*5.2. Validation of the Implementation of the Bi-CITSGM Method on the Dowlati's Experiment*

The Bi-CITSGM method is solved at each REV of the tube bundle and each iteration in order to compute the Forchheimer term. In addition, the void fraction model of Feenstra et al. is always implemented. Figure 17 compares the total two-phase pressure drop of Dowlati's tube bundle given by the experiment, the Zukauskas correlation implementation and the Bi-CITSGM method implementation in the Forchheimer term. Total pressure drops well fit with the experiment results at low void fractions. Errors become higher when the void fraction increases; however, the post-processing of the pressure drop from the paper of Dowlati et al. [26] is not immediate, nor is it very accurate. The post-process of the pressure drops resulting from the implementation of the Zukauskas correlation and that of the Bi-CITSGM method in the source term are superimposed on the graph. These results validate the suggested approach. Likewise, in Figure 18, the two-phase frictional pressure drops are plotted for different mass flux and are compared to the experiment results. We note that the two-phase frictional pressure drop is highly dependent on the mass flux and less on the void fraction. On the contrary, the gravitational pressure drop decreases when the void fraction increases and is barely dependent on the mass flux. These results are consistent with the physical phenomena that occur in a two-phase flow through a tube bundle. Consequently, the aim to determine the momentum sink by the non-intrusive reduced-order model, Bi-CITSGM, is validated by the results presented in this subsection.

**Figure 17.** Total pressure drop of Dowlati's tube bundle (×: *Zukauskas*; ◦: *Bi-CITSGM*; -: *experiment*).

**Figure 18.** Frictional pressure drop of Dowlati's tube bundle (×: *Zukauskas*; ◦: *Bi-CITSGM*; -:*experiment*).

#### **6. Conclusions**

In order to predict the thermal–hydraulic performance of an adiabatic upward airwater flow through a horizontal tube bundle, two approaches are suggested. They are validated with the experimental results of Dowlati et al. First, the prediction of the void fraction on the tube bundle was improved by using the mixture model and rewriting the drift velocity as a function of slip. Two correlations coming from the literature, Hibiki et al. [17] and Feenstra et al. [16], are compared to the experimental results. They are given similar results with a relative error about 20%. Moreover, we showed that the definition of the Capillary number with the upstream mass flux in Feenstra's correlation significantly improves the void fraction prediction with a relative error under 10%. Second, the CFD porous media approach used implies adding a momentum sink to the governing momentum equation named the Darcy–Forchheimer term. Usually, pressure drop correlations coming from the literature have been used to compute the Forchheimer term except for complex and non-usual geometry for which there is no correlation. In this instance, we demonstrate that it is possible to determine a numerical pressure drop correlation by solving a non-intrusive parametric reduced-order model of the flow through a Representative Elementary Volume of the tube bundle. In the case of the straight tube bundle of Dowlati et al., the Bi-CITSGM method is consistent with the Zukauskas correlation [21]. Moreover, there is a short gap with the experimental results despite a significant possible post-processing error. The two proposed methods that yield satisfactory results need to be expanded. For instance, it would be interesting to simulate a two-phase parallel-flow in a staggered vertical tube bundle with the mixture model modified by the rewriting of the drift velocity. Moreover, the use of a non-intrusive reduced-order model applied to a non-usual geometry of REV in order to compute the Forchheimer term could be an axis of development.

**Author Contributions:** We would like to declare that all authors have the same contribution listed as follows: C.D., C.A., V.M., C.B. (Claudine Béghein), M.O. and C.B. (Clément Bonneau): Conceptualization; methodology; software; validation; formal analysis; investigation; resources; data curation; writing—original draft preparation; writing—review and editing; visualization; supervision; project

administration; funding acquisition. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** The authors wish to express their thanks to ANRT (Association Nationale Recherche Technologie) as part of the CIFRE convention N°2018/0443.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **Appendix A. Algorithm of the ITSGM Method**

Let Φ*<sup>γ</sup>*<sup>1</sup> , Φ*<sup>γ</sup>*<sup>2</sup> , ... , <sup>Φ</sup>*<sup>γ</sup>Np* be a set of POD basis and Φ*<sup>γ</sup>*<sup>1</sup> , Φ*<sup>γ</sup>*<sup>2</sup> , ... , <sup>Φ</sup>*<sup>γ</sup>Np* the subvector space associated belonging to the Grassmann manifold. By using the definitions of the geodesic path, exponential application, and logarithm application, the aim is to approximate the subspace [Φ*<sup>γ</sup>*<sup>1</sup> ] amount for a new parameter *<sup>γ</sup>*<sup>1</sup> = *γi*. The different steps of the ITSGM method come hereafter.

**Algorithm A1** ITSGM Algorithm.


$$\mathcal{X}\_{i} = \mathcal{U}\_{i}\operatorname{arctan}(\Sigma\_{i})\mathcal{V}\_{i}^{T}, i = 1, \dots, N\_{p} \tag{A1}$$

where *Ui*Σ*iV<sup>T</sup> <sup>i</sup>* is the truncated SVD decomposition of (*I* − Φ*γi* Φ*γi <sup>T</sup>*)Φ*<sup>γ</sup><sup>i</sup>* (Φ*<sup>γ</sup><sup>i</sup> <sup>T</sup>*Φ*<sup>γ</sup><sup>i</sup>* )−1.

0 0 0 (c) Interpolate <sup>X</sup><sup>1</sup> , <sup>X</sup><sup>2</sup> , ... , <sup>X</sup>*Np* and obtain the initial velocity <sup>X</sup>*<sup>γ</sup>*<sup>1</sup> linked with the new parameter *<sup>γ</sup>*1. As the tangent space <sup>T</sup>[Φ*<sup>i</sup>* 0 ] G(*q*, *N*) is a vector space, the interpolation

standard technique can be used like Lagrange or RBF.

(d) Determine the interpolate sub-vector space

$$\Phi\_{\bar{\gamma}} = \Phi\_{\gamma\_{\bar{i}\_0}} \vec{V} \cos(\hat{\Sigma}) + \vec{\mathcal{U}} \sin(\hat{\Sigma}) \tag{A2}$$

where *<sup>U</sup>*1Σ1*V*1*<sup>T</sup>* is the SVD decomposition truncated of <sup>X</sup>*<sup>γ</sup>*<sup>1</sup> .

#### **Appendix B. Algorithm of the Bi-CITSGM Method**

Let the POD decomposition of order *<sup>q</sup>* of the matrices *<sup>S</sup><sup>γ</sup><sup>i</sup>* linked to the parameter *<sup>γ</sup><sup>i</sup>* such as

$$\mathbf{S}\_{\gamma\_i} \approx L L\_{\gamma\_i} \Sigma\_{\gamma\_i} V\_{\gamma\_i}^T$$

where *<sup>U</sup>γ<sup>i</sup>* <sup>∈</sup> <sup>R</sup>*Nx*×*<sup>q</sup>* et *<sup>V</sup>γ<sup>i</sup>* <sup>∈</sup> <sup>R</sup>*Ns*×*<sup>q</sup>* are, respectively, the spatial and temporal bases, and <sup>Σ</sup>*<sup>γ</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>*q*×*<sup>q</sup>* is the singular value matrix. Obtaining the solution *<sup>S</sup><sup>γ</sup>*<sup>1</sup> for a new parameter, *γ*1 = *γ<sup>i</sup>* is given by the online step of the Bi-CITSGM method defined below.

#### **Algorithm A2** Bi-CITSGM Algorithm.

*Offline step (do this step only once):*

(a) For *<sup>i</sup>* <sup>=</sup> 1, ... , *Np*, to approximate the snapshot matrices *<sup>S</sup><sup>γ</sup><sup>i</sup>* , use the truncated SVD decomposition of order *q* as follows:

$$\mathbf{S}\_{\gamma\_i} \approx \mathcal{U}\_{\gamma\_i} \Sigma\_{\gamma\_i} V\_{\gamma\_i}^T \tag{A3}$$

where *<sup>U</sup>γ<sup>i</sup>* <sup>∈</sup> <sup>R</sup>*Nx*×*<sup>q</sup>* and *<sup>V</sup>γ<sup>i</sup>* <sup>∈</sup> <sup>R</sup>*Ns*×*<sup>q</sup>* are, respectively, the spatial and time basis and <sup>Σ</sup>*<sup>γ</sup><sup>i</sup>* <sup>∈</sup> <sup>R</sup>*q*×*<sup>q</sup>* is the singular values matrix.

*Online step:*


$$||\Phi\_{\gamma\_{k\_0}}{}^j - \Phi\_{\gamma\_k}{}^j||\_2 > ||\Phi\_{\gamma\_{k\_0}}{}^j + \Phi\_{\gamma\_k}{}^j||\_2 \text{ with } k\_0 = \underset{i \in \{1, \dots, N\_{\mathcal{P}}\}}{\text{argmin}} \; dist\_{\mathcal{G}}(\Phi\_{\tilde{\gamma}}, \Phi\_{\gamma\_i}).$$

(e) Calculate the considered coefficients *ω<sup>i</sup>* and *κ<sup>i</sup>*

$$\omega\_{i} = \frac{\text{dist}\_{\mathcal{G}}(\boldsymbol{\mathcal{U}}\_{\overline{\gamma}^{\prime}}, \boldsymbol{\mathcal{U}}\_{\gamma\_{i}})^{-m}}{\sum\_{k=1}^{N\_{p}} \text{dist}\_{\mathcal{G}}(\boldsymbol{\mathcal{U}}\_{\overline{\gamma}^{\prime}}, \boldsymbol{\mathcal{U}}\_{\gamma\_{k}})^{-m}} \qquad \qquad \kappa\_{i} = \frac{\text{dist}\_{\mathcal{G}}(\boldsymbol{V}\_{\overline{\gamma}^{\prime}}, \boldsymbol{V}\_{\gamma\_{i}})^{-1}}{\sum\_{k=1}^{N\_{p}} \text{dist}\_{\mathcal{G}}(\boldsymbol{V}\_{\overline{\gamma}^{\prime}}, \boldsymbol{V}\_{\gamma\_{k}})^{-1}} \tag{A4}$$

(f) Calculate *λ* the diagonal matrix of eigenvalues, and *P* the matrix of eigenvectors by verifying the following eigenvalue decomposition:

$$\sum\_{i=0}^{N\_p} \sum\_{j=0}^{N\_p} \omega\_i \omega\_j \mathcal{U}\_{\gamma\_i}^T \mathcal{U}\_{\overline{\gamma}} \mathcal{U}\_{\overline{\gamma}}^T \mathcal{U}\_{\overline{\gamma}} = P \lambda P^T \tag{A5}$$

(g) Calculate *η* the diagonal matrix of eigenvalues, and *H* the matrix of eigenvectors by verifying the following eigenvalue decomposition:

$$\sum\_{i=0}^{N\_p} \sum\_{i=0}^{N\_p} \kappa\_i \kappa\_{\bar{\gamma}} V\_{\gamma i}^T V\_{\bar{\gamma}} V\_{\bar{\gamma}}^T V\_{\gamma j} = H \eta H^T \tag{A6}$$

(h) Calculate the orthogonal matrices *K* and *Q* given by:

$$K = \mathcal{U}\_{\overline{\mathcal{V}}} \left( \sum\_{i=0}^{N\_p} \omega\_i \mathcal{U}\_{\mathcal{V}\_i} \right) P \lambda^{-\frac{1}{2}} P^T \tag{A7}$$

$$Q = V\_{\overline{\gamma}} \left( \sum\_{i=0}^{N\_p} \kappa\_i V\_{\gamma\_i} \right) H \eta^{-\frac{1}{2}} H^T \tag{A8}$$

(i) Build the interpolate snapshot matrix defined by:

$$\mathcal{S}\_{\bar{\gamma}} = \mathcal{U}\_{\bar{\gamma}} \mathcal{K} \hat{\Sigma} \mathcal{Q}^T V\_{\bar{\gamma}}^T \tag{A9}$$

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