Anisotropy and relaminarisation of the turbulent flow near a rotating cylindrical cavity wall

Rotor internal cooling is a new concept for high power density electric vehicle drives. A turbulent pipe flow is injected into a cylindrical cavity in the rotor shaft of the motor. The flow is deflected in the cavity, accelerated in circumferential direction by the rotor wall and exits through an annular duct with the outer wall rotating. Due to the opposing effects of rotation on turbulence, a complex transitional flow develops. The strong shear layer in the jet region causes high turbulence production. On the other hand, these fluctuations are damped by the centrifugal forces due to the flow rotation. To investigate the influences of the rotation on the turbulence properties and the mean flow, highly resolved large eddy simulations are performed. It is shown that the turbulence production and attenuation due to rotation affect different components of the Reynolds shear stress tensor. This results in highly anisotropic turbulence. In certain areas, where the turbulence attenuation is strongest, the flow even relaminarises. Since the cooling efficiency depends on the turbulent heat transfer of the flow, the local turbulence characteristics are key quantities for the cooling application.


INTRODUCTION
Asynchronous machines can be used in electric vehicle drives to avoid the use of permanent magnets and rare earths.A considerable amount of heat is generated in the rotor windings and it is difficult to transfer the heat from the centre of such a highly integrated drive motor.Rotor internal cooling is a new concept for cooling of electric vehicle drives with high power density and able to solve this issue.A turbulent pipe flow enters a cylindrical cavity in the rotor, forming a jet.Within the cavity the flow is deflected and exits through an annular duct formed by the stationary pipe wall on the inside and the rotor wall on the outside.This creates a complex transitional flow.The basic opposing effects of turbulence intensification and damping by flow rotation are known in simple configurations, such as a rotating pipe.In an earlier study [1] the laminarisation of a turbulent pipe flow with a Reynolds number of  = 10 000 entering a rotating pipe was studied by velocity measurements with a laser Doppler velocimeter (LDV).Laminarisation was observed in the rotating layer near the entrance of the rotating pipe, resulting in a significant reduction of drag.However, further downstream the flow was destabilised again by bursts of turbulence.LDV measurements of Reynolds stresses in a turbulent rotating pipe flow at  = 20 000 were presented in a later work [2].Turbulent fluctuations were found to decrease gradually with an increasing rotation rate and the turbulent shear stresses were suppressed even more strongly than the turbulent kinetic energy.Direct numerical simulations of an axially rotating pipe flow at  = 5000 were carried out in literature [3], and in a further study [4] for a slightly higher Reynolds number.It was found that the reduction of drag and turbulence was not only caused by the centrifugal force of the swirling flow, but also by a modification of the near wall vortical structures.More recently, the suppression of turbulence by streamwise-varying wall rotation was studied using direct numerical simulations [5].With a sufficiently high velocity magnitude and wavelength, relaminarisation of the flow was achieved.Examining the energy fluxes it was shown that the rotational effect caused part of the energy of wall normal fluctuations to be absorbed by the circumferential mean flow.This was assumed to be the main cause of the decay of turbulence.
These effects of rotation have not yet been investigated in more complex flow situations, such as the present one.In this study, the general structure of the flow and the turbulence properties are analysed for a given rotation rate.Turbulent fluctuations are enhanced or reduced in specific areas, which has significant impact on the heat transfer in the cooling application.It is investigated, how the rotation of the flow induces highly anisotropic states of turbulence that vary widely across the domain.

SETUP
Figure 1 shows the considered configuration, which was developed according to the patents [6,7], but with reduced complexity.The flow enters the domain through a stationary pipe with diameter  and a length of  p = 15.At the inlet, the velocity distribution is prescribed by a 1∕7 power law.A tripping force is used to activate the turbulence [8], resulting in a developed turbulent pipe flow at the outlet of the pipe with Re = 17 640 based on the bulk velocity  b .The cylindrical cavity has a length of  c = 5 and a diameter of  = 2 = (7∕3).The annular duct leading to the outlet of the domain has a width of  = (7∕12) and a length of  a = 12.At the end of the annular duct, the outlet of the domain is located in the outer wall and has an axial extent of  o = 2∕3.A short fixed wall section with an axial extent of  f = ∕2 is positioned between the outlet and the rotating wall section for reasons of numerical stability.The ratio of the circumferential velocity of the outer wall to the bulk velocity in the pipe is  = Ω∕ b = 2.An open boundary condition [9] is used at the outlet.Although the simulation typically shows no backflow and the level of turbulence at the outlet is at most moderate, this boundary condition ensures energy stability of the system even when strong vortices or backflow occur.In this first simulation, heat transport is not included, so the viscosity and density of the fluid are set constant.Time averaging was started when a statistically stationary flow was reached.Then, the flow was averaged for about 18.5 revolutions of the rotor wall, which is 4100 viscous time units of the pipe flow.

Solver and discretisation
To simulate the flow, the spectral element solver Semtex [10] was employed.It solves the incompressible Navier-Stokes equations in an axisymmetric domain using a cylindrical coordinate system (, , ), with , ,  the axial, radial and circumferential velocity components, respectively.The domain was discretised by 5253 elements with a polynomial order of 10 in the meridional plane.With such a grid the wall boundary layers are fully resolved for the present physical parameters.In the circumferential direction a Fourier expansion with 240 modes was used.Time integration was carried out with a semi-implicit scheme of third order.

Spectral vanishing viscosity
A spectral vanishing viscosity (SVV) approach was used to stabilise the solver and account for the smallest turbulent eddies.The radially shifting SVV kernel of Koal et al. [11] was adapted and applied in the circumferential direction, otherwise the standard kernel of Maday et al. [12] was applied.With the SVV method, the viscosity is increased for the smallest resolved wavelengths, which can be considered similar to the use of a subgrid scale model in classical large eddy simulations.Local maxima of the additional dissipation generated by the SVV remained in the same order of magnitude as the resolved dissipation and the volume-averaged ratio of additional to resolved dissipation was approximately 12 %.With this approach, the turbulence was well resolved in most parts of the domain, except in the highly turbulent jet region.corresponding Reynolds normal stresses ⟨ ′  ′ ⟩ and ⟨ ′  ′ ⟩ in radial and circumferential direction.Angular brackets ⟨ ⟩ indicate averaging in time and circumferential direction.Near the outer wall (∕ = 0.98) a clear peak of TKE is observed, resulting from the even higher peak of the circumferential Reynolds normal stress ⟨ ′  ′ ⟩.This is caused by the steep radial gradient of the circumferential velocity and a negative Reynolds shear stress ⟨ ′  ′ ⟩, leading to an high production of turbulent fluctuations in the circumferential direction.In contrast to this, the radial Reynolds normal stress ⟨ ′  ′ ⟩ is relatively small.Although there is some redistribution of turbulent fluctuations from the circumferential to the radial direction by pressure and viscous diffusion, the stabilising effect of the rotation overwhelms it and the radial fluctuations remain small.This leads to a highly anisotropic, almost one-component state of turbulence in the near wall area of the jet region [13].

Jet region
The jet shear layer can be seen approximately between ∕ = 0.3 and ∕ = 0.7.In this region, the TKE is dominated by the axial fluctuations, which are not shown separately here, while radial and circumferential fluctuations are of similar size.This creates an axisymmetric state of turbulence with one large eigenvalue of the anisotropy tensor [13, pp.393 ff.].

Intermediate region
Close to the pipe outlet (and annulus inlet) an intermediate region is created, which is shown in Figure 4 in an enlarged view.At the top the averaged velocity magnitude is shown with the corresponding in-plane streamlines, and at the bottom the instantaneous velocity field.In this region, the influence of the jet decreases and the rotating boundary layer grows rapidly around  ≈ 0.2  c .The turbulent structures in this section of the rotating layer are comparatively big, which can be seen in the instantaneous velocity field, and a large amount of circumferential momentum is transferred radially inwards.The broader rotating layer causes an increase of pressure along the rotating wall due to the centrifugal force.Consequently, the axial flow towards the annular duct is deflected radially inwards by the high pressure area.This can be seen from the streamlines in Figures 2 and 4, top.A small recirculation area appears in the rotating boundary layer at −0.1 ≤  ≤ 0.2  c , since the flow detaches due to the pressure increase.Both effects seem to reinforce each other.As a result, the rotating boundary layer grows faster because the axial flow is deflected inwards, leading to even higher pressure near the rotating wall, which then leads to stronger deflection of the axial flow until saturation.

Annular region
In the annular duct the situation changes completely.The flow rotates within the entire cross-section and the axial bulk velocity is lower than in the pipe ( b,a = 0.25  b ), which results in a lower bulk Reynolds number of  = 5150.The TKE and the Reynolds normal stresses shown in Figure 3A and c are overall approximately one order of magnitude lower than in the jet region.This large difference to the jet region is mainly due to the absence of the jet shear layer in the annulus and the lower bulk Reynolds number.But the stabilising effect of the stronger further amplifies the difference.Near the rotating wall (∕ > 0.8), the circumferential velocity component is dominating the flow, leading to a strong reduction of the TKE, as can be seen in Figure 3A.Particularly close to the rotating wall (∕ > 0.94) the flow begins to laminarise and the radial Reynolds normal stress is reduced to almost zero as the centrifugal forces in the flow stabilise the radial fluctuations.This becomes more pronounced further downstream as the rotating layer continues to grow.Even the circumferential Reynolds stress is reduced in the outer region ∕ > 0.8.At ∕ = 0.95 a small peak is visible due to the production caused by the high radial gradient of the circumferential velocity and a small negative Reynolds shear stress ⟨ ′  ′ ⟩.The anisotropy of the turbulence reaches an almost one-component state here, which is even more pronounced than in the jet region.For smaller radial positions, either the axial or the circumferential fluctuations are dominant in certain places.Both are of equal size close to the inner wall, where a two-component state of turbulence is reached.Together, the jet region and the annular region perfectly show the opposing effects of the rotation.In the jet region the production of turbulence is the main effect of the rotation and only partially reduced by stabilisation.In contrast, the stabilising effect of the rotation dominates in the annular region, and there is only a minor increase of fluctuations caused by the production of turbulence.

Heat transfer
Although the heat transfer is not included in this simulation, some conclusions can still be drawn in this respect.The wall normal turbulent heat flux depends mainly on the radial velocity fluctuations close to the rotating wall.These are mostly unaffected by the rotation in the jet region, but strongly reduced in the annular region.Therefore, it is likely that the heat transfer in the annular region would also be reduced to a similar extent, whereas it would hardly be affected in the jet region.Both regions form the main surface for heat transfer with similar proportions of it.Overall, the rotation would, therefore, have an unfavourable effect on the heat transfer.However, this could change depending on the rotation rate.Otherwise, design measures would have to be taken to increase the turbulence and thus the heat transfer for the cooling application.

CONCLUSIONS
A high-resolution large eddy simulation was performed to investigate the effects of flow rotation on the turbulence in a flow configuration that occurs in the application of rotor internal cooling of electric vehicle drives.The general appearance of the flow was shown and discussed for three characteristic regions, the jet region, the intermediate region and the annular region.In the intermediate region, a recirculation area was found in the rotating layer along outer wall where two effects reinforce each other.Further properties of the flow and the turbulence were analysed by studying radial profiles in the jet and the annular region.In both regions the opposing effects of turbulence intensification and damping can be seen.On the one hand, the steep radial gradient of the circumferential velocity near the rotating wall leads to an increase of turbulence production in the highly turbulent jet region.The stabilisation has little effect on the radial fluctuations and is overwhelmed by the redistribution of fluctuations from the circumferential direction.On the other hand, the stabilising effect is dominant in the stronger rotating annular region.Even the circumferential fluctuations are significantly reduced close to the rotating wall.However, in a small radial area they are increased again due to the production of turbulence.Meanwhile, the radial fluctuations disappear almost completely in the vicinity of the rotating wall and the flow begins to laminarise in this region.The states of turbulence anisotropy vary widely as the dominating component of the Reynolds stress tensor changes multiple times across the domain.In the strongly rotating layer of both regions, the turbulence becomes increasingly anisotropic and reaches an almost one component state close to the rotating wall.Although it was not considered in the simulation, it can be concluded that the heat transfer, especially in the annular region, is significantly reduced due to the stabilising effect of the rotation.The stabilising effect is expected to be even stronger for higher rotation rates.In a vehicle drive, this would correspond to high travelling speeds typically reached after periods of acceleration when heat release is greatest.This would be highly detrimental to the cooling application of the flow.
Ongoing studies include the heat transport in the flow to verify the conclusion for the influence of the rotation on the cooling application of this configuration.Furthermore, the budget terms of the Reynolds stress transport and turbulent heat flux equations will be evaluated and analysed to better understand the redistribution of turbulence between the different directions as well as the influence on the heat transport.

A C K N O W L E D G M E N T S
This research is funded by the European Social Fund and co-financed by tax funds based on the budget approved by the members of the Saxon State Parliament.The authors also gratefully acknowledge the GWK support for funding this project by providing computing time through the Center for Information Services and HPC (ZIH) at TU Dresden.
Open access funding enabled and organized by Projekt DEAL.

F I G U R E 1
Simulation domain.Blue walls indicate rotation around the -axis.The vertical red lines A and B mark the position of the profiles in Figure 3 at  = −0.25  c and  = 0.25  c , respectively.

F I G U R E 2
Flow in rotating cavity with  = 2. Top: Streamlines and velocity magnitude of mean flow.Bottom: Snapshot of the instantaneous velocity magnitude.Inflow pipe not shown.

Figure 2 ,F I G U R E 3
Figure2, top, shows the mean flow, averaged in time and circumferential direction.A snapshot of the instantaneous flow is presented in the lower part of the figure.The turbulent pipe flow entering the rotating cylinder creates a highly turbulent jet with a strong shear layer visible in the mean flow graph.The radial expansion of the jet is restricted by the deflected flow in the negative axial direction near the rotating outer wall.When the turbulent jet reaches the back wall of the cavity, the flow stagnates and starts to rotate with the wall.Centrifugal forces accelerate the fluid in radial direction along the back wall.The rotating boundary layer along the outer wall remains very thin in the jet region ( ≥ 0.25  c ) since the incoming jet flow inhibits the rotation.High momentum exchange is caused by the turbulent shear layer of the jet, which results in a steep velocity gradient and enhanced turbulence production near the rotating wall.In the following, different properties of the flow are discussed on the basis of the profiles in Figure3taken along two radial lines marked as A and B in Figure1.Radial profiles of the circumferential velocity ⟨⟩ as well as the turbulent kinetic energy (TKE)  = ⟨    ⟩∕2 in the jet region at  = 0.25  c are shown in Figure3B, while Figure3Ddepicts the

F I G U R E 4
Enlarged view of the flow in the intermediate region close to the end of the inlet pipe.Top: Streamlines and velocity magnitude of mean flow.Bottom: Snapshot of the instantaneous velocity magnitude.The rotation axis is marked by the dash-dotted lines.