Enhanced Turbomachinery Capabilities for Foam-Extend: Development and Validation

Abstract

Turbomachinery simulations represent one of the most challenging fields in Computational Fluid Dynamics (CFD). In recent years, the general CFD capabilities of foam-extend have been extended by introducing and maintaining additional features specifically needed for turbomachinery applications, with the aim of offering a high-quality CFD tool for the study of rotating machinery. This work presents the implementation and validation of new capabilities for turbomachinery with foam-extend, a community-driven fork of OpenFOAM$^{\textregistered }$. The formulation of an energy equation more convenient for compressible turbomachinery applications has resulted in the rothalpy equation. Rothalpy is a physical quantity conserved over a blade row, stator or rotor, but not over a stage, both stator and rotor. It is fundamental to take into account that the value of rothalpy is not continuous across the rotor–stator interface, due to the change of rotational speed between zones. The rothalpy equation has been derived for both relative and absolute frames of reference, showing that additional terms appear in the absolute frame of reference. Moreover, additional functionality has been added to the rotor–stator interface boundary conditions’ General Grid Interface (GGI), partial Overlap GGI and Mixing Plane Interface, in order to account for the rothalpy jump. The development of these new capabilities and their validation are shown, as well as industrial applications of compressible turbomachinery flows.

Appendix

The MRF may be formulated using the relative or the absolute velocity formulation. In the first case, the conservation of rothalpy in a moving reference frame is derived directly from the conservation of energy formulated in terms of relative internal energy. It is

$$\begin{aligned} \dfrac{\partial (\rho i)}{\partial t} + \nabla \!{\scriptscriptstyle \bullet }(\rho i \mathbf{W}) = \dfrac{\partial p}{\partial t} + \nabla \!{\scriptscriptstyle \bullet }(k \nabla T + \bar{\bar{\tau }} \cdot \mathbf{V}) + S_H. \end{aligned}$$

(5)

When the absolute formulation is used for the MRF, the conservation of rothalpy equations for a steadily moving frame is

$$\begin{aligned} \dfrac{\partial (\rho i)}{\partial t} + \nabla \!{\scriptscriptstyle \bullet }(\rho i \mathbf{W})&=\dfrac{\partial p}{\partial t} - \dfrac{\partial (\rho \omega R V_\theta )}{\partial t}\nonumber \\&\quad - \nabla \!{\scriptscriptstyle \bullet }(\rho \omega R V_\theta \mathbf{W}) - \nabla \!{\scriptscriptstyle \bullet }(p \mathbf{U}) + \nabla \!{\scriptscriptstyle \bullet }(k \nabla T + \bar{\bar{\tau }} \cdot \mathbf{V}) + S_H. \end{aligned}$$

(6)

This equation has been derived starting from the well-known conservation of energy equation in MRF for absolute velocity formulation [7]:

$$\begin{aligned} \dfrac{\partial (\rho e)}{\partial t} + \nabla \!{\scriptscriptstyle \bullet }(\rho e \mathbf{W}) = - \nabla \!{\scriptscriptstyle \bullet }(p \mathbf{V}) + \nabla \!{\scriptscriptstyle \bullet }(k \nabla T + \bar{\bar{\tau }} \cdot \mathbf{V}) + S_H. \end{aligned}$$

(7)

And substituting the energy expression

$$\begin{aligned} e = h_0 - \frac{p}{\rho } = i + \omega R V_\theta - \dfrac{p}{\rho } \end{aligned}$$

(8)

leads to

$$\begin{aligned}&\dfrac{\partial (\rho i)}{\partial t} + \dfrac{\partial (\rho \omega R V_\theta )}{\partial t} - \dfrac{\partial p}{\partial t} + \nabla \!{\scriptscriptstyle \bullet }(\rho i \mathbf{W}) + \nabla \!{\scriptscriptstyle \bullet }(\rho \omega R V_\theta \mathbf{W}) - \nabla \!{\scriptscriptstyle \bullet }(p \mathbf{W}) =\nonumber \\&\quad - \nabla \!{\scriptscriptstyle \bullet }(p \mathbf{V}) + \nabla \!{\scriptscriptstyle \bullet }(k \nabla T + \bar{\bar{\tau }} \cdot \mathbf{V}) + S_H. \end{aligned}$$

(9)

Reorganizing the final terms, Eq. 6 is obtained.