A virtual flow meter downstream of various elbow configurations

Three different 90^∘ elbow configurations of circular pipes with a pipe diameter of D = 0.1 m are studied: a single elbow (SE), a double elbow out-of-plane (DE) and a double S-elbow (DSE), compare figure 1. The geometrical degrees of freedom are the curvature radius $R_\textrm c$ (for all cases) and the distance between the two elbows $d_\textrm l$ (for the geometries with two elbows DE and DSE). Multiple geometries with different parameter realizations are generated. For the SE case, the curvature radius $R_\textrm c$ is sampled between $R_\textrm {c.min} = 0.51 \,D$ and $R_\textrm {c.max} = 10 \,D$ in $N_{R_c} = 21$ steps. Note that $R_\textrm {c.min}$ is chosen to be $0.01 \,D$ higher than the theoretical minimum ($0.5\,D$) to avoid a sharp edge, which would lead to grid elements with low quality. As the dynamics are expected to be higher at lower values of $R_\textrm c$, the sample positions are distributed according to

$$\begin{align} R_\textrm c & = R_\textrm {c.min} + \left(R_\textrm {c.max} - R_\textrm {c.min} \right)\left(\frac{i}{N_{R_\textrm c}-1}\right)^2 \nonumber\\ & \quad \textrm{with } i = 0,1, \dots N_{R_\textrm c}-1. \end{align} \tag{ 1 }$$

For the other two cases, values of $R_\textrm {c.max} = 5.51 \, D$ and $N_{R_\textrm c} = 11$ are chosen. The distance between the two elbows $d_l \in [d_\textrm {l.min},d_\textrm {l.max}]$ is distributed according to

$$\begin{align} d_\textrm l & = d_\textrm {l.min} + \left(d_\textrm {l.max}-d_\textrm {l.min}\right) \left(\frac{i}{N_{d_\textrm l}-1}\right)^2 \nonumber\\ & \quad\, \textrm{with } i = 0,1, \dots N_d-1, \end{align} \tag{ 2 }$$

where the minimum and maximum distances are $d_\textrm {l.min} = 0.01 \, D$ and $d_\textrm {l.max} = 10.01\,D$, respectively. The number of parameter values is $N_{d_\textrm l} = 11$.

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Altogether, a total of 263 geometries are processed. The geometries and meshes are generated automatically using the BlockMesh implementation from [24]. In the computational domain, the length from the inlet to the entry of the upstream elbow is $7\,D$, while the straight pipe section between the exit of the downstream elbow and the outlet has a length of $100\,D$.

The flow conditions in the simulations are selected in conformity with the measurement setup described in Straka et al [28, 29]. The water temperature is 30 ^∘C, which results in a kinematic viscosity of $0.8 \cdot 10^{-6}$ m² s⁻¹. Furthermore, the flow rate is 11.32 m³ h⁻¹, which corresponds to a volumetric velocity of $u_\textrm {vol} = 0.4$ m s⁻¹. This results in a diameter-based Reynolds number of $Re = 5 \cdot 10^4$ and a friction Reynolds number of $Re_\tau = 1288$ for a smooth pipe. At the inlet, a fully developed velocity profile and turbulent viscosity taken from a precursor simulation of a straight pipe is prescribed. At the outlet zero pressure, and zero velocity at the pipe walls, are prescribed.

In order to perform a CFD simulation, it is necessary to solve mathematical equations that describe the behavior of the fluid. The fluid motion of water can be described by the incompressible Navier–Stokes equations. As the resolution of all turbulent fluctuations is computationally too expensive, additional equations can be included to model the effects of turbulence. There are different types of turbulence models for transient simulations such as large eddy simulation (LES) or detached eddy simulation (DES), see, e.g. Ferzinger and Peric [7], Wilcox [38]. These models resolve large turbulent flow structures and model the effect of the smallest turbulent scales, yet they are still too computationally expensive, especially for high Reynolds numbers. In this study a large number of simulations needs to be performed at a comparatively high Reynolds number. Therefore, RANS models are employed, which are less computationally expensive by several orders of magnitude. RANS models rely on a time-averaged version of the Navier–Stokes equations and additional transport equations to calculate an artificial local eddy viscosity that accounts for all turbulent fluctuations. Among the RANS type models, the most common approaches are the k-ε, k-ω and SST model. As of today, these models are still the industry standard. In this study, the Spalart–Allmaras turbulence model [26] is used, which consists of one partial differential equation for calculating the eddy viscosity $\tilde{\nu}_t$. Although being less computationally expensive, it has been demonstrated to be equally or even more precise compared to other RANS models in pipe flow applications, see Straka et al [29].

Second order discretization schemes are chosen, for divergence of the velocity bounded Gauss linear and bounded Gauss linearUpwind grad(U) for the divergence of the turbulent viscosity. Moreover, Gauss linear and Gauss linear corrected are used for the gradient and Laplacian schemes, respectively. For the DE and DSE the solver OpenFOAM version 1812 is used, whereas for the SE, RapidCFD is used. Both solvers are based on the finite volume method. RapidCFD is a fork of OpenFOAM (version 2.3) operating on GPU, which produces similar results for the Spalart–Allmaras turbulence model, see Ekat et al [4].

A grid-independence study is conducted to ensure the accuracy of the numerical solution of the mathematical equations. Block-structured hexahedral O-grids are created using BlockMesh with increasing grid densities. The study is conducted for a DE with $R_\textrm c = 1.425 \, D$ and $d_\textrm l = 0.01 \,D$. Five grids are created, each with an inlet length of $7 \,D$ between the inlet and the elbow entrance, and an outlet length of $42\, D$ downstream of the last elbow. The coarsest grid, labeled G₀, contains 160 000 elements. Subsequent grids G₁ and G₃ have twice and quadruple the number of grid cells in all three dimensions compared to G₀. Grid G₂ is created by refining G₁ slightly, and the finest grid, G₄, is generated by moderately refining G₃. The grid parameters are summarized in table 1. A quarter of the pipe cross section for G₀, G₁, G₃ and G₄ as well as the full cross section of G₂ are presented in figure 2.

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Table 1. The parameters of the computational grids used for the simulations in the grid study, where $N_\textrm {elem}$ is the number of grid elements in millions, $y^+$ the dimensionless distance of the first grid cell to the pipe wall (mean, minimum and maximum value over the whole fluid domain), N_xy the number of discretization points in the x and y direction in the inner core of the O-grid, N_o the number of radial elements of the outer ring and N_z the number of grid elements in flow direction per diameter D.

Grid Name	$N_\textrm {elem}/10^6$	$y^+$ (min, max)	Nxy	No	Nz per D
G4	35	0.2 (0.03, 0.3)	60	75	24
G3	9	0.3 (0.04, 0.5)	40	50	16
G2	2	0.5 (0.08, 1)	26	32	10
G1	1	0.8 (0.1, 1.3)	20	25	8
G0	0.1	2 (0.5, 4)	10	12	4

To compare the performance of the grids, the magnitude of the difference between the velocity vectors of the results with each grid u _i and the finest grid $-{u}_\textrm{ref}$ is calculated according to $\varepsilon(x,y,z) = \Vert\boldsymbol{u}_i(x, y, z) - \boldsymbol{u}_\textrm{ref}(x, y, z)\Vert_2$. After that, the L²-norm over the cross sections Ω is evaluated at multiple downstream positions z:

$$\begin{align} \mathcal{E} = \left(\frac{1}{A}\int\limits_\Omega \varepsilon^2 \, \textrm{d}(x,y) \right)^\frac{1}{2}. \end{align} \tag{ 3 }$$

Here, A is the area of the cross section Ω. All velocities $\boldsymbol{u}^*$ are normalized with the bulk velocity u_vol , so that $\boldsymbol{u} = \frac{\boldsymbol{u}^*}{u_{vol}}$. The result of each grid is interpolated onto the cell centers of the coarsest grid G₀. All integrals are calculated using piecewise linear shape functions on a Delaunay triangulation of the x–y coordinates. It can be observed that for all grids the highest deviations appear near the elbow exit and decline with further distance, compare figure 3 (left). In figure 3 (right), the maximum of $\mathcal{E}$ over all evaluated cross sections for each grid, $\max_z \mathcal{E}(z)$, is shown over the number of grid elements. As a compromise between accuracy and efficiency, the parameters of G₂ are chosen for the simulations of the elbows with the geometry variation. Other than in the grid study, the length of the straight pipe downstream the elbow is extended to $100\,D$. This leads to a number of grid elements between $2 \cdot 10^6$ and $5 \cdot 10^6$.

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In general, the flow downstream of all geometries is highly disturbed which manifests in asymmetric axial velocity profiles and enhanced secondary flow motions. A brief overview of the resulting velocity profiles at $z = 6 \, D$ with different values of $R_\textrm c$ and $d_\textrm l$ is shown in figure 4. For the SE (top row), the typical sickle-shaped axial velocity profile can be observed, where the conformation of the shape differs with different values of $R_\textrm c$. The secondary flow motions, which are represented by arrows, show two counter-rotating swirls. Downstream of the DE (center row) the profile is also sickle-shaped but, in contrast to the SE, it is rotating clockwise due to the centered single swirl. The variation of shapes downstream of the DSE (bottom row) is more divers. The position of the highest axial velocities varies between the upper, lower, and center part of the pipe cross section. The secondary flow motions build two counter-rotating swirls, similar to the SE but in opposite direction. With higher values of $R_\textrm c$ and $d_\textrm l$, the strength of the swirl gets weaker for the depicted velocity profiles. Note that the plots represent only a small part of the results, but illustrate the complexity of the dynamics in the parameter space.

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For each simulation, the velocity field at each cross section is sampled in cylindrical coordinates $(r,\varphi,z)$. Whereas the angular positions are chosen uniformly, the radial positions are chosen to be more dense at the wall according to

$$\begin{equation} r_i = \left( \frac{i}{N_r-1}\right)^{0.6} \, , i = 0 \dots N_r. \end{equation} \tag{ 4 }$$

Similarly, the downstream positions z are sampled with a higher density close to the elbow exit with

$$\begin{equation} z_i = \left( \frac{i}{N_z-1}\right)^2 100 D \, , \, i = 0 \dots N_z. \end{equation} \tag{ 5 }$$

The corresponding number of sampled data points are summarized in table 2.

Table 2. The number of data points sampled from the simulation results with regard to the velocity components (N_u ), curvature radii ($N_{R_\textrm c}$), in-between distances ($N_{d_\textrm l}$), downstream positions (N_z ), radial positions (N_r ), angular positions (N_ϕ ), and the overall data points in millions ($N_\textrm {data}$).

Case	Nu	$N_{R_\textrm c}$	$N_{d_\textrm l}$	Nz	Nr	Nϕ	$N_\textrm {data}$/106
SE	3	21	—	101	41	81	21
DE	3	11	11	200	41	81	241
DSE	3	11	11	151	41	81	182

The axial velocity profile u_z can be decomposed into a fully developed part $u_\textrm{fully}$ and a disturbed part $u_\textrm{dis}$ according to

$$\begin{align} u_z\left(r,\varphi,z,Re,k_\textrm s \right) = u_\textrm{fully}(r,Re,k_\textrm s) + u_\textrm{dis}(r,\varphi,z). \end{align} \tag{ 6 }$$

This decomposition offers the benefit of utilizing an analytical approach for the fully developed part of the profile. Consequently, varying the Reynolds number (Re) or sand grain roughness ($k_\textrm s$) can be achieved without the requirement of performing a new simulation. By replacing $u_\textrm{fully}$ with modified parameters of Re and $k_\textrm s$ in equation (6), a new profile u_z is created. In figure 5, the decomposition of the velocity profile is illustrated on the example of a vertical line at $z = 6\,D$ downstream of the SE with $R_\textrm c = 1.1$ D with original parameters $(Re = 5\cdot10^4, \, k_\textrm s = 0)$ and with modified parameters $(Re = 5\cdot10^5,k_\textrm s = 0)$ and $(Re = 5\cdot10^4,\, k_\textrm s = 0.01 \, D)$. As an important property, this decomposition conserves the flow rate. This approach implies that $u_\textrm{dis}$ is independent of the Reynolds number and the pipe wall roughness. A similar approach can be found in Gersten and Papenfuss [9]. This approach is reasonable for moderately high Reynolds numbers ($Re \gt 5\cdot10^4$). Note that for the decomposition, $u_\textrm{fully}$ is taken from a simulation of a long straight pipe, whereas for the reconstruction, $u_\textrm{fully}$ is calculated with the semi-analytical formulation from Gersten [8]. This alteration of the profile is insignificant in regions of strong disturbance (close to the elbows), but eliminates systematic errors which are inherent in simulation results of fully developed pipe flow. Even though the theoretical profile is also an approximation, it is in better agreement with measurement data than simulation results with RANS models, see, e.g. Tawackolian [30].

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To reduce the amount of data, the velocity profiles $\vec{u} = \left(u_x, u_y, u_\textrm{dis} \right)^\textrm T$ can be further decomposed with the method of POD. While POD is extensively used in the analysis of temporal fluctuations in fluid dynamics, see, e.g. Hellstroem et al [11] or Sieber et al [25], it can also be used to reduce the dimensions of a data set, e.g. Brunton and Kutz [2]. To accomplish that, the data matrices need to be reorganized such that the data matrix of each of the $N_\textrm s$ cross sections is flattened as a row vector $\boldsymbol{b_i}, i = 1, \dots, N_\textrm s$. These vectors are treated as snapshots, similar to transient data, and are stored in the rows of the matrix $\boldsymbol{B} = \left[ \boldsymbol{b_1} , \boldsymbol{b_2}, {\ldots} \boldsymbol{b_{N_s}}\right]$. Then, the mean vector $\boldsymbol{\bar{b}}$ over all rows $\boldsymbol{b_i}$ is subtracted to receive the matrix

$$\begin{equation} {X} = \boldsymbol{B} - \boldsymbol{\bar{b}} \end{equation} \tag{ 7 }$$

and its covariance matrix

$$\begin{equation} \boldsymbol{C} = \frac{1}{N_\textrm s-1} \vec{X}^T \vec{X}. \end{equation} \tag{ 8 }$$

Its eigenvalues λ and modes Φ are calculated by an eigendecomposition

$$\begin{equation} \lambda , \boldsymbol{\Phi} = \textrm{eig}(\boldsymbol{C}). \end{equation} \tag{ 9 }$$

The modes are sorted in descending order of their corresponding eigenvalues, so that the first modes contain most of the energy. The coefficient matrix A can be determined by the scalar product of the data matrix $\vec{X}$ and the modes Φ according to

$$\begin{equation} \boldsymbol{A} = \boldsymbol{X} \boldsymbol{\Phi}. \end{equation} \tag{ 10 }$$

The original data matrix B can be approximately reconstructed with

$$\begin{align} {B} \approx \boldsymbol{\bar{b}} + \sum_{i = 1}^{N_\textrm m} \boldsymbol{a_i} \otimes \boldsymbol{\phi_i}, \end{align} \tag{ 11 }$$

where $\boldsymbol{a_i}$ and $\boldsymbol{\phi_i}$ are the ith column of A and Φ, respectively, and $N_\textrm m$ is the number of modes used for the reconstruction. Furthermore, $N_\textrm c = N_u N_r N_\varphi$ is the number of data in one cross section and $N_\textrm s$ the number of sampled planes ('snapshots'). Instead of storing the data matrix $\boldsymbol{B} \in \mathbb{R}^{N_\textrm s \times N_\textrm c}$, only the mean data vector $\bar{b} \in \mathbb{R}^{N_\textrm c}$, the coefficient matrix $\boldsymbol{A} \in \mathbb{R}^{N_\textrm s \times N_\textrm m}$ and the modes $\boldsymbol{\Phi} \in \mathbb{R}^{N_\textrm c \times N_\textrm m}$ need to be stored. If the number of used modes $N_\textrm m$ is small enough, the amount of data is reduced, since $N_\textrm s N_\textrm m + N_\textrm c(N_\textrm m+1) \lt N_\textrm s N_\textrm c$ if $N_\textrm m\lt\frac{N_\textrm s N_\textrm c - N_\textrm c}{N_\textrm s+N_\textrm c}$. In this paper, the POD is not just done for one set of parameters but for all parameter combinations and cases together. Hence, the data matrix B is constructed by stacking the velocity profiles of all cases, resulting in $N_\textrm s = \sum_\textrm{cases} N_{R_\textrm c} N_{d_\textrm l} N_z$.

Contour plots of $u_\textrm{dis}$ and vector plots of the secondary flow motions u_x and u_y of the mean profile $\boldsymbol{\bar{b}}$ and the first eight modes are shown in figure 6. The typical sickle-shaped asymmetry of the velocity profiles can be observed, especially in the mean profile. Also, different flow structures such as single swirl (mean profile and mode 2), typically for the DE case, as well as pairs of counter-rotating swirls, so called Dean vortices [15], (mode 4 and mode 7) can be observed.

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To determine the number of modes that have to be considered, the velocity field u is reconstructed for an increasing number of modes. The L²-error $\mathcal{E}(R_\textrm c,d_\textrm l,z)$ from equation (3) is calculated, with $\varepsilon(R_\textrm c,d_\textrm l,x,y,z) = \Vert\boldsymbol{\tilde{u}_i}(R_\textrm c,d_\textrm l,x,y,z) - \boldsymbol{u}(R_\textrm c,d_\textrm l,x,y,z)\Vert_2$ being the magnitude of the difference of the approximated velocity field $\boldsymbol{\tilde{u}_i}$ ($\tilde{\vec{B}}_i$) reconstructed with i modes and u ($\vec{B}$) the original simulation data. Both the mean and maximum of $\mathcal{E}(R_\textrm c,d_\textrm l,z)$ over the whole data set for each case and over the number of used modes $N_\textrm m$ is shown in figure 7. It can be observed that the errors decline with an increasing number of modes, and that the mean errors are smaller than the maximum errors by a factor of about 10. A total of $N_\textrm m = 400$ modes are chosen as a good balance between accuracy and efficiency. As a result, the maximum error $\mathcal{E}(R_\textrm c,d_\textrm l,z)$ is 0.35% and the mean error of $\mathcal{E}(R_\textrm c,d_\textrm l,z)$ is smaller than 0.025%. This means that the data can be reduced by a factor of 20. In other words, it is possible to reconstruct the entire data with an error of less than 0.4% by storing 5% of the data.

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Predicting the velocity distribution within the three dimensional (physical) space and for arbitrary values in the space of geometry parameters requires an interpolation of the POD coefficients. In the x–y (or r–ϕ) space, the interpolation can be performed within each of the POD modes. To obtain the velocity profile at any spatial position z, curvature radius $R_\textrm c$ and in-between distance $d_\textrm l$, the POD coefficients $\boldsymbol{a_i}$ of each mode (compare equation (11)) need to be interpolated. The high dynamics of $\boldsymbol{a_1}$ for low parameter values can be observed in the contour plots of figure 8. To perform the interpolation, $\boldsymbol{a_i}(R_\textrm c,d_\textrm l,z)$ must be reshaped to restore its original dimensionality $\mathbb{R}^{N_{R_c}\times N_{d_\textrm l} \times N_z}$. This corresponds to a 2D interpolation for the SE and 3D interpolation for the two other cases. Here, a tensor-product spline interpolation implemented in the python package scipy.interpolate called RegularGridInterpolator is used, see Weiser and Zarantonello [33]. This method enables an interpolation on n-dimensional rectangular shaped grids.

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To check the accuracy of the interpolation, a so called cross validation procedure is performed, in which one of the sample positions in $\boldsymbol{a_i}(R_\textrm c,d_\textrm l,z)$ is left out. To maintain the rectangular grid structure of the samples, a whole slice (row/column in 2D) of values is removed. Afterwards, the interpolation is performed with the remaining samples and the velocities $\tilde{\vec{u}}$ are calculated for the previously eliminated parameters. The difference to the original simulation results is calculated as $\varepsilon = \Vert\tilde{\vec{u}} - \vec{u}\Vert_2$ and assessed by means of the L²-error $\mathcal{E}$, compare equation (3). The procedure is repeated consecutively for each value of the sampled parameters except for the upper and lower parameter values, which are always included. Interpolations with spline functions of polynomial degree p = 1 (linear), p = 3 (cubic) and p = 5 (quintic) are compared in figure 9(a), where the mean L²-error over all R_c and d_l , $\mathcal{E}(R_c,d_l)$, is shown for the interpolation at each downstream distance z. For the linear interpolation, errors with a maximum of 2% occur near $z = 0.1 \, D$. For the higher polynomial spline interpolations, the errors are distinctively lower.

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Figure 9(b) shows the mean of $\mathcal{E}(z,d_\textrm l)$ over all z and $d_\textrm l$ at each curvature radius $R_\textrm c$ for the SE, DE and DSE cases. With linear interpolation, the values rise up to 6% at $R_\textrm c\approx 1$, whereas the errors of the cubic spline polynomials are in general slightly higher. Quintic interpolation produces significantly high errors for both the DE and DSE cases.

The mean of $\mathcal{E}(z,R_\textrm c)$ over all z and $R_\textrm c$ is shown at each in-between distance $d_\textrm l$ for the DE and DSE in figure 9(c). Errors with piecewise linear interpolation show a maximum of 4% at $d_\textrm l\approx 1$ for the DE and below 1% for the DSE. For the DE, the cubic interpolation errors are higher than the linear, and the quintic ones are beyond 10% for $d_\textrm l\gt5.5$.

In conclusion, the cross validation study is useful to identify parameter regions worth of refining, as well as for comparing the performance of different interpolation methods. In certain cases, the failure of spline interpolation with higher order polynomials can be attributed to its oscillating nature, which has a tendency to over- and undershoot, especially if the interpolation points are not perfectly smooth. Moreover, the steep gradients concentrated at low values of $R_\textrm c$, z and $d_\textrm l$ as well as the unsteadiness of the sampled data are challenging for any interpolation method. The high interpolation errors are located at positions in proximity to the elbow exit $z\lt 1 \, D$ and small values of $R_\textrm c$, where recirculating flow is present. Overall, the comparison between the spline interpolations with different polynomial degrees yields that piecewise linear interpolation provides the most reliable method with mean errors of less than 6%.

The most common type of ultrasonic flow meter uses the time-of-flight principle. A transducer (A) induces a sonic wave into the pipe, which is detected by another transducer (B) installed further downstream. The time between sending and receiving the signal is measured. This procedure is repeated, whereas the downstream transducer is sending and the upstream transducer is receiving. Since the fluid flow is carrying the sound waves, the measured time is proportional to the mean flow velocity along its traveled path $u^\textrm{path}$. The transducers can be aligned to directly send the sonic waves to each other or by using one or more reflections at the pipe wall. A common configuration with one reflection is the so-called V-path configuration, compare figure 10. If the velocities in the pipe are known, the measurement value of an ultrasonic flow meter can be predicted by evaluating the integral

$$\begin{align} u^\textrm{path} = \frac{1}{L e_z} \left( \sum_{i = 1}^N \int\limits_{0}^{L_i} \boldsymbol{u} \cdot \boldsymbol{e}_i \; \mathrm{d}l_i \right)\,. \end{align} \tag{ 12 }$$

Here, $L = \sum_{i = 1}^N L_i$ denotes the length of the ultrasonic path from one transducer to the other, e _i is the direction vector parallel to the ultrasonic beam, and $e_z = \frac{1}{N} \sum_{i = 1}^N e_{i.z}$ the mean axial share of the direction vectors e _i .

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To determine the volume flow Q of the fluid inside the pipe, the measured path velocity $u^\textrm{path}$ must be multiplied by the area of the pipe cross section A and a calibration factor k that assigns the mean velocity over the path to a mean velocity over the pipe cross section:

$$\begin{align} Q = A \, k \, u^\textrm{path}. \end{align} \tag{ 13 }$$

The calibration factor k is dependent on the velocity profile. For fully developed flow conditions, it only depends on the Reynolds number Re and the wall roughness $k_\textrm s$ as $u_\textrm{fully}$ only depends on r, Re, and $k_\textrm s$ (cp. equation (6)):

$$\begin{align} k_\textrm{fully} = \frac{1}{u_\textrm{fully}^\textrm{path}}. \end{align} \tag{ 14 }$$

For disturbed flow conditions, an additional calibration factor

$$\begin{align} k_d = \frac{u_\textrm{fully}^\textrm{path}}{u_\textrm{fully}^\textrm{path} + u_\textrm{dis}^\textrm{path}}, \quad u_\textrm{fully}^\textrm{path} + u_\textrm{dis}^\textrm{path} \neq 0\, , \end{align} \tag{ 15 }$$

can be introduced. The fluid-specific calibration factor is then calculated as $k = k_\textrm{fully} \, k_d$. In a long straight pipe, k_d is approaching a value of one, as $u_\textrm{dis}^\textrm{path}$ is converging to zero.

To predict the calibration factor k_d for any geometric parameter ($R_\textrm c$ or $d_\textrm l$), interpolation can be performed on the flow meter values directly (instead of interpolating the POD coefficients, see section 3.4). A cross validation study similar to the POD coefficients in section 3.4 is performed for interpolating $u_\textrm{dis}^\textrm{path}$. The mean errors resulting from leaving out one value of $R_\textrm c$ and $d_\textrm l$ are shown in figures 11 and 12, respectively. Similar to the results in section 3.4, the spline interpolations with higher polynomial degrees lead to higher errors, and piecewise linear interpolation is the most reliable. Here, the errors are below 1.2% for interpolating $R_\textrm c$ and below 3% for d_l . Furthermore, the mean errors resulting from leaving out a value of z are below 0.2% and therefore not shown here. Note that maximum errors of more than 60% occur in regions with recirculating flows, i.e. in the immediate vicinity to the elbow exit ($z \lt 1 \,D$), especially for small values of $R_\textrm c$. Therefore, the predicted calibration factors might not be meaningful in this region. Likewise, flow rate measurement in separated flows might not be meaningful as the measurement value could approach zero, which is the denominator in the calibration factor, compare equation (15). In the following, the values of k_d are calculated using the velocities reconstructed from the POD results at the sample positions and be interpolated with piecewise linear splines.

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Measurements with ultrasonic clamp-on flow meters (FLEXIM FLUXUS F721) in V-path configuration have been conducted for the SE and DE with a curvature radius of $R_\textrm c = 1.425\,D$, in-between distances $d_\textrm l \in [0; 3; 4; 6] \, D$, Reynolds numbers $Re = 5\cdot10^4$ and $5\cdot10^5$, downstream distances $z \in [2.45; 5.45; 10.45; 15.45; 22.49; 25.49; 30.49; 35.49] \, D$ and angular orientations of $\varphi \in [18^{\circ}; 63^{\circ}; 108^{\circ}; 153^{\circ}]$. The expanded measurement uncertainty is stated as 1%. For a detailed description of the measurement setup and the experimental determination of the calibration factors k_d , see Straka et al [29].

The surrogate model is evaluated with the parameters present in the measurement setup and compared to the experimental data. Figure 13 shows the calibration factor k_d over the distance z for four different parameter sets. In figure 13(a), the measurement values are in close agreement with the predicted ones, which is also true for $z\lt 22.5\, D$ as shown in figure 13(b). The oscillations of k_d are stemming from the nature of the flow structure. That is, the centered single swirl, causing a rotation of the velocity profile in every downstream position and thus, the value of k_d . In figures 13(c) and (d), the prediction follows the tendency presumed by the experimental data, but misses the exact values.

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The mean of the difference between the predicted $(k_d)$ and measured $(k\,_d^\textrm{meas})$ calibration factors $\delta_1 = k_d - k\,_d^\textrm{meas}$ is −0.77% and the standard deviation is 1.36%. The mean of the absolute difference $\mathord{\left\lvert\delta_1\right\rvert}$ is 1.26% and the maximum absolute deviation is 4.65%. All statistics are summarized in table 3.

Table 3. Statistics of the differences between the predicted and experimental values of k_d , $\delta_1 = k_d - k\,_d^\textrm{meas}$.

Case	SE	DE				All
$d_\textrm l/D$ Re	—	0	3	4	6	—
	Mean of δ1 %
$5 \cdot 10^4$	−1.16	−1.78	−0.77	−0.58	−0.71	−0.77
$5 \cdot 10^5$	−0.58	−0.79	−0.71	−0.40	−0.10
	STD of δ1 in %
$5 \cdot 10^4$	1.40	1.58	1.16	1.14	1.25	1.36
$5 \cdot 10^5$	1.00	1.26	1.39	1.42	1.25
	Maximum of $\mathord{\left\lvert\delta_1\right\rvert}$ in %
$5 \cdot 10^4$	3.86	4.65	2.93	2.96	3.2	4.65
$5 \cdot 10^5$	1.62	3.07	3.20	3.71	3.42

It is well known that for rotating flows, such as downstream of a DE, the strength of the swirl cannot be accurately predicted by RANS turbulence models, compare, e.g. [3, 27]. This becomes even more evident if the inflow conditions upstream of the elbows are disturbed. In consequence, the phase of the profile cannot be predicted and the evaluation of k_d for a specific angle might become biased. To overcome this issue, the angular orientation of the velocity field, respectively the angular orientation of the flow meter, can be taken as random as proposed in earlier works [34, 35]. With this approach, expected values, standard deviations and maximum/minimum bounds of k_d can be determined and used as a pragmatic prediction and estimation of the uncertainty. In figure 14, the angular mean $\overline{k_d} = \frac{1}{N_\varphi}\sum_{i = 1}^{N_\varphi} k_d(\varphi_i)$ with its corresponding standard deviation and maximum/minimum bounds downstream of the SE is compared to the measurement data for two Reynolds numbers. It can be observed that $\overline{k_d}$ is a good estimation for all angular orientations of the measurements, as most of the measurement data coincide within the standard deviation, all within the upper and lower bounds. In figures 15 and 16, the same evaluation is performed for the other parameter sets and plotted beside the measurement data for $Re = 5\cdot10^4$ and $Re = 5\cdot10^5$, respectively. Although the expected (mean) calibration factors follow the trend of the measurement data, they slightly underestimate them. Most data coincides with the maximum/minimum bounds and the uncertainty of the measurement values. An exception is the DE with $d_\textrm l = 0 \, D$ and $Re = 5\cdot 10^4$, see figure 15(a), where the underestimation of the experimental data is more significant.

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Altogether 91.25% of the measurement data including its uncertainty range are inside the standard deviation of the model and 97.2% of the data are between the maximum and minimum bounds. The mean of the difference $\delta_2 = \overline{k_d} - k\,_d^\textrm{meas}$ is −0.78% with a standard deviation of 1.14%, whereas the mean of the absolute difference is 1.06% with a maximum of the absolute deviation of 5%. All statistics are summarized in table 4.

Table 4. Statistics of the differences between the predicted and experimental values of k_d , $\delta_2 = \overline{k}_d - k\,_d^\textrm{meas}$.

Case	SE	DE				All
$d_\textrm l/D$ Re	—	0	3	4	6	—
	Mean of δ2 in %
$5 \cdot 10^4$	−1.03	−1.85	−0.78	−0.63	−0.80	−0.78
$5 \cdot 10^5$	−0.44	−0.86	−0.72	−0.54	−0.19
	STD of δ2 in %
$5 \cdot 10^4$	1.00	1.21	1.18	1.10	1.41	1.14
$5 \cdot 10^5$	0.34	1.07	1.03	0.90	1.01
	Maximum of $\mathord{\left\lvert\delta_2\right\rvert}$ in %
$5 \cdot 10^4$	3.73	4.07	4.07	3.85	5.01	5.01
$5 \cdot 10^5$	1.18	2.40	3.36	3.19	4.18

In all parameter variations, the measurement data at the higher Reynolds number ($5\cdot 10^5)$ are closer to the model predictions. This is remarkable, as the model is based on the simulations with the lower Reynolds number ($5\cdot 10^4$). In general, the Reynolds number dependency of k_d is low.