2 MR velocimetry

The principles of MR velocimetry have been covered in detail by Callaghan (Reference Callaghan2011). For a review on the applications of MR velocimetry to fluid mechanics problems in particular or to industrially important systems in general, the reader is referred to Fukushima (Reference Fukushima1999) and Gladden et al. (Reference Gladden, Akpa, Anadon, Heras, Holland, Mantle, Matthews, Mueller, Sains and Sederman2006), respectively. Unlike conventional velocity measuring techniques of laser Doppler velocimetry (LDV) or anemometry (LDA), hot-wire anemometry (HWA) and Pitot tubes, MR velocimetry is able to measure velocities at different spatial locations simultaneously and is routinely used to measure all three components of the flow field in two-dimensional slice images, which can be acquired in any direction, as well as three-dimensional volume images. MR velocimetry is non-invasive in the sense that no measuring devices or tracer particles are inserted in the flow, as is the case with the most prominent technique of particle image velocimetry (PIV) or with Doppler ultrasound velocimetry (DUV). MR velocimetry can be used to study optically opaque systems, such as the velocity inside monolith channels, and gives chemically selective information. The main disadvantages include the inability to study ferromagnetic materials. The historical limitations of MR velocimetry in terms of spatial and temporal resolution have been significantly reduced by advances in hardware, development of ultrafast imaging techniques (Ahn, Kim & Cho Reference Ahn, Kim and Cho1986; Hennig Reference Hennig1986) and acquisition methods (Candés, Romberg & Tao Reference Candés, Romberg and Tao2006; Donoho Reference Donoho2006). It is now possible to acquire sub-millimetre resolution two-dimensional MR velocity images in ${<}$ 10 ms (Gladden & Sederman Reference Gladden and Sederman2013). Applications of MR velocimetry to fluid mechanics problems include the study of flow past obstructions and in complex geometries (Xia, Callaghan & Jeffrey Reference Xia, Callaghan and Jeffrey1992; Elkins et al. Reference Elkins, Markl, Iyengar, Wicker and Eaton2004; Newling et al. Reference Newling, Poirier, Zhi, Rioux, Coristine, Roach and Balcom2004; Mullin et al. Reference Mullin, Seddon, Mantle and Sederman2009), as well as multi-phase flow (Tayler et al. Reference Tayler, Holland, Sederman and Gladden2012, Reference Tayler, Benning, Sederman, Holland and Gladden2014; Boyce et al. Reference Boyce, Rice, Ozel, Davidson, Sederman, Gladden, Sundaresan, Dennis and Holland2016).

In its simplest form, MR velocimetry is a combination of an MR imaging technique with a method of encoding velocity in the pixels of the image. MR imaging consists of applying a sequence of radio frequency (RF) pulses and imaging magnetic field gradients, $\boldsymbol{G}_{im}$ , which encode the spatial resolution in the MR signal (i.e. the image) such that the acquired signal is given by:

(2.1) $$\begin{eqnarray}S(t)=\iiint M(\boldsymbol{r})\text{exp}[\text{i}\unicode[STIX]{x1D6FE}\boldsymbol{G}_{im}\boldsymbol{\cdot }\boldsymbol{r}t]\,\text{d}\boldsymbol{r},\end{eqnarray}$$

where $\boldsymbol{r}$ is the position vector, $M(\boldsymbol{r})$ is the magnitude image, $\unicode[STIX]{x1D6FE}$ is the gyromagnetic ratio of the nucleus being imaged and the integration is performed over all space. In the present work, and in the majority of studies, the nucleus observed is $^{1}\text{H}$ . With the definition of an inverse space variable, $\boldsymbol{k}=(2\unicode[STIX]{x03C0})^{-1}\unicode[STIX]{x1D6FE}\boldsymbol{G}_{im}t$ , $S(\boldsymbol{k})$ and $M(\boldsymbol{r})$ become a Fourier pair, and the magnitude image can be obtained by Fourier transformation of the signal acquired.

Velocity in a given direction is encoded in the phase of the image by a applying a second set of magnetic field gradients, $\boldsymbol{G}_{vel}$ , in the direction of the velocity. The phase imparted due to the velocity-encoding gradients is given by:

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FE}\left(\boldsymbol{r}\boldsymbol{\cdot }\int \boldsymbol{G}_{vel}(t)\,\text{d}t+\frac{\text{d}\boldsymbol{r}}{\text{d}t}\boldsymbol{\cdot }\int t\boldsymbol{G}_{vel}(t)\,\text{d}t+\frac{1}{2}\frac{\text{d}^{2}\boldsymbol{r}}{\text{d}t^{2}}\boldsymbol{\cdot }\int t^{2}\boldsymbol{G}_{vel}(t)\,\text{d}t+\cdots \,\right).\end{eqnarray}$$

The velocity-encoding gradients typically consist of a pair of bipolar gradient pulses (or, equivalently, a pair of gradient pulses of the same polarity separated by a $180^{\circ }$ RF pulse) with magnitude $g$ , duration $\unicode[STIX]{x1D6FF}$ and time between the centre of bipolar gradient pulses $\unicode[STIX]{x1D6E5}$ . The bipolar gradient pulses have the property that the zeroth moment of $\boldsymbol{G}_{vel}$ (the first term in (2.2)) is zero while the second and higher moments of $\boldsymbol{G}_{vel}$ (the third and higher terms in (2.2)) can be neglected compared to the first moment of $\boldsymbol{G}_{vel}$ (Sederman et al. Reference Sederman, Mantle, Buckley and Gladden2004). Therefore, the phase of the image is related to the velocity of the fluid in the direction of the velocity-encoding gradients, $u$ , by:

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FE}g\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D6E5}u.\end{eqnarray}$$

3 Experimental

Experiments were performed on a flow loop comprising of a vertical cylindrical pipe of inner diameter $D=16$  mm. The flow loop was designed such that a uniform velocity profile at the entrance to the cylindrical pipe was produced by flowing the fluid through a regular square-channel monolith. Water with 0.36 mM $\text{GdCl}_{3}.6\text{H}_{2}\text{O}$ added ( $T_{1}\approx T_{2}=80$  ms) was used as the flowing fluid. Flow was driven upwards by a Watson–Marlow 505s peristaltic pump, with a dampener used to mitigate flow fluctuations. A 100 mm long cylindrical cordierite monolith with $0.4~\text{mm}^{2}$ square channels running throughout its length was inserted in the pipe to obtain a uniform velocity profile. All experiments were conducted on an AV-400 Bruker spectrometer, operating at a resonant frequency of 400.25 MHz for $^{1}\text{H}$ observation, with a radio frequency (RF) coil of 25 mm diameter. The maximum magnetic field gradient amplitude available in each spatial direction was $146~\text{G}~\text{cm}^{-1}$ .

Figure 2. (a) Schematic of the positions inside the monolith (position A) and at the exit of the monolith, or equivalently the entrance to the pipe ( $z/D=0.25$ , 0.5, 1, 2, 4, 6) where MR velocimetry acquisitions of the axial velocity in a transverse plane were obtained. (b) MR velocimetry pulse sequence used in this work, with the important acquisition parameters identified. Flow is in the positive $z$ -direction.

MR velocimetry acquisitions of the axial velocity in a transverse slice were performed at a distance of 10 mm before the exit of the monolith (position A) in order to establish whether the velocity profile at the exit of the monolith is uniform and at distances $z$ from the exit of the monolith, $z/D=0.25,0.5,1,2,4,6$ , in order to investigate the development of the velocity profile beyond the monolith. The position $z/D=0$ which identifies the exit of the monolith, also identifies the entrance to the pipe. These are illustrated in figure 2(a). Four Reynolds numbers, $Re_{p}$ , were investigated in this study: $120\pm 10$ , $250\pm 10$ , $500\pm 20$ and $1100\pm 50$ . The quoted uncertainty, which will be dropped when referring to the Reynolds number in the manuscript, is the calculated standard deviation of the Reynolds numbers measured at the different $z/D$ positions set, and arises from experimental variation in the flow rate with axial position in the pipe. The corresponding mean velocities to the Reynolds numbers studied are 0.8, 1.6, 3.1 and $6.9~\text{cm}~\text{s}^{-1}$ . The Reynolds number $Re_{p}$ relates to the flow at the exit of the monolith and uses the diameter of the pipe as the characteristic length. The MR velocimetry technique used was a combination of a conventional spin-echo based imaging technique with velocity-encoding magnetic field gradients, as illustrated in figure 2(b). For velocity imaging inside the monolith, the transverse slice thickness was 5 mm, images comprised of $512\times 512$ pixels and were of resolution $0.039~\text{mm}\times 0.039~\text{mm}$ , $\unicode[STIX]{x1D6FF}=0.25$  ms, $\unicode[STIX]{x1D6E5}=0.5$  ms, $\unicode[STIX]{x1D70F}=0.51$  ms; $g$ was optimised to achieve a high signal-to-noise ratio, and the time between complex data points acquired in the read ( $y$ ) direction was $1~\unicode[STIX]{x03BC}\text{s}$ . The uncertainty in the pixel-wise velocity measurement inside the monolith was $1.4~\text{mm}~\text{s}^{-1}$ , as calculated from the standard deviation of the velocity image acquired under zero flow conditions. The spatial resolution of the velocity images inside the monolith was high in order to resolve velocities inside individual channels. For all velocity images beyond the exit of the monolith (i.e. within the pipe), the transverse slice thickness was 1 mm, images comprised of $64\times 64$ pixels and were of resolution $0.313~\text{mm}\times 0.313~\text{mm}$ , $\unicode[STIX]{x1D6FF}=0.5$  ms, $\unicode[STIX]{x1D6E5}=1$  ms, $\unicode[STIX]{x1D70F}=1.28$  ms; $g$ was optimised to achieve a high signal-to-noise ratio, and the time between complex data points acquired in the read ( $y$ ) direction was $20~\unicode[STIX]{x03BC}\text{s}$ . The uncertainty in the pixel-wise velocity measurement at the exit of the monolith was $0.3~\text{mm}~\text{s}^{-1}$ , as calculated from the standard deviation of the velocity image acquired under zero flow conditions. The phase image used to calculate the velocity in (2.3) was obtained from the phase difference between two images with positive and negative velocity-encoding gradients ( $+g$ and $-g$ ) and with correction from a zero velocity image (Sederman et al. Reference Sederman, Mantle, Buckley and Gladden2004).

4 Results and discussion

First, the results of velocity measurements inside the monolith are presented, which show that the velocity profile immediately after the exit of the monolith is well approximated by a uniform velocity profile. Then, the results of the velocity profile development at the entrance to the pipe are given, which show clearly a maximum in the velocity profile whose position moves from the wall to the centre of the pipe with distance along the pipe. These data are then compared to available results from numerical finite-difference solutions of the full Navier–Stokes equations in the literature and the differences are critically discussed.

Figure 3. (a) Cross-section magnitude image of the fluid in the monolith, composed of parallel square channels, at a distance of 10 mm before the exit of the monolith (position A in figure 2 a). The fluid is MR-active while the monolith is MR-inactive and does not contribute to the MR signal. The image comprises of $512\times 512$ pixels and is of resolution $0.039~\text{mm}\times 0.039~\text{mm}$ . The thickness of the wall separating the channels is approximately 0.15 mm. (b) Axial velocity image of a single representative channel highlighted in (a) at $Re_{p}=500$ , $Re_{c}=21$ . The image comprises of $16\times 16$ pixels and is of resolution $0.039~\text{mm}\times 0.039~\text{mm}$ . The maximum uncertainty in the pixel-wise reported velocity is $1.4~\text{mm}~\text{s}^{-1}$ . The axial velocity in the monolith is denoted by $v$ in order to distinguish it from the axial velocity at the exit of the monolith, denoted by $u$ .

4.1 Velocity profile inside the monolith

In this section, results of the axial velocity measurements on a transverse slice positioned at a distance of 10 mm before the exit of the monolith are presented. Figure 3(a) displays the magnitude image of the fluid in the monolith, which shows that the square channels are of similar size and they are distributed evenly throughout the cross-section of the pipe. Each channel comprises of ${\sim}16\times 16$ pixels, which should give sufficient information about the velocity profile inside an individual channel. The velocity profile distribution for an expanded region corresponding to a single representative channel at $Re_{p}=500$ is given in figure 3(b). The Reynolds number in each individual channel, $Re_{c}$ , is estimated to be ${\sim}\,6,10,21$ and 46 when the corresponding Reynolds number at the exit of the monolith, $Re_{p}$ , is 120, 250, 500 and 1100, respectively. $Re_{c}$ is calculated based on the hydraulic diameter of the channel. At these low Reynolds numbers, the entrance length required for fully developed flow in an individual square channel is ${<}\,5$  mm (Sparrow, Hixon & Shavit Reference Sparrow, Hixon and Shavit1967). Therefore, at the imaging section situated 10 mm before the exit of the monolith (position A in figure 2(a)), fully developed laminar flow is expected in each channel, even at the highest Reynolds number studied. At this distance before the exit of the monolith, the mean velocities in each channel were then radially averaged to give the velocity profile across the whole cross-section of the monolith. The velocity profiles obtained at each of the Reynolds numbers investigated are shown in figure 4. It is observed that the velocity profile across the cross-section of the monolith is well approximated by a uniform velocity profile at all Reynolds numbers investigated. The slight deviation from the uniform velocity profile close to the wall of the pipe is caused by the non-ideality of the channels close to the wall of the pipe. The non-ideality of the channels extends for approximately one channel side length from the wall of the pipe. This region corresponds to $r/R$ in the range 0.9–1.0, where $r$ is the radial distance from the centre of the pipe and $R$ is the radius of the pipe. It is in this region that the slight deviation from the uniform velocity profile is observed in figure 4. The effect that the deviation from the uniform velocity profile close to the wall of the pipe has in the development of the velocity profile at the exit of the monolith is discussed in § 4.2. Since the velocity profile at each individual channel is fully developed at 10 mm before the exit of the monolith, it is expected that the velocity profile across the cross-section of the monolith shown in figure 4 remains the same until the exit of the monolith. Therefore, the velocity profile immediately after the exit of the monolith and entrance to the pipe is well approximated by a uniform velocity profile.

Figure 4. Radially averaged axial velocity profile at a position of 10 mm before the exit of the monolith (position A in figure 2 a) at the four Reynolds numbers studied. $r$ refers to the radial distance; $R$ refers to the radius of the pipe; and $u_{mean}$ refers to the mean velocity. The lines are included to guide the eye. The graphs were constructed by radial averaging the mean velocities of the fluid within each channel. The uncertainty related to each data point is dominated by the circular asymmetry of the velocity profile, rather than the uncertainty related to pixel-wise velocity measurements. The maximum uncertainty of the reported $u/u_{mean}$ values is 0.05. For reasons of clarity, error bars are not shown.

Figure 5. Radially averaged axial velocity profiles at positions $z/D=0.25$ , 0.5, 1, 2, 4 and 6 from the entrance to the pipe and Reynolds numbers, $Re_{p}$ , of (a) 120, (b) 250, (c) 500 and (d) 1100. The lines are included to guide the eye. For each Reynolds number, the entrance length required for fully developed laminar flow, $z_{ent}/D$ , is included in the figure. The uncertainty related to each data point is dominated by the circular asymmetry of the velocity profile, rather than the uncertainty related to pixel-wise velocity measurements. The maximum uncertainty of the reported $u/u_{mean}$ values is 0.03. For reasons of clarity, error bars are not shown.

4.2 Velocity profile development at the entrance to the pipe

The development of the axial velocity profile at the entrance to the pipe from the initial profile shown in figure 4 is now reported, with a particular focus on the location and magnitude of the maximum velocity in the profile.

The experimental results for the axial velocity profile at distances of $z/D=0.25$ , 0.5, 1, 2, 4 and 6 from the entrance to the pipe and Reynolds numbers, $Re_{p}$ , of 120, 250, 500 and 1100 are presented in figure 5. The key observation from these results is that for a short distance from the entrance to the pipe (the extent of which depends on the Reynolds number), a maximum in the velocity profile which is not positioned at the centre of the pipe is clearly distinguished at all Reynolds numbers investigated. These experimental data therefore strongly support the predictions of the numerical finite-difference solutions of the full Navier–Stokes equations and do not agree with the predictions of analytical solutions of approximated Navier–Stokes equations.

Considering the distance from the entrance of the pipe, $z/D$ , over which the phenomenon is visible, it is seen that the phenomenon extends to greater distances down the pipe as $Re_{p}$ increases; the phenomenon is visible until $z/D\sim 0.5$ for $Re_{p}=120$ , $z/D\sim 2$ for $Re_{p}=250$ , $z/D\sim 4$ for $Re_{p}=500$ and $z/D\sim 6$ for $Re_{p}=1100$ . This observation is in qualitative agreement with the observations of all the works reported which use numerical finite-difference methods to solve the full Navier–Stokes equations and can be explained in terms of the boundary-layer theory (Schlichting Reference Schlichting1969); the boundary layer develops faster for low Reynolds number, which, as is well known, is why the entrance length for fully developed laminar flow is smaller for low Reynolds numbers.

Sufficient literature data for the quantitative comparison of the present experimental results are only available at $Re_{p}=500$ . Figure 6 shows (a) the variation of the position of the maximum on the velocity profile, $r_{c}/R$ , (b) the variation of the maximum velocity, $u_{max}/u_{mean}$ , and (c) the variation of the velocity at the centre of the pipe, $u_{centre}/u_{mean}$ , with the distance from the entrance to the pipe, $z/D$ , at $Re_{p}=500$ for the experimental data presented in this study and the results from the numerical finite-difference solutions of the full Navier–Stokes equations of Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993) at $Re_{p}=500$ and Wagner (Reference Wagner1975) at $Re_{p}=400$ .

Considering the results from the numerical finite-difference solutions of the full Navier–Stokes equations, a major difference is observed between the results of Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993), as compared to the results of Wagner (Reference Wagner1975). This is attributed to the use of different inlet boundary conditions; Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993) used a uniform velocity profile while Wagner (Reference Wagner1975) used inlet boundary conditions corresponding to a piston moving at constant speed and solved the problem in a frame of reference in which the piston is stationary. The observation that the use of different inlet boundary conditions leads to significant quantitative differences in the measured $r_{c}/R$ , $u_{max}/u_{mean}$ and $u_{centre}/u_{mean}$ is the main reason for not including in figure 6 the data from the numerical work of dos Santos & Figueiredo (Reference dos Santos and Figueiredo2007). A minor difference is observed even between the results of Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993), showing that even with the use of the same inlet boundary conditions, different implementations of numerical methods can lead to different results, in terms of the predicted values of $r_{c}/R$ , $u_{max}/u_{mean}$ and $u_{centre}/u_{mean}$ .

Figure 6. Comparison of the present experimental results with the results from finite-difference numerical methods solution of the full Navier–Stokes equations from Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993) at $Re_{p}=500$ and Wagner (Reference Wagner1975) at $Re_{p}=400$ . The graphs show the variation of (a) the critical radius $r_{c}/R$ at which the maximum of the velocity occurs, (b) the maximum of the velocity $u_{max}/u_{mean}$ and (c) the velocity at the centre of the pipe $u_{centre}/u_{mean}$ with the distance from the entrance to the pipe $z/D$ . The results are only shown for values of $z/D$ at which the maximum in the velocity profile is not at the centre of the pipe. The uncertainty associated with the estimation of $r_{c}/R$ is dominated by the spatial resolution of the MR image; the uncertainty associated with $u_{max}/u_{mean}$ and $u_{centre}/u_{mean}$ is dominated by the circular asymmetry of the velocity profile, rather than the uncertainty related to pixel-wise velocity measurements.

The inlet boundary condition in the present experimental work is closer to the inlet boundary conditions used by Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993), and it is observed from figure 6 that there is good agreement between the present experimental data and the results of those workers. However, the experimentally observed $r_{c}/R$ is consistently lower, while $u_{max}/u_{mean}$ and $u_{centre}/u_{mean}$ are consistently higher than the values predicted by Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993). These discrepancies can be explained by the fact that the inlet boundary condition in the experiment was not a perfect uniform velocity profile (as it is assumed in the work of Friedmann et al. (Reference Friedmann, Gillis and Liron1968) and Dombrowski et al. (Reference Dombrowski, Foumeny, Ookawara and Riza1993)) but was as shown in figure 4. The consistently lower $r_{c}/R$ observed experimentally is consistent with the flow already spreading inwards from the wall towards the centre of the pipe before the flow exits the monolith and enters the pipe; this is supported by the data shown in figure 4. The consistently higher $u_{max}/u_{mean}$ and $u_{centre}/u_{mean}$ follow from continuity.