14. Jump conditions

Continuous medium. Jump conditions for a continuous medium read

\[\begin{split}\begin{aligned} 0 & = [ \rho u^{rel}_n ] \\ \mathbf{0} & = \dot{m}[ \mathbf{u} ] - [ \mathbf{t}_{\mathbf{n}} ] \\ 0 & = \dot{m}[ e^t ] + [ \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u} - \hat{\mathbf{n}} \cdot \mathbf{q} ] \ , \end{aligned}\end{split}\]

being \(\dot{m} = \rho_1 u^{rel}_{n,1} = \rho_2 u^{rel}_{n,2}\).

Inviscid flows, w/o heat conduction (Euler equations). If viscous stress and heat conduction are negligible (as they’re in Euler equations), jump conditions become

\[\begin{split}\begin{aligned} 0 & = [ \rho u^{rel}_n ] \\ \mathbf{0} & = \dot{m}[ \mathbf{u} ] + [ p \mathbf{n} ] \\ 0 & = \dot{m}[ h^t ] + [ p ] \hat{\mathbf{n}} \cdot \mathbf{u}_s \ . \end{aligned}\end{split}\]

Details

Starting from integral equations

\[\begin{split}\begin{aligned} & \dfrac{d}{dt} \int_{v_t} \rho + \oint_{\partial v_t} \rho \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = 0 \\ & \dfrac{d}{dt} \int_{v_t} \rho \mathbf{u} + \oint_{\partial v_t} \rho \mathbf{u} \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = \int_{v_t} \rho \mathbf{g} + \oint_{\partial v_t} \mathbf{t}_{\mathbf{n}} \\ & \dfrac{d}{dt} \int_{v_t} \rho e^t + \oint_{\partial v_t} \rho e^t \mathbf{u}^{rel} \cdot \hat{\mathbf{n}} = \int_{v_t} \rho \mathbf{g} \cdot \mathbf{u} + \oint_{\partial v_t} \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u} - \oint_{\partial v_t} \mathbf{q} \cdot \hat{\mathbf{n}} \\ \end{aligned}\end{split}\]

and collapsing the volume \(v_t\) on one of its dimension, volume contributions become negligible when compared to surface contributions. Jump conditions immeadiately follow, setting \(\hat{\mathbf{n}} = \hat{\mathbf{n}}_2 = - \hat{\mathbf{n}}_1\),

\[\begin{split}\begin{aligned} 0 & = [ \rho u^{rel}_n ] \\ \mathbf{0} & = [ \rho \mathbf{u} u^{rel}_n - \mathbf{t}_{\mathbf{n}} ] \\ 0 & = [ \rho e^t u^{rel}_n + \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u} - \hat{\mathbf{n}} \cdot \mathbf{q} ] \\ \end{aligned}\end{split}\]

Thus \(\rho_1 u_{n,1}^{rel} = \rho_2 u_{n,2}^{rel} = \dot{m}\), and

\[\begin{split}\begin{aligned} 0 & = [ \rho u^{rel}_n ] \\ \mathbf{0} & = \dot{m}[ \mathbf{u} ] - [ \mathbf{t}_{\mathbf{n}} ] \\ 0 & = \dot{m}[ e^t ] + [ \mathbf{t}_{\mathbf{n}} \cdot \mathbf{u} - \hat{\mathbf{n}} \cdot \mathbf{q} ] \\ \end{aligned}\end{split}\]

If viscous streass and conduction heat flux are negligible,

\[\begin{split}\begin{aligned} 0 & = [ \rho u^{rel}_n ] \\ \mathbf{0} & = \dot{m}[ \mathbf{u} ] + [ p \mathbf{n} ] \\ 0 & = \dot{m}[ e^t ] + [ p \hat{\mathbf{n}} \cdot \mathbf{u} ] = \\ & = \dot{m}[ e^t ] + \left[ \frac{p}{\rho} \underbrace{\hat{\mathbf{n}} \cdot \rho ( \mathbf{u} - \mathbf{u}_s )}_{ \rho u_{n}^{rel} = \dot{m}} + p \hat{\mathbf{n}} \cdot \mathbf{u}_s \right] = \\ & = \dot{m} \left[ e^t + \frac{p}{\rho} \right] + [ p \hat{\mathbf{n}} \cdot \mathbf{u}_s ] = \\ & = \dot{m} \left[ h^t \right] + \left[ p \right] \, \hat{\mathbf{n}} \cdot \mathbf{u}_s \\ \end{aligned}\end{split}\]

Two main cases are determined by the value of the mass flow \(\dot{m}\):

• if \(\dot{m} = 0\),

  \[\begin{split}\begin{aligned} & \dot{m} = 0 \\ & [ p ] = 0 \ , \end{aligned}\end{split}\]

  while nothing can be said about the tangential component of the velocity field and the total enthalpy. These conditions hold contact discontinuities.

• if \(\dot{m} \ne 0\)

  \[\begin{split}\begin{aligned} 0 & = [ \dot{m} ] = [ \rho u_{n}^{rel} ] \\ \mathbf{0} & = \dot{m} [ \mathbf{u}_{\mathbf{t}} ] \\ 0 & = \dot{m} [ u_n ] + [ \, p \, ] = \dot{m} [ u_n^{rel} ] + [ \, p \, ] \\ 0 & = \dot{m} [ h^{t,rel} ] = \dot{m} [ h^{t,rel}_n ] \end{aligned}\end{split}\]

  These conditions hold for shocks (usually compression shocks, e.g. for PIG, but for heavy fluids expansion shocks may exist as well).

Details

\[\begin{split}\begin{aligned} 0 & = [ \dot{m} ] = [ \rho u_{n}^{rel} ] \\ \mathbf{0} & = \dot{m} [ \mathbf{u}_{\mathbf{t}} ] \\ 0 & = \dot{m} [ u_n ] + [ \, p \, ] = && ([u_s] = 0) \\ & = \dot{m} [ u_n - u_s ] + [ \, p \, ] = \\ & = \dot{m} [ u_n^{rel} ] + [ \, p \, ] \\ 0 & = \dot{m} [ h^t ] + u_{s,n} [ \, p \, ] = \\ & = \dot{m} \left[ h^t \right] - u_{s,n} \, \dot{m} \left[ u_n \right] = \\ & = \dot{m} \left[ h + \frac{u_n^2 + u_t^2}{2} - u_{s,n} u_n \right] = \\ & = \dot{m} \left[ h + \frac{u_n^2 - 2 u_{s,n} u_n + u_{s,n}^2}{2} - \frac{u_{s,n}^2}{2} + \frac{u_t^2}{2} \right] = \\ & = \dot{m} \left[ h + \frac{(u_n - u_{s,n})^2}{2} - \frac{u_{s,n}^2}{2} + \frac{u_t^2}{2} \right] = \\ & = \dot{m} \left[ h^{t,rel} \right] = \\ & = \dot{m} \left[ h^{t,rel}_n \right] \ , \end{aligned}\end{split}\]

as \(\left[ u_{s,n}^2 \right] = 0\) being \(u_{s,n}\) the velocity of the «discontinuity» surface, \([ \mathbf{u}_t ] = 0\) from the tangential component of the momentum equation, and with the definition of the relative total enthalpy

\[h^{t,rel} := h + \frac{1}{2} \left( (u_n - u_{s,n})^2 + u_t^2 \right) = h + \frac{1}{2} \left( \left( u_{n}^{rel} \right)^2 + u_t^2 \right) \ , \]

and \(h^{t,rel}_n\) the relative total enthalpy built with the normal component of the relative velocity only.