Structure of wall-bounded flows at transcritical conditions

At transcritical conditions, the transition of a fluid from a liquidlike state to a gaslike state occurs continuously, which is associated with significant changes in fluid properties. Therefore, boiling in its conventional sense does not exist and the phase transition at transcritical conditions is known as “pseudoboiling.” In this work, direct numerical simulations (DNS) of a channel flow at transcritical conditions are conducted in which the bottom and top walls are kept at temperatures below and above the pseudoboiling temperature, respectively. Over this temperature range, the density changes by a factor of 18 between both walls. Using the DNS data, the usefulness of the semilocal scaling and the Townsend attached-eddy hypothesis are examined in the context of flows at transcritical conditions—both models have received much empirical support from previous studies. It is found that while the semilocal scaling works reasonably well near the bottom cooled wall, where the fluid density changes only moderately, the same scaling has only limited success near the top wall. In addition, it is shown that the streamwise velocity structure function follows a logarithmic scaling and the streamwise energy spectrum exhibits an inverse wave-number scaling, thus providing support to the attached-eddy model at transcritical conditions.

Figure 1

State diagram and thermotransport properties of nitrogen (critical pressure of 3.40 MPa and critical temperature of 126.2 K) from NIST database [15]. (a) Pressure-temperature state diagram with the compressibility factor contours shown in gray and specific-heat capacity at constant pressure shown as red contours in units of J/$\left(\mathrm{g}\mathrm{K}\right)$. The blue solid line is the coexistence line and the blue dashed line is the pseudoboiling line defined by the peak of specific-heat capacity. The black dot represents the critical point. (b) Density, specific-heat capacity, and dynamic viscosity plotted as a function of temperature at three different pressures, emphasizing the drastic property changes in the pseudoboiling region.

Figure 2

(a) A sketch of hypothesized boundary-layer structure. The attached eddies are space filling. On a vertical plane cut as shown here, the number of eddies doubles as the size halves. An eddy affects the shaded region below it. The velocity at a generic point in the flow field is a result of the additive superposition of all the eddy-induced velocity fields there. (b) A sketch of the additive cascade in wall-bounded flows. The sketch is the same as (a), but we have retained only the hierarchical organization of the attached eddies, schematically indicated by color lines. The two black squares indicate two points in the flow field. For these particular points under consideration (at the same wall-normal height), eddies in blue affect the two points simultaneously, eddies in orange can only affect one of the two points, and eddies in gray affect neither of them.

Figure 3

Schematic of the transcritical turbulent channel flow. The top and bottom walls are isothermal kept at 300 K and 100 K, respectively; $p_0$ is the bulk pressure.

Figure 4

Favre- and Reynolds-averaged (a) axial velocity and (b) temperature as a function of the wall-normal coordinate. $T_{\mathrm{pb}}$ is the pseudoboiling temperature, where the specific-heat capacity peaks. The dashed line is at wall-normal coordinate $y=y_{\delta }$, where $\tilde{u}$ is at its maximum.

Figure 5

Ensemble-averaged density $\left(\rho \right)$, compressibility factor $\left(Z\right)$, Mach number $\left(M\right)$, dynamic viscosity $\left(\mu \right)$, and heat conductivity $\left(\lambda \right)$ as (a) a function of the normalized mean temperature and (b) a function of the wall-normal coordinate. Density, dynamic viscosity, and heat conductivity are normalized by their respective values at the bottom wall.

Figure 6

(a) Ensemble-averaged heat capacity and the heat capacity computed using ensemble-averaged temperature and density as a function of the wall-normal coordinate. (b) Ensemble-averaged local Prandtl number and $\overline{c_p}\overline{\mu }/\overline{\lambda }$ as a function of the wall-normal coordinate.

Figure 7

Isosurfaces of constant density, colored by streamwise velocity. (a) $\rho =60$ kg/${\mathrm{m}}^3\left(y_s/L_y=0.98\right)$, (b) $\rho =100$ kg/${\mathrm{m}}^3\left(y_s/L_y=0.9\right)$, and (c) $\rho =200$ kg/${\mathrm{m}}^3\left(y_s/L_y=0.6\right)$. The fluid density at the top and bottom walls is 44 and 785 kg/${\mathrm{m}}^3$, respectively; $y_s$ denotes the mean location of the isosurface of density.

Figure 8

Instantaneous contours on a $z$–$y$ plane for (a) reduced pressure, (b) temperature (the black line indicates $T/T_{\mathrm{pb}}=1)$, (c) density $({\rho }_0$ is the volume-averaged bulk density), and (d) streamwise velocity.

Figure 9

Instantaneous contours of the streamwise velocity fluctuations at (a) $y_{\mathrm{SL}}=80$ from the bottom wall and (b) $y_{\mathrm{SL}}=80$ from the top wall. The velocity fluctuations are normalized using the friction velocity at the respective wall.

Figure 10

Turbulence-invariant map, showing $\eta$ as a function of $\xi$ for $b_{ij}$ near the bottom cooled wall $\left(y<y_{\delta }\right)$ and near the top heated wall $\left(y>y_{\delta }\right)$. Results for a channel flow in which both walls are set to a constant and equal temperature (denoted as “Const. Prop., ${\mathrm{Re}}_{\tau }=390$”; Appendix pp1) are included for comparison. The Lumley triangle [76] corresponds to $\eta =\xi ,\eta =-\xi$, and $\eta =\sqrt{1/27+2{\xi }^3}$.

Figure 11

$y_{\text{SL}}=y\sqrt{{\tau }_w\overline{\rho }}/\overline{\mu }$ as a function of $y^{+}$ near the two walls. $y_{\text{SL}}=y^{+}$ for constant-property boundary-layer flows, and is included for comparison. $y_b$ is the distance from the bottom wall, and $0<y_b<y_{\delta }+L_y$. $y_t$ is the distance from the top wall and $0<y_t<L_y-y_{\delta }$.

Figure 12

Kolmogorov length scale ${\eta }_u$ and ${\eta }_T={\eta }_u/\sqrt{{\text{Pr}}^{*}}$ as a function of $y_{\mathrm{SL}}$. ${\mathrm{Re}}_{\tau }^{*}=\delta \sqrt{{\tau }_w\overline{\rho (y=\delta )}}/\overline{\mu (y=\delta )}$ is the friction Reynolds number defined based on semilocal quantities at a distance $y=\delta$ from the wall. $\delta$ is an outer length scale and is $L_y+y_{\delta }$ for the boundary layer near the bottom wall and is $L_y-y_{\delta }$ for the boundary layer near the top wall. Results near the bottom cooled wall are shown as solid lines and results near the top heated wall are shown as dashed lines. ${\eta }_u$ and ${\eta }_T$ are shown in different colors. “Expt.” are experimental results reported in Ref. [77] for two constant-property boundary layers at ${\mathrm{Re}}_{\tau }=2800$ and ${\mathrm{Re}}_{\tau }=19000$. Patel et al. (2016) are DNS results reported in Ref. [20] for a constant-property channel at ${\mathrm{Re}}_{\tau }=395$.

Figure 13

Mean velocity profiles as a function of wall-normal distance near (a) the bottom cooled wall and (b) the top heated wall. VD: van Driest transformation; TL: transformation by Trettel and Larsson [35]. The law of the wall corresponds to the established logarithmic scaling $\overline{u^{+}}=log\left(y^{+}\right)/\kappa +B$. $\kappa =0.41$ and $B=5$ for the solid black line. $\kappa =0.38,B=5.2$ for the solid gray line. The viscous scaling $\overline{u^{+}}=y^{+}$ is also included for comparison.

Figure 14

(a) $R_{ij}^{\prime \prime }$ and (b) $R_{ij}^{\prime }$ as a function of $y_{\mathrm{SL}}$. Different components are color coded, and we use different line types to differentiate between the statistics near the bottom wall and the statistics near the top wall. $R_{ij}^{\prime \prime }$ in (a) are shown using thin lines and $R_{ij}^{\prime }$ in (b) are shown using bold lines. We have shown only data in the near-wall regions. Results of incompressible flows at ${\mathrm{Re}}_{\tau }=395$ and ${\mathrm{Re}}_{\tau }=2000$ are included for comparison. The solid black line indicates the expected scaling of $\overline{{u^{\prime +}}^2}$ at high Reynolds numbers, $1.26log(\delta /y)+2.0$.

Figure 15

Computed damping functions near the walls as functions of $y_{\mathrm{SL}}$ and $y^{+}$. $D_{\mathrm{VD}}$ is the van Driest damping function.

Figure 16

$\overline{\mathrm{\Delta }{u}^{\prime 2}}=\overline{{[u^{\prime }(x+r,y)-u^{\prime }(x,y)]}^2}$ as a function of the two-point distance at $y^{+}=80$ and $y_{\mathrm{SL}}^{+}=60,80,100$ from the two walls. The same statistics in a canonical low-speed turbulent channel flow at a Reynolds number of ${\mathrm{Re}}_{\tau }=390$ is included for comparison (solid red line). The statistics near the bottom wall are shown in (a) and the statistics near the top wall are shown in (b). Statistics at a distance $y^{+}=80$ from the two walls are normalized using $u_{\tau }$ and statistics at distances $y_{\mathrm{SL}}$ from the two walls are normalized using $u_{\tau }^{*}$.

Figure 17

Streamwise energy spectra as a function of the streamwise wave number at different wall-normal distances from (a) the bottom cooled wall and (b) the top heated wall.

Figure 18

Probability density function of the density at several wall-normal distances from (a) the bottom wall and (b) the top wall. The fluid density at top and bottom walls is 44 and 785 kg/${\mathrm{m}}^3$, respectively. The volume-averaged bulk density is 365 kg/${\mathrm{m}}^3$.

Figure 19

(a) Standard variance of the density fluctuations, (b) skewness, and (c) kurtosis of the density fluctuations as a function of the distance from the two walls. The Gaussian corresponds to skewness being 0 and kurtosis being 3. We plot from both walls $\left(y=0\right)$ towards $y=y_{\delta }$.

Figure 20

(a) Mean velocity profiles and (b) Reynolds stresses as a function of the wall-normal distance. A comparison between the present DNS (lines) and the DNS by Moser et al. [74] (symbols).

Figure 21

Profiles of (a) mean velocity and (b) mean temperature as a function of the wall-normal coordinate. The symbols are shown every five grid points for better visualization.

Figure 22

Density, compressibility factor, Mach number, viscosity, and heat conductivity as a function of mean temperature. Symbols are plotted the same way as in Fig. 21. Different quantities are color coded.

Figure 23

(a) Mean velocity profiles as a function of the wall-normal coordinate. Symbols are plotted the same way as in Fig. 21. (b) $y$ coordinate normalized using the local Kolmogorov length.