Heat transfer from an array of resolved particles in turbulent flow

The physalis method for resolved numerical simulation of particulate flows, recently extended to include particles-fluid heat transfer, is applied to the turbulent flow past a planar particle array perpendicular to the incoming mean flow. The array consists of nine equal spheres. Periodicity boundary conditions are imposed on the boundaries of the computational domain parallel to the mean flow. The Reynolds number based on the particle diameter and mean incident flow is 120, the Taylor-scale Reynolds number is close to 30, and the ratio of particle radius to the Kolmogorov length is about 10. A detailed characterization of the flow and heat transfer is given including probability distribution functions of temperature and streamwise velocity, contour maps of the temperature fluctuations, diagonal Reynolds stresses, turbulent heat flux, and the various contributions to the energy budget. Turbulence moderately increases the heat transfer and considerably shortens the thermal wake of the particles. Temperature and streamwise velocity develop very differently downstream of the spheres in spite of the fact that the Prandtl number equals 1, because of the blockage by the spheres, which has no counterpart for the temperature.

Figure 1

Snapshot of the normalized temperature $T_{*}=(T-T_p)/(T_i-T_p)$ in the flow studied in this paper; the isosurfaces correspond to $T_{*}=0.8$.

Figure 2

Comparison between the time-mean normalized temperature distribution on a plane parallel to the mean velocity through the centers of three contiguous particles for turbulent flow (left) and for laminar flow.

Figure 3

Examples of the instantaneous Nusselt number vs time for two different spheres; ${\tau }_E$ is the eddy turnover time.

Figure 4

Average local Nusselt number over the spheres' surface; $\theta =0$ and $\pi$ are the front and rear stagnation points, respectively. The thick dashed line is for laminar flow; the lightly dashed line is the pure conduction limit $\text{Nu}=2$.

Figure 5

Three examples of the instantaneous local Nusselt number. The solid line is the mean value shown in Fig. 4, and the lightly dashed line the pure conduction limit $\text{Nu}=2$.

Figure 6

Contour plot of the normalized average temperature field $(T-T_p)/(T_i-T_p)$ on a plane parallel to the flow direction through the particle center.

Figure 7

Contour plot of the normalized average streamwise velocity field $u_z/U$ on a plane parallel to the flow direction through the particle center.

Figure 8

Average normalized temperature deficit $(T_i-T)/(T_i-T_p)$ (upper diagram) and velocity deficit $(U-u_z)/U$, vs distance along a line through the particle center parallel to the mean flow; $z/a=1$ is the rear stagnation point of the particle.

Figure 9

Probability density function of the temperature along a line through the sphere center parallel to mean flow; the sphere center is at $z=0$. From left to right the curves are for $z/a=1.1$, 1.25, 1.5, 2, 3, 8, 17.5.

Figure 10

Probability density function of the streamwise velocity along a line through the sphere center parallel to mean flow; the sphere center is at $z=0$. From left to right the curves are for $z/a=1.1$, 1.25, 1.5, 2, 3, 8, 17.5.

Figure 11

Contour plots of the RMS normalized temperature fluctuations defined in (23).

Figure 12

Dependence of the RMS temperature (upper diagram) and velocity fluctuations vs distance along a line through the particle center parallel to the mean flow; $z/a=1$ is the rear stagnation point of the particle. In the lower diagram the upper two lines show the fluctuations of the two cross-stream velocity components; the thick line is for the streamwise velocity.

Figure 13

Normalized temperature variance ${\sigma }_T=\overline{T^{\prime 2}}/{(T_i-T_p)}^2$ in the cross-stream direction at different downstream distances from the sphere. In descending order of the maxima, the lines are for $z/a=1.25$, 1.5, 2, 3, and 5; the particle center is on the plane $x=0$.

Figure 14

Normalized diagonal turbulent Reynolds stress $u_x^{\prime }{u}_x^{\prime }/U^2$ (upper diagram) and $u_z^{\prime }{u}_z^{\prime }/U^2$ in the cross-stream planes at downstream distances from the sphere $z/a=1.25$, 1.5, 2, 3, and 5; the particle center is on the plane $x=0$.

Figure 15

Dependence of the average $x$ (solid lines) and $y$ components of the turbulent heat flux on distance from the sphere axis at downstream locations $z/a=1.2$, 1.5, 2, and 3; the particle center is on the plane $x=0$.

Figure 16

Dependence of the average $z$-components of the turbulent heat flux on distance from the sphere axis at downstream locations $z/a=1.2$, 1.5, 2, and 3; the particle center is on the plane $x=0$.

Figure 17

The three terms in the energy budget equation (24) as functions of the cross-stream coordinate at $z/a=1.2$ (upper diagram) and 1.5.

Figure 18

Mechanical (upper two lines) and thermal time scales, defined in (28), as functions of the downstream distance; the dashed line is for turbulent flow without the particles.