In this study, flow across a circular cylinder confined in an inclined channel is considered, with account taken of mixed convection. Cases with opposing or aiding buoyancy that is not in alignment with the incoming flow are considered, and it is found that the flow around and in the wake of the cylinder is affected. The effects of inclination and buoyancy on the flow and heat transfer patterns are investigated, with a focus on the stabilization and destabilization of flow by these two factors. Simulation results are analyzed with regard to the flow field, lift and drag forces, heat transfer, the deviation of separation and through-flow in the channel, and the temperatures on the channel walls. The flow is more strongly destabilized by an opposing buoyancy, but can be fully stabilized by an aiding buoyancy in an inclined channel. The drag on the cylinder is minimized around a critical Richardson number *Ri*_{cr}. The inclination of the channel has significant effects on the local heat transfer on the cylinder and on the characteristics of the through-flow. The pattern of the wake flow, i.e., whether it is steady or unsteady, affects the heating of the channel walls.

## I. INTRODUCTION

Flow across one or multiple cylinders is one of the most fundamental problems in civil, ocean, and wind engineering, in petrochemical and process engineering, and, in particular, in heat exchangers. For Reynolds numbers below a critical value *Re*_{cr}, the flow past a cylinder is steady, as manifested by the presence of recirculating vortices downstream of the cylinder; however, the flow becomes unsteady once the Reynolds number exceeds *Re*_{cr}, and this results in pulsating forces being exerted on the cylinder, possibly leading to structural failure. Flow across a single cylinder has been studied in Refs. 1–6. In cases in which the fluid is heated or cooled, buoyancy forces due to density variations resulting from temperature variations affect both thermal and flow behaviors, and this has been investigated using a simplified model in which the cylinder is heated or cooled by the fluid. Most research to date has focused on the flow around one or multiple cylinders in a confined channel or unconfined medium, in which the effects of inertial and buoyancy forces on thermal and flow behaviors are represented respectively by the Reynolds and Richardson numbers. In general, for a buoyancy force that is aligned with the direction of flow, there is a critical value of the Richardson number, *Ri*_{cr}, characterizing steady and unsteady flow as affected by buoyancy: the flow is unsteady for *Re* > *Re*_{cr} and *Ri* < *Ri*_{cr}, but transitions to steady for *Ri* > *Ri*_{cr} even when *Re* > *Re*_{cr}. The opposing buoyancy destabilizes the flow separating from the cylinder, and transition to unsteady flow occurs at a lower *Re*_{cr}, as manifestted by the appearance of initial vortex shedding (IVS) adjacent to the cylinder and a reduction in the distance between neighboring vortices in the wake. In addition, the buoyancy also influences heat transfer. As well as situations where buoyancy acts in the same or opposite direction to that of the flow, there are also circumstances where the buoyancy force is not aligned with the flow direction. In these cases, the buoyancy drives the fluid to deviate from the vertical direction in which the force of gravity acts, and thus the flow in the channel is asymmetric.

Low-*Re* flow in an inclined channel has been investigated numerically by considering a local thermal source imposed on the channel walls with constant temperature or heat flux. Marroquín-Desentis *et al.*^{7} studied transient laminar mixed convective flow with opposing buoyancy in an inclined channel at *Re* = 500, *Pr* = 7, and inclination angle 0° ≤ *α* ≤ 90°. The fluid was abruptly heated by two discrete isothermal flush-mounted heat sources on the channel walls. It was found that the recirculation flow downstream of the heated region occurred at a much higher value of *Ri* for a horizontal channel, and as *α* increased, the transition from steady to unsteady flow occurred at a value of the Richardson number that depended on *Re* and *α*. The same configuration was also studied by García *et al.*^{8} at *Re* = 10^{5} by considering finite-thickness heat sources. These heat sources induced oscillatory flow and an oscillatory temperature distribution in the channel, as manifested by a single but complex vortical structure in the adjacent region. As the thickness of the heat sources was increased, the thermal and flow behaviors transitioned from oscillatory to steady. Guimarães and Menon^{9} investigated the effect of enhancing buoyancy with heat sources on the bottom wall of the channel imposing constant heat flux on the fluid. The effect of inclination was pronounced in the low-*Re* regime, especially at 0° ≤ *α* ≤ 45°, for which recirculating flow could be observed at a very low Reynolds number *Re* = 10.

In addition to the effects of zero- and small-thickness flush-mounted heat sources, those of thermal ribs and cavities have also been studied. Yang and Yeh^{10} explored the effects of thermal conductivity of solid material and inclination angle in a configuration consisting of a channel with transverse fins attached in a staggered pattern on both surfaces. They observed that the recirculation vortices downstream of the last fin diminished as the flow became dominated by natural convection. The optimum geometry of the fins was found to be shorter and fatter at low thermal conductivity ratios, but longer and thinner at large thermal conductivity ratios. Pérez-Flores *et al.*^{11} experimentally investigated transient mixed convective flow in a channel with two wall-mounted semicircular cavities at 500 ≤ *Re* ≤ 1500. Three thermal and flow regimes were identified: steady, oscillatory, and irregular. Increasing the values of *Re* and *Ri* enhanced heat transfer at a fixed inclination angle. A similar geometry with square cavities was studied at 100 ≤ *Re* ≤ 1000 by García *et al.*,^{12} who found that the aspect ratio of the cavity had crucial effects on heat transfer.

Mixed convective flow in a three-dimensional duct has been considered under conditions of isothermal or constant heat flux imposed on the walls. Ichimiya and Matsushima^{13} analyzed mixed convective flow in a square duct that was three-dimensionally inclined with respect to the horizontal plane. Constant temperature was imposed for both inflow and walls. Two recirculating vortices formed in the inclined duct, and heat transfer was pronounced in the case of opposing buoyancy at 15° ≤ *α* ≤ 60°. Pérez-Flores *et al.*^{14} performed an experimental investigation of transient flow in the case of opposing buoyancy in an inclined duct with two flush-mounted heat sources. Heat transfer was found to be enhanced in the high-*Re* regime, and the inclination was found to have its most substantial effects at *α* = 60° in the low-*Ri* regime. For a horizontally placed duct, buoyancy acts in the vertical direction and has an indirect effect on the flow, and thus flow instability is affected only in the high-*Ri* regime. Three-dimensional effects enhance the local heat transfer close to the corners of a channel. Tian *et al.*^{15} studied mixed convection in the entrance region of a channel at 1000 ≤ *Re* ≤ 3000 and 0° ≤ *α* ≤ 30°. Secondary flow was observed to increase with increasing *α*, and the thermal instability induced by the secondary flow was enhanced with increasing *α* and reduced with increasing *Re*, with consequent enhancement of heat transfer.

The above literature review has considered mainly investigations of mixed convection heat transfer for flow past cylinders and other obstacles or for flow heated by walls in a two- or three-dimensional channel. In contrast to such cases, the flow over a cylinder in a channel is characterized by the interaction between bluff body flow and wall confinement. Owing to its inherent instability, the flow separating from the cylinder is destabilized by environmental perturbations, and this affects the development and recovery of the wake flow. In the case of flow in an inclined channel, the destabilization, development, and recovery of the flow are dependent on the inclination, and thus are very different from those in the case of unheated flow or flow in a vertical channel. To date, there has been a lack of studies focusing on mixed convective flow across a cylinder in an inclined channel with the aim of investigating the suppression of flow instability and the effects on heat transfer. In the present work, we perform a numerical study of low-*Re* mixed convective flow across a circular cylinder in an inclined channel to explore the influences of inclination on fluctuations of wake vortices, suppression of flow instability, the forces acting on the cylinder, fluid–solid heat transfer, and heating of the channel walls.

## II. NUMERICAL DETAILS

The physical model is depicted in Fig. 1. A circular cylinder of diameter *D* is located in an inclined straight channel of width *W* = 4*D* such that the interactions between the boundary flows of the cylinder and channel walls and the through-flow are significant. The inlet and outlet of the channel are far away from the cylinder, at distances of *L*_{u} = 10*D* and *L*_{d} = 50*D*, respectively, to permit full development of the inflow and outflow and prevent the occurrence of reversed flow at the outlet boundary. The inclination angle is *α*, and the channel is positioned vertically when *α* = 0° and horizontally when *α* = 90°. At the inflow boundary, the streamwise velocity exhibits a parabolic profile with average value *v*_{in}. The temperatures of the inflow and cylinder are *θ*_{in} and *θ*_{w}, respectively, and the channel walls are assumed to be adiabatic. The Reynolds number of the inflow is 100 as scaled by *D* and *v*_{in}, and the Prandtl number *Pr* = 0.7 (air).

*Re*=

*ρv*

_{in}

*D*/

*μ*is the Reynolds number and

*Ri*=

*Gr*/

*Re*

^{2}is the Richardson number, where

*Gr*is the Grashof number.

*Ri*is positive or negative when the fluid is heated or cooled by the cylinder, respectively (generating an aiding or opposing buoyancy, respectively). The last term in the momentum Eq. (2) represents the buoyancy arising from temperature variation, which is derived from the Boussinesq approximation. The variables in the governing equations are scaled by

*D*,

*v*

_{in}, $\rho vin2$, and

*D*/

*v*

_{in}. The temperature is nondimensionalized as

*θ*= (

*θ** −

*θ*

_{in})/(

*θ*

_{w}−

*θ*

_{in}), and thus

*θ*= 0 for the inflow and

*θ*= 1 for the cylinder. The boundary conditions are expressed mathematically as

The governing equations are discretized on a multiblock grid, and example of which is illustrated in Fig. 2. A 256 × 768 grid is used in this study, with 256 cells in the transverse direction and 768 cells in the streamwise direction, and 512 cells are placed around the cylinder with a grid size in the wall-normal direction of about 0.005*D*. The discretized governing equations are solved using our in-house code based on the fractional step method.^{16} The size of the physical time step is chosen as 0.002*D*/*V*_{in} to resolve the unsteady patterns. The temporal integration is performed for 500*D*/*V*_{in} until good periodicity of the wake flow is reached.

The in-house code employed in the numerical simulation has been well validated in our earlier works on flow around bluff bodies in a wall-confined channel with forced or mixed convection.^{17–19} For mixed convective flow across a cylinder, the code is validated for forced convective flow in a vertical channel; the parameters are *Ri* = 0.0, *α* = 0°, *Re* = 100, *Pr* = 0.7, and blockage ratio *BR* = 25%. The computed Strouhal number and mean Nusselt number are 0.29 and 6.35, which are in agreement with the values of 0.32 and 6.4 in Prasad *et al.*^{20} We doubled the number of cells in each direction, and found that the mean Nusselt number varied by about 0.9%, which confirms that the present grid is sufficient.

## III. RESULTS AND DISCUSSION

### A. Flow physics

The inclination angle of the channel determines the deviation between the buoyancy and streamwise directions and thus affects the instability and asymmetry of the wake flow. Figures 3 and 4 show the instantaneous vorticity fields for vertical (*α* = 0°) and horizontal (*α* = 90°) channels. For the vertical channel, since the buoyancy and flow are aligned in the same direction, the wake flow sheds transversely in a manner that is symmetric about the centerline *x* = 0. The wake flow is steady or unsteady, depending on *Ri*. Shedding of wake vortices is clearly observed for opposing- and zero-buoyancy cases (i.e., *Ri* ≤ 0), and the decay of the vorticity magnitude in the wake is slower for strong opposing buoyancy; thus, the unsteady wake persists over a rather large region. The transverse motion of wake vortices occupies most of the channel for large negative *Ri*, and the vortices are stretched along the transverse direction, which enhances the interaction between the flow in the wake and the boundary layers of the channel walls. For *Ri* = 0.6 and 1.0, the aiding buoyancy accelerates the near-wall and wake flows as a body force, and lowers the local shearing and flow instability; thus, there is a transition to steady flow.

For flow in the horizontal channel (*α* = 90°), the direction of the aiding buoyancy is toward the left/top wall of the channel, and the opposing buoyancy is in the opposite direction, and both are perpendicular to the walls of the channel. Although the flow is affected by buoyancy, it never transitions to steady. It can be seen from Figs. 3 and 4 that the wake flow does not obviously deviate from the centerline *x* = 0, and the sizes, vorticity magnitudes, and distances of the wake vortices are quite similar for the different Richardson numbers. The shedding flow, as represented by the vorticity with positive and negative magnitudes, exhibits a quasi-symmetric distribution similar to the case of *α* = 0°. This observation demonstrates that the Richardson number investigated here is still small in magnitude, and would not produce a significant effect on the flow patterns.

### B. Global quantities

To quantitatively analyze the effect of buoyancy, in terms of both its magnitude and direction, on the flow patterns, flow-induced forces acting on the cylinder, and heat transfer, this subsection presents and analyzes the variations of characteristic global quantities with the buoyancy. Figure 5 shows the Strouhal number obtained for various values of (*Ri*, *α*) to demonstrate the frequency of fluctuation of the wake flow. The Strouhal number remains at around 0.3 for the *α* = 90° channel and exhibits almost no variation with *Ri*, indicating that the buoyancy perpendicular to the incoming flow has no effect on the fluctuation of the wake flow. For the vertical and inclined channels, there exists a critical Richardson number *Ri*_{cr} at which the flow transitions from unsteady, with opposing and weakly aiding buoyancy, to steady, with medium and strongly aiding buoyancy. The value of *Ri*_{cr} increases with increasing *α* from 0.2 at *α* = 0° to 0.5 at *α* = 30° and 0.8 at *α* = 60° for the several representative Richardson numbers discussed in this work. In the case of unsteady flow, *St* increases monotonically with *Ri*, although the variation is minor; it increases with *α* in the case of opposing buoyancy, but decreases in the case of an aiding buoyancy, and the variation can be up to 11% at *Ri* = −1.0.

*x*) and streamwise (

*y*) directions, respectively. The forces are notably affected by

*α*and

*Ri*. The mean lift increases monotonically with

*Ri*for the inclined and horizontal channels, but remains at a constant value of zero for the vertical channel. The effect of

*Ri*on the mean lift is most pronounced in terms of the amplitude of variation for the horizontal channel, and it gradually becomes weaker as

*α*decreases. However, it can be seen that the mean lift does not vary monotonically with

*α*for the inclined channels at

*α*= 30° and

*α*= 60°, i.e., it increases from around −0.4 at

*Ri*= −1.0 to 0.7 at

*Ri*= 1.0 for

*α*= 30°, but from −0.8 to 1.2 for

*α*= 60°. This observation reflects the complexity of the combined effects of the channel walls and the buoyancy on the near-wall flow, as a consequence of which the boundary layer flow deviates from the incoming flow, and this deviation results in the differences in the behavior of the lift under the actions of an aiding and an opposing buoyancy, respectively.

The mean drag first decreases and then increases with increasing *Ri* for all values of *α*. For the vertical and inclined channels at 0° ≤ *α* ≤ 60°, there is a noticeable variation in mean drag with *Ri*, with a maximum variation of about 10%. The highest drag is observed at *Ri* = −1.0, where the opposing buoyancy is the strongest, while the minimum drag does not occur at *Ri*_{cr}. It is interesting to note that both aiding and opposing buoyancy, as long as they are strong, will always enhance the drag acting on the cylinder. For the horizontal channel at *α* = 90°, the mean drag varies slightly with *Ri*, with a relative difference of about 2%. The drag is relatively large for strongly aiding and opposing buoyancy, indicating that the deviated boundary layer flow somewhat enhances the drag. It can also be noted from Fig. 6 that the effect of *α* on the mean drag is most substantial for the regimes of strongly aiding and opposing buoyancy, but relatively slight for the weakly-aiding buoyancy regime at 0.0 ≤ *Ri* ≤ 0.5.

The variation of mean Nusselt number with *Ri* can be generally divided into two categories depending on *α*.

At

*α*= 0° and 30°,*Nu*_{avg}increases almost linearly with*Ri*in the regimes of strongly aiding and opposing buoyancy. There is a minor decrease at 0.2 ≤*Ri*≤ 0.4; as a consequence mainly of the increased stability of the wake flow, the local heat transfer on the leeward side is reduced (see the discussion in Sec. III C).At

*α*= 60° and 90°, the effect of*Ri*on*Nu*_{avg}is relatively small. At*α*= 60°, the magnitude of*Nu*_{avg}remains almost constant for −1.0 ≤*Ri*≤ 0.2 and then slightly varies by less than 2%. The variation of*Nu*_{avg}in the case of*α*= 90° is close to that of the mean drag, with the magnitude being larger under conditions of strongly aiding and opposing buoyancy and smaller for buoyancy-free cases, although the variation is quite minor; this is a consequence of the local heat transfer on the leeward side, as will be discussed below.

The fluctuations of the forces are inherently dependent on the unsteady–steady transition of the boundary layer and wake flows with increasing *Ri*: they are normally large under opposing- and weakly aiding-buoyancy conditions, but fall to zero as the flow transitions to steady beyond *Ri*_{cr}. The fluctuating forces decrease with increasing *α* under opposing-buoyancy conditions, and the decrease is clearly more rapid for near-vertical channels. For the horizontal channel at *α* = 90°, the effect of *Ri* on the fluctuating forces is minor in magnitude, but the variation can still be up to around 10% for the fluctuating drag. The fluctuating Nusselt number is influenced by *Ri* and *α* in a more complex manner. For channels at *α* = 0°–60°, *Nu*_{rms} decreases with increasing *Ri* as the flow is stabilized and reaches zero at *Ri*_{cr}; however, the deviated wake flow enhances the transverse shedding of wake vortices, and *Nu*_{rms} is finite although small.

### C. Heat transfer on the cylinder

Buoyancy causes the wake flow to deviate, accelerates or decelerates the boundary layer flow, and affects the heat transfer between fluid and cylinder. This subsection analyzes the characteristics of local heat transfer on the cylinder.

The near-wall flow on the windward side of the cylinder is not significantly determined by the shedding wake in the time-averaged sense, and thus the mean local heat transfer is affected by *Ri* and *α* in a consistent manner, i.e., it increases with increasing *Ri* owing to the acceleration of boundary layer flow by the aiding buoyancy. Therefore, we do not present a detailed discussion here. Figure 7 shows the distribution of mean *Nu* on the leeward side of the cylinder (90° ≤ *φ* ≤ 270°) for various values of *Ri* and *α*, in which *φ* = 180° corresponds to the rear stagnation point. The distribution is symmetric about *φ* = 180° for the vertical channel (*α* = 0°) and buoyancy-free case (*Ri* = 0.0), but is nonsymmetric for the inclined channels, where *Ri* has an obvious effect. For the *α* = 0° channel, it can be seen that *Ri* has a notable effect on the magnitude of *Nu* for the whole leeward surface: the magnitude decreases from the lateral side to the leeward side and reaches a local minimum at a certain position, after which it recovers to a local maximum at the rear stagnation point. In the region from *φ* = 90° to a certain point (*φ* = 120° for *Ri* = −1.0 and *φ* = 150° for *Ri* = 1.0), the magnitude increases with *Ri*, which is the same behavior as on the windward side. This is due to the accelerated boundary layer flow before separation, as a consequence of which convective heat transfer becomes strong. In the adjacent region downstream of the cylinder, however, the magnitude of *Nu* decreases with increasing *Ri*, since the wake flow is stabilized, and the size of the wake region and the velocity of the reversed flow are reduced. The distributions for the *α* = 30° channel are quite similar to those for the *α* = 0° channel. However, owing to the inclination of the channel, the curves are not symmetric about the streamwise centerline: the magnitude is large on the left/top section of the cylinder surface for opposing-buoyancy conditions, and on right/bottom section for aiding-buoyancy conditions. The circumferential difference is not obvious; for example, the magnitudes of the local minima on the lateral sides of the cylinder show a relative difference of about 15% for *Ri* = −1.0% and 12% for *Ri* = 1.0.

As the channel inclines further to *α* = 60° and 90°, the nonsymmetric distribution on the leeward surface becomes more obvious. For *α* = 60°, the relative difference reaches 20% and 15% for *Ri* = −1.0 and 1.0, respectively. It can also be noted that the circumferential position of the minimum Nusselt number moves under the effect of variations in buoyancy; for example, compared with the *α* = 0° channel, the two local minima are observed to move from *φ* = 120°/140° to *φ* = 125°/230° at *Ri* = −1.0, and from *φ* = 155°/205° to *φ* = 145°/220° at *Ri* = 1.0. The local maximum does not necessarily occur at the rear stagnation point. For *α* = 90°, the direction of the buoyancy force is perpendicular to the direction of the incoming flow, and the curves representing aiding and opposing buoyancies of the same magnitude are symmetric about *φ* = 180°. As *α* increases, the values on the lateral sides of the cylinder gradually become closer, indicating that the effects of buoyancy have become weaker. This can be explained by the fact that there is negligible unsteady shedding of flow in this region, owing to the quasi-steady boundary layer flow.

The effects of buoyancy and channel inclination on the unsteady characteristics of local heat transfer are revealed in Fig. 8 by the fluctuating amplitude of the local Nusselt number. In general, the fluctuation is zero only at the front stagnation point and nonzero over the rest of the circumstance. There is a local minimum at 120° ≤ *φ* ≤ 130° as the boundary layer flow separates and another at the rear stagnation point where the detached vortices interact with each other and a quasi-steady flow forms. For *α* = 0°, the distribution of the fluctuating amplitude is symmetric about *φ* = 180°. The amplitude decreases monotonically with *Ri* over almost the whole circumstance, reflecting the fact that the boundary layer flow is less destabilized by the weakening opposing buoyancy. The strongest fluctuation is observed within the separated flow at around *φ* = 155°. It can be seen that with increasing *Ri*, the circumferential position of the minimum fluctuation on the lateral side of the cylinder gradually moves downstream from around *φ* = 120° at *Ri* = −1.0 to *φ* = 130° at *Ri* = 0.0 as a result of the weakened opposing buoyancy and the narrowing wake.

As the channel is inclined at *α* = 30°, the variation of the fluctuation with *Ri* is generally the same as for *α* = 0°, i.e., monotonically decreasing with *Ri*. The distribution of the fluctuation upstream of the local minimum is roughly symmetric about the streamwise centerline, which indicates the negligible influence of the channel inclination on the forced convection on the windward side of the cylinder. A notable nonsymmetric distribution is observed in the region 120° ≤ *φ* ≤ 240°, and although there are still two maxima of the fluctuating amplitude, that at *φ* = 160° (right/bottom side of the cylinder) is much stronger than the other. The nonsymmetric distribution becomes less remarkable with increasing *Ri* in the case of opposing buoyancy, but a contrasting distribution is found in the case of weakly aiding buoyancy at *Ri* = 0.3, where the fluctuation on the left/top side of the cylinder is stronger than that on the right/bottom side. When the inclination angle is increased to *α* = 60°, it is notable that for *Ri* = −1.0, there is just a single maximum in the fluctuation, and the differences between the two local maxima for *Ri* = −0.6 and −0.3 are larger than those at *α* = 30°. The same distinctions between the channels with *α* = 60° and *α* = 30° also arise in the case of an aiding buoyancy, although they are less pronounced. For the horizontal channel at *α* = 90°, the fluctuation is substantially reduced, and it is less affected by buoyancy, especially on the windward and lateral sides of the cylinder, as shown by the overlapping curves in Fig. 8. The fluctuation on the leeward side also greatly decreases in magnitude, and a single-peaked pattern is observed for the cases of strongly aiding or opposing buoyancy, with *Ri* = ±0.6 and ±1.0, while the maximum value still appears around *φ* = 160°, as for *α* = 30° and 60°. In conclusion, therefore, the buoyancy and channel inclination mainly affect the fluctuating heat transfer on the leeward side: the distribution of the fluctuation changes from a double-peaked to a single-peaked one as the channel is inclined, and the fluctuating amplitude becomes smaller with increasing *Ri*, owing to the stabilization of the boundary layer flow.

### D. Nonuniform through-flow

The misalignment between the directions of the buoyancy force and the incoming flow will reduce the nonsymmetric flow about the centerline *x* = 0 for an inclined channel. The flows between the left/top and right/bottom walls of the cylinder and the respective nearby channel walls have different velocity distributions and flow rates, and the boundary layer flow separates from the cylinder at different circumferential positions. The spatially nonuniform wake flow also exhibits different and complex patterns of shear and decay compared with those in the absence of buoyancy. In this subsection, we analyze and discuss the characteristics of nonuniform through-flow next to and downstream of the cylinder.

Figure 9 gives the circumferential position of the separation point, as measured by the time-averaged separation angle from the front stagnation point, on the cylinder for various values of (*Ri*, *α*). Since there are two separation points and these are not symmetric about the streamwise centerline, both of their positions are shown on the figure. For *α* = 0°, the angles corresponding to the two separation points coincide with each other, and both increase with increasing *Ri* in an almost linear way, namely, from around *φ* = 100° at *Ri* = −1.0 to *φ* = 119° at *Ri* = 1.0. The downstream movement of the separation points results from the boundary layer flow becoming less destabilized or more stabilized owing to weakening of the opposing buoyancy or strengthening of the aiding buoyancy, respectively, with a consequent reduction of the shearing in the separated shear layer flow and a delay in separation. For *α* = 30°, the separation position on the left/top side of the cylinder is almost unaffected by the inclination of the channel, as can be seen from the corresponding curve, which nearly coincides with that for *α* = 0° case. However, the separation point on the right/bottom side is notably affected; the separation of the boundary layer flow is delayed to around *φ* = 102° for *Ri* = −1.0 and advanced to *φ* = 116° for *Ri* = 1.0. As *α* increases further, the separation point continues to move and becomes less affected by *Ri*, while the difference between the two angles grows. For *α* = 60°, the two separation angles vary substantially with *Ri*; the one on the left/top side moves from *φ* = 102° to *φ* = 115°, and the one on the right/bottom side moves from *φ* = 106° to *φ* = 110° as *Ri* increases from −1.0 to 1.0. Considering the large angle between the buoyancy force and the incoming flow, this pattern of variation indicates that the boundary layer flow to the left/top side is significantly affected by the buoyancy, while the flow to the right/bottom side is less so. For the horizontal channel (*α* = 90°), the effects of buoyancy on the two separation angles exhibit entirely opposite trends: separation is delayed on the side of the cylinder on which the buoyancy force is exerted, while it is advanced on the other hand. Although the direction of the buoyancy force is perpendicular to the direction of flow here, the conclusion is consistent with that in the *α* = 0° case, namely, that an aiding buoyancy will delay separation. It is also demonstrated that the separation point of the boundary layer flow on the cylinder surface is not fixed, but varies with the shedding of flow. However, the position of the separation point varies only slightly, given the low Reynolds number of the present configuration, and thus it will not be discussed further here.

*y*= 0 for a vertical channel in the absence of buoyancy and varies with both

*Ri*and

*α*for inclined channels with nonzero buoyancy. Figure 10 shows the variation of the mass flow rate (MFR) with

*Ri*and

*α*at

*y*= 0, i.e., the position where the size of the gap between cylinder and channel is the smallest. The value of the MFR is exactly 0.5 for the vertical channel regardless of

*Ri*, since the flow is symmetric in a time-averaged sense. For inclined and horizontal channels, the MFR decreases monotonically with

*Ri*under an aiding-buoyancy condition and is smaller than 0.5, reflecting the fact that less than 50% of to fluid flow moves through the gap to the left of the cylinder; this can be attributed to the aiding buoyancy, which causes the flow to deviate toward the negative-

*x*direction even upstream of the windward side of the cylinder. In contrast, an opposing buoyancy increases the MFR. It can be noted that for

*α*= 30° and 60°, the two curves in the aiding- and opposing-buoyancy regimes are not antisymmetric about the point (

*Ri*, MFR)=(0.0, 0.5), which is due to the finite angle between the buoyancy force and the incoming flow.

The deviation of wake flow is quantitatively represented in Fig. 11 by the MFR along the streamwise direction. The value of the MFR is dependent on the streamwise position, owing to the convection of the deviated wake flow, and is exactly 0.5 for the buoyancy-free case and the vertical channel (which is not shown here). For *α* = 30°, it can be seen that the MFR exhibits entirely different distributions in the cases of aiding and opposing buoyancy: it is greater than 0.5 until the far-wake region for opposing buoyancy (−1.0 ≤ *Ri* ≤ −0.3), but its variation is minor. The aiding buoyancy lowers the MFR in the near-wake region *y* < 5 such that a smaller portion of fluid moves through the gap to the left side of the cylinder; the MFR then recovers to become greater than 0.5, especially in the strong-buoyancy cases *Ri* = 0.6 and 1.0, indicating that the fluid is driven by the aiding buoyancy and approaches the left wall of the channel. For *α* = 60°, the value of MFR is increased more by an opposing buoyancy in both the near- and far-wake regions, whereas an aiding buoyancy produces different distribution patterns depending on *Ri*, as indicated by the smaller value just downstream of the cylinder. The distributions of the MFR for the horizontal channel are totally symmetric about the line of MFR = 0.5, since the flow is deviated only in the vertical direction.

The effect of *Ri* and *α* on the shearing of the wake flow is reflected by the magnitude of the vorticity, which indicates the velocity gradient. Figure 12 shows the distribution of the maximum instantaneous vorticity in the streamwise direction. In general, the vorticity magnitude decreases as the flow develops downstream owing to viscous diffusion, and approaches a constant value in the far-wake region. An opposing buoyancy produces stronger vorticity for channels of all inclinations, and the vorticity magnitude decreases with increasing *Ri* in almost the whole of the channel until the far wake. At *α* = 30° and 60°, there are significant differences between the curves for different values of *Ri*, but at *α* = 90°, the curves are all close together, since the unsteadiness of the flow is generated mainly by shear layer instability rather than buoyancy.

### E. Heating of channel walls

The transverse motion of the wake vortices and thermal diffusion heat the channel walls and determine the distribution of temperature and its fluctuation. Figure 13 shows the distribution of the time-averaged temperature on both left/top and right/bottom walls. In general, the temperatures of both walls start to increase at around *y* = 3, since the spatial motion of wake vortices takes some time, and the increase in temperature is much more rapid for unsteady wake flow than for steady flow. At *α* = 0°, the mean temperatures of both walls are the same for the whole channel. The increase in temperature is more rapid for opposing-buoyancy conditions, and the rate decreases with increasing *Ri*. For *Ri* = −1.0 and −0.6, the temperature approaches a constant value at around *y* = 30, whereas it exhibits a moderate increase at around *y* = 15 for *Ri* = 0.0. For the steady flow cases at *Ri* = 0.6 and 1.0, the wall temperature increases rather slowly, because the heating of the walls is due to heat diffusion rather than shedding vortices; the temperature still does not reach a constant value at the outlet of the channel at *y* = 50. The stronger aiding buoyancy at *Ri* = 1.0 results in more efficient heating, which is attributed to the higher flow velocity and thus enhanced convection. For *α* = 30°, the temperature distributions are different for the two walls, and the influence of *α* on the temperature distributions has quite a strong dependence on *Ri*. For *Ri* = −1.0 and −0.6, the temperature of the right/bottom wall increases more rapidly in the near-wake region *y* < 10, owing to the deviated wake flow toward the wall, whereas the left/top wall has a higher temperature in the region 10 < *y* < 20 as a result of the shedding wake vortices. For the aiding-buoyancy cases with steady wake flow, since the buoyancy-driven flow deviates upward, the temperature of the left/top wall is higher than that of the other wall. The temperature distribution for *α* = 60° exhibits several differences compared with that for *α* = 30°. The first is that for an opposing buoyancy, there is a local peak in the region 10 < *y* < 15, owing to the stronger transverse shedding in heated wake flow. The second difference is that the temperature of the right/bottom wall is higher in the far-wake region for opposing-buoyancy cases; this is the result of the combined effects of deviated wake flow and vortex shedding. The third difference is that for steady flow cases with aiding buoyancy, the increase in temperature for *Ri* = 0.6 is even more rapid than that for *Ri* = 1.0, especially for the left/top wall, and the wall temperature is even higher than in cases of unsteady flow as far as the outlet of the channel. The temperature of the channel walls for *α* = 90° is positively correlated with the direction and intensity of buoyancy, i.e., the temperature is higher for the wall toward which the buoyancy acts.

When the flow is unsteady, the channel walls are perturbed by shedding wake vortices, and the temperature also fluctuates. Figure 14 shows the fluctuating amplitude of the channel wall temperatures. Since the separated flow and wake flow are destabilized via the Kelvin–Helmholtz mechanism and exhibit periodic shedding, multiple local peaks form as a result of transverse shedding vortices. For *α* = 0°, the fluctuating amplitude generally decreases with increasing *Ri*, while its peak position varies slightly in the streamwise direction. As *α* increases, the fluctuation decreases, and it is smallest for the horizontal channel. For *α* = 30°, two local peaks are clearly seen, with the upstream one at around 4 < *y* < 6 and the downstream one at around *y* = 10; there are still peaks in the downstream region, but their amplitude is quite small. Inclination of the channel produces different fluctuations on the two walls of the channel. The fluctuation on the right/bottom wall is more notable for *y* < 6, while it is higher for the left/top wall in the region *y* > 8. As a consequence of the deviated wake flow, the heated flow impinges on the channel walls at different positions; in the opposing-buoyancy cases, the fluctuation on the right/bottom wall occurs earlier. The downstream peak around *y* = 10 for the left/top wall is stronger than the upstream one for *α* = 60°, and the amplitude decreases further for *α* = 90°.

## IV. CONCLUSIONS

We have performed a numerical study of flow across a circular cylinder in an inclined channel to investigate the effects of inclination and buoyancy on flow and heat transfer characteristics. Vertical (*α* = 0°), inclined (*α* = 30° and 60°), and horizontal (*α* = 90°) channels have been considered for cases with aiding (*Ri* > 0) or opposing (*Ri* < 0) buoyancy.

The flow across the cylinder is destabilized and stabilized by opposing and aiding buoyancy, respectively. Full stabilization can be achieved for Richardson numbers above a critical value *Ri*_{cr} that exists for vertical and inclined channels and increases with increasing *α*.

The mean drag and heat transfer rate are strongly dependent on *Ri* and *α*. The drag is minimized around *Ri*_{cr} for channels of all inclinations, and local or global minima of the mean Nusselt number are also observed. The fluctuations of lift, drag, and Nusselt number normally decrease to zero as the flow is stabilized.

The mean local heat transfer rate is higher on the windward side of the cylinder and decreases downstream. Heat transfer on the leeward side of the cylinder is weakened with increasing *Ri*, and the differences between the heat transfer curves for different values of *Ri* decrease with increasing *α*. The fluctuating heat transfer rate decreases in amplitude with increasing *Ri* and *α*, and a transition from a double-peaked to a single-peaked distribution occurs as *α* increases.

The separation of the boundary layer flow from the cylinder is dependent on both *Ri* and *α*. Separation is generally prone to occur with opposing buoyancy. Buoyancy and channel inclination also result in a nonuniform through-flow as far as the outlet of the channel.

The walls of the channel are heated mainly by the shedding wake flow. Heating is more effective for unsteady flow, owing to the impingement of transverse shedding vortices on the walls, while heating by steady flow is also significant far from the cylinder, especially for the left/top wall, where the wake flow is driven by an aiding buoyancy.

The physical problem investigated in this work is a simplified model of many engineering applications related to heat transfer between fluid and solid. The stabilization and destabilization of the flow by buoyancy are potentially useful in situations where it is desirable to reduce forces acting on solid objects or to increase or reduce solid–fluid heat transfer.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Wei Zhang**: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). **Ravi Samtaney**: Conceptualization (equal); Formal analysis (equal); Methodology (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.