19.2. Non-ideal quasi-1 dimensional flows

Details

3-dimensional flows. Equations for a generic volume \(v_t\),

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

For a volume \(V\) at rest, \(\mathbf{u}^{rel} = \mathbf{u}\).

Quasi-1 dimensional approximation. Introducing the quasi-1 dimensional model, the contributions of heat conduction through lateral walls and heat source can be collected in one term,

\[-\int_{\partial \Delta S^{lat}} \hat{\mathbf{n}} \cdot \mathbf{q} + \int_{\Delta V} \rho r = \rho r A \Delta z \ ,\]

while the conduction along the axis of the stream-tube reads

\[-\int_{\partial A(z)} \hat{\mathbf{n}} \cdot \mathbf{q} - \int_{\partial A(z+\Delta z)} \hat{\mathbf{n}} \cdot \mathbf{q} \simeq A(z) q(z) - A(z+\Delta z) q(z+\Delta z) = - \partial_{z} ( q A ) \Delta z + o(\Delta z) \ .\]

With a hydraulic approximation1, the contribution of viscous stress in the momentum equation can be written as

\[\oint_{\partial v_t} \mathbf{s}_{\mathbf{n}} = - \frac{f_d A}{2 D_h} \rho \mathbf{u} |\mathbf{u}|\]

Quasi-1 dimensional model: differential equations. The conservative form of governing equations of compressible flows with friction and heat coduction reads

\[\partial_t \mathbf{u} + \partial_z \mathbf{F}(\mathbf{u}) + \partial_z \mathbf{F}^d(\mathbf{u}) = \mathbf{s} \ ,\]

or explicitly

\[\begin{split} \partial_t \begin{bmatrix} \rho A \\ \rho u A \\ \rho e^t A \end{bmatrix} + \partial_z \begin{bmatrix} \rho u A \\ \rho u^2 A + p A \\ \rho u h^t \end{bmatrix} + \partial_z \begin{bmatrix} 0 \\ 0 \\ - q A \end{bmatrix} = \begin{bmatrix} 0 \\ p \partial_z A - \frac{f A}{2 D_h} \rho u |u| \\ \rho r A \end{bmatrix} \ , \end{split}\]

where the axial heat conduction is logically serapated from the «convective» flux, as it’s usually a diffusive term containing 2-nd order spatial derivative of primary variables, and changing the mathematical nature of the equations. As an example, using Fourier’s law, \(q = -k \partial_z T\), this term becomes

\[\begin{split}\begin{bmatrix} 0 \\ 0 \\ \partial_z ( k \partial_z T ) \end{bmatrix} \ .\end{split}\]

If axial conduction is negligible if compared to convection and heat conduction through the walls of the streamtube, this contribution can be set to zero.

19.2.1. Fanno

With non-negligible friction…

todo

• analytical solution and diagrams for steady flows

• simulation for unsteady flows

19.2.2. Rayleigh

With non-negligible heat conduction through lateral walls…

todo

• analytical solution and diagrams for steady flows

• simulation for unsteady flows

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1

This is a high-Reynolds model, with viscous stress proportional to dynamic pressure through a friction coefficient and the ration of the area of the section and the hydraulic diameter, \(\frac{A}{D_h}\). As a reference see chapter about Similitude and in particular the exexricse and the discussion about Moody’s diagram.