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.
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  and Wendt ). 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  and OIML R 49-2:2013 , 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  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 , and Steenbergen and Voskamp . More recent investigations by Müller et al. , Tawackolian , Eichler and Lederer , Graner et al. , and Turiso et al. , 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.  and Turiso et al. . 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 . 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.  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) , ISO 21748 , and ASME V&V 20-2009 ). 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  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  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.  and Han et al. [20,21].
The polynomial chaos method has also found applications to study three-dimensional pipe flow. Schmelter et al.  investigate the influence of skew inflow profiles. The impact of symmetric swirling inflow conditions is considered by Congedo et al. . Furthermore, Weissenbrunner et al.  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  and Xiu and Karniadakis .
where a superposed dot denotes the material time derivative following the velocity is the three-dimensional Laplacian, ρ is the density, ν is the kinematic viscosity, is the eddy-viscosity, and p is the pressure.
Uncertainty Quantification and Simulation Process.
Modeling Error and Validation Uncertainty
where with coverage factor k = 2.0 at the 95% confidence level.
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  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 , and are summarized in Table 2.
Numerical Uncertainty unum.
Additional data are summarized in Table 2.
Input Uncertainty uinput.
The input uncertainty 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 for a large input uncertainty with , 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 standard measurement uncertainties with k = 2 at other downstream locations are computed analogously and summarized in Table 2.
Conclusions From the Validation Uncertainty.
At , the swirl angle has a numerical uncertainty of , an input uncertainty of , and an experimental uncertainty of , resulting in an expanded validation uncertainty of . 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
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 , 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 . 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 . 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 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 , 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 and , 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 . These asymmetric flow patterns become more and more symmetric further downstream. The expectation of the axial velocity is symmetric at 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 downstream. With a maximum standard deviation of approximately 11% at 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 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 . 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 . Finally, to study small input uncertainty, twelve profiles that are nearly fully developed are sampled for X with a narrow, uniform PDF of . 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: , and . 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 in diameter. For a water temperature of 25 °C the Reynolds numbers correspond to flows of approximately and .
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 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 from the asymmetric swirl generator. This suggests that flow disturbance tests with the asymmetric swirl generator are more reproducible for distances at approximately and onward. Similarly, experiments with intermediate input uncertainty show better reproducibility for distances of approximately and onward. Finally, low input uncertainty results in better reproducibility from approximately 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 , and 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 than for . For , 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 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 , all standard deviations are below 5% at 80D and for 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. ). However, for and high input uncertainties, both disturbers show higher standard deviation than for the case. This indicates that the reproducibility is weaker at intermediate Reynolds numbers around . 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 than for flows with .
However, as discussed in Sec. 3, the change of wall boundary conditions across different Reynolds numbers may affect the simulation results significantly, especially at . Furthermore, it is important to take into account that pipe flow with 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 . 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 , 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 , 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. ). 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.