An elementary observation of a laminar cylindrical Poiseuille–Couette flow profile reveals no distinction in the parabolic streamwise profile, from one without a cross-stream flow, in whatever reference frame the observation is made. This is because the laminar flow is in solid-body rotation and there is no fluid intrinsic rotation. Hence, the main streamwise Poiseuille flow is unaffected. On the contrary, in turbulent (unsteady) cylindrical axial-Couette flow, the rotational reference frame must be considered, and any observation from an external inertial reference frame can give outright incorrect results. However, even in axial turbulent pipe flow with axial rotation, the resultant effect of azimuthal velocity on the flow profile is still usually too low. Hence, the importance of consideration in the rotational frame is often overlooked. A common misconception in the study of fluid mechanics is that the position of the observer does not matter. In this direct numerical simulation study, firstly turbulent flow in a pipe with axial rotation is established. Then turbulent flow in the concentric pipe, with inner wall rotation, is used to show how tilted wall streak direction is oriented by the rotational reference frame and not the inertial reference frame.

## I. INTRODUCTION

### A. Wall bounded rotational turbulence

For high frequency rotating pipes, axial flow tends toward a parabolic Poiseuille laminar profile where *U*^{+} = *y*^{+}. Hence, a higher axial velocity *U*^{+} is obtained per *Re*_{τ} equivalent in the non-rotating pipe (Brehm , 2019; Davis , 2020; Kikuyama , 1983; Kitoh, 1991; Nygard and Andersson, 2010; Orlandi and Fatica, 1997; Reich and Beer, 1989; Speziale , 2000; and White, 1964). Either due to the relaminarization of the flow or the symmetry of the streamwise flow, the effect of axial rotation (azimuthal velocity) on streamwise turbulent flow in a pipe is insignificant or non-existent. In the turbulent pipe flow by Nygard and Andersson (2010), an objective of imposing an artificial azimuthal pressure gradient at some distance from the wall was so that azimuthal velocity had an effect and tilted wall streaks on the pipe wall were able to be studied. In turbulent flow in a concentric pipe with either inner or outer wall rotation, the azimuthal velocity naturally does have a significant effect on the streamwise flow, and this causes very obvious tilted wall streaks (Dunstan, 2020). Throughout this paper, the rotation rate is quantified by *N*, the ratio of mean azimuthal velocity (*U*_{θ}) to mean streamwise velocity (*U*).

### B. Work on the concentric pipe

When a three dimensional turbulent boundary layer (3DTBL) forms over a convex surface, turbulence production increases, but when it forms over a concave surface, turbulence suppression occurs (Chung and Sung, 2005; Hadziabdic , 2013; Jung and Sung, 2006; Orlandi and Fatica, 1997; Strutt, 1917; and Taylor, 1923). There are also important practical examples where related flows are encountered. Such as in the rotor-stator scenario of pumps and motors, around drill pipes, rifled projectiles, and the curved surface of swept wings.

Without rotation, turbulent flow in a concentric pipe, or turbulent annular pipe flow, provides an interesting case to study asymmetric flow, which occurs when the streamwise mean velocity profile is not symmetric. Asymmetric flows also occur elsewhere, including in channel flow with unequal wall roughness and in a wall jet. When one or both of the walls of the concentric pipe are rotated, a 3DTBL is formed, which further complicates. When wall rotation in a concentric pipe is allowed to dominate flow momentum, this can present interesting scenarios for Taylor-Poiseuille-Couette flow. Such emphasis was made by Meseguer and Marques (2002), Meseguer (2009), and Moser (1999). The current research is confined to: turbulent axial flow; wall rotation, which is confined to only the inner wall; only anticlockwise (right-handed) rotation with respect to a consistent inertial reference; and at no time does the Rossby number decrease below unity. The last condition ensures turbulent axial flow is always prominent and resultant flow is always oriented less than 60° to the axial.

### C. Motivation

The importance of considering globally axially rotating turbulent systems (having a mean azimuthal velocity *U*_{θ}) in the rotational reference frame needs to be emphasized. Many turbulent energy budget calculations in such systems require consideration of the mean azimuthal velocity [for example, see Eggels (1994) and Facciolo (2006)]. When considering *U*_{θ} in the inertial reference frame, it also has a solid body rotation added, which creates so-called orbital rotation. Neglecting this solid body influence results in intrinsic turbulent fluid rotation (spin rotation). This results in consideration in a Coriolis or rotational reference frame. When *U*_{θ} is considered as such, turbulent energy budgets can now be calculated accurately. This paper has not included energy budgets.

The flow orientation angle is given as $\gamma s=tan\u22121U\theta \u2212U\theta wallU$. It has been used in earlier papers such as Littell and Eaton (1994) and Schwarz and Bradshaw (1994) for 3DTBL flows. It has also been used in concentric pipe studies by Hadziabdic (2013), Jung and Sung (2006), and Poncet (2014), among others. However, this use of *γ*_{s} has perhaps been rather indiscriminate, as the actual wall streak orientation would seem to indicate otherwise. Essentially, this letter seeks to demonstrate and ensure that rotating turbulent flows are correctly considered in the rotational reference frame and not the inertial reference frame, regardless of the perspective of the observer. Hence, for turbulent flows, the wall velocity is inconsequential. It is the fluid velocity relative to a static wall that matters.

Relatively recent large eddy simulation (LES) and direct numerical simulation (DNS) studies have considered the friction Reynolds number (*Re*_{τ}) in terms of the characteristic length, $Re\tau =u\tau \xd7\delta \nu $. The characteristic length in these investigations is half the distance between inner and outer walls, *δ* = 0.25*R*. Hence, mesh construction has been based on 0.25*Re*_{τ} instead of actual *Re*_{τ}. With such coarse mesh, near-wall and small-scale turbulence is unresolved, and graphs such as the semi-log plots for the law of the wall and the structure parameter, *a*_{1}, among others, cannot be plotted. The finer mesh used for the concentric pipe in this research is used to re-examine these quantities as well as other fluctuating velocity quantities.

### D. Frame invariant quantities

Given the comparative assessment of reference frames with rotation, the objectivity of streak identification quantities is best considered *a priori*. When fluid is in rotation, it is subject to a non-inertial reference frame, which is additional to steady flow Galilean invariance. Pope (2003) explains that the Navier–Stokes equations require transformation to the non-inertial frame by addition of the so-called *fictitious* terms represented by the centrifugal force, and the Coriolis force, and the angular acceleration force. That is, the Navier–Stokes equations are not material frame indifferent. Ariki (2015) mentions that Reynolds stresses and other higher order correlations are form invariant under arbitrary time dependent rotations and translations. These quantities constitute turbulent velocity fluctuations. Years earlier, Speziale (1979) had mathematically demonstrated that velocity fluctuations are material frame indifferent. Speziale (1979) also showed that the Reynolds stress transport equations are frame-dependent because they also constitute of mean velocity quantities, which are frame dependent. In this study, we utilize the invariance of velocity fluctuations to identify wall streaks.

## II. COMPUTATIONAL DETAILS

Nek5000 solves the above in Cartesian coordinates, where **f** is the body force term, which in these simulations adds the rotational component, angular velocity Ω, after multiplying to respective point-wise Cartesian velocities. The Coriolis force contained in **f** is −2**u** × Ω, and the arising centrifugal force 1/2(Ω × **r**)^{2} is absorbed in the pressure term (reduced pressure) with rotation. The reference length is *R* = 1. The reference mean velocity is always *U* = 1 in the axial *z*-direction, and the reference density is *ρ* = 1; hence, the term 1/*ρ* in front of ∇*p* in Eq. (2) is left out. Here, kinematic viscosity *ν* represents the Reynolds number, user input parameter, as *ν* = 1/*Re*, which is also in dimensionless parameters. In addition, Δ = ∇^{2} and **u** = (*v*_{x}, *v*_{y}, *v*_{z}).

Nek5000 is written in *Fortran*77 and *C* and is parallelized by MPI (message passing interface). Spatial discretization is achieved by the spectral element method, and temporal discretization is resolved by a third-order semi implicit method. Nek5000 is widely used where there is access to high performance computing infrastructure. Chen (2020), El-Khoury (2013), Hufnagel (2017), Noorani and Schlatter (2015), Wang (2018), and Rinaldi (2019), among others, have used turbulent pipe flow studies. Each structured element in Nek5000 is further refined by subgrid nodes numbering $N$ (polynomial order), where the grid point spacing is smaller toward the element boundaries [see Fischer (2008)]. Hence the total number of grid points is $numberofelements\xd7N3$ [see Dunstan (2020) from p. 28].

The visualization software VisIt by Childs (2012) was used throughout this research to generate such images as velocity fluctuations and the *λ*_{2} coherent structure identification criterion (Jeong and Hussain, 1995). VisIt images are generated in Cartesian coordinates from the raw data produced by Nek5000.

### A. Geometry and flow parameters

#### 1. Coordinate transformation

#### 2. Pipe mesh template and flow conditions

The 2D pipe geometry consists of an annular region with equally spaced azimuthal elements and a central region with elements arranged within an octagonal-like shape. The use of Cartesian coordinates in this central octagonal region eliminates issues with the singularity at the center encountered when using cylindrical coordinates.

The streamwise axial Reynolds number based on diameter was *Re*_{D} = 5300 and *Re*_{D} = 4900, with a corresponding *Re*_{τ} = 180. The optimal grid point spacing based on the use of polynomial order $N=7$ and domain length *L*/*R* is as detailed in Table I. The values in brackets are the maximum and minimum grid point spacing in wall units.

Re_{D}
. | Re_{τ}
. | L/R
. | Δz^{+}
. | Δr^{+}
. | RΔθ^{+}
. | $N$ . |
---|---|---|---|---|---|---|

5300 | 180 | 30 | (3.05, 9.96) | (0.14, 4.38) | (1.52, 4.96) | 7 |

Re_{D}
. | Re_{τ}
. | L/R
. | Δz^{+}
. | Δr^{+}
. | RΔθ^{+}
. | $N$ . |
---|---|---|---|---|---|---|

5300 | 180 | 30 | (3.05, 9.96) | (0.14, 4.38) | (1.52, 4.96) | 7 |

From Table I, for a 30*R* long pipe, there are 432(*r* × *R*Δ*θ*) × 114*z* = 49 248 elements, and the number of grid points is $49248\xd7N3\u223c16.9$M. Streamwise grid spacing is rounded down to a rational number (four elements per *R* unit streamwise length) so that the number of elements is 432(*r* × *R*Δ*θ*) × 120*z* = 51 840 and the number of grid points is $51840\xd7N3\u223c17.8$M.

#### 3. Pipe boundary conditions

Two possible methods of carrying out global rotation in Nek5000 are carried out in the pipe. First, a Coriolis body force can be incorporated into the governing equations of the solver as described in **f** in Eq. (2), in which case the wall boundary is no-slip. The effect is that which is referred to as swirling, where the fluid flow is now in a rotational reference frame.

The other method of imposing rotation is by setting the wall boundary condition to velocity inflow and imposing a tangential velocity to each grid point on the circumference of the pipe cross-section, i.e., effectively making the lengthwise circumference a slip wall boundary condition. The result is that there is a no-slip surface rotating at some tangential velocity. In this case, there is no need for **f** in Eq. (2). The effect of imposing this wall tangential velocity is that the rotation is now in an inertial, external, or laboratory reference frame. In this scenario, a linear solid body rotation component is now added to the mean azimuthal velocity *U*_{θ}.

Both methods of rotation were applied only to the simulations of the pipe with rotation *N* = 2. Here the pipe ends were set to periodic. Pipe simulations for *N* = 0, 0.5, and 1 were of the same pipe geometry, but with only the Coriolis body force applied so the wall boundary was no-slip. A summary of the five pipe cases is given in Table II, where *Nel* is the number of elements and BC is the boundary condition.

Case . | N
. | Re_{D}
. | Design Re_{τ}
. | ∼actual Re_{τ}
. | Nel
. | BC . | Rotation method . | Development time (R/U)
. |
---|---|---|---|---|---|---|---|---|

1 | 2.0 | 4900 | 180 | 100 | 49 248 | Periodic | Wall | $>$300 |

2 | 2.0 | 4900 | 180 | 160 | 49 248 | Periodic | Coriolis | $>$300 |

3 | 0.0 | 5300 | 180 | 170 | 51 480 | Recycle | Nil | $>$700 |

4 | 0.5 | 5300 | 180 | 167 | 51 840 | Recycle | Coriolis | $>$600 |

5 | 1.0 | 5300 | 180 | 165 | 51 840 | Recycle | Coriolis | $>$700 |

Case . | N
. | Re_{D}
. | Design Re_{τ}
. | ∼actual Re_{τ}
. | Nel
. | BC . | Rotation method . | Development time (R/U)
. |
---|---|---|---|---|---|---|---|---|

1 | 2.0 | 4900 | 180 | 100 | 49 248 | Periodic | Wall | $>$300 |

2 | 2.0 | 4900 | 180 | 160 | 49 248 | Periodic | Coriolis | $>$300 |

3 | 0.0 | 5300 | 180 | 170 | 51 480 | Recycle | Nil | $>$700 |

4 | 0.5 | 5300 | 180 | 167 | 51 840 | Recycle | Coriolis | $>$600 |

5 | 1.0 | 5300 | 180 | 165 | 51 840 | Recycle | Coriolis | $>$700 |

#### 4. Concentric pipe mesh template

The 2D concentric pipe template is built as separate annular sections, similar in form to the pipe annular sections (see Fig. 1). In this instance, the spanwise coordinate bases are *x* and *y*, and streamwise is taken to be *z*. In all concentric pipe simulations, the outer radius *R*_{o} = 1 and the radius ratio *α* = *R*_{i}/*R*_{o} = 0.5. Previous similar investigations using this median *α* were Chung and Sung (2003), (2005), Chung (2002), Hadziabdic (2013), Jung and Sung (2006), Meseguer and Marques (2002), Moser (1999), Nouri and Whitelaw (1994), and Nouri (1993).

#### 5. Concentric pipe flow conditions

Investigations into the concentric pipe were focused on the following inner wall rotation rates: *N* = 0.0, 0.2145, 0.429, 0.858, and *N* = 2.0.

In all concentric pipe simulation cases, the pipe ends were set to periodic boundary conditions. For the concentric pipe without wall rotation, the wall boundary condition was set to no-slip. Rotation was applied to the inner wall by numerically setting the slip wall. This means the inner wall boundary condition was set to velocity inflow and a tangential velocity was imposed to each grid point on the circumference of the inner wall cross-section, i.e., effectively making the lengthwise circumference a slip wall boundary condition. The equivalent result is that there is a no-slip surface rotating at some tangential velocity (Dunstan, 2020, p. 29). Hence, there is no need for **f** in Eq. (2). The outer non-rotating wall of the concentric pipe was set to no-slip. Like in the pipe with wall rotation applied, in this scenario, a linear solid body rotation component is now added to the mean azimuthal velocity *U*_{θ}, and the rotation is now in an inertial, external, or laboratory reference frame.

Following the two-point correlation of Chung and Sung (2005), the minimum streamwise length for the concentric pipes was set at 18*δ*. Table III details the number of elements, and Table IV shows the grid point spacing for the five different concentric pipe simulations. Values in brackets indicate maximum and minimum grid spacing (non-uniform mesh). Streamwise (Δ*z*) and azimuthal (*R*Δ*θ*) elements are of uniform dimensions. The radial element (Δ*r*) arrangement is as detailed in Fig. 2. The radial elements are spaced out in a roughly tanh profile so that there is no abrupt change in element width, and the radial element numbering arrangement is such that the limits of grid point spacing never exceed those given for Δ*r*^{+} in Table IV. Here, the red line is for the pipe, and the green ones are for the three mesh constructions for the concentric pipe simulations. Other lines are for the evolving mesh during mesh development. Flow development (stationarity) and grid independence are demonstrated in the dissertation by Dunstan (2020) for all cases of the pipe and concentric pipe.

N
. | Re_{H}
. | Design Re_{τ}
. | No. of elements Δr × RΔθ ×Δz
. | No. of grid points ×10^{6}
. | Streamwise length . |
---|---|---|---|---|---|

0 | 8900 | 600 | 22 × 162 × 70 | 85.57 | 22δ |

0.2145 | 8900 | 600 | 22 × 162 × 70 | 85.57 | 22δ |

0.429 | 8900 | 600 | 22 × 162 × 70 | 85.57 | 22δ |

0.858 | 8900 | 680 | 24 × 182 × 80 | 119.85 | 22δ |

2.0 | 8900 | 800 | 28 × 214 × 77 | 158.25 | 18δ |

N
. | Re_{H}
. | Design Re_{τ}
. | No. of elements Δr × RΔθ ×Δz
. | No. of grid points ×10^{6}
. | Streamwise length . |
---|---|---|---|---|---|

0 | 8900 | 600 | 22 × 162 × 70 | 85.57 | 22δ |

0.2145 | 8900 | 600 | 22 × 162 × 70 | 85.57 | 22δ |

0.429 | 8900 | 600 | 22 × 162 × 70 | 85.57 | 22δ |

0.858 | 8900 | 680 | 24 × 182 × 80 | 119.85 | 22δ |

2.0 | 8900 | 800 | 28 × 214 × 77 | 158.25 | 18δ |

## III. RESULTS AND DISCUSSION

### A. The pipe under rotation

#### 1. Pipe flow mean velocity profiles

In the current research, it has been observed that the flow development needs sufficient time to evolve from the inertial (laboratory) reference frame to a fully rotational reference frame, also pointed out by Orlandi and Fatica (1997). This is seen for the wall rotation method, for the higher rotation rate of *N* = 2, where rotation is performed in the inertial reference frame. The transient flow first becomes fully laminar before breaking down into its steady turbulent state at around time *t* = 200*R*/*U*, where the turbulent structures are larger and more coalesced (see Fig. 3). This coalescence increases with rotational frequency, and eddy elongation is clearly visible. The objectivity of the *λ*_{2} (Jeong and Hussain, 1995) vortex identification criterion is slightly ambiguous when comparing Figs. 3(d) and 3(e) (Haller, 2004). In the pipe, while there are clear effects of rotation, the lowest Rossby number obtained in all pipe cases would be in the *N* = 2 rotation rate case and would only be around *Ro* = *U*_{max}/|*V*_{θmin}| ≈ 1.5/0.3 = 5. Here, $V\theta min$ is the minimum azimuthal velocity in the rotating reference frame. Since this is a long way from unity, the linear effects of rotation via the Coriolis effect, as well as diffusion, are not prominent in these cases of turbulent pipe rotation. Hence, tilted wall streaks cannot be observed in turbulent axial pipe flow with axial rotation. This does not matter if the rotation applied is in the inertial or rotational reference frame.

#### 2. Comparison of reference frames in the pipe flow

Figure 4(b) shows the azimuthal velocity plots *W*_{θ}, given in the inertial reference frame. Only the *N* = 2 case was rotated via wall rotation in the inertial reference frame, as well as by Coriolis rotation. The mean azimuthal velocity for the other rotation rates was only for Coriolis rotation [shown in Fig. 4(b)]. However, the Coriolis rotation *V*_{θ} can be transformed to the inertial reference frame coordinate basis by adding the component for linear tangential velocity in solid body rotation (Ω*r*), which is the straight diagonal line in Fig. 4(b). This line is also the profile for rotational pipe flow, which is laminar (Brehm , 2019; Davis , 2020) and is the tangential (azimuthal) velocity solid body relation to the radius, where *V*_{θ} = Ω*r*. Vice versa, the inverse transform can be performed by removing this function (straight line) from the mean azimuthal velocity plots in the inertial reference frame to give the mean azimuthal velocity plots in the rotational reference frame, as shown in Fig. 4(c). Following Orlandi and Fatica (1997), the azimuthal profile in the inertial reference frame is here denoted *W*_{θ} [Fig. 4(b)], and the azimuthal profile in the rotating reference frame is denoted *V*_{θ} [Fig. 4(c)]. *W*_{θ} is normalized by dividing by the rotation rate *N*, which is the same as the Ω angular velocity at the wall and the radius at the wall, *R* = 1. Note that here and in ensuing figures, the radial distance from the pipe center to the pipe wall is given as *y*; this is in line with the convention of previous papers. Henceforth, this denotation is no longer used for the Cartesian *y* axis unless otherwise indicated (i.e., the VisIt images showing 3D representations in Figs. 3, 9, 10, and 12).

The parabolic plot given as *theoretical r*^{2} in Fig. 4(b) is the mean azimuthal velocity profile of a fluid in solid body rotation and is a consequence of the decrease in centripetal acceleration (*a* = *r*Ω^{2}) as one moves from the rotating wall, where tangential or azimuthal velocity is maximum at *R* = 1, to the center, where it decreases. This calculation does not consider the effects of turbulence and, in particular, the importance of viscosity in turbulence closer to the wall.

#### 3. Pipe flow validation

Validation for this research has been performed with the pipe flow simulations. This has been extensively performed and covered in Dunstan (2020) (from p. 65). For the non-rotating turbulent axial pipe flow, there is an exact comparison with the $virms\u2032$ plots of Wang (2018) [Fig. 4.6 of Dunstan (2020)]. For the rotating pipe, there is an exact comparison with the $virms\u2032$ plots for *N* = 1 of Ould-Rouiss (2010) [Fig. 4.8 of Dunstan (2020)]. In Fig. 4(b), the plots do not match with the plots of Orlandi and Fatica (1997). However, the plots match to within fine tolerance with the data of Ould-Rouiss (2010) for *N* = 1. This is also the case in Figs. 4(a) and 4(c).

The pipe geometry of Orlandi and Fatica (1997) contained at most 1.57 × 10^{6} grid points. The current pipe simulations have 11.85 × 10^{6} grid points for a 20*R* length-equivalent and for the same *Re*_{τ}. Where there has been deviation from the plots of Orlandi and Fatica (1997) for the validation plots of the present simulations, the reasons have been attributed to the difference in mesh resolution as well as the difference in the DNS numerical method; essentially, the second-order finite difference method for Orlandi and Fatica (1997) vs the spectral element method of Nek5000. Where there are differences, attempts are made to further verify them with those of Ould-Rouiss (2010). Ould-Rouiss (2010) use 4.28 × 10^{6} grid points in their rotating pipe, which is a closer comparison with the 8.89 × 10^{6} grid points used in the 15*R* length-equivalent of the present simulations.

Unlike Orlandi and Fatica (1997), with increased rotation, the profile moves away from the theoretical curve of *r*^{2} and toward solid body rotation (Ω*r*), but the viscous effects of turbulence are reduced, and hence the profile tends toward being more parabolic. In the low rotation rates, the curve is closer to the theoretical *r*^{2}, but near the wall, it is further away due to lower turbulence and higher viscous effects. This confirms the findings of Mullyadzhanov (2017).

### B. The concentric pipe under rotation

Previous DNS studies of the concentric pipe are limited to Chung (2002) and Chung and Sung (2003) for the non-rotating (*N* = 0) case. There has only been one previous DNS study of the concentric pipe with the inner wall rotating at *N* = 0.429 by Jung and Sung (2006). As such, larger turbulent scale mean and fluctuating velocity comparisons are restricted to these two. There have been several LES studies of the concentric pipe with inner wall rotation. The LES study by Chung and Sung (2005) has four rotating rate cases: *N* = 0, 0.2145, 0.429, and 0.858. Therefore, comparisons have also been made against this research where necessary. There are inevitable discrepancies due to the difference in mesh resolution, as mentioned at the end of Sec. I C.

#### 1. Concentric pipe mean streamwise velocity

The streamwise velocity, as shown in Fig. 5, is in good agreement with both Chung (2002) for the *N* = 0 case and Jung and Sung (2006) for the *N* = 0.429 case. Note here that the left side is toward the inner wall at *y* = 0, and the right side is toward the outer wall at *y* = 0.5. This also goes for all ensuing plots. The streamwise velocity profile becomes less biased toward the inner wall, and the asymmetry moves toward the outer wall with inner wall rotation. Even with rotation, at no time do the mean flow profiles become symmetric, as such, mean flow profiles are usually plotted across the entire gap span 2*δ*. The exception is when plotting in wall units, as the friction velocity is different for either wall, and likewise for the extent of the boundary layer from either wall in wall units, *Re*_{τ}.

#### 2. Concentric pipe azimuthal velocity and reference frames

The mean azimuthal velocity for the concentric pipe with inner wall rotation is given in Fig. 6, in both the inertial and rotational reference frames. From the previous arguments established in the pipe with axial rotation, mean azimuthal velocity in the rotating reference frame can be ascertained by removing the linear azimuthal velocity, *V*_{θ} = Ω*r*, which is the solid body rotation of laminar Couette flow, from the turbulent mean velocity, which is essentially turbulent Couette flow. This *deviation* gives azimuthal velocity in the rotating reference frames. This is shown in Fig. 6. An illustration of this is given in Tuckerman (2020) as deviations of turbulent Couette flow from laminar Couette flow.

Unlike in the pipe with axial rotation, the azimuthal velocity in the concentric pipe with the inner wall rotating in the rotating reference frame is quite large. For the *N* = 2 case, the maximum *V*_{θ} is around 1. At the corresponding radial position, the streamwise velocity is also around 1, hence *Ro* = 1 in this region. From literature review, Davidson (2013), Godeferd and Lollini (1999), etc., inertial waves can be sustained around *Ro* ≤ 1.

Furthermore, in the rotational reference frame, there will be counter-rotating flows generated. Negative clockwise across most of the radial gap in the region 0 < *y* ≤ 0.35 and anticlockwise positive from *y* ≥ 0.35. Similar flow segregation was observed by Taylor (1923). This rotational segregation in orientation is consistent for all rotation rate cases, hence appearing to be independent of *N*. Other two parameters that might influence this consistency are the radius ratio *α*, the mean streamwise velocity *U*, or some function of it (e.g., the ratio of inner to outer *Re*_{τ}).

#### 3. Concentric pipe mean velocity fluctuations

Reasonable comparisons are made in mean velocity fluctuation *rms* plots, with the plots of Chung (2002) for the *N* = 0 case and Jung and Sung (2006) for the *N* = 0.429 case. This can be seen in Fig. 7. In these figures, it can be seen that there is a marked increase in fluctuations with rotation, which is indicative of an increase in turbulence with inner wall rotation.

The half sum of the diagonal of the Reynolds shear stress (RSS) tensor components (the $vii\u2032$ variances) gives the turbulent kinetic energy (TKE). There is an increase in TKE with rotation, as shown in Fig. 8(a). This is expected, as indicated in the previous figure (Fig. 7). Figure 8(b) shows the structure parameter. The structure parameter is given as $a1=[vr\u2032vz\u2032\u03042+vr\u2032v\theta \u2032\u03042+vz\u2032v\theta \u2032\u03042]1/2vr\u20322\u0304+v\theta \u20322\u0304+vz\u20322\u0304$ (Townsend, 1961). Unlike in other 3DTBLs, in the concentric pipe, *a*_{1} increases with rotation above the *N* = 0 case. On the inner rotating wall, this increase continues to *N* = 0.858 but drops for the *N* = 2 case. This can be attributed to a very high TKE in comparison to RSS. The trend on the outer non-rotating wall continues to increase with rotation. High *a*_{1} at the rotating wall was also observed by Poncet (2014) in the concentric pipe with a high radius ratio *α*. High *a*_{1} has also been observed in the pipe with wall rotation in the current research (Dunstan, 2020), by Orlandi and Fatica (1997), and for flow in the concentric pipe with the outer wall rotating by Hadziabdic (2013). The energy partition parameter *K** is given as $K*=2vz\u20322\u0304vr\u20322\u0304+v\theta \u20322\u0304$ (Chung , 2002; Poncet , 2014). *K** is shown in Fig. 8(b). As can be anticipated, this reduces with increased rotation rate as the radial and azimuthal velocity fluctuations become more prominent. The plots are reflective of Rossby number across the gap span. As can be seen for the *N* = 2 case, the value of *K** is not much above unity across most of *y* from the rotating inner wall. In the pipe with rotation, *K** around 1 results in eddy elongation (Dunstan, 2020). However, here *Ro* is also around one, and therefore an inertial wave appears to take precedence over eddy elongation (Sec. III B 5).

#### 4. Concentric pipe instantaneous radial velocity fluctuations

Figure 9 shows 3D *shells* taken from the inner wall, to a distance of *y*^{+} = 5 away from the inner wall, for the *N* = 0, 0.429, and 2 cases. In the obvious wall streaks seen, splatting is apparent with the negative $vr\u2032$ fluctuations being converted to $vz\u2032$ and $v\theta \u2032$ fluctuations. There is an obvious tilting of the wall streaks here, which is very clear. The adjacent hemispherical figure is a half section taken from the middle of the concentric pipe and gives a perspective of the fluctuations across the section. There is a strong indication of increased turbulence with right-handed inner wall rotation. For the *N* = 2 case, the streaks appear to be tilted at −55° to the streamwise axial *z*-direction [Fig. 9(c)].

#### 5. Concentric pipe *λ*_{2} visualizations

The instantaneous *λ*_{2} plots in Fig. 10 are *shell* regions taken in the concentric pipe from near the rotating inner wall up to *y*^{+} = 20 away from the wall, which is further into the gap-span. The *shells* show *λ*_{2} at various contour levels and are colored by streamwise velocity. In these figures, as with the $vr\u2032shell$ images, streamwise flow is from left to right, and the positive anticlockwise wall rotation is right-handed, therefore, up around the back of the curvature of the *shell*. For the lower/right inner view of the *shell*, wall streaks can be discerned. These *λ*_{2} wall streaks are reflective of the figures shown in Fig. 9 for $vr\u2032$ streaks and, therefore, allay any concerns from Haller (2004) regarding the objectivity of *λ*_{2} for visualizations in these simulations. With rotation, it is quite obvious that the entire flow becomes tilted, and the flow actually becomes corrugated in helical wave-like forms. For the *N* = 2 case in Fig. 10(c), the flow is clearly high speed at the corrugation crests and low speed in the troughs. This image should be examined concurrently with Fig. 6, where the largest |*V*_{θ}| of 1 occurs at *y* ≈ 0.02, which is equivalent to *y*^{+} ≈ 15 (at *Re*_{τ} = 800). Clearly, a wave is discernible at *N* = 2, precisely where *V*_{θ} (in the rotational reference frame) approaches one, at *y*^{+} ≈ 15, and so *Ro* = 1. Therefore, an inertial wave is sustained.

#### 6. Concentric pipe flow orientation angle and resultant velocity

The flow angle for flow direction *γ*_{s} commonly given for any 3DTBL is calculated here and plotted. The angles on the vertical axis are in degrees (°), and the distance across the gap span is given on the horizontal axis. For flow direction, the equation for *γ*_{s} is commonly given as $\gamma s=tan\u22121(W\theta \u2212W\theta wall)U$. In the rotational reference frame, *γ*_{s} is here modified as *γ*_{s} = tan^{−1} *V*_{θ}/*U*. This modification gives the true resultant flow orientation all across the concentric pipe gap.

Also in this presentation, the following calculations for resultant velocity are made: $|V|=U2+U\theta 2$, $V\tau =\nu \u2202U\theta \u2202rw+\u2202U\u2202rw$, *V*^{+} = |*V*|/*V*_{τ}, and *y*^{+} = *y*_{w}*V*_{τ}/*ν*.

By taking the arctan of *W*_{θ}/*U* with an unmodified mean azimuthal velocity, we can get the flow direction in the inertial reference frame; this is shown in Fig. 11(a). The magnitude of this is shown in Fig. 11(b). Note that the integral of each rotation rate curve will not give a consistent mass flow rate as their orientations (to axial) are not equal. If the azimuthal velocity in the rotating reference frame (*V*_{θ}) is used instead, as was obtained in Fig. 6, the resultant direction is now shown in Fig. 11(c), and the resultant magnitude of the azimuthal velocity in the rotating reference frame is shown in Fig. 11(d).

Evidently, from the velocity fluctuation wall streaks and the instantaneous *λ*_{2} 3D plots [e.g., Fig. 10(c)], the flow is in the rotating reference frame, regardless of the static outer wall. Besides, in Figs. 11(a) and 11(c), at *y* = 0.5, both plots converge on an equivalent angle at the outer wall (e.g., ∼ +27° for *N* = 2). Take, for instance, the *N* = 2 case at *y* = 0 the flow is oriented at around −55° to the axial. This is obviously the case in the streak and instantaneous *λ*_{2} images [Figs. 9(c) and 10(c)]. Figure 12, for the outer nonrotating wall of the *N* = 2 case, shows that streak tilting is in accordance with the given angle in Fig. 11(c) at *y* = 0.5, which is around +27°. The right handed convention must be considered throughout.

Having established that the actual flow orientation and magnitude are in the rotating reference frame, the effective semi-log plot needs to be considered with respect to the effective friction velocity in this orientation. In Fig. 13, *V*^{+} and *y*^{+} are obtained using the friction velocity *V*_{τ}. The azimuthal velocities used are from both the inertial (dashed lines) and rotating reference frames (solid lines). First, it can be seen that the plots in the inertial reference frame (dashed line) are distorted in the viscous sublayer with respect to the *N* = 0 case and, therefore, should not be considered. The viscous regions for the plots in the rotational reference frames (solid line) do coincide with the *N* = 0 case and are therefore in an equal basis. For the *N* = 2 case on the inner wall [Fig. 13(a)], it can be seen that the boundary layer is very thin. For drag reduction studies, the question is: how is it possible to shift the *V*^{+} plot of Fig. 13(a) vertically up by a large measure? One answer is: since an inertial wave is formed close to the rotating inner wall (∼*y*^{+} = 15) at high *Re*_{τ}, is it possible to accommodate the low velocity troughs in physical grooves? Such physical grooves have been observed in nature. In the not clearly discernible herringbone pattern on birds’ feathers (Chen , 2013) and the cutaneous ridges in dolphin skin (Lisi, 1999). Larger grooves are more clearly visible on the throat area of humpback whales, etc.

In the concentric pipe, to compensate for the short boundary layer on the rotating inner wall, on the non-rotating outer wall the boundary layer is elongated [Fig. 13(b)].

## IV. CONCLUSIONS

This research has performed DNS simulations of turbulent axial flow in the rotating pipe and concentric pipe with inner wall rotation, with a better resolved mesh than most earlier comparable investigations. Preliminary examination of the pipe with rotation in different reference frames has confirmed quasi-relaminarization, accompanied by increased TKE. These preliminary checks in the pipe with rotation have shown how the concentric pipe with inner wall rotation can also be similarly examined in a rotating reference frame by removing the solid body rotation component, leaving only the fluid intrinsic rotation, which is independent of its center of mass.

In a concentric pipe with inner wall rotation, there is always counter-rotation in the flow. This will occur at the same radial position in the gap span regardless of the rotation rate and is likely to be a function of the radius ratio *α*. Taylor (1923) observed the same phenomenon in his experiments with both walls of a concentric pipe rotating in opposite directions. Hence, an affirmation of fluid behavior in the rotational frame. In the investigations into the turbulent jet by Facciolo and Alfredsson (2004), an absence of a wall in the ambient jet may have resulted in inadvertent observation of flow in a rotational frame, hence their reported counter-rotation in the jet.

In studying the concentric pipe with inner wall rotation in the rotational reference frame, it has been seen that wall streak orientation, and indeed flow orientation, can now be correctly predicted. The prominence of azimuthal velocity relative to streamwise velocity is now accurately determined. It is also essential that turbulent flow in the rotational reference frame be always considered in material frame dependent calculations, for example, in modeling analysis, such as in the Reynolds stress transport equations and for energy budgets.

## ACKNOWLEDGMENTS

The author is grateful to the University of Warwick, Centre for Scientific Computing, for use of the HPC facilities *Tinis* and *Orac*, and to HPC West Midlands for use of *Athena*. The author also thanks Professor Yongmann Chung for supervising the dissertation research at the University of Warwick, and Professor Rustam Mullyadzhanov for sharing data on the rotating pipe. Last but not least, the author thanks Professor Augustine Moshi, former Pro-VC of PNG Unitech, for his invaluable support and mentoring.

The Ph.D. research was funded by the Commonwealth Scholarship Commission and the Papua New Guinea University of Technology.

## AUTHOR DECLARATIONS

### Conflict of Interest

The author has no conflicts to disclose.

### Author Contributions

**Samuel D. Dunstan**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Resources (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## REFERENCES

*Turbulence in Rotating, Stratified and Electrically Conducting Fluids*