Much faster heat/mass than momentum transport in rotating Couette flows

Heat and mass transport is generally closely correlated to momentum transport in shear flows. This so-called Reynolds analogy between advective heat or mass transport and momentum transport hinders efficiency improvements in engineering heat and mass transfer applications. I show through direct numerical simulations that in rotating plane Couette and Taylor-Couette flow wall-to-wall passive tracer transport, representative of heat mass/transport, can be much faster than momentum transport, clearly in violation of the Reynolds analogy. This difference between tracer and momentum transport is observed in steady flows with large counter-rotating vortices at low Reynolds numbers as well as in fully turbulent flows at higher Reynolds numbers. It is especially large near the neutral (Rayleigh's) stability limit. The rotation-induced Coriolis force strongly damps the streamwise/azimuthal velocity fluctuations when this limit is approached while tracer fluctuations are much less affected. Accordingly, momentum transport is much more reduced than tracer transport, showing that the Coriolis force breaks the Reynolds analogy. At higher Reynolds numbers this strong advective transport dissimilarity is accompanied by a limit cycle dynamics with intense low-frequency bursts of turbulence when approaching the neutral stability limit. The study demonstrates that body forces can induce highly efficient heat/mass transport in shear flows.


INTRODUCTION
Advective transport of heat and mass by fluid motions is fundamental to planetary and astrophysical processes and many engineering applications [1,2]. Efficient advective transport contributes to energy savings in buildings [3], process industry [4] and data centers [5], and can be obtained by e.g. applying wall roughness [6] and flow-control [7]. Especially optimal transport given a minimal power input generates energy savings in applications [5,8], but optimization is challenging since flow vortices and eddies generally transport momentum and heat/mass at similar rates. This so-called Reynolds analogy between transport of momentum and heat/mass was postulated by [9], and applies to many shear flows [2,10] including astrophysical flows [11]. The Reynolds analogy is used for modelling advective transport in engineering [2], geophysical [12] and astrophysical flows [13], but implies that higher heat/mass transfer goes together with higher momentum transfer and thus power input.
In recent theoretical studies, incompressible steady flows are computed that maximize heat transfer for a given power input [5,8,14]. [8] consider plane Couette flow and show that the optimized flow has a much higher heat transfer for a given power input than the ordinary turbulent flow. The computed optimized flows are not required to obey known momentum equations in these theoretical studies, that is, these optimal flows can be obtained applying a body force, but the body force can be arbitrary and does not (necessarily) have a familiar form. It is therefore not clear if this optimal transport is realizable, although [5] and [8] suggest that optimal flows can be approached by applying smart forcing or control techniques. I show through direct numerical simulations (DNSs) that in existing flows, namely incompressible plane Couette flow (PCF) and Taylor-Couette flow (TCF) subject to a Coriolis force, passive tracer transport can be much faster than momentum transport, in violation of the Reynolds analogy. It is therefore not only theoretically but also actually possible to optimize wall-to-wall transport by body forces.

GOVERNING EQUATIONS AND NUMERICAL PROCEDURE
TCF is a shear flow created between two rotating concentric cylinders while PCF is the smallgap limit d/r i → 0 (η = r i /r o → 1) of TCF, where d is the gap between the cylinders/walls and r i and r o the inner and outer radius, respectively. Many flow properties of TCF are similar to these of PCF for η 0.9 [15]. Momentum transport in Couette flows has been explored extensively [16,17] owing to its relevance for e.g. astrophysics. In these flows I study passive tracer transport mimicking heat and mass transport when the temperature/mass does not affect the flow. Hereafter, the passive tracer is called temperature for convenience but the only body force affecting the flow is the Coriolis force that does not perform any work.
Fluid motion and passive tracer transport in the PCF and TCF DNSs are governed by the Navier-Stokes and advection-diffusion equation together with ∇ · U = 0. The imposed azimuthal (streamwise) velocity and temperature at the inner and outer no-slip and iso-thermal walls ±U w and ±T w , respectively, are constant. Velocity U is normalized by U w , temperature T by T w , and length by d. The modified non-dimensional pressure P includes the centrifugal force [16]. The rotation axis, defined by the unit vector e z , is the spanwise and central axis in PCF and TCF, respectively, as in [15], and is parallel with the mean flow vorticity. Sketches of the flow geometries are presented in the supplementary material.
A Reynolds number Re = ∆Ud/ν and rotation number Ro = 2Ωd/∆U, where ∆U = 2U w , ν the kinematic viscosity and Ω the imposed system rotation, characterize the flow. Ro is defined such that it is negative for cyclonic flows (same sign for shear and rotation) and positive for anticyclonic flows. These parameters are equivalent to the shear Reynolds and rotation numbers used by [15,18]. The rotating reference frame for TCF can naturally be translated back to a laboratory reference frame, see e.g. [19]. Pr = ν/α is the Prandtl number with α the thermal diffusivity.
From (1) and (2) follows that in PCF the wall-to-wall mean dimensionless momentum J m = UV − ∂ y U /Re and heat fluxes J h = VT − ∂ y T /(RePr) are conserved, and in TCF the are conserved [15]. Here, ω = U/r is the angular velocity, U and V the streamwise (azimuthal) and wall-normal (radial) velocity, and ... denotes averaging over time and area at constant wall-normal (radial) distance in PCF (TCF). The Nusselt numbers Nu m = J m /J m lam and Nu h = J h /J h lam quantify the wall-to-wall (angular) momentum transport [15] and heat transport, respectively. The subscript To study the influence of Coriolis forces on momentum and heat transfer I have carried out The governing equations for PCF are solved with a Fourier-Fourier-Chebyshev code, with periodic boundary conditions in the streamwise and spanwise directions [20]. The computational domain size is 6πd and 2πd in the streamwise and spanwise direction, respectively, which is large enough to accommodate several pairs of counter-rotating large-scale vortices. The governing equations for TCF in cylindrical coordinates are solved with a Fourier-Fourier-finite-difference code [21,22], with periodic boundary conditions in the axial and azimuthal directions. In the radial direction, a sixth-order compact-finite-difference scheme is used. Like others, I do not simulate the flow around the entire cylinder but use a domain with reduced size in the azimuthal direction. Previously, it has been verified that changing the domain size has little effect on the computed torque [15,23]. The computational domain size in the DNSs of TCF, listed in Table   I, is basically the same as in the DNSs of [23] up to Re = 29 531 and wide enough to capture at least one pair of counter-rotating Taylor vortices. In the DNSs at Re = 40 000 the domain is significantly larger. is linearly unstable and Nu m > 1 if Ro = 0 at all Re [18]. In both flows Nu m first grows with Ro due to destabilization by anticyclonic rotation and then declines towards unity for Ro → 1 when the flow approaches the linearly stability limit Ro c and relaminarizes [18]. Disturbances and turbulence cannot sustain beyond Ro c , even at higher Re [26]. Momentum transport is maximal around Ro = 0.2 in PCF and around Ro = 0.3 to 0.1 at low to high Re in TCF, consistent with previous numerical [15,16,23] and experimental observations [27]. This broad maximum is linked to intermittent bursts in the outer layer in TCF and to strong vortical motions in PCF [15,23].

The resolution increases with
Another narrow maximum in Nu m caused by shear instabilities appears in PCF at Ro ≈ 0.02 at sufficiently high Re [23], but my DNSs do not cover this narrow region near Ro = 0 and therefore do not reveal this second maximum. With increasing Re this narrow maximum overtakes the broad maximum, which disappears if Re is higher than in my DNSs and η ≥ 0.9, as shown by [19].
Heat transfer in terms of Nu h behaves similarly as Nu m at low Ro but differently at higher Ro ( figure 1.c,d). Its maximum is higher and at higher Ro for almost all Re, demonstrating that flow structures causing optimal momentum transport do not necessarily cause optimal heat transport.
At higher Re, Nu h is maximal near Ro = 0.5 in both PCF and TCF and then sharply declines when Ro → 1 and the flow relaminarizes. This means that in higher Re TCF maximal momentum and heat transport happens with moderately counter-rotating and co-rotating inner and outer cylinders, respectively, in a laboratory frame of reference. The growth of the maximum Nu m and Nu h with Re show similar trends in PCF and TCF and follows Nu m , Nu h ∼ Re 0.6 at higher Re ( Fig. 2.a).
Experiments show that at high Re the maximum Nu m ∼ Re 0.77 [23,27], suggesting that at high Re also the maximum Nu h follows a similar scaling.
The ratio HTE = Nu h /Nu m , shown in figure 3, is a measure of heat transfer efficiency since Nu m is proportional to the power input [27]. A high similarity between momentum and heat transport can be expected at Ro = 0 in PCF because Pr = 1 and momentum and heat transport are similarly forced. This is vindicated by the DNSs; the difference between Nu h and Nu m is not more than 2% at all Re and accordingly HTE ≃ 1, meaning that the Reynolds analogy perfectly applies. HTE is somewhat smaller in TCF at Ro = 0 because Pr < 1 but still near unity so that the Reynolds analogy practically holds. Clear differences in heat and momentum transport emerge for increasing Ro. HTE rapidly grows with Ro in PCF and TCF and reaches a maximum around Ro ≈ 0.85 − 0.99 at low to high Re before abruptly dropping to unity for Ro ≥ 1. Its maximum grows from about two at the lowest Re to eight and more than six at Re = 40 000 in PCF and TCF, respectively ( figure 3 and figure 2.b). In TCF the maximum HTE seems to level off at higher Re while in PCF it still grows. Note that Couette flow is linearly unstable very near Ro = 1 [28,29], so flow motions can be sustained even very near Ro = 1. Figure 4 shows    where u and v is the streamwise and wall-normal velocity fluctuation, respectively; if Ω > 0 the Coriolis force reduces production of u by mean shear when v 0. Note that the absolute mean vorticity ∂ y U − 2Ω ≈ 0 about the channel centre at sufficiently high Ro [15,30] and in the whole channel if Ro → 1. The Coriolis term in the Reynolds stress transport equation of uu counterbalances then the production term and the only term producing uu is the pressure-strain correlation [31]. If a fluid particle is displaced in the wall-normal direction by vortical motions the Coriolis force basically accelerates or decelerates the particle so that its streamwise velocity approaches the local mean velocity.  figure 6.(a,b). These vortices appear above the stability limit [28,29] and echo structures producing optimal heat transport in theoretical studies of PCF [8]. They transport considerable heat but little momentum since streamwise velocity fluctuations are small when Ro → 1, as discussed above. For streamwise-invariant PCF with Pr = 1 one can further quantify this and derive from (1) and (2)û whereû andθ are the streamwise velocity and temperature deviations from the laminar situation, respectively. Further, using a variable transformation as in [32] gives These relations are exact as long as Re ≤ 800 and PCF is streamwise-invariant. Equation The growth rate of the most unstable mode follows a similar trend as the burst frequency ( figure   7.c), suggesting that the bursts are related to linear instabilities.

CONCLUDING REMARKS
The key conclusion of my study is that heat and mass transfer can be optimized by imposing a body force on the flow, as indicated theoretically recently [5,8]. Optimization of heat/mass transfer by body forces is thus a promising avenue for further research. The mechanism of momentum transport reduction by the Coriolis force does not depend on Re, implying that the observed dissimilarity between momentum and heat transfer persists at higher Re. The highest dissimilarity happens in rotating Couette flows close to the inviscid neutral stability state. Also other rotating shear flows tend to evolve towards this state [33,34], suggesting that heat and mass are transported much faster than momentum in such flows. Dissimilarity between momentum and heat transport is also found in rotating channel flow [35][36][37], albeit in a limited region where the flow approaches the zero-absolute-mean-vorticity state, and in shear flows with buoyancy forces [38,39]. In DNS and rapid distortion theory of rotating uniformly sheared turbulence [40] observed turbulent Prandtl numbers much smaller than one when the zero-absolute-mean-vorticity state is approached, which also implies fast heat transport. This all suggests that more engineering and astrophysical flows display dissimilarities between heat or mass transfer and momentum transfer. Another implication of the present study is that heat and mass transfer modelling in flows with body forces requires careful considerations since the Reynolds analogy can fail. * geert@mech.kth.se