Achieving good reproducibility in fluid flow experiments can be challenging, in particular in scenarios where the experimental boundary conditions are obscure. We use computational uncertainty quantification (UQ) to evaluate the influence of uncertain inflow conditions on the reproducibility of experiments with swirling flow. Using a nonintrusive polynomial chaos method in combination with a computational fluid dynamics (CFD) code, we obtain the expectation and variance of the velocity fields downstream from symmetric and asymmetric swirl disturbance generators. Our results suggest that the flow patterns downstream from the asymmetric swirl disturbance generator are more reproducible than the flow patterns downstream from the symmetric swirl disturbance generator. This confirms that the inherent breaking of symmetry eliminates instability mechanisms in the wake of the disturber, thereby creating more stable swirling patterns that make the experiments more reproducible.

Introduction

Swirling flows are generated in various technical configurations, for example to increase mixing in combustion chambers and furnaces. However, swirling flow may also be a spurious effect, for example in flow measurement applications where an undisturbed fully developed flow profile may be favorable to achieve a small measurement error and good repeatability. Flow sensors are commonly tested with swirling flow conditions to assess their robustness (see, for example, Carlander and Delsing [1] and Wendt [2]). In such experiments, the flow patterns introduced by disturbance generators emulate flow disturbances caused by installation effects often found in pipe configurations, such as bends, junctions, and valves. The flow measurement industry generally uses the standardized swirl disturbance generator (EN ISO 4064-2:2014 [3] and OIML R 49-2:2013 [4], Fig. 1(a)) in tests to determine the impact of nonideal inflow profiles on the measurement error.

The fluid mechanics of swirling flow has been investigated for many decades. In the 50 s, Talbot [5] conducted a theoretical analysis of laminar flow with a weakly swirling velocity profile. Turbulent swirling flow was analyzed in the 90 s, for example, by Kitoh [6], and Steenbergen and Voskamp [7]. More recent investigations by Müller et al. [8], Tawackolian [9], Eichler and Lederer [10], Graner et al. [11], and Turiso et al. [12], focus specifically on swirling flow generated by standardized swirl disturbers used in flow disturbance tests. For a more complete review of swirling flow, we refer to Graner et al. [11] and Turiso et al. [12]. A typical experimental setup of a flow disturbance test is illustrated in Fig. 2. The purpose of such a test is to introduce standardized disturbances in the flow that can be reproduced in different laboratories. Yet, the reproducibility of the disturbance experiment may be affected by many factors, including manufacturing tolerances in the disturber specimens, assembly tolerances, and boundary conditions such as the flow profile upstream from the swirl generator. Experience suggests that in particular the inlet velocity profile upstream from the swirl disturber is expected to influence the downstream swirling patterns. But these upstream boundary conditions are often difficult to assess, control, and reproduce, resulting in poor overall reproducibility of the experiment.

Currently, the asymmetric swirl generator (Fig. 1(b)) is under consideration to replace or complement the standardized (symmetric) swirl generator in the current standards for flow sensor testing [9]. The aim is to standardize more reproducible and realistic tests of installation effects. One possible idea is to introduce an intrinsic symmetry breaking in the flow patterns and thereby make them less susceptible to instabilities due to upstream flow profiles. Turiso et al. [12] show that the asymmetric swirl generator creates disturbances with similar features as the flow downstream from a double-bend out of plane. In consequence, the asymmetric swirl generator is considered to be able to realistically emulate installation effects. However, the performance regarding reproducibility remains unclear.

Uncertainty budgets of experiments usually include contributions due to systematic and random errors from different parameters that impact the measurement outcome. In bottom-up approaches, the different contributions are estimated and combined in a propagation equation (see, for example, guide to the expression of uncertainty in measurement (GUM) [13], ISO 21748 [14], and ASME V&V 20-2009 [15]). Since reproducibility contributions of the specific experimental setup are unknown without conducting a large set of experiments, it is often not possible to include them in bottom-up uncertainty budgets. In this particular example, the combined uncertainty of single-point laser-Doppler velocimetry (LDV) measurements is well established [9,16] and confirmed with experiments [17] for fully developed pipe flow. Yet, for measurements in disturbed conditions, the reproducibility of the flow patterns is unknown. With an alternative top-down approach to assess measurement uncertainty, the variation in experimental outcomes can be observed directly in reproduced experiments. The total variation is usually reported as a standard uncertainty across the reproduced measurement results and naturally includes reproducibility contributions.

There are two main difficulties in investigating the reproducibility experimentally: (1) In practice, it is difficult to separate reproducibility contributions from the swirling flow field from intrinsic reproducibility contributions of the flow sensor under test and (2) to reliably assess reproducibility of such tests, a large set of experiments would be required.

In this paper, we address these two issues by using a computational uncertainty quantification (UQ) approach. Within this framework, we can directly assess the reproducibility of the flow patterns instead of the reproducibility of the sensor reading. Hence, intrinsic reproducibility from the sensor is removed. Further, the virtual experiments allow for efficient testing of a much wider range of boundary conditions without conducting a large set of experiments.

For the reproducibility study, we use a nonintrusive polynomial chaos method coupled with a computational fluid dynamics (CFD) solver. We investigate the spatial development of velocity profiles downstream from the symmetric and asymmetric swirl disturbance generators subject to arbitrary inlet profiles. The variation in the flow patterns downstream from the disturbers as a consequence of the arbitrary inflow conditions gives an objective measure for the expected reproducibility of the fluid flow experiment. The statistical quantities of the velocity field downstream from the disturber are computed by a polynomial chaos method for flows with different Reynolds numbers and different levels of input uncertainty. The variation in the axial velocity and the secondary velocity quantifies the reproducibility of flow disturbance tests. We aim to give insight regarding the suitability of the asymmetric swirl generator to replace or complement the symmetric swirl generator for more reproducible and realistic tests of installation effects. In addition, the results of this computational UQ study can be used to create an overall uncertainty budget of this particular experimental setup that includes the repeatability and reproducibility contributions along with the systematic and random errors of the LDV measurement method. Hence, we establish a practical method to substitute a large set of experiments with a computational UQ approach.

Nonintrusive Polynomial Chaos Method

To analyze the propagation of uncertainty from model inputs to outputs, polynomial chaos methods have been applied to numerous CFD models. Najm [18] provides an overview of early applications including flow in porous media, incompressible and compressible flow, thermodynamics, and reacting flow. Specifically, the influence of uncertain inflow conditions, mostly in two-dimensional cases, is investigated by Ko et al. [19] and Han et al. [20,21].

The polynomial chaos method has also found applications to study three-dimensional pipe flow. Schmelter et al. [22] investigate the influence of skew inflow profiles. The impact of symmetric swirling inflow conditions is considered by Congedo et al. [23]. Furthermore, Weissenbrunner et al. [24] study the influence of a double elbow on flow sensor measurements.

In this paper, we choose a nonintrusive polynomial chaos method, which gives the advantage that the original CFD code can be treated as a black box without code modifications. We briefly review the nonintrusive polynomial chaos method in this section. For a more elaborate explanation, see, for example, Le Maître and Knio [25] and Xiu and Karniadakis [26].

We consider the general framework
$Y=F(X)$
(1)
where $F$ represents the model, X is a vector of n uncertain input parameters, and Y is an uncertain output quantity of interest. The model contains the physics of the underlying problem, often in a simplified mathematical formulation. For example, the model could be a set of algebraic and differential equations. In the present case, the model is constituted by the Reynolds–averaged Navier–Stokes equations (RANS). The concept of the polynomial chaos method is to approximate the probability density function (PDF) of uncertain model parameters with a series expansion of orthogonal polynomials. Each uncertain input parameter of Eq. (1) can be approximated by
$X≈∑i=0paiΩi(ξ)$
(2)
where Ωi are orthogonal polynomial functions of the one-dimensional (1D) random parameter ξ, ai are known spectral mode strengths of the input parameter, i denotes the mode of the polynomial function and p the order of the one-dimensional basis. We choose the type of polynomials according to the PDF of the uncertain input parameters (Table 1). For example, for X sampled from a uniform PDF, Ωi are one-dimensional Legendre polynomials
$Ωi(ξ)={1for i=0ξfor i=112(3ξ2−1)for i=2⋯$
(3)
where ξ is a standardized uniform random parameter with minimum −1 and maximum 1: $U[−1,1]$. An expansion of a uniform PDF with Legendre polynomials up to order p = 1 is exact (Karniadakis et al. [27]) such that
$X=x0+x1ξ$
(4)
where $x0−x1$ is the minimum and $x0+x1$ is the maximum of the PDF. Analogous to Eq. (2), each uncertain output quantity of the model (1) can be approximated by an expansion. In general, the uncertain output quantities depend on all uncertain input parameters, yielding the expansion
$Y≈∑i=0PbiΨi(ξ)$
(5)
where bi are the unknown spectral mode strengths of the output quantity, P is the order of the n-dimensional polynomial basis, and $Ψi$ are orthogonal polynomial functions of the n-dimensional vector of random input parameters $ξ$. In general, the type of output distribution is unknown. Hence, a high enough order of polynomials is required to provide reasonable approximations. The number of expansion terms in Eq. (5) is determined by
$P+1=(p+n)!p!n!$
(6)
Note that the model output is expanded on an n-dimensional basis and will generally depend on all uncertain input parameters. In this paper, we consider a special case with only one uncertain input parameter such that n = 1 and, hence p = P. In this special case, the expansion of an output quantity reduces to the one-dimensional polynomials such that
$Ψi(ξ)=Ωi(ξ)$
(7)
In view of Eq. (7), the expansion in Eq. (5) reduces to
$Y≈∑i=0PbiΩi(ξ)$
(8)
The spectral mode strengths bi of Eq. (8) are unknown. The model output Y is evaluated numerically at a discrete set of collocation points. P + 1 samples of the random input parameter ξ are generated according to the sampling strategy of choice. For each sample ξj, the model output is evaluated through a CFD simulation such that
$Yj=F(Xj)$
(9)
and, hence
$Yj≈∑i=0PbiΩi(ξj)$
(10)
Note that Eq. (10) is the discrete counterpart of Eq. (8) evaluated at the collocation points. Rearranging Eq. (10) yields a linear system
$(Ω0(ξ0)Ω1(ξ0)⋯ΩP(ξ0)Ω0(ξ1)Ω1(ξ1)⋯ΩP(ξ1)⋮⋮⋮Ω0(ξP)Ω1(ξP)⋯ΩP(ξP))(b0b1⋮bP)=(Y0Y1⋮YP)$
(11)
with P + 1 unknown spectral mode strengths $b0,…,bP$. The right-hand side vector contains the model outputs from Eq. (9). After solving Eq. (11) for the spectral mode strengths, the mean μ and variance $σ2$ of the PDF of Y are computed following Kewlani et al. [28]
$μ=b0$
(12)

$σ2=∑i=1Pbi2$
(13)

Numerical Simulations

The model $F$ in the present study is the stationary, incompressible RANS equation (see, for example, Pope [29])
$u˙=(ν+νT)△u−1ρgrad pdivu=0}$
(14)

where a superposed dot denotes the material time derivative following the velocity $u=(u,v,w)⊤, △$ is the three-dimensional Laplacian, ρ is the density, ν is the kinematic viscosity, $νT$ is the eddy-viscosity, and p is the pressure.

We simulate the flow at three different Reynolds numbers: $Re=4×102, Re=4×103$, and $Re=4×104$, with the Reynolds number
$Re=wvolDν$
(15)
based on the pipe diameter D and the volumetric velocity $wvol=4Q/πD2$, where Q is the volumetric flow rate. The open-source CFD solver OpenFOAM is used to perform the numerical simulations. The RANS equations are solved along with the k-ω SST turbulence model [30], which was previously found to provide the best agreement between simulations and experimental results of swirling flows [31,32]. The k-ω SST model is a combination of the k-ϵ model for regions outside the boundary layer and the k-ω model for regions close to the boundary layer. Blending functions are used to switch between the k-ϵ and k-ω transport equations and the associated constants. The k-ω SST model implementation includes a wall function treatment that is dependent on the y+ values and, hence, the grid resolution close to the wall. This ensures flexibility such that the model including the wall functions can be used for a wide range of y+ values. For the uncertainty quantification, we use a structured mesh of approximately 1.3 × 106 cells as shown in Fig. 3. (Note that different mesh sizes are used in Sec. 4 for assessing mesh convergence.) The mesh is created with the tool Pointwise. In the present case, at $Re=4×102$ ($y+<1$) and $Re=4×103$ ($y+≈2$), the gradients near the wall region are resolved and the wall function is switched off. Note that although the nominal pipe Re number is low, there are still turbulent regions to be expected in the swirl disturber and the associated wake. Hence, the SST model is used for all nominal Reynolds numbers. For $Re=4×104$, the flow is turbulent with $y+≈17$ resulting in the use of wall functions. Besides the available references pointing toward reasonable performance, our choice for the k-ω SST model is also driven by the motivation to use a well-established standard model that reflects current simulation practice, also in industry. Yet, it is reasonable to expect that the k-ω SST model as a semi-empirical turbulence model that was not necessarily tailored for swirling flows will include potentially significant modeling errors. As discussed by Jakirlić et al. [33], various standard assumptions in typical RANS closures are not expected to hold for swirling flows. The resulting modeling errors will be included in all simulation results and are addressed in Sec. 4 following the framework of the ASME V&V 20-2009 [15] standard.

Uncertainty Quantification and Simulation Process.

We investigate how uncertain flow conditions upstream from the symmetric and asymmetric swirl disturbance generator affect the downstream outcome. The setup of the UQ and simulation process is shown in Fig. 4. We start with a baseline simulation of the asymmetric swirl disturbance generator with a fully developed inflow obtained from separate simulations of a straight pipe for the different Reynolds numbers. The velocity profiles downstream from the asymmetric swirl disturber have an axially asymmetric shape and a swirl decreasing in strength further downstream. These profiles are used as a library of arbitrary inflow profiles to create inlet boundary conditions for the CFD simulations, as illustrated in Fig. 4. For an UQ with polynomials up to order P, the flow profile is sampled at P + 1 arbitrary downstream distances and used to define the inflow conditions of the CFD simulations. Notice that boundary conditions could be sampled from alternative baseline cases or from an arbitrary selection or artificially generated flow profiles. The advantage of sampling from a baseline simulation case is to ensure that a large range of flow conditions is captured by the sampling. The downstream distance of the sampling plane is defined as uncertain input parameter with a uniform PDF. The discrete sampling locations are determined by the efficient collocation method introduced by Isukapalli [34]. In contrast to Monte Carlo methods, this deterministic sampling method draws collocation points that cover the parameter domain in a more uniform way and thereby minimizes bias when using only a small set of collocation points. The variation in the flow patterns downstream from the symmetric or asymmetric swirl disturbance generator gives a measure for the reproducibility of the experiment. The PDFs of the uncertain model outputs such as velocity are approximated by the orthogonal Legendre polynomials (Table 1). Two polynomial chaos studies are performed to investigate clockwise rotating inflow profiles as well as counter-clockwise rotating inflow profiles. The second study (A2) is a mirror image of the first study (A1). The orientation is defined as discrete uncertain input parameter. Following Schmelter et al. [22], the results of the two studies are combined by computing the expectation μ and variance $σ2$ with
$μ(A)=12(μ(A1)+μ(A2))$
(16)

$σ2(A)=12(σ2(A1)+σ2(A2))+14(μ(A1)−μ(A2))2$
(17)

Modeling Error and Validation Uncertainty

The main objective of the present paper is to estimate the reproducibility of swirling flow patterns. This is done with a polynomial chaos method accounting for variation in input parameters. This method requires much less individual simulations than a Monte Carlo approach, but still, a significant set of simulations for different input parameters is required. In consequence, the computational resources for a single simulation needs to be kept at a reasonable level. At the same time, it is desirable to minimize the numerical error by using sufficiently fine grids. Most importantly, the numerical error needs to be quantified to find a reasonable trade-off between simulation time and accuracy required to reach the objectives of the simulations. It is also important to note that the objectives of the simulation might differ from application to application and set different requirements on the uncertainties. In particular, the objective of the present study is not to obtain the most accurate model, but rather to quantify reproducibility characteristics that are very hard to obtain in actual experiments. This objective will be reflected in the discussion of the uncertainties below. To put the procedure into perspective, we follow the ASME V&V 20-2009 [15] framework to assess all contributions to the validation uncertainty. The validation uncertainty
$uval=unum2+uinput2+udata2$
(18)
contains contributions from the numerical uncertainty $unum$ of the simulations, contributions due to variation in input parameters $uinput$, and contributions due to the uncertainty of the experimental data $udata$. The validation comparison error E is defined as
$E=S−De$
(19)
where S denotes the CFD results and De is experimental data. Assuming a Gaussian distribution, the model error $δmodel$ falls within the interval
$δmodel∈[E−Uval,E+Uval]$
(20)

where $Uval=kuval$ with coverage factor k = 2.0 at the 95% confidence level.

For the validation procedure, we choose the baseline model with flow at $Re=4·104$ downstream from a symmetric swirl disturber with fully developed inflow conditions. We evaluate all contributions in Eq. (18) for the swirl angle
$φ=arctan|v¯xy|maxwvol$
(21)
where $|v¯xy|max$ is the maximum of the magnitude of the secondary velocity
$v¯xy=u¯2+v¯2$
(22)

The swirl angle (21) is a commonly used integral metric to measure the intensity of swirling flow [8,10,11,35]. Experimental data from LDV measurements [11] in our laboratory are used as reference results to determine the comparison error (19). A comparison of the downstream decay of the swirl angle for different mesh resolutions with experimental measurements is shown in Fig. 5(a). All contributions to the validation uncertainty of the swirl angle at downstream distances $z=10D, z=50D$, and $z=105D$ are summarized in Table 2.

Numerical Uncertainty unum.

The numerical uncertainty $unum$ of the swirl angle is estimated with the least squares version [36,37] of the grid convergence index [38]. This approach is considered the most robust method available for prediction of the numerical uncertainty [15]. The method proved suitable for reliable uncertainty estimates of simple problems; however, uncertainty estimation of more complex turbulent flows can introduce challenges [39,40] including scatter and nonmonotonic convergence with grid refinement. The grid convergence of the swirl angle at downstream distance $z=10D$ is shown in Fig. 5(b). The representative cell size h is the average edge length of a cell in the mesh, where the finest mesh corresponds to h1 and the coarsest mesh to h5. By Richardson extrapolation (RE), the assumed one-term expansion of the discretization error $δRE$ is
$δRE=φi−φext=αhiq$
(23)
where $φext$ is the estimated exact solution, α is a constant, and q is the observed order of convergence. The relative residual error of the order $10−4$ for the velocity is considered negligible in the computation of the numerical uncertainty. With k = 2.0, the standard numerical uncertainty of the swirl angle at $z=10D$ for the coarsest mesh of 1.3 × 106 cells (h5) is
$unum,φ=±2.06 deg$
(24)

Additional data are summarized in Table 2.

Input Uncertainty uinput.

The input uncertainty $uinput$ for the swirl angle is evaluated by the polynomial chaos method and is a measure for reproducibility. For a conservative estimate in the validation procedure, we compute $uinput$ for a large input uncertainty with $X∈[5D,105D]$, as summarized in Table 2. The simulation process and level of input uncertainty for the polynomial chaos method are discussed in more detail in Sec. 3.1.

Uncertainty of the Experimental Data udata.

The uncertainty of the test result $udata$ used for validation includes random and systematic errors from the experiments. The measurement uncertainty in the swirl angle depends on the measurement uncertainty in the local single-point LDV measurements of the maximum of the secondary flow $|v¯xy|max$. For a detailed uncertainty budget of LDV measurements, we refer to Tawackolian [9] and Büker [16]. LDV measurements of axial velocity components can reach uncertainties of approximately 0.5% for fully developed, undisturbed flows [9,16,17]. Yet, the measurement of disturbed flows with radial components is associated with much higher uncertainties, mainly due to geometrical tolerances, lower signal, and, hence, a larger statistical uncertainty. Additionally, the repeatability and reproducibility of the experiment itself, for example due to boundary conditions is usually not included in such uncertainty budgets, as discussed by Hinz [17]. In view of existing uncertainty budgets accounting for all uncertainty contributions through a propagation equation, it is reasonable to assume an uncertainty of approximately $10.0 %$ for $|v¯xy|max$ such that
$Udata,|v¯xy|max≈10 %|v¯xy|max$
(25)
A calculation propagating the uncertainty through the definition in Eq. (21) yields
$Udata=11+(|v¯xy|maxwvol)2Udata,|v¯xy|maxwvol360deg2π$
(26)
With $wvol=2.6794 m/s$ and $|v¯xy|max=0.7170 m/s$, the experimental measurement uncertainty for the swirl angle at $z=10.0D$ downstream becomes
$udata,φ=±0.72deg$
(27)

The standard measurement uncertainties with k = 2 at other downstream locations are computed analogously and summarized in Table 2.

Conclusions From the Validation Uncertainty.

At $z=10D$, the swirl angle has a numerical uncertainty of $±2.06 deg$, an input uncertainty of $±4.21 deg$, and an experimental uncertainty of $±0.72 deg$, resulting in an expanded validation uncertainty of $±9.48 deg$. Since the input uncertainty is approximately twice as large as the numerical uncertainty, the choice of the coarsest mesh with approximately 1.3 × 106 cells is sufficient for the present purpose. Further, the modeling error is contained within the interval (20). In conclusion, the uncertainty interval of the modeling error is much larger than the actual comparison error. Formulating model improvements based on the present expanded validation uncertainty would not be possible because the validation uncertainty is too large to point toward a specific reliable value of the modeling error. To make more precise statements on the modeling error, mainly the numerical uncertainty and the input uncertainty would have to be reduced to get a lower noise level. On one hand, the numerical uncertainty could be minimized by choosing a finer mesh. However, this would result in significantly increased simulation times. On the other hand, it is important to note that the expanded validation uncertainty is significant mainly due to the large contribution of the input uncertainty. This confirms the importance of the scope of the present study.

Convergence With Polynomial Order

Besides the validation within the ASME V&V 20-2009 [15] framework, we also validate the convergence of the statistics with increasing polynomial order. The polynomials are an approximation to an unknown exact PDF. Hence, it is reasonable to expect that the level of approximation depends on the order of the polynomial chaos method. To determine which polynomial order is sufficient to represent the PDFs of the uncertain input and output parameters, we study the convergence of cases with 3, 6, 9, and 12 collocation points, corresponding to probability distributions of order 2, 5, 8, and 11, respectively. The random input parameter has a uniform PDF with $X∈[5D,105D]$. We run the polynomial chaos method in the UQ postprocessing as illustrated in Fig. 4 to compute the expectation and the variance of the axial velocity $w¯$ and the magnitude of the secondary velocity (22) at different cross sections throughout the pipe. Each cross section consists of approximately $N=105$ data points. To have an integral measure at each cross section, the averaged standard deviation per cross section is computed as
$σ¯=1N∑i=1Nσi$
(28)

The downstream development of the standard deviation of the axial and the secondary velocity is shown in Fig. 6. In general, the standard deviation of the axial and secondary velocity increases with the number of collocation points. Considering the downstream location at $z=50D$, the standard deviation of the axial velocity increases from 7% for three collocation points to 16% for 12 collocation points. Similarly, the standard deviation of the secondary velocity increases from 1.2% for 3 collocation points to 2.4% for 12 collocation points. Importantly, the difference between the solutions with 9 and 12 collocation points gives an indication on the level of accuracy of the predicted standard deviation. The standard deviation of the axial velocity increases by 2% going from 9 to 12 collocation points. This indicates that the solution with 12 collocation points provides an approximation of the variance that is accurate to approximately 2%. Since the solution will also contain numerical errors and modeling errors that are larger than 2% as shown in Sec. 4, a polynomial chaos method with 12 collocation points provides suitable results for the present study. All results presented in the remainder of the paper are computed with an order 11 polynomial chaos method using 12 collocation points. Finally, it should be noted that the numerical solution of Eq. (11) contains round-off errors on the order of machine precision that can be neglected.

Expectation and Variance in the Wake of the Swirl Generator

First, we explore the development of the velocity fields downstream from the symmetric and asymmetric swirl generators with $Re=4×104$. The expectations of the axial velocity at several downstream distances are shown in Fig. 7. The characteristic geometries of the disturbers are recognizable in the velocity patterns at $z=1D$. The asymmetric disturber produces a region of high variance that coincides with the location of the half-open section of the disturber. This region, with a maximum standard deviation of 70%, contributes to an elevated averaged standard deviation of 21% at $z=1D$ downstream from the asymmetric swirl generator. In contrast, the symmetric swirl generator exhibits an averaged standard deviation of 15% at the same cross section. This indicates that at $z=1D$, the axial velocity component is more reproducible for tests with the symmetric swirl generator. The better reproducibility may be caused by a constriction effect. The symmetric swirl generator constricts the flow more than the asymmetric swirl generator. This constriction results in a very stable, well-conditioned flow field that is not susceptible to small variations in the boundary conditions.

Further downstream, at $z=20D$ and $z=50D$, both swirl disturbers provide asymmetric expectations. While the asymmetric disturber provides inherently asymmetric flow patterns, the patterns from the symmetric disturber are initially symmetric and asymmetric patterns only emerge at around $z=20D$. These asymmetric flow patterns become more and more symmetric further downstream. The expectation of the axial velocity is symmetric at $z=80D$ downstream from both swirl generators. Nevertheless, the variance in Fig. 7(o) and 7(p) indicates that the velocity field is asymmetric in individual realizations of the experiments. Figures 7(o) and 7(p) show that both cases exhibit high standard deviations near the walls and a minimum standard deviation at the pipe center. The maximum axial velocity is off-center for the individual realizations of the experiment. The resulting skewed velocity profiles cause a high standard deviation of the axial velocity near the wall of the pipe. Yet, the symmetric disturber provides much higher standard deviations at $z=80D$ downstream. With a maximum standard deviation of approximately 11% at $z=80D$ downstream, an individual realization of an experiment with the symmetric disturber is much more likely to provide skewed profiles than the same experiment with an asymmetric swirl disturber. Overall, the velocity field at $z=80D$ downstream from the symmetric swirl generator has a higher variance, and hence is less reproducible than the counterpart downstream from the asymmetric swirl generator.

Reproducibility of Flow Disturbance Tests

To reach a more detailed conclusion on the reproducibility of these experiments, we investigate the dependency of the standard deviation on two parameters: (1) the level of input uncertainty and (2) the Reynolds number of the flow. For this purpose, we change the width of the uniform PDF of the input parameter X by adjusting the sampling range of X. The results discussed in Sec. 6 are based on a large input uncertainty where the twelve collocation points are sampled from a wide, uniform PDF of $X∈[5D,105D]$. These input conditions cover the entire range of flow profiles from very skewed with swirling components to fully developed. The effect of intermediate input uncertainty is emulated by sampling twelve profiles for X with $X∈[55D,105D]$. Finally, to study small input uncertainty, twelve profiles that are nearly fully developed are sampled for X with a narrow, uniform PDF of $X∈[80D,105D]$. The input uncertainties in the axial velocity of the input profiles of the different studies are summarized in Table 3. From an experimental perspective, the level of input uncertainty describes how well the boundary conditions of an experiment are known and controllable in consecutive realizations of the experiment on different equipment or in different laboratories. In addition, each study is performed for three different Reynolds numbers: $Re=4×102, Re=4×103$, and $Re=4×104$. These Reynolds numbers are chosen based on typical test points for water, heat, and cooling meters with rated dynamic range R100. For example, a meter with a G3/4B threaded connection will typically be tested in pipes with approximately $15 mm$ in diameter. For a water temperature of 25 °C the Reynolds numbers correspond to flows of approximately $15 l/h, 150 l/h,$ and $1500 l/h$.

Dependency on the Input Uncertainty.

The downstream development of the standard deviation of the axial velocity at different Reynolds numbers and different input uncertainties is shown in Fig. 8. In general, the standard deviation vanishes at a long enough downstream distance. This indicates that, as disturbances relax toward a fully developed profile, the variations also relax and become independent of the input uncertainty.

The symmetric and asymmetric swirl disturbers exhibit some distinctive mechanisms of how the averaged standard deviation of their flow patterns evolves downstream. The symmetric swirl disturber starts out with a low standard deviation very close to the disturber followed by a steep increase to reach a peak in standard deviation at around 5D to 10D downstream. For the highest input uncertainty, there is a second peak located at around 50D. In contrast, the asymmetric swirl disturber starts out with a higher standard deviation very close to the disturber. For low and intermediate input uncertainties, there are a few small oscillations in the region around 5D to 20D, but no pronounced peaks. Hence, for low and intermediate input uncertainties, the standard deviation decreases almost monotonically for the asymmetric swirl disturber. In contrast, the symmetric swirl generator starts low, increases, and decreases again. This indicates that the velocity field becomes unstable through an instability mechanism that kicks in at around 10D downstream. The presence of a second peak suggests that there might be several instability mechanisms or complex nonlinear interactions causing large standard deviations at different locations in the wake.

For the symmetric swirl disturber, the cases with intermediate and low input uncertainty show very similar standard deviations. This suggests that the standard deviation saturates at this level and a further decrease in input uncertainty does not result in improved reproducibility of the experiments. From an experimental perspective, this means that even small deviations from reference boundary conditions will result in significant spread in the wake of the symmetric swirl disturber. Hence, the experimental boundary conditions can only be improved up to a certain threshold where further improvements will not yield better reproducibility. In contrast, the asymmetric swirl disturber shows a significant decrease in standard deviation between the three input uncertainty levels. In consequence, the asymmetric swirl disturber provides superior reproducibility in particular for small input uncertainties.

Figure 9 shows the standard deviation of the axial velocity field at downstream distance $z=50D$ for different levels of input uncertainty. For large input uncertainty, both disturbers exhibit peaks of high standard deviation along the pipe walls. These peaks attain higher values across larger regions for the asymmetric swirl disturber. The intermediate input uncertainty, and the differences between symmetric and asymmetric swirl disturber become even more pronounced. In the wake of the symmetric swirl disturber, patches of very high standard deviation cover almost the entire cross section whereas the asymmetric disturber shows an overall lower standard deviation. For low input uncertainty, the standard deviation in the wake of the asymmetric swirl disturber decreases to values below 10%. In contrast, the symmetric swirl disturber exhibits patches of standard deviation above 20%, even for low input uncertainty.

In conclusion, a smaller input uncertainty increases the reproducibility of the velocity close to the walls, either by a decrease in maximum standard deviation, a decrease in the region along the wall with relatively high standard deviation, or a combination of the two. This mechanism is more pronounced for the asymmetric swirl disturber, which confirms that these experiments are expected to be more reproducible.

By comparing Figs. 8(a) and 8(b), we obtain the location from which the flow downstream from the asymmetric swirl generator is more reproducible than the flow downstream from the symmetric swirl generator. For experiments with a large input uncertainty, the standard deviation of the axial and the secondary velocity is lower at downstream locations $z>5D$ from the asymmetric swirl generator. This suggests that flow disturbance tests with the asymmetric swirl generator are more reproducible for distances at approximately $z>5D$ and onward. Similarly, experiments with intermediate input uncertainty show better reproducibility for distances of approximately $z>10D$ and onward. Finally, low input uncertainty results in better reproducibility from approximately $z>15D$ and onward. The secondary flow shown in Figs. 10(a) and 10(b) shows analogous reproducibility.

Overall, experiments with the asymmetric swirl disturber show superior reproducibility, in particular with large input uncertainty. This suggests that the symmetry breaking in the asymmetric swirl disturber makes the flow patterns in the wake less susceptible to perturbations from input uncertainties. In practice, that means that tests with the asymmetric swirl disturber have clear benefits in particular in scenarios where the experimental boundary conditions are unknown or difficult to control.

Dependency on the Reynolds Number.

The standard deviation of the axial velocity for flows with Reynolds numbers $Re=4×104, Re=4×103$, and $Re=4×102$ at different levels of input uncertainty is shown in Fig. 8. The influence of the level of input uncertainty on the reproducibility is more pronounced for $Re=4×103$ than for $Re=4×104$. For $Re=4×103$, the asymmetric swirl disturber has consistently lower standard deviation for all levels of input uncertainty. In particular, at intermediate input uncertainty, the asymmetric swirl disturber shows values below 5% in standard deviation, whereas the symmetric swirl disturber still shows significant spread. This indicates that the reproducibility of experiments with the asymmetric swirl generator is significantly better. For example, at $z=20D$ and intermediate input uncertainty, the standard deviation of the axial flow is reduced from 17% to 3% by application of the asymmetric swirl generator instead of the symmetric swirl generator.

In all cases, the standard deviation decays faster for smaller Reynolds numbers. For $Re=4×103$, all standard deviations are below 5% at 80D and for $Re=4×102$ all standard deviations are below 5% at 20D. This is consistent with disturbances decaying faster for lower Reynolds numbers (see, for example, Graner et al. [11]). However, for $Re=4×103$ and high input uncertainties, both disturbers show higher standard deviation than for the $Re=4×104$ case. This indicates that the reproducibility is weaker at intermediate Reynolds numbers around $Re=4×103$. This example illustrates that the reproducibility depends very much on the Reynolds number and the application of the asymmetric swirl generator instead of the symmetric swirl generator is more effective in improving the reproducibility of flow disturbance tests for flows with $Re=4×103$ than for flows with $Re=4×104$.

However, as discussed in Sec. 3, the change of wall boundary conditions across different Reynolds numbers may affect the simulation results significantly, especially at $Re=4×104$. Furthermore, it is important to take into account that pipe flow with $Re=4×103$ is nominally in the transition region where various phenomena are known to impact flow patterns. Mechanisms include turbulent puffs and slugs and coexistence of laminar and turbulent regions. Yet, the swirl disturbance generator is a rather large perturbation to the pipe flow producing a turbulent wake, even at $Re=4×103$. As discussed in Sec. 3, the standard RANS model used in this study does not capture any phenomena associated with transitional flow. Further, the k-ω SST turbulence model is calibrated for fully turbulent flows with high Reynolds numbers which might result in erroneous behavior at the lower end of Reynolds numbers. Hence, the actual reproducibility behavior of experiments with transitional Reynolds numbers might be much more complex than these RANS simulations.

In the laminar case with $Re= 4×102$, both swirl disturbers show similar performance. For intermediate and small input uncertainties, the input profiles of the individual simulations are close to fully developed and hence the variation between the outputs is negligible. This results in standard deviations below 1%. Only a large input uncertainty results in significant standard deviation close to the swirl disturbers. For these cases, the asymmetric swirl disturber provides slightly lower standard deviations. Hence, experiments with low Reynolds number are expected to show weaker reproducibility for experiments very close to the swirl generator and experiments with the asymmetric disturber are expected to be slightly more reproducible.

Summary and Discussion

In this paper, we investigated the reproducibility of flow disturbance tests with symmetric and asymmetric swirl generators that are commonly used in flow sensor testing. We use a nonintrusive polynomial chaos method in combination with a CFD code to obtain the expectation and variance of the flow field downstream from the disturber. Through this computational approach, conclusions on the reproducibility of flow patterns and, hence, the reproducibility of an experimental setup can be reached without a large set of expensive experiments. A convergence study of the polynomial order shows that 12 collocation points are sufficient to predict the standard deviation of the flow field with a convergence level of 2%. Our study of this practically relevant example illustrates how computational UQ can be applied to complement experimental investigations to create a stronger and more meaningful connection between simulations and experiments.

Our results suggest that experiments with the asymmetric swirl generator are more reproducible than experiments with the symmetric swirl generator. The reproducibility of the velocity patterns downstream from the asymmetric swirl generator is better over almost the entire range of downstream locations. An exception are the flow patterns directly downstream from the disturber. At $z=1D$, the symmetric swirl generator provides better reproducibility. This is due to a constriction effect resulting in stable velocity profiles that are not affected by variations in the boundary conditions.

The asymmetric swirl disturber was also found to produce realistic disturbances that are close to real installation effects (see, for example, Turiso et al. [12]). In view of the present results on reproducibility, the asymmetric swirl generator is a good candidate for more reproducible and more realistic standardized flow disturbance tests at flow laboratories. However, an additional modification of the present standard should be taken into account. In the presently established method of standardized testing, the symmetric swirl disturber is directly connected to the meter under test. Hence, there is almost no downstream distance. Our present results suggest that the use of the asymmetric swirl disturber as an alternative with better reproducibility only makes sense in combination with an increased downstream distance for standardized testing.

Practical Relevance and Implications for Experiments.

The present study illustrates how a computational uncertainty quantification framework can be used to provide valuable insight regarding an experimental setup without actually performing a large set of experiments. The outlined method can be applied as a tool that could prove valuable in industry. As discussed in Sec. 4.3, reproducibility contributions from lack of knowledge in the boundary conditions are typically not included in a classical bottom-up measurement uncertainty quantification. In particular for standardized tests in type approval evaluations, it is extremely important to choose experimental setups that provide sufficient reproducibility for meaningful comparisons. Yet, it is often impossible to assess reproducibility before actually building an experimental setup and performing a significant amount of experiments. The method used in the present study provides a practically minded pathway toward getting insight on quantities of interest that are inaccessible in experiments. The outlined uncertainty quantification process can be applied in the design phase of experimental setups to achieve superior reproducibility of the experiments by complementing classical measurement uncertainty budgets.

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