**2. Results**

The set of two-phase air-water flow experiments that were performed is shown in Figure 4. Flow experiments have been performed for free flow and for flow disturbed by a downstream valve. This downstream ball valve closes in the vertical direction and was for 25% opened in the disturbed flow experiments. Flow regimes were identified for each flow experiment based on the multiphase flow profiles observed through a transparent pipe section. Stratified, wavy, and slug flow regimes were observed during the flow experiments and snapshots of typical gas and liquid phase distributions in these flows are indicated in Figure 2. Based on these visual identifications flow regime transition boundaries could be identified and these are indicated by the solid lines in Figure 4. Dashed lines indicate the gas volume fraction (GVF) of the multiphase flows, i.e., GVF = Qgas/(Qgas + Qliq). Four experiments are highlighted by a black circle. These experiments are discussed in more detail in this article as examples. Video footage is made available in the supplementary information for some example experiments to illustrate dynamic liquid holdup variations occurring in gas-liquid multiphase flow.

**Figure 4.** The measured single phase reference flow rates for all two-phase air-water flow experiments performed in this study indicated as dots. The dashed lines indicate the gas volume fraction (GVF) of the multiphase flows. Flow experiments have been performed for free flow and for flow disturbed by a downstream valve. Approximate flow regime transition boundaries were derived from visual inspection and are indicated by the solid lines. Four experiments that are discussed in more detail in this article are highlighted by a black circle.

In each flow, experiment data were acquired for 30 min using the low-field MR technology-based multiphase flow measurement method that is discussed in detail in Section 4. This measurement method uses a broadband excitation, constant-gradient LFA-CPMG pulse sequence to derive liquid holdup information from 1D liquid distribution images obtained from frequency encoded spin-echo signals, while liquid flow velocity information is derived from the convective amplitude decay of the LFA-CPMG signals with time as induced by the outflow of the excited sample volume. The frequency distribution of each spin-echo that is induced by the gradient field along the vertical direction represents a distribution image along the height of the pipe of the liquid portion of the flow, as air does not give an MR signal. This imaging functionality can be used to determine the multiphase flow profile in a given flow experiment from the combined liquid distribution images acquired during the 30 min of data acquisition.

## *2.1. Liquid Distribution Image Interpretation*

Figure 5 shows the set of liquid distribution images acquired for the four experiments that are marked by a black circle in Figure 4. A surface representation of the liquid distribution images is used in which the images are sorted from the highest to lowest measured holdup to create a smooth surface that is more easily compared between experiments. For a full pipe of water, the liquid distribution image would take on the form of a semicircle and these conditions occur for about 25% of the time in the slug flow experiment (Qgas = 11.5 m3/h and Qliq = 7.3 m3/h) shown in Figure 5a. The remainder of the time corresponds to a steady flow situation in which the pipe is partially filled with a constant liquid fraction. Slug flow can thus be envisioned as a binary flow system with two main events: Short bursts of liquid slugs with *h*liq ≈ 1, and longer events in which gas is accumulated at the top of the pipe and liquid at the bottom. This latter phase is very much comparable to the stratified flow experiment (Qgas = 46.6 m3/h and Qliq = 1.1 m3/h) presented in Figure 5d and is often referred to as the film phase of the slug flow.

**Figure 5.** The set of liquid distribution images acquired in four different flow experiments represented as a surface plot in which liquid distribution images are sorted from the highest to lowest measured holdup. The axis labeled as *z* indicates the height along the flow tube with the pipe axis located at *z* = 0 mm. Each experiment corresponds to a unique multiphase flow profile: (**a**) Slug flow (Qgas = 11.5 m3/h and Qliq = 7.3 m3/h); (**b**) slug flow, disturbed by valve (Qgas = 10.4 m3/h and Qliq = 7.3 m3/h); (**c**) wavy flow (Qgas = 46.1 m3/h and Qliq = 2.6 m3/h); and (**d**) stratified flow (Qgas = 46.6 m3/h and Qliq = 1.1 m3/h).

The slug flow experiment was repeated with the flow disturbed by the downstream valve (Qgas = 10.4 m3/h and Qliq = 7.3 m3/h), in order to induce a more unstable flow profile. Figure 5b shows that although the flow profile can in general still be classified as slug flow, the valve disturbance leads to a considerably altered liquid distribution image surface, especially in the film phase. Although still about 15% of the time slugs with *h*liq ≈ 1 are observed, there is no longer a steady flow situation in the film phase. The film phase consequently has a liquid fraction in the pipe that changes continuously in time. This situation can be compared to the flow experiment labeled as wavy flow (Qgas = 46.1 m3/h and Qliq = 2.6 m3/h) that is presented in Figure 5c. The closing of the downstream valve in this flow experiment thus reduced the fraction of liquid slugs and induced wavy flow in the film phase of the slug flow.

## *2.2. Liquid Holdup and Velocity Correlations*

As mentioned in the introduction, the accurate calculation of the liquid flow rate in multiphase flow comes down to the task of acquiring the correlations between the instantaneous liquid holdup and liquid flow velocity that are characteristic for a given flow profile. Figure 6 shows these correlations as derived from our low-field MR-based

flow measurements for the same four flow experiments as for which the liquid distribution images were presented in Figure 5.

**Figure 6.** Measured liquid flow velocity as a function of measured liquid holdup for the same four flow experiments as for which the liquid distribution images were presented in Figure 5. These experiments correspond to: (**a**) Slug flow (Qgas = 11.5 m3/h and Qliq = 7.3 m3/h); (**b**) slug flow, disturbed by valve (Qgas = 10.4 m3/h and Qliq = 7.3 m3/h); (**c**) wavy flow (Qgas = 46.1 m3/h and Qliq = 2.6 m3/h); and (**d**) stratified flow (Qgas = 46.6 m3/h and Qliq = 1.1 m3/h).

Starting with the simplest case, stratified flow as presented in Figure 6d, a single point correlation is observed. This means that a given liquid holdup is directly related to a given liquid flow velocity. In such cases, the sampling rate and measurement time of the flow measurement method has little influence on the measurement results, as a single measurement already represents a representative sample of the multiphase flow. More structure is visible in the correlation plot for slug flow shown in Figure 6a. The binary character of slug flow is clearly represented by the two main concentrations of data points around *h*liq = 0.3 (film phase) and around *h*liq = 1 (slug phase). Note the higher flow velocity of about 2.5 times in the slug phase of the flow. Recalling that about 25% of the time the flow can be associated with the slug phase, most of the liquid flow is transported by the slug phase. This shows the importance of representative sampling of the flow, as even the minor under sampling of the slug phase may lead to large flow measurement errors.

The disturbed slug flow (Figure 6b) and wavy flow (Figure 6c) experiments exhibit a large spread in the flow velocities that are observed at a given liquid holdup. This spread consequently signals that complex stochastic processes are describing the correlations between the instantaneous liquid holdup and liquid flow velocity. Sufficiently fast sampling is expected to be very important for the accurate measurement of the liquid flow rate for these seemingly chaotic flow profiles. The fact that even for these flows clusters of data points are clearly observable in Figure 6, provides an indication that the statistics of these flows is sufficiently sampled, thus ensuring a representative sampling set of the liquid holdup and flow velocity correlations in the flow.
