$ ${\ displaystyle D [\ rho u] = [p + \ rho u ^ {2}],}$

$D[\rho (\varepsilon +{\frac {u^{2}}{2}})]=[\rho u(\varepsilon +{\frac {p}{\rho }}+{\frac {u^{2}}{2}})]$ ${\ displaystyle D [\ rho (\ varepsilon + {\ frac {u ^ {2}} {2}})] = [\ rho u (\ varepsilon + {\ frac {p} {\ rho}} + {\ frac {u ^ {2}} {2}})}}$ ${\ displaystyle D [\ rho (\ varepsilon + {\ frac {u ^ {2}} {2}})] = [\ rho u (\ varepsilon + {\ frac {p} {\ rho}} + {\ frac {u ^ {2}} {2}})}}$

which connect jumps of hydrodynamic quantities on the discontinuity line $x=x(t)$ ${\ displaystyle x = x (t)}$ $x = x (t)$ and speed $D=x^{\prime }(t)$ ${\ displaystyle D = x ^ {\ prime} (t)}$ ${\ displaystyle D = x ^ {\ prime} (t)}$ break lines.
The latter relations are called the conditions of hydrodynamic compatibility of the gap or the Hugoniot conditions .

Hugoniot Condition

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