Segmental orifice plates and the emulation of the 90°-bend

2.2.1 Definition of the mean profile factor

The mean profile factor ⟨KP⟩ is given by

and constitutes the arithmetic mean of the individual values of KP for the Nφ=20 radial paths of the LDA measurement grid (see Fig. 4). For a wide range of disturbed flow conditions, it describes the profile shape of a velocity distribution, that can be either pointed (⟨KP⟩>1), flattened or concave (⟨KP⟩<1) or equal to a fully developed reference (⟨KP⟩=1). (Due to the definition with the velocity in the pipe center, these correlations are not necessarily applicable at the presence of a strong asymmetry.)

In the equation, w and ws denote the local velocities of the measurement data and the reference profile in the axial direction, whereas wm and ws.m designate the respective values in the pipe center. As a reference, the analytical profile described by Gersten and Herwig [14] with updated constants given in Merzkirch [17] was used. The terms in the numerator and denominator placed in brackets are integrated over the radial position r in the closed interval of r=[−R,R], where R denotes the pipe radius. Both integrals can be solved analytically after replacing the LDA data points with piecewise sub-splines as described by Akima [8]. Moreover, the wall function defined by Straka et al. [22] is used to interpolate the velocity distribution between the outermost measurement points and the wall. (The representation with sub-splines and the wall function are also used below for the calculation of KA, QUFM and QEFM.) A comprehensible derivation of KP including a graphical representation is given in Müller [19].

2.2.2 Definition of the mean asymmetry factor

The mean asymmetry factor ⟨KA⟩ is given by

and corresponds to the arithmetic mean of the absolute values of KA in the grid paths. It describes the radial displacement of the profile’s velocity centroid expressed in percent and is therefore a measure for the degree of asymmetry. In the equation, the first term within the vertical bars is equivalent to the mathematical definition of a geometrical centroid. A division by the pipe diameter D yields a relative difference with respect to the path length.

2.2.3 Definition of the maximum turbulence factor

The maximum turbulence factor KTu.max is given by

and corresponds to the maximum value of KTu. It describes the ratio between the highest degree of turbulence Tu within a specified core region of the pipe (r≤0.2·R) and the theoretical degree of turbulence Tus in the center, using an approximation described by Durst et al. [11]. Tu is the coefficient of variation for the velocity samples at each grid point calculated as the ratio between the standard deviation to the mean.

2.2.4 Definition of the maximum horizontal velocity

In contrast to the performance indicators representing the axial component, the maximum horizontal velocity umax defined as

is based on the horizontal velocity component u in the direction of x. It describes the absolute maximum of u on the horizontal grid path, where y=0 and φ=0° (highlighted red in Fig. 4), normalized with the volumetric velocity wvol. Although a complete visualization of the two-vortex Dean structure as illustrated in Fig. 7 (i) requires the measurement of both secondary components u and v, umax allows for a good approximation of the swirling velocity.

2.3.1 Ultrasonic flow meter

The flow rate QUFM of a transit-time UFM with the transducers positioned at an angle of φj (see Fig. 5) can be modeled using the equation given by

where k is a Re-dependent correction factor and A=π·R2 the pipe’s cross section. Moreover, wp is the averaged path velocity along the ultrasonic beams calculated from the difference of the transit times tAB and tBA, which correspond to the signal times from transducers A to B and B to A, respectively.

Figure 5

Illustration of a transit-time ultrasonic flow meter in reflection mode. The transducers A and B have an axial distance of Δz and are located at φj (here: 63°). tAB and tBA correspond to the ultrasonic transit times from A to B and B to A, whereas ur and w denote the radial and axial velocity components.

For a derivation of wp in a direct path arrangement, see Moore et al. [18]. In a reflection alignment as illustrated in Fig. 5, the radial velocity component ur can be neglected (assuming dur/dz=0 along the axial distance between the transducers Δz), as its influence on the propagation of the ultrasonic signal is compensated after the reflection on the wall. If it is further assumed that dw/dz=0 along Δz, wp is independent from the inclination angle α and can be calculated representatively along a radial path with length D given by

In Straka et al. [23], the same model calculated from LDA measurements was compared to clamp-on UFMs in reflection mode (Δz≈0.5·D) exposed to the flow disturbance downstream of an SOP33. For the measurement distances greater than one diameter, the comparison resulted in deviations of ≈1%.

2.3.2 Electromagnetic flow meter

An EFM measures the voltage induced across the fluid through a magnetic field, which is proportional to the fluid’s average velocity. For a circular cross section and the electrodes positioned perpendicular to the magnetic field at angles of φj and φj+180°, the flow rate QEFM can be calculated according to

Here, k is a Re-dependent correction factor, whereas W(φj) denotes the weight function as described by Shercliff [21] given by

and depicted in Fig. 6, which accounts for the different weighting of the flow field as a function of the distance to the electrodes. In proximity to the point electrodes, W(φj) increases without bound and is therefore cut off at a maximum value of 2.5.

Figure 6

Contours of the Shercliff weight function [21] given by Eq. (9) used for the model of the electromagnetic flow meter. The point electrodes A an B are located at the pipe wall with angles of φj and φj+180° (here: 27°, 207°) perpendicular to the magnetic field. W(φj) is cut off at a maximum value of 2.5 in proximity to the electrodes.

In contrast to UFMs, an EFM is generally not affected by non-axial velocity components, as only the flow perpendicular to the magnetic field contributes to the creation of the potential difference. However, the model ignores the axial extent of the magnetic field and thus also assumes dw/dz=0 along the sphere measurable for the electrodes. Other 2D and 3D based weight functions for circular and rectangular cross sections and different meter designs have been derived theoretically and numerically. An overview is given by Yin et al. [28].

3.1.1 Segmental orifice plate

As depicted in Fig. 8, the orifice geometry of the SOP33 located on the left results in a primary displacement of the core flow towards the opposite (right) side of the pipe. At a distance of 2.0D, the velocity values are highly accelerated and show a maximum of 1.9·wvol. At Re=5×104, the recirculation zone behind the orifice is still visible through the negative velocities, while the flow at the higher Re has already reattached on the wall. Looking at the non-axial component u in Fig. 7 (a), the dominant horizontal movement is directed towards the left, realizing the expansion of the flow over the entire pipe’s cross section as the recirculation zone decreases.

Following a singular symmetric state measured at a distance of 5.1D, the secondary motion (now in accordance with the Dean vortex structure) results in a shift of the core flow to the left, similar to the observations made by Eichler [13] and Straka et al. [23] for the SOP7 and SOP33 in DN 80 and DN 200, respectively. After falling to a minimum at 15.2D, the velocity values of the asymmetric profiles increase again, and the radial distance of the peak velocity region gets smaller. Meanwhile, the values of the secondary component u and accordingly the rotational velocity of the Dean vortices constantly decreases. Comparing the different Re numbers, the (normalized) peak velocity values at Re=5×105 are smaller starting at 10.2D, which complies with the characteristics of a fully developed turbulent pipe flow.

3.1.2 90 ° - bend

Figure 9

Contours of the axial velocity component w normalized with the volumetric velocity wvol for the 90°- bend at Re=5×104 (a–f) and 5×10⁵ (g–l), presented at measurement positions between 2.4 and 30.8 times the pipe diameter (D).

In accordance with the flow phenomena inside the 90°- bend, the LDA velocity profiles measured at a distance of 2.4D (see Fig. 9 a/g) are highly asymmetric and have the shape of a crescent. The peak velocity regions are located at the outer left wall with maximum values of 1.3·wvol and an area with low values of 0.8·wvol is enclosed within the pipe center. At this position, the strength of the Dean vortices superimposing the flow have a maximum of u=0.25·wvol (see Fig. 7 e).

Further downstream, the axial velocity values converge, resulting in a flat profile at 10.7D with velocities below 1.1·wvol. As the disturbance relaxes towards the fully developed state, the velocities rise again, whereas the radial distance of the peak velocity and the values of u constantly decrease. Again, the (normalized) velocity values at the higher Reynolds number of Re=5×105 are smaller throughout all the measurement positions.

3.1.3 Qualitative comparison

With the orifice and outer curvature wall located at the left, both disturbances are oriented so that the Dean vortices downstream are equally aligned, rotating in accordance with the illustration in Fig. 7 (i). In the case of the 90°- bend, its magnitude (≈0.25·wvol at 2.4D) correlates with the development of the crescent-shaped asymmetric profile, while the SOP initially causes a displacement to the right.

However, in between a distance of ≈5 and 10D, the flow development downstream of the SOP33 shows similarities with the close-range effects downstream of the 90°- bend, as the core flow is heading left, displacing the slower flow on the orifice side. Since the rotational momentum of the secondary motion is far below the level of the 90°- bend (≈0.12·wvol at 5.1D), the mechanism creates an asymmetry but is insufficient to develop the crescent-shaped profile. As the velocity values downstream of the 90°- bend converge, the profiles optically show a high degree of resemblance. This means, that the SOP33 reproduces the 90°- bend starting at ≈10D, but does not reproduce the disturbance in the close-range distance.

In comparison to the SOP33, the degree of asymmetry downstream of the SOP7 observed by Eichler [13] is much weaker, especially with respect to the radial distance of the core velocity region. No optical similarity with the 90°- bend can be determined in neither the near- nor the far-field range.

3.2.1 Mean profile factor

The similarities in the courses of the mean profile factor ⟨KP⟩ (describing the peakness/flatness of the profile) illustrated in Fig. 10 (a) in between ≈10 and 31 D quantitatively confirm the optical resemblance between the 90°- bend and the SOP33. Within that range, ⟨KP⟩ rises from approximately 0.1 to 0.8, meaning that the profiles are comparatively flat in comparison to the fully developed reference conditions, where ⟨KP⟩=1. In the case of the SOP33, the relaxation takes place at a noticeably faster rate, indicated through the slightly higher slope.

At a 2.4D distance to the 90°- bend, the negative values reaching a minimum of ⟨KP⟩≈−0.9 comply with the low values of the velocity in the pipe center and the concave shape of the profile. In contrast, the peak values up to ⟨KP⟩≈1.8 in the case of the SOP33 at 5.1D correlate with the steep profiles and the high velocities in the pipe center. Despite the differences 2.0D downstream of the SOP33 (again due to the variation of the velocity in the center), the courses of ⟨KP⟩ exhibit no clear Re-related effects.

As for the SOP7, the development of ⟨KP⟩ is clearly different. While the missing peak observed for the SOP33 at the temporal symmetric state is simply not covered by the measurement positions, the values are much higher throughout the entire range, meaning that the profiles are much less flat or less disturbed.

3.2.2 Mean asymmetry factor

Starting with the 90°- bend, the mean asymmetry factor (describing the radial displacement of the velocity centroid) depicted in Fig. 10 (b) initially falls to an intermediate low of ⟨KA⟩≈0.2% at 5.6D, as the velocity values of the crescent-shaped profile converge. Further downstream, the peak values and thus, also ⟨KA⟩, rises to a maximum of 0.5 % at 15.7D (Re=5×105) and subsequently falls towards the end, because the radial distance of the velocity peak decreases.

As for the SOP33, ⟨KA⟩ has a very high value of 10.4 % at the strongly asymmetric profile at the 2.0D measurement position (Re=5×105) and reaches an intermediate low of 0.3 % at the (nearly symmetrical) state at a distance of 5.1D. With the core velocity region shifted to the left, ⟨KA⟩ increases to a maximum of 1.1 % at 10.2D and subsequently falls. In comparison to the 90°- bend, there is a similar progression starting at ≈15D. Once again, the relaxation towards fully developed conditions emerges at a higher pace, expressed through the lower ⟨KA⟩ at the last measurement position. For both the SOP33 and the 90°- bend, the lower Re is connected to higher values of ⟨KA⟩, presumably due to the constantly higher (normalized) velocity values.

The SOP7 creates comparatively high values of ⟨KA⟩ throughout all measurement positions, exceeding the values of the 90°- bend and the SOP33 starting at ≈15D. In the case of the SOP7, the displacement of the velocity centroid is connected to the large peak velocities. Furthermore, the development of ⟨KA⟩ demonstrates the much slower abatement of the flow disturbance.

3.2.3 Maximum turbulence factor

A significant increase of the maximum turbulence factor KTu.max (describing the extent of velocity fluctuations) illustrated in Fig. 10 (c) can be observed in proximity to all the disturbances, relaxing towards a nearly constant level at a distance of ≈15D. A comparison between the SOPs reveals the correlation between the height of the orifice geometry and the turbulence intensity, connected to the production of turbulence kinetic energy in the recirculation zone. With a maximum value of 13.2 at 2.0D (Re=5×105), KTu.max for the SOP33 exceeds the value of the SOP7 and the 90°- bend by a factor of 4.4 and 3.3. It can be concluded, that the higher values of KTu.max also correlate with the faster transition of the flow disturbance towards reference conditions, expressed through the relaxation of ⟨KP⟩ and ⟨KA⟩.

3.2.4 Maximum horizontal velocity

As observed qualitatively, umax (describing the maximum non-axial velocity in the horizontal direction) depicted in Fig. 10 (d) has higher values in the case of the 90°- bend, meaning that the rotation of the Dean vortices in the case of the SOP33 is comparatively lower. Throughout the measurement range, the values decrease in an exponential manner.

As for the SOP7, umax is smaller in proximity to the disturbance (by a factor of 3 in comparison to the SOP33), but abates at a much smaller rate, exceeding the SOP33 and the 90°- bend at distances of approximately ≈9 and 16D, respectively.

3.3.1 Ultrasonic flow meter

In the case of the 90°- bend, the error calculated for an ultrasonic flow meter εUFM (see Fig. 11 a) has a value of −11.1(±0.5)% at 2.4D (Re=5×105) and decays exponentially to a minimum of −1.6(±0.2)% at the measurement position at 30.8D. Throughout the range, the mean error has a negative value and the angular dependency expressed by the standard deviation is below 0.7 %.

The negative sign can be linked to the consistently flattened and partially concave shape of the velocity profiles, quantified also by ⟨KP⟩ in Fig. 10 (a). At fully developed reference conditions, the integral in Eq. (7) yields values for the averaged path velocity wp in the range of wp≈1.05·wvol, because the radial distance is not weighted in the line integral, resulting in an over-representation of the high velocities in the center region. Compensated by a Re-dependent correction factor k in the range of ≈0.95, k·wp equals wvol and the correct flow rate can be calculated by multiplication with the pipe’s cross section. When exposed to a comparatively flattened profile, wp takes on values closer to wvol, resulting in k·wp<wvol and a negative deviation of the calculated flow rate with a maximum of −5 % for a theoretically uniform profile, where wp=wvol. In the case of a concave profile as observed up to a distances of 5.6D to the 90°- bend, wp takes on values smaller than wvol and the flow rate deviation falls below −5 %.

At 2.4 and 5.6D, the crescent-shaped profiles in Fig. 9 (a/b, g/h) feature high and low velocities in the wall-near region opposing each other, which to some extent balance out in the path integrals calculated from r=−R to R. For this reason, the standard deviations of 0.5 and 0.3 % (Re=5×105) are relatively low.

In comparison, the errors downstream of the SOP33 are initially positive, showing a maximum of 9.2(±3.6)% (Re=5×105) and a higher standard deviation, especially at a distance of 2.0D. Beginning at ≈10D, the course of εUFM is in line with the 90°- bend. This is different in case of the SOP7, which causes a much lower mean error ranging from 1.7 % to −1.9 % within the measurement range. Since the disturbance abates at a slower rate, εUFM intersects with the courses of the 90°- bend and SOP33 at ≈25D.

3.3.2 Electromagnetic flow meter

As depicted in Fig. 11 (b), the profiles downstream of the 90°- bend cause an exponentially decaying negative mean error εEFM and a low standard deviation for the model of the electromagnetic flow meter. In comparison to the UFM, the error is smaller, reaching a maximum deviation of −2.1(±0.3)% at 2.4D (Re=5×105).

In contrast to the evident correlation between εUFM and ⟨KP⟩, the measurement deviation of the EFM is influenced by a variety of effects related to asymmetry (⟨KA⟩), shape (⟨KP⟩) and the peak velocity. In the presence of an asymmetry, the velocities in proximity to one or both electrodes may be increased, increasing the value of the surface integral in this strongly weighed area, whereas secondly, the velocities in the greater part of the cross section are reduced, partially reducing the surface integral.

In case of the 90°- bend and the SOP33 starting at 10.2D, the second effect is predominant, resulting in a negative error. In the case of the SOP33 at 2.0D, the first effect outweighs the second for most angular positions, because the peak velocities are very high. Here, the mean error is positive, reaching a maximum of 12.1(±1.2)% at Re=5×105. As for the (nearly symmetrical) pointed profile at 5.1D, the velocity values in proximity to both electrodes are comparatively low, resulting in a negative error with a value of −5.2(±0.2)%.

Following the fluctuations at 2.0 and 5.1D, the error curve of the SOP33 is similar to the 90°- bend starting at ≈10D. In comparison, the SOP7 causes a maximum error of only 0.4(±1.2)% at 1.0D, falling towards −0.0(±0.1)% and −0.1(±0.3)% at 13 and 25D, respectively.