Mixture model

The mixture simulation framework in OpenSTREAM provides a simplified yet powerful approach to modeling two-phase flows. Based on a three-equation formulation, it treats the phases as a single continuum with averaged properties. This abstraction enables efficient simulation of systems where phase separation is minimal or where a fully resolved multi-field model is not required.

The mixture model serves two key roles within OpenSTREAM:

  • Robust initialization for more complex solvers, ensuring robust starting conditions.

  • Efficient simulation of flows with hydrodynamically well-coupled phases, where the mixture approximation remains physically meaningful.

  • Industry relevance due to its simplicity and robustness, offering a practical approach for simulating averaged flow behavior in complex thermal-hydraulic systems.

In addition, a Homogeneous Relaxation Model (HRM) is currently under development. This extended formulation introduces vapor mass and energy conservation equations, allowing for thermal non-equilibrium between phases and expanding the model’s applicability to more dynamic flow regimes.

By reducing complexity while preserving essential dynamics, the mixture model offers a practical entry point for both model development and exploratory analysis in thermal-hydraulic systems.

An overview of the mixture model implemented in OpenSTREAM is provided below. A more detailed derivation and theoretical background can be found in Le Corre et al. [2025].

Governing equations

The mixture model solves the following conservation equations:

1. Mass conservation

\(\frac{\partial}{\partial t}(\frac{W}{u}) + \frac{\partial W}{\partial z} = 0\)

where:

  • \(W\) is the mixture mass flow rate

  • \(u\) is the mixture velocity

2. Momentum conservation

\(\rho A (\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial z}) = -A (\frac{\partial p}{\partial z} + \frac{\partial p_K}{\partial z} + \cos\theta g \rho) - \sum \Pi_{wall}^n {\tau}_{wall}^n\)

where:

  • \(\rho\) is the mixture density

  • \(A\) is the cross-sectional area

  • \(p\) is the pressure

  • \(K\) relates to obstruction form loss

  • \(\theta\) is the inclination angle

  • \(g\) is the gravitational acceleration

  • \(n\) is the wall index

  • \(\Pi_{wall}^n\) is the wall perimeter for wall index \(n\)

  • \(\tau_{wall}^n\) is the wall shear stress for wall index \(n\)

3. Energy conservation

\(\rho A (\frac{\partial h}{\partial t} + u \frac{\partial h}{\partial z}) = \sum \Pi_{wall}^n {q^{\prime\prime}}_{wall}^n\)

where:

  • \(h\) is the mixture specific enthalpy

  • \({q^{\prime\prime}}_{wall}^n\) is the wall heat flux for wall index \(n\)

Homogeneous relaxation model

Under development

Closure relations

To complete the conservation equations, several closure relations are required:

  • Wall shear stress: \(\tau_w^n\)

  • Form pressure losses: \(\frac{\partial p_K}{\partial z}\)

  • Wall heat transfer models

For each simulation, the selected closure models are defined in the OpenSTREAM model file, chosen from the available options listed in InputEnums. If not explicitly specified by the user, default models are applied as defined in Inputs.Model. All relevant closure relations are implemented in Solvers.Mixture.Mixture, which the users can modify to suit specific simulation needs.

In addition, the thermodynamic properties for each phase are computed using CoolProp, an open-source thermophysical property library that provides accurate equations of state and transport properties for a wide range of fluids.

Features and assumptions

  • Supports both steady-state and transient simulations in straight channels

  • Supports uniform and non-uniform wall heat flux distribution

  • Can include thermal non-equilibrium modeling (subcooled boiling or post Critical Heat Flux) via constitutive models or HRM

  • Allows phase velocity slip using drift flux models

  • Neglects minor contributions such as frictional heating and temporal pressure gradient contributions

Role in OpenSTREAM

The mixture model provides a robust tool for simulating homogeneous two-phase flow, including cases with thermal non-equilibrium. It also provides an initialization framework for more complex solvers (i.e., two-fluid, three-field, four-field). Currently, its pressure gradient solution is reused across all solver frameworks to enhance robusness and numerical stability.

Implementation notes

Package

Module

Mixture solver class

Field and phase classes:

Key solver methods

Key field properties: