Microconfined high-pressure transcritical fluid turbulence

Microﬂuidics technology has grown rapidly over the past decades due to its high surface-to-volume ratios, ﬂow controllability, and length scales efﬁciently suited for interacting with microscopic elements. However, as a consequence of the small rates of mixing and transfer they achieve due to operating under laminar ﬂow regimes, the utilization of microﬂuidics for energy applications has long been a key challenge. In this regard, as a result of the hydrodynamic and thermophysical properties they exhibit in the vicinity of the pseudo-boiling region, it has recently been proposed that microconﬁned turbulence could be achieved by operating at high-pressure transcritical ﬂuid conditions. Nonetheless, the underlying ﬂow mechanisms of such systems are still not well characterized, and, thus, need to be carefully investigated. This work, consequently, analyzes supercritical microconﬁned turbulence by computing direct numerical simulations of high-pressure ( P = P c ¼ 2) N 2 at transcritical conditions imposed by a temperature difference between the bottom ( T = T c ¼ 0 : 75) and top ( T = T c ¼ 1 : 5) walls for a friction Reynolds number of Re s ¼ 100 (bottom wall). The results obtained indicate that microconﬁned turbulence can be achieved under such conditions, leading to mixing and heat transfer increments up to 100 (cid:3) and 20 (cid:3) , respectively, with respect to equivalent low-pressure systems. In addition, it is found that the near-wall ﬂow physics deviates from a single-phase boundary layer theory due to the presence of a baroclinic instability in the vicinity of the hot/top wall. This instability is generated by the combination of the external force driving the ﬂow and the large variation of density across the pseudo-boiling region, which strongly modiﬁes the ﬂow behavior in the vicinity of the wall and renders present “law of the wall” transformation models inaccurate.


I. INTRODUCTION
Standard microfluidic systems are typically limited to operate under laminar flow regimes due to their small characteristic hydraulic diameters [D h $ 1-1000 lm] and bulk velocities [u b $ oð1Þ m/s]. In particular, at atmospheric pressure conditions, the Reynolds numbers encountered in microfluidics are in the order of Re b ¼ u b D h = $ 0:1-100, where $ 10 À6 -10 À4 m 2 /s are typical values for the kinematic viscosity and consequently flows tend to remain laminar. 1 The smooth nature of this flow regime does not provide good mixing and transfer efficiencies in comparison to the performances obtained if turbulence, which is characterized by rapid fluctuations of flow variables in space and time, is present. 2 In this regard, an important body of research in microfluidics is focused on improving molecular diffusion in laminar flows through different strategies, 3 like, for example, fabricating serpentine-like microchannels. However, these techniques typically impose important construction complexities without exceptional increases in mixing and transfer rates. Other strategies significantly explored (i) try to achieve enhanced chaotic mixing through electrokinetic forcing 4,5 or (ii) reach incipient turbulent flow conditions by extraordinarily increasing the volumetric flow rates (large sizes/velocities), [6][7][8] which is in contradiction to the small values typically sought in microfluidic applications. Finally, an interesting approach recently proposed to achieve turbulent regimes in microdevices is the utilization of flexible microfluidics. 9 However, this technology has not yet been demonstrated at the microscale since the experiments were performed utilizing microchannels with widths in the millimeter-scale range.
A novel potential approach to achieve microconfined turbulence, which is studied in this work, is based on operating under highpressure supercritical conditions to leverage the hybrid thermophysical properties of supercritical fluids. High-pressure supercritical fluids are used in a wide range of engineering applications, like, for example, in gas turbines, supercritical water-cooled reactors, and liquid rocket engines. 10 They operate within high-pressure thermodynamic spaces in which intermolecular forces and finite packing volume effects become important. In this regard, it is important to distinguish between supercritical gas-like and liquid-like fluids separated by the pseudo-boiling line: 11,12 (i) a supercritical liquid-like fluid is one whose density is large and whose transport coefficients behave similar to a liquid; whereas (ii) the density of supercritical gas-like fluids is smaller, and their transport coefficients vary similar to gases. In particular, the strategy proposed makes use of the rapid smooth transition when crossing the pseudo-boiling line to tune supercritical fluids to present liquid-like densities [q $ 10 3 kg/m 3 ] and gas-like viscosities [l $ 10 À5 Pa s], and therefore, achieve Re b $ 10 3 -10 4 for typical microfluidic velocities and channel sizes, and favoring, in this manner, inertial over viscous forces and resulting in turbulent flow. Focusing on the thermophysical properties of different supercritical fluids, this approach has been recently explored by Bernades and Jofre. 1 The theoretical estimations presented in their work indicate that microconfined turbulent flow regimes can be potentially achieved by operating in the vicinity of the pseudo-boiling region for a wide range of popular working fluids, like, for example, carbon dioxide, methane, nitrogen, oxygen, and water. In connection with this strategy, Zhang et al. 13 explored mixing intensification for antisolvent processes by operating at high pressures in free-shear coflow configurations at isothermal conditions. 13 Nonetheless, the overall strategy is significantly different to the one studied in this work as (i) jet flows are inherently unstable, and consequently laminar-to-turbulent transition occurs in the range Re b % 30-2000; 14 (ii) coflows require complex microfluidic configurations; and (iii) isothermal conditions are not generally suitable for energy-related applications.
The experimental study of supercritical fluids is notably challenging due to the high pressures typically involved, which significantly limit the quantity and quality of data that can be extracted from laboratory experiments and/or industrial applications. 15 This intricacy is further increased by the interest in working at microfluidics conditions. Consequently, this work will utilize computational approaches. In that sense, the inherent physics complexity of supercritical fluids across the pseudo-boiling region typically leads to poor prediction capabilities of Reynolds-averaged Navier-Stokes (RANS) simulation approaches. 10 Therefore, scale-resolving methodologies based on direct numerical simulation (DNS) and large eddy simulation (LES) are typically required to properly characterize and/or predict highpressure transcritical turbulent flows. Selected examples of scaleresolving studies include (i) DNS of supercritical heat transfer in pipe flows by Bae et al., 16,17 (ii) DNS of transcritical flows close to the critical point by Sengupta et al., 18 (iii) LES and DNS of transcritical channel flows by Doehring and Adams 19 and Ma et al., 20 respectively, and (iv) DNS of transcritical turbulent boundary layers by Kawai. 21 Nonetheless, scale-resolving computational studies of high-pressure transcritical flows are significantly challenging due to the appearance of spurious pressure oscillations and amplification of aliasing errors, 22 mostly as a result of the large density gradients across the pseudoboiling region. In consequence, straightforward discretizations of the compressible equations of supercritical fluids motion typically turn out to be unsuitable. On this subject, the computational study presented in this work is based on a novel discretization approach grounded on extending a stable and non-dissipative formulation of the kinetic-energy preserving scheme introduced by Kennedy and Gruber 23 and later shown to be energy-preserving by Pirozzoli 24 and Coppola et al. 25 (referred to as KGP), to high-pressure transcritical flow problems. 26 Therefore, the objectives of this study are to computationally (i) demonstrate the potential of supercritical fluids to achieve wallbounded turbulent flow regimes at the microscale and (ii) characterize the corresponding flow physics. In this regard, the work is organized as follows. First, in Sec. II, the flow physics modeling of supercritical fluids is described together with the novel numerical scheme utilized. Next, the computational results are presented and analyzed in Sec. III. Finally, the work is concluded and future directions are proposed in Sec. IV.

II. PHYSICS MODELING AND NUMERICAL METHOD
The framework utilized for studying supercritical fluids turbulence in terms of (i) equations of fluid motion, (ii) real-gas thermodynamics, (iii) high-pressure transport coefficients, and (iv) numerical method is described below.

A. Equations of fluid motion
The turbulent flow motion of supercritical fluids is described by the following set of governing equations for mass, momentum, and pressure: where q is the density, v is the velocity vector, P is the pressure, s ¼ lðrv þ rv T Þ À ð2l=3Þðr Á vÞI is the viscous stress tensor with l as the dynamic viscosity and I as the identity matrix, c ¼ 1= ffiffiffiffiffiffiffi qb s p is the speed of sound with b s ¼ Àð1=vÞð@v=@PÞ s as the isentropic compressibility and v ¼ 1=q as the specific volume, b v ¼ ð1=vÞð@v=@TÞ P is the volume expansivity with T as the temperature, c v is the isochoric specific heat capacity, b T ¼ Àð1=vÞð@v=@PÞ T is the isothermal compressibility, and q ¼ ÀjrT is the Fourier heat conduction flux with j as the thermal conductivity.

B. Real-gas thermodynamics
The thermodynamic space of solutions for the state variables pressure P, temperature T, and density q of a single substance is described by an equation of state. One popular choice for systems at high pressures, which is used in this study, is the Peng-Robinson 27 equation of state written as where R u is the universal gas constant, v ¼ W=q is the molar volume, and W is the molecular weight. The coefficients a and b take into account real-gas effects related to attractive forces and finite packing volume, respectively, and depend on the critical temperatures T c , critical pressures P c , and acentric factors x. They are defined as where coefficient c is provided by ( The Peng-Robinson real-gas equation of state needs to be supplemented with the corresponding high-pressure thermodynamic variables based on departure functions 28 calculated as a difference between two states. In particular, their usefulness is to transform thermodynamic variables from ideal-gas conditions (low pressure-only temperature dependent) to supercritical conditions (high pressure). The ideal-gas parts are calculated by means of the NASA 7-coefficient polynomial, 29 while the analytical departure expressions to high pressures are derived from the Peng-Robinson equation of state as detailed in Jofre and Urzay. 12 C. High-pressure transport coefficients

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The high pressures involved in the analyses conducted in this work prevent the use of simple relations for the calculation of the dynamic viscosity l and thermal conductivity j. In this regard, standard methods for computing these coefficients for Newtonian fluids are based on the correlation expressions proposed by Chung et al. 30,31 These correlation expressions are mainly function of critical temperature T c and density q c , molecular weight W, acentric factor x, association factor j a and dipole moment m, and the NASA 7-coefficient polynomial; 29 further details can be found in dedicated works, like, for example, Poling et al. 32 and Jofre and Urzay. 12

D. Numerical method
Simulations of high-pressure transcritical turbulence are strongly susceptible to numerical instabilities due to the presence of nonlinear thermodynamic phenomena and large density gradients, which can trigger spurious pressure oscillations that may contaminate the solution and even lead to its divergence. Consequently, it is highly beneficial that the numerical schemes utilized, in addition to being kinetic-energy preserving (KEP), attain the so-called pressureequilibrium-preservation (PEP) property. 33,34 The numerical scheme utilized in this work has been developed specifically to be simultaneously KEP and PEP. The latter property is achieved by solving a pressure evolution equation. A thorough description and validation of this method can be found in Bernades et al. 26,35,36 In brief, the transport equations are numerically solved by adopting a standard semidiscretization procedure; viz., they are first discretized in space and then integrated in time. In particular, spatial operators are treated using second-order central-differencing schemes, and timeadvancement is performed by means of a third-order strong-stability preserving (SSP) Runge-Kutta explicit approach. 37 The convective terms are expanded according to the Kennedy-Gruber-Pirozzoli (KGP) splitting, 25,38 which has been recently assessed for highpressure supercritical fluids turbulence. 26 As a result, the method utilized (i) preserves kinetic energy by convection, (ii) is locally conservative for mass and momentum, (iii) preserves pressure equilibrium, and (iv) yields stable and robust numerical simulations without adding any numerical diffusion to the solution or stabilization procedures.

III. RESULTS AND DISCUSSION
Microconfined high-pressure transcritical turbulence is studied by means of DNS strategies based on the flow physics framework described in Sec. II. Data are obtained utilizing the in-house compressible flow solver RHEA. 39 The problem setup and discussion of results are described below.

A. Problem setup
The fluid selected to study and characterize microconfined highpressure transcritical fluids turbulence is N 2 , whose critical pressure and temperature are P c ¼ 3:4 MPa and T c ¼ 126:2 K. The fluid system is at a supercritical bulk pressure of P b =P c ¼ 2 and confined between cold/bottom (cw) and hot/top (hw) isothermal walls, separated at a distance H ¼ 2d with d ¼ 100 lm the channel half-height, at T cw =T c ¼ 0:75 and T hw =T c ¼ 1:5, respectively. Consequently, the boundary conditions imposed correspond to no-slip for the wall normal-direction and periodic for the streamwise and spanwise directions. This problem setup imposes the fluid to undergo a transcritical trajectory by operating within a thermodynamic region across the pseudo-boiling line. Based on preliminary theoretical estimations, 1 the friction Reynolds number selected at the cold/bottom wall is Re s;cw ¼ q cw u s;cw d=l cw ¼ 100, with u s ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðl=qÞðdhui=dyÞ p being the friction velocity, to ensure fully developed turbulent flow conditions; the q cw and l cw values are obtained from the thermophysical model described in Sec. II. The mass flow rate in the streamwise direction is imposed through a body force controlled by a proportional feedback loop (gain k p ¼ 0:1) aimed at reducing the difference between the desired (Re s;cw ¼ 100) and measured (numerical) Re s;cw values. In addition, gravity forces are not considered as the resulting Froude number of the problem is Fr ¼ u b = ffiffiffiffiffiffi gH p % 55, and consequently inertial forces are roughly 3000Â more important; viz., the importance of gravity scales as 1=Fr 2 . The computational domain is 4pd Â 2d Â 4=3pd in the streamwise (x), wall-normal (y), and spanwise (z) directions, respectively, which is large enough to represent the largest flow scales of the problem. 40 The mesh resolution is selected based on preliminary studies. In particular, the monocomponent single-phase channel flow DNS performed by Chevalier et al. 41 at a same Re s ¼ 100 is considered as an initial reference. That study utilized 64 Â 64 Â 64 grid points and corresponded to a fine-resolved DNS. For the problem of interest in this work, in addition to the classical consideration of the Kolmogorov scale, the length scales related to density gradients must also be considered. To this extent, based on the estimations provided by Jofre and Urzay, 11 the characteristic length scale for density gradients is approximately 10Â larger than the Kolmogorov scale, thereby confirming that the latter is the driving factor to select mesh resolution. Nonetheless, the mesh selected is 2Â finer in each direction with respect to Chevalier et al. 41 Consequently, this grid arrangement corresponds to a DNS of size 128 Â 128 Â 128 grid points. The grid is uniform in the streamwise and spanwise directions with resolutions in wall units (based on cw values) equal to Dx þ % 9:8 and Dz þ % 3:3, and stretched toward the walls in the vertical direction with the first grid point at y þ ¼ yu s;cw = cw % 0:1 and with sizes in the range 0:2 Շ Dy þ Շ 2:3, which are smaller than any of the reference DNS channel flow cases considered by Lee and Moser. 40 The simulation strategy starts from a linear velocity profile with random fluctuations, 42   scitation.org/journal/phf In this regard, flow statistics are collected for roughly 10 FTTs once steadystate conditions are achieved. Finally, a snapshot of the instantaneous streamwise velocity in wall units u þ on a x-y slice is displayed in Fig. 1 to provide qualitative information of the microconfined wall-bounded turbulence studied in this work.

B. Thermophysical analysis
The configuration of the problem, as previously introduced, forces the high-pressure fluid to undergo a second-order phase transition from supercritical liquid-like to supercritical gas-like by operating across the pseudo-boiling region. This phase transition is quantified in Fig. 2 by plotting the joint probability density function (PDF) of ensemble-averaged instantaneous values of reduced temperature T=T c and density q=q c together with the pseudo-boiling line (dasheddotted curve). The results indicate that the most probable phase states correspond to supercritical liquid-like in the vicinity of the pseudoboiling region (T=T c % 0:9-1:1 and q=q c % 1-2), followed by supercritical liquid-like at relatively cold temperatures (bottom wall region) and supercritical gas-like near the pseudo-boiling line, whereas supercritical gas-like states at relatively hot temperatures (top wall region) are significantly less probable. Consequently, at steady-state operating conditions, the system is dominated by supercritical liquid-like states near the pseudo-boiling line, which are characterized by densities similar to those of a liquid and presenting viscosities similar to a gas. 1 It can be also observed in the figure the rapid and smooth variation of density (gradient) across the pseudo-boiling line along the transcritical temperature trajectory, which, as will be studied in the subsections below, induces a notably important baroclinic torque when combined with the external force moving the fluid.
The thermophysical properties, as depicted in Fig. 3, can be further characterized by analyzing the compressibility factor Z ¼ PW= ðqR u TÞ and Prandtl number Pr ¼ lc P =j along the wall-normal direction. It can be observed, focusing on Z, that the fluid behaves very differently from an ideal gas, i.e., Z 6 ¼ 1. In particular, the supercritical fluid markedly deviates toward low values (Z % 0:3) in the vicinity of the cold/bottom wall, which is indicative of large densities similar to those of a liquid. The Z values slightly increase monotonically, remaining below 0.5, when moving away from the cold wall up to y=d % 1:9. At that point, the curve rapidly increases to reach Z % 0:8 at the hot/ top wall, which is still a value below 1 and corresponding to a relatively dense supercritical gas. The Prandtl number also depicts a very interesting behavior: (i) it presents values of Pr % 2:2 in the vicinity of the cold/bottom wall, indicating that momentum diffusivity dominates over thermal diffusivity, i.e., the momentum boundary layer is thicker than the thermal one; which is typical for liquid-like fluids; (ii) it rapidly reduces to values of approximately 1 close to the region of the hot/ top wall as a result of crossing the pseudo-boiling line, denoting that momentum and thermal diffusivities are of the same importance; and (iii) it follows an oscillatory trajectory across the channel height with valley (cold/bottom side of the channel) and peak (hot/top side of the channel) values of Pr % 1:9 and Pr % 2:4, respectively. This behavior can be also observed from the instantaneous snapshot of Pr on a x-y plane shown in Fig. 4; note the relatively low values found in the vicinity of the walls (especially at the hot/top wall), and the higher values    Table I, together with the quantities obtained for a similar system at low-pressure conditions. The low-pressure setup is defined by providing the same volumetric power input [i.e., u b ðÀdP=dxÞ ¼ u b ðs w;cw þ s w;hw Þ=2d with s w denoting the wall-shear stress] to the fluid as in the high-pressure case to compare equivalent energized flows. In this regard, the mass flow rate in the streamwise direction is imposed through a body force controlled by a feedback loop, which in this case aims at reducing the difference between the desired and measured volumetric power input. However, the main difference is introduced by reducing the operating bulk pressure to P b =P c ¼ 0:03. Therefore, a low-pressure framework is utilized based on the ideal-gas equation of state and Sutherland's law 43 for the dynamic viscosity and thermal conductivity, which are temperature-dependent in the following form:

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where l 0 and j 0 are evaluated at a reference temperature T 0 ¼ 273 K, and S l ¼ 107 K and S j ¼ 150 K are fitting parameters calibrated for N 2 . It can be noted the five following observations. First, as characterized by the friction Reynolds number Re s , the high-pressure channel flow presents higher levels of inertia with respect to viscous forces at the hot/top wall in comparison to the cold/bottom wall, which, as will be analyzed in Subsection III C, corresponds to larger turbulent flow fluctuations. Instead, at low-pressure conditions, the flow becomes laminar as studied by Bernades and Jofre. 1 Second, the skin-friction coefficient C f is an order of magnitude smaller at the top wall with respect to the bottom wall for the high-pressure case, while it is of the same order for the low-pressure system. Third, the Prandtl number Pr is approximately 2Â larger at the bottom wall at high pressures, and presents virtually no difference between walls at low pressure conditions. Fourth, the Nusselt number Nu, which quantifies the heat transfer enhancement by convection with respect to conduction, presents an increase in roughly 20Â for the supercritical system with respect to the low pressure case. Finally, for the high-pressure setup, the Stanton number St, which measures the ratio of heat transferred to a fluid with respect to its thermal capacity, at the top wall is an order of magnitude larger than at the bottom wall, while it presents similar values at lowpressure conditions. These results are very important as they computationally corroborate the hypothesis that microconfined turbulent flow can be achieved when operating at high-pressure transcritical regimes, and consequently significantly larger values of flow mixing and heat transfer rates with respect to similar microchannels at atmospheric pressure conditions can be potentially obtained. For example, the size of the cooling systems of present-day microchips 44 could be significantly reduced (by a factor of oð10Þ as theoretically estimated by Bernades and Jofre 1 ) using microfluidic channels at high-pressure supercritical fluid conditions.

C. Flow statistics
The time-averaged streamwise velocity u þ and Favre-averaged root-mean-squared (rms) velocity fluctuations u þ rms ; v þ rms and w þ rms along the wall-normal direction y þ in viscous units for the cold/ bottom and hot/top walls are depicted in Figs. 5 and 6. It is important

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ARTICLE scitation.org/journal/phf to note that for the high-pressure results the maximum u þ is located at y=d ¼ 1:07 from the cold wall. Consequently, flow statistics are plotted from each wall to this particular position. Two main observations can be inferred from these plots. First, as shown in Fig. 5, the shapes of the time-averaged u þ profiles for the high-pressure case present topologies similar to turbulent boundary layers, and collapse to the same curve for the two walls. Instead, at low-pressure conditions, the profiles present parabolic-like shapes reminiscent of laminar flow and with the line corresponding to the cold wall presenting larger velocity values toward the center. Second, focusing on the Favre-averaged rms velocity fluctuations for the high-pressure system shown in Fig. 6, turbulence intensity is significantly different between the hot/top (gas-like) and cold/bottom (liquid-like) walls in the streamwise direction. In particular, the hot wall presents larger fluctuations in the viscous sublayer and buffer region, while turbulence intensity grows following a concave parabola from the cold wall to the center of the channel. An equivalent, although significantly less pronounced, behavior is observed for the fluctuations in the spanwise direction, whereas they virtually collapse to the same curve for the wall-normal velocity. The quantitative differences between the high-pressure and lowpressure cases discussed above can be undoubtedly visualized by examining the instantaneous snapshots of streamwise velocity shown in Fig. 7. From the figure, it can be easily observed the turbulent nature of the velocity field at high-pressure conditions, whereas the flow regime becomes laminar at low-pressure conditions. In particular, at high pressure, the flow presents a chaotic organization with a wide range of spatiotemporal scales characteristic of turbulence. Additionally, the boundary layer at the cold/bottom wall is relatively thicker than at the hot/top wall as it can be inferred from the size of the flow structures emanating from both sides. However, as shown in Fig. 5, the two boundary layers virtually collapse to the same curve when they are normalized, respectively, by each wall friction velocity.
Focusing on the temperature distribution, different behaviors can be observed. In particular, Fig. 8 depicts the time-averaged temperature in viscous units T þ ¼ ðT À T w Þ=T s , with T w being the wall temperature and T s ¼ ½j=ðqc P u s ÞðdhTi=dyÞ, along the wall-normal direction y þ , and Fig. 9 presents the Favre-averaged temperature fluctuations. As indicated by Fig. 8, the profiles collapse for the most part to a single curve for the low-pressure case, which is a consequence of the dominance of molecular diffusion over convective mixing resulting from the laminar regime developed. Instead, at high-pressure conditions, the time-averaged temperature values at the cold/bottom wall are roughly 2-4Â larger than at the hot/top wall. This behavior is a direct consequence of the higher thermal inertia of the supercritical liquid-like fluid found at the cold/bottom part of the channel with

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scitation.org/journal/phf respect to the supercritical gas-like fluid occupying the hot/top part. Additionally, as shown in Fig. 9, the temperature fluctuations are larger at the cold/bottom side of the channel in comparison to the hot/ top region. Consequently, the degree of convective-dominated mixing is higher at the cold part, which enhances the transport of supercritical liquid-like fluid toward the hot side. In detail, the fluctuations at the hot/top wall follow a convex trajectory peaking at y þ % 50 with the value T þ % 1:3, whereas for the cold wall the fluctuations present a maximum of T þ % 6:3 at y þ % 15 followed by an oscillatory shape with a minimum of T þ % 5:0 at y þ % 60. As depicted in Fig. 5, the streamwise velocity along the wallnormal direction roughly follows the topology of turbulent boundary layers. However, the inherent flow physics is significantly different from incompressible wall turbulence. In detail, as shown in Fig. 10, the u þ profile as a function of y þ does not exactly collapse to the standard "law of the wall" curve since, as initially proposed by van Driest 45 for non-isothermal regimes with variable density and viscosity, the variables need to be transformed accounting for the variation in mean quantities. However, as depicted in Fig. 10, the profile does not collapse either when utilizing an alternative transformation specifically suited for non-isothermal compressible boundary layers derived by Trettel and Larsson. 46 This transformation calculates y þ and u þ as where subindex q w is the density at the walls, and q and l are mean quantities averaged over each y-plane. In particular, for both walls, the transformed results collapse to the standard profile along the viscous sublayer, but they deviate starting from the buffer layer especially for the hot/top wall. It is observed by performing inspectional analysis that the term dominating the time-averaged streamwise velocity in Eq. 9 is ffiffiffiffiffiffiffiffiffiffi ffi q=q w p . As a result, the significantly strong density variations across the channel largely influence the shape of the u þ profiles. In addition, due to the relatively small Reynolds numbers of the resulting turbulent flow, viscosity effects are still important which, since the transformation assumes that viscous stresses are negligible within the stress balance, further contribute to the deviation from the standard profile in the log-law region; especially at the hot/top wall where viscosity variation is the largest. Furthermore, the transformed friction Reynolds number 46 designed to incorporate the asymmetry of the solution in the wall-normal direction, which is defined as where q Ã and l Ã are evaluated at the y-position with maximum streamwise velocity, d Ã is the distance from the wall to this maximum position, and u Ã s ¼ ffiffiffiffiffiffiffiffiffiffiffiffi , correspond to Re s Ã % 382 and Re s Ã % 132 for the cold/bottom and hot/top walls, respectively. In particular, the transformed friction Reynolds number is used to compare a boundary layer with respect to the incompressible case at the same Re s value. Consequently, these values indicate that, as a result of the thermophysical and flow distribution along the wall-normal direction, the turbulent intensity is significantly larger at the cold/bottom wall with respect to the hot/top wall, which further supports the hypothesis that the large deviation from the standard "law of the wall" observed at the hot/top wall is caused by strong variations in density combined with the resulting relatively low friction Reynolds number achieved. Similarly, the recent data-driven transformation proposed by Volpiani et al., 47 which was validated for a wide range of Reynolds and Mach numbers, has been also analyzed. However, the transformation has not been adequate for the this problem either. To this end, Subsection III D will investigate in more detail the underlying turbulent flow mechanisms responsible for the differences observed.

D. Turbulence sources
As previously introduced in Sec. III C, turbulent flow statistics differ substantially from the standard incompressible turbulent channel flow case. One of the underlying physical mechanisms responsible for such differences is the presence of a significantly localized baroclinic torque close to the hot/top wall. In particular, the torque, which is generated by the combination of (i) the external force driving the flow in the streamwise direction equivalent to an imposed pressure gradient of value rP ¼ ½Àðs hw þ s cw Þ=2d; 0; 0, and (ii) the density gradient across the pseudo-boiling region in the wall-normal direction, is responsible for injecting vorticity with a peak value located in the vicinity of the hot/top wall as will be discussed below. As a result, notably higher levels of enstrophy are observed near the hot/top wall (values approximately 50Â larger) in comparison to the cold/bottom wall as quantified in Fig. 11. To further quantify this enhanced level of enstrophy especially near the hot/top wall, the high-pressure (HP) case is compared against the (i) low-pressure (LP) case defined in Sec. III B, and (ii) a low-pressure isothermal turbulent case at Re s ¼ 100 (LP isoT). This latter setup is similar to the LP case, but at an isothermal temperature of value T=T c ¼ 0:75 to achieve Re s ¼ 100 at both walls through the proportional controller. The enstrophy of this additional cases is also quantified in Fig. 11, indicating that the levels of kinetic energy corresponding to dissipation is significantly higher in the HP system with respect to the LP and LP isoT cases; especially in the region of the hot/top wall. In particular, the enstrophy levels near the hot/top wall for the HP case are roughly 100Â and 10Â larger than for LP and LP isoT, respectively, and these are responsible for altering the standard behavior of boundary layers as observed in Fig. 10.

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scitation.org/journal/phf In order to characterize the intrinsic flow mechanisms responsible for this enhanced enstrophy levels, which can be linked to vorticity generation and distribution, the transport equation of vorticity is considered as follows: where x is the vorticity, ðx Á rÞv and Àxðr Á vÞ represent, respectively, vortex stretching due to velocity gradients and compressibility effects, 1=q 2 ðrq Â rPÞ corresponds to the baroclinic torque, and r Â ½ð1=qÞr Á s is the vorticity diffusion term. In this regard, the different terms of the vorticity transport equation normalized by u 2 s =d 2 are shown in Fig. 12  FIG. 14. Ensemble-averaged values of baroclinic torque in wall units along the wall-normal direction for the low-pressure (LP) and high-pressure (HP) cases; the curve labels are defined in Fig. 11.

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ARTICLE scitation.org/journal/phf values are four orders of magnitude smaller. Second, the baroclinic torque responsible for generating flow rotation, which is generated by the combination of the external force driving the flow and the variation of density, is negligible for the LP case as depicted in Fig. 14, but is significantly important (approximately 500Â larger) for the HP system as a result of the large density gradients obtained across the pseudo-boiling region. Third, these notable differences in terms of laminar/turbulent regime between HP and LP systems are further noted by the amount of vorticity dissipation shown in Fig. 15. As it can be observed from the figure, the vorticity dissipation is almost two orders of magnitude larger for the HP and LP isoT cases with respect to the LP case, which is indicative of the turbulent fluctuations present in the former systems and not present in the latter.

E. Small-scale flow topology
As discussed in Subsection III D, the baroclinic torque is the mechanism responsible for enhancing the flow rotation, whereas the vortex stretching acts as a turbulence generator by tilting and stretching the vortices. Hence, to further characterize the flow physics of the high-pressure system, the topology of the small scales is analyzed. In particular, the topological flow features are examined by considering the discriminant D A of the velocity-gradient tensor A ij ¼ @u i =@x j for compressible flows, which can be decomposed into symmetric S ij (rate-of-strain) and skewsymmetric X ij (rate-of-rotation) parts, and calculated as 48 where P A , Q A , and R A are the first, second, and third invariants of the velocity-gradient tensor expressed as 49 In this regard, the ensemble-averaged values of normalized D þ A ¼ D A ðd=u s Þ 6 along the wall-normal direction for the cold/bottom and hot/top walls of the high-pressure case are shown in Fig. 16. As it can be observed, the discriminant presents larger positive values for the hot/top region (approximately 100Â) with a maximum D þ A $ 10 9 at y þ % 15, whereas the maximum is at y þ % 30 for the cold/bottom wall. Therefore, the flow is dominated by enstrophy along the entire channel, which is characteristic of turbulent flow systems. In more detail, Fig. 17 depicts the ensemble-averaged values of the three invariants in wall units along the wall-normal direction. The following observations can be obtained. First, P þ A is small and positive in the vicinity of the cold/bottom wall indicating that the flow tends to slightly expand, while it becomes large and negative near the hot/top wall corresponding to compression states in which the flow follows stable trajectories. 48 Second, Q þ A , which measures the relative Physics of Fluids ARTICLE scitation.org/journal/phf importance of straining (Q þ A < 0) and rotational (Q þ A > 0) parts of the velocity-gradient tensor, fluctuates between negative and positive values for both walls, and presents a positive maximum around y þ % 20-30. Hence, the flow is dominated by straining motions (especially for the hot/top wall) characterized by large rates of kinetic energy dissipation 50 close to the walls, whereas vorticity is prevalent toward the center of the channel. Similarly, R þ A also transitions from negative to positive values along the wall-normal direction for both walls. As a result, when moving slightly away from the walls, the flow tends to reverse direction from stable focus to unstable due to the dominance of vorticity in those regions.

IV. CONCLUSIONS
Direct numerical simulations have been performed to analyze and characterize the flow physics of microconfined high-pressure transcritical fluids turbulence. The problem of interest consists in a threedimensional channel flow with (i) a friction Reynolds number of Re s ¼ 100 imposed at the cold/bottom wall, and (ii) a temperature difference between bottom and top walls to force the system cross the pseudo-boiling region. Such operating conditions result in turbulent flow at the microscale with a subsequent increase in fluid mixing and heat transfer. As it is studied and discussed, the achievement of turbulent flow is a result of the combination of (i) the thermophysical properties (relatively large density and small viscosity) of supercritical fluids across the pseudo-boiling region and (ii) the presence of a localized baroclinic torque close to the hot/top wall introducing vorticity into the system. To further characterize these effects, two additional cases at low-pressure (atmospheric) conditions have also been considered, which consist in a similar computational setup with same: (i) volumetric input power and (ii) same friction Reynolds number at isothermal conditions. The flow regime becomes laminar in the former case, whereas it remains turbulent (without heat transfer) in the latter.
The computational results demonstrate the feasibility of achieving microconfined turbulence by leveraging the properties of supercritical fluids. However, the resulting flow physics differs significantly from the typical behavior of turbulent boundary layers. Particularly, the first-order velocity statistics do not collapse to the standard "law of the wall" presenting the following features: (i) the hot/top wall deviates, especially in the log-law region, when transformed using nonisothermal models and (ii) the streamwise velocity fluctuations differ significantly between the cold/bottom and hot/top walls. In addition, the time-averaged temperature distribution also differs between walls, with the cold/bottom wall achieving slightly larger values due to the higher thermal inertia of supercritical liquid-like fluids. Moreover, the temperature fluctuations are higher at the cold/bottom wall as a result of the higher degree of convective mixing, which enhances the transport of supercritical liquid-like fluid toward the hot region. These differences between walls are mainly driven by a baroclinic torque localized in the vicinity of the hot/top wall, which is generated by the combination of the external force driving the flow in the streamwise direction and the density gradient across the pseudo-boiling region in the wall-normal direction, and yielding an increment of flow rotation in comparison to the isothermal low-pressure turbulent case also considered. As studied in detail, this baroclinic torque injects additional flow rotation into the system that is subsequently transformed into a wide range of scales (i.e., turbulent flow motions) through vortex stretching mechanisms. This behavior is further confirmed by the peak of velocity-gradient tensor discriminant in the vicinity of the hot/ top wall, which is indicative of the prevalence of enstrophy over dissipation and characteristic of turbulent flow systems.
Finally, the rich phenomena of the results presented motivate the authors to continue studying the complex flow physics of microconfined turbulence at high-pressure transcritical conditions. Particularly, the focus in the short-term will be placed on (i) analyzing the flow behavior in the vicinity of the hot/top wall, especially in the log-law region where current non-isothermal transformation models fail; (ii) further characterizing the baroclinic torque identified; (iii) studying higher-order flow statistics for which the sensitivity to mesh resolution is currently unknown by the community; (iv) investigating the flow structures and patterns driving the enhanced mixing levels observed; and (v) expanding the knowledge of this problem by studying other fluids and operating conditions.