Modeling of Turbulent Heat-Transfer Augmentation in Gas-Droplet Non-Boiling Flow in Diverging and Converging Axisymmetric Ducts with Sudden Expansion

Abstract

The effect of positive (adverse) and negative (favorable) longitudinal pressure gradients on the structure and heat transfer of gas-droplet (air and water) flow in axisymmetric duct with sudden expansion are examined. The superimposed pressure gradient has a large influence on the flow structure and heat transfer in a two-phase mist flow in both a confuser and a diffuser. A narrowing of the confuser angle leads to significant suppression of flow turbulence (more than four times that of the gas-drop flow after sudden pipe expansion without a pressure gradient at φ = 0°). Recirculation zone length decreases significantly compared to the gas-droplet flow without a longitudinal pressure gradient (by up to 30%), and the locus of the heat-transfer maximum shifts slightly downstream, and roughly aligns with the reattachment point of the two-phase flow. Growth of the diffuser opening angle leads to additional production of kinetic energy of gas flow turbulence (almost twice as much as gas-droplet flow after a sudden pipe expansion at φ = 0°). The length of the flow recirculating region in the diffuser increases significantly compared to the separated gas-droplet flow without a pressure gradient (φ = 0°), and the location of maximum heat transfer shifts downstream in the diffuser.

1. Introduction

Two-phase flows in pipes or channels with a backward-facing step (BFS) are often used in energy and chemical equipment. They have a rather simple flow geometry and are one of the classical types of shear flows, but their flow structure is quite complex. A flow detaches from the sharp edge at the flow SE station, thus forming a region of shear mixing layer. A large recirculation flow region (a few step heights) develops (see comprehensive reviews [1,2]).

The complexity of modeling flow and heat transfer is exacerbated after BFS in the presence of a longitudinal pressure gradient (LPG) in an expanding (diffuser) or narrowing (confuser) subsonic turbulent two-phase flow (see Figure 1). An overview of the state of research on flows in a diffuser or confuser without sudden expansion of a pipe [3] or a channel [4,5] has been presented. The study of the effect of LPG behind a pipe or channel with SE on mean and fluctuational flow and heat transfer is an important for mechanical engineering. There are several studies on the development of separated flows with the influence of longitudinal pressure gradient for a single-phase flow, yet only a few of these experimental works concerned the flow in diffusers and confusers with a BFS [6,7,8,9].

An effect of flow separation in the field of LPG was experimentally evaluated in said studies. The position of an “upper” duct wall was changed, which caused narrowing or expansion of a cross-section, whereas the “lower” wall with the SE remained unchanged. The most detailed structures of the turbulent flow were assessed in [6] using the LDA method along the length of the diffuser channel. The authors measured the profiles of averaged longitudinal and transverse velocities and their fluctuations, Reynolds stresses, length of the recirculation region, triple correlations, and turbulent viscosity. The authors then compared their experimental and numerical data.

The experimental results on the effect of favorable pressure gradient (FPG) and adverse pressure gradient (APG) in a channel behind a BFS on heat transfer and wall pressure distributions at Reynolds numbers Re_H = U_m1H/ν = (0.4–1.2) × 10⁴ were presented in [10]. The diffuser opening angle varied in the range of φ = 0–4°, and confuser narrowing angle was varied within φ = 0–−7.5°. The magnitude of the Nusselt number increases as the LPG increases for a narrowing channel, and it decreases for the diffuser. The locus of the heat-transfer maximum moves downstream with diffuser expansion, and shifts upstream towards the step as the confuser narrows. In [11], a quantitative study assessed the effect of an APG on mean flow, turbulence, and heat transfer in an axisymmetric diffuser in a pipe with SE. The literature also presents experimental [12] and numerical [13,14,15,16,17,18] studies of fluid flow and heat transfer in single-phase turbulent flows without SE of a pipe or channel in the presence of APG and FPG for a single-phase flow.

Solid particles addition to a turbulent flow in a BFS have large effect on reduction of turbulent kinetic energy (TKE) in backward-facing step flow [19]. Droplets evaporation in turbulent flow behind a BFS [20] or after a pipe with SE [21] causes significant intensification of heat transfer (by several times in comparison with a single-phase flow). Authors of this work have published numerical investigations of heat-transfer augmentation in gas-droplet flows behind a pipe with SE [21]. There are few papers concerning numerical simulation of gas-liquid flow [22,23] and droplet-laden [24] flows in a converge or divergent channel without sudden expansion; we know of only one work on the numerical study of heat transfer in two-phase flows after pipe sudden expansion with LPG [25], where the effect of evaporation of water droplets on heat transfer in an axisymmetric diffuser was studied. Heat transfer in turbulent droplet-laden flow with SE with APG and FPG has not been previously performed. The influence of LPG on flow and heat transfer in the confuser and diffuser after pipe SE is evaluated in the present study.

2. Mathematical Methods and Numerical Solution

The motion and heat transfer of a two-phase turbulent gas-droplet flow in a pipe with SE is numerically considered. A sketch of the flow is given in Figure 1. To simulate the dispersed phase dynamics, the Eulerian approach [26,27,28] is used. The Eulerian approach is widely used for the simulations of two-phase confined flows [21,22,29,30]. The system of axisymmetric stationary Reynolds-averaged Navier–Stokes (RANS) equations accounts for the effect of vaporizing drops on mean and fluctuational transport processes [21,25]. The set of governing equations both for gas and dispersed phases have been provided in detail [21,25]. The volume fraction of the droplets is low (Φ₁ = M_L1ρ/ρ_L < 1.2 × 10⁻⁴ for the highest mass fraction studied M_L1 = 10%). Drops are rather small (d₁ < 100 μm), so effects of their collisions can be neglected. Gas phase turbulence is predicted using the elliptical Reynolds stress model [31] by taking the dispersed phase influence on TKE [32]. Break-up and coalescence of droplets in flow is not taken into account due to their rarity (Φ₁ < 1.2 × 10⁻⁴) [33]. The Weber number $\mathrm{We}=\rho {\left({\mathbf{U}}_S-{\mathbf{U}}_L\right)}^{2}d/\sigma$ << 1 and the bag break-up are ignored [33,34]. Here, ${\mathbf{U}}_S=\mathbf{U}+〈u_S^{\prime }〉$ is the gas velocity seen by the droplet, and $〈u_S^{\prime }〉$ is the drift velocity between fluid flow and drops [35]. This assumption is applicable when the pipe cross-section expands for a diffuser. The use of this approach seems less obvious for a confuser, even when taking into account the preliminary pipe with SE. Effect of break-up and coalescence in the flow can be neglected due to a low droplet volume fraction at the inlet according to preliminary author’s estimations.

The technique for numerical implementation of the Eulerian approach for two phases is described in detail in [21,25]. The numerical solution was obtained using the finite volume method on staggered grids. The QUICK scheme of third-order ode accuracy was utilized for solution of convective terms. Central differences of second-order accuracy were evaluated for diffusion fluxes. Pressure–velocity fields were corrected according to SIMPLEC procedure.

All simulations were carried out on a “basic” mesh containing 550 × 200 control volumes (CV) for the diffuser with the largest opening angle, and for the confuser with the largest convergence angle of 550 × 100. The information about meshes for the confuser and diffuser is summarized in

Table 1. Meshes for two-phase flow in the confuser (FPG) and diffuser (APG).

Flow Type	“Basic”	“Coarse”	“Fine”
Confuser	550 × 100	300 × 50	850 × 150
Diffuser	550 × 200	300 × 100	850 × 300

. The difference in calculations of the Nusselt number for the two-phase gas-droplet flow did not exceed 0.1%. A further increase in their number does not significantly affect the results of numerical calculations. The grid verification for the case of droplet-laden flow in pipe with SE was presented in [21]. The grid independence tests for two-phase flows in APG and FPG are given in Figure 2 for the smallest constriction angle in the confuser and for the largest opening angle in diffuser. The Nusselt number at a constant wall temperature is determined by dependence:

$$\mathrm{Nu}=-{\left(\partial T/\partial y\right)}_{W}H/\left(T_W-T_m\right),$$

where T_m is mass-averaged temperature of gas in the considered cross-section.

The convergence criteria for all residual levels in this study were up to 10⁻⁵. The differences in Nusselt number and gas-phase kinetic energy of turbulence for gas-droplet APG and FPG flows were up to 10⁻⁶.

The model was validated against experimental results on the flow and heat transfer for the single-phase axisymmetric diffuser downstream of a pipe with SE. The difference between our predictions and measured results of previous experiments did not exceed 15%. These results were given in our previous paper [25], but this comparison is not presented here. We did not find measured or numerical results concerning the study of an APG or FPG of gas-droplet flow in a pipe or duct with sudden expansion. We performed the comparisons with experimental two-phase droplet-laden mist and solid particle-laden turbulent flow behind the BFS and a pipe with SE. These results were published in a previous paper [21] but are not included here. We believe that the validation analysis of two-phase solid particle-laden and droplet-laden flows behind backward-facing step or pipe sudden expansion without LPG have been fully completed.

3. Results and Discussion

The primary concern of this study was shown the effect of diffuser opening and confuser narrowing angles on the characteristics and heat transfer in the two-phase mist with vaporized water droplets after the pipe with SE. Drop diameter and mass fraction decreased due to evaporation both in the axial and radial directions after the flow detachment section.

The diffuser expanding angle was φ = 0–5° and the confuser convergence angle φ = 0–−3°. The pipe diameter before SE was 2R₁ = 20 mm, after SE it was 2R₂ = 60 mm, and the step height was H = 20 mm. The computational domain after pipe expansion was 25H = 0.5 m. Mass-average air velocity before separation was U_m1 = 15 m/s, and the Reynolds number was Re_H= HU_m1/ν ≈ 2 × 10⁴. The wall temperature was T_W = const = 373 K, and the temperatures of air and droplets at the inlet were T₁ =T_L1 = 293 K. Water droplets were added to a single-phase air turbulent flow at the inlet, and their initial velocity was set constant over pipe cross-section: U_L1 = 0.8U_m1. Inlet droplet size was constant d₁= 1–100 µm, and mass fraction M_L1 = 0.01–0.1. The Stokes number in mean motion was Stk = τ/τ_f = 0.03–3, where τ_f = 5H/U_m1 is the turbulent time macroscale [19,20]. Here,$\tau ={\rho }_L{d}^2/(18\rho \nu W)$ is the particle relaxation time, $W=1+{\mathrm{Re}}_L^{2/3}/6$ and ${\mathrm{Re}}_L=\left|{\mathbf{U}}_S-{\mathbf{U}}_L\right|d/\nu$ is the dispersed-phase Reynolds number. The Stokes number Stk_K = τ/_fK = 0.2–20, where τ_K is the Kolmogorov timescale. While the value of interfacial velocity in our previous works [21,25] was based only on the average velocity of the carrier phase, it is based on the actual value in the present study.

3.1. The Wall Friction and Pressure Coefficients

The distributions of wall friction coefficient $C_f/2={\tau }_W/\left(\rho U_{m1}^2\right)$ and pressure coefficient $C_P=2\left(P_W-P_1\right)/\left(\rho U_{m1}^2\right)$ along the length of diffuser and confuser in gas-droplet flow with variation of expansion and contraction angles are shown in Figure 3. Here, τ_W is the wall friction; P_W, P₁ are the mean static pressures on the wall in considered and inlet cross-sections. The data for the flow after a pipe with SE, without an effect of LPG (φ = 0°), are also shown in this figure for comparison.

The distributions of non-dimensional pressure coefficients along axial coordinates with the development of a separated flow in the diffuser and confuser are shown in Figure 3a. In the diffuser, directly behind the flow separation point, a negative pressure region is formed, with a length of x/H ≈ 7. The pressure coefficient also increases with an increase in the diffuser opening angle, which can be attributed mainly to flow deceleration. The zone with pressure attenuation is formed in the confuser directly behind the flow detachment cross-section, and its length axial direction is x/H = 5–7. With growth of the confuser convergence angle, the presence of a significant region of pressure attenuation is observed, and the absolute value of pressure attenuation increases noticeably as the convergence angle increases (more than 5.5 times at φ = −2°). Obviously, the main reason for a significant pressure decrease in confuser is flow acceleration. The wall friction coefficient C_f decreases significantly (several times over) with growth of APG, and a sharp increase in the flow recirculation zone is observed (see Figure 3b). With the increase in the magnitude of FPG, the wall friction coefficient increases noticeably (almost doubling) after the zone of flow relaxation.

3.2. The Flow Structure in Confuser (FPG) and Diffuser (APG)

Profiles of the mean axial velocity, temperature, and turbulent kinetic energy of the gas phase in a cross-section at x/H = 15 are shown in Figure 4. The predictions are carried out for different values for the diffuser (APG, φ > 0°), the confuser (FPG, φ < 0°), and in the separated flow behind sudden expansion of the pipe (φ = 0°). A large effect of two-phase flow detachment with a zero pressure gradient (ZPG) and with FPG and APG on the mean axial velocity distributions is revealed in two-phase flow. Obviously, the increase in the diffuser opening angle leads to a reduction of gas velocity in the core zone (see Figure 4a). It should be noted that in a cylindrical duct, as well as at small diffuser opening angles (φ ≤ 2°), the separated flow is reattached in cross-section (x/H = 15) and the flow is relaxed. Air velocity and the velocity gradient in the radial direction in the core region increase in the confuser.

Gas temperature distributions $\mathsf{\Theta }=\left(T-T_W\right)/\left(T_0-T_W\right)$ over the pipe radius depend, to a lesser extent, on the longitudinal pressure gradient rather than on distributions of the axial gas velocity (see Figure 4b). Here, T₀ and T_W are gas phase temperatures on a pipe axis and on a wall. The slightly changing diverging angle of the diffuser (φ ≤ 2°) and the converging angle of the confuser (φ ≥ −2⁰) have little effect on the gas phase temperature in droplet-laden flow. The temperature increases for the diffuser and decreases for the confuser. This leads to heat-transfer enhancement in the confuser and heat-transfer suppression in the diffuser. These conclusions qualitatively concur with the results of simulations [11] for a single-phase flow in a diffuser behind a pipe with SE. Gas temperature becomes lesser for gas-droplet flow compared to the case at φ = 0°.

Turbulent kinetic energy (TKE) of the gaseous phase is significantly enhanced (by up to two times over) by an increase in the diffuser opening angle (see Figure 4c). The TKE of the gas phase is calculated for an axisymmetric flow using a known formula: $2k=〈u^{\prime 2}〉+〈v^{\prime 2}〉+〈w^{\prime 2}〉\approx 〈u^{\prime 2}〉+2〈v^{\prime 2}〉$. This is not an effect of the dispersed phase; it is known that the presence of a finely dispersed phase suppresses the carrier-phase turbulence in the separated flow, both behind the BFS [19,20] and with sudden expansion of the pipe [21,22]. Particles or droplets are involved in the mean gas movement and a part of the turbulent energy of a carrier flow is spent on this process [19,32]. The maximum kinetic energy of turbulence is observed in the mixing layer, and the same phenomena were found for the gas-droplet flow in the pipe with sudden expansion at ZPG (φ = 0°) [21]. This effect was shown previously, in our recent study of an axisymmetric diffuser with a sudden pipe expansion [25]. An increase in the LPG in an axisymmetric diffuser with pipe SE causes additional flow turbulization. The maximum value of the turbulent kinetic energy of the carrier phase for a confuser decreases almost twice as much compared to the turbulence level of a separated two-phase flow at ZPG. Such a significant TKE suppression of the carrier phase cannot be explained only by the effect of the dispersed phase.

The transverse distributions of mean axial water droplet velocity U_L/U_L1 (a), drops in temperature ${\mathsf{\Theta }}_L=\left(T_L-T_{L,\max }\right)/\left(T_{L,0}-T_{L,\max }\right)$ (b), and the mass fraction M_L/M_L1 (c) of dispersed phase in the confuser and diffuser in a pipe with SE are presented in Figure 5. Here T_L, T_L, and T_L,_max are the droplet temperature, the droplet temperature on pipe axis, and maximum droplet temperature in corresponding cross-section, respectively.

With growth of the confuser convergence angle, a significant increase in the longitudinal averaged velocity of droplets occurs (by more than double at φ = −2° as compared to the separated flow at ZPG) (see Figure 5a). The droplet temperature profile has a qualitatively similar form for all three types of ducts (ZPG, APG, and FPG) studied previously (see Figure 5b). On the whole, droplet temperature distributions are similar to those for the gas phase. The maximum value of droplet mass fraction is obtained in the axial region of the pipe, and the minimum value is obtained in its near-wall region (see Figure 5c). The simulations for droplets’ mass fractions M_L1 > 10% were not successful due to the possible effect of droplets deposition in reality. Most likely, the distribution of the mass fraction of droplets is qualitatively similar to those for M_L1 = 10%, but there are quantitative differences. It is also necessary to take into account the effect of droplet deposition on the wall from a two-phase flow, and the possible formation of liquid spots and films on the wall surface. The influence of droplet deposition on transport processes and heat transfer are not taken into account for our numerical results obtained for M_L1 = 5%. Obviously, for high values of the droplets’ mass fraction at the inlet, it is necessary to account for the influence of the deposition process and the entrainment of liquid droplets into the droplet-laden flow from the liquid film or spots.

3.3. The Effect of LPG on the Mean Parametrs of the Two-Phase Mist Flow

A significant increase in the length of recirculating area x_R is observed in two-phase flow in diffuser (see Figure 6). The locus of the heat-transfer peak x_max moves in the downstream direction. A slight increase in the flow recirculation region is shown for small expanding angles (φ ≤ 1°), and the position of the heat-transfer maximum is close to the locus of the reattachment point of two-phase flow. The coordinate of x_max moves downstream by almost double (φ = 5°) in comparison with the case of φ = 0°. The presence of FPG leads to a reduction in flow recirculation area and the coordinates of x_max move upstream by about 30–35% compared to the case of φ = 0°. The significant displacement of flow reattachment points in the diffuser and confuser is caused by deformation of the gas phase velocity profile due to the effect of LPG.

The TKE of the carrier phase increases almost two times over in the diffuser at φ = 5° as compared with the case without longitudinal pressure gradient φ = 0°. Changing the confuser convergence angle causes suppression of the level of turbulence more than three times over. The heat transfer decreases significantly with expanding of diffuser opening angle (almost by a factor of 1.5 as compared to the separated flow in the pipe at φ = 0°). For the confuser, an increase in the relative value of the maximum heat transfer is observed at φ = −3° (by approximately 20%). The heat transfer for the confuser (FPG) case has the greatest value, and the diffuser (APG) has the smallest.

The effect convergence (confuser) and divergence (diffuser) on Nusselt number distributions along the axial coordinate for the separated flow (ZPG), confuser (FPG), and diffuser (APG) are shown in Figure 7. Initially, for two-phase mist flows with APG and FPG, the attenuation of heat transfer rate is observed. This is typical both for both types of flows and for the case of gas-droplet flow in pie with SE at φ = 0°. Then there is a sharp increase in heat transfer with the achievement of a maximum heat transfer. In the zone of flow relaxation, the observed decrease in Nusselt number is similar to a single-phase flow.

The influence of water droplets’ mass concentration on the maximal magnitude of heat transfer Nu_max for a diffuser and confuser after pipe SE is presented in Figure 8. For all types of flow behind the pipe with SE at φ = 0°, in the diffuser (φ = 2°), and in the confuser (φ = −2°), an increase in the maximum heat-transfer value (up to 75% in a single-phase airflow) was obtained with increasing mass fraction of droplets to M_L1 = 10%. The heat transfer in confuser enhances compare to the diffuser and for two-phase separated flow with ZPG at φ = 0°.

4. Conclusions

The numerical results of the effects of favorable and adverse longitudinal pressure gradients on the flow and heat transfer augmentation, in a droplet-laden flow in a pipe with SE, are presented. Elliptical second-moment closure was used to predict the gas phase turbulence with taking into account the effect of droplets presence. While this study does not have a direct application, it shows potential ways to control the turbulence level and to enhance heat transfer performance in APG and FPG flow behind a backward-facing step. Thus, these data may be of interest for various practical applications. The scope of the model’s use is limited the inlet droplet diameter d₁ = 100 µm and their initial mass fraction M_L1 ≤ 10%. This can be explained by noting that the model does not take into account the formation and evolution of a liquid film on the pipe wall, as drops break up and coalesce.

The presence of flow expansion (diffuser) and constriction (confuser) of pipe with SE shows significant effect on the mean and fluctuational flow characteristics, and heat transfer in an axisymmetric. The increase of the confuser constriction angle causes considerable reduction of the pressure coefficient. The length of the flow recirculating area noticeably shortens compared to the gas-droplet flow behind the pipe with SE at angle φ = 0°, and the point of maximum of heat transfer slightly shifts downstream. The heat transfer augmentation and the suppression of turbulence in a two-phase flow in a confuser are mainly due to the FPG. The large growth of flow recirculating area (up to 3.5 times at φ = 5°) compared to the gas-droplet flow downstream of pipe SE at φ = 0° is obtained. The expansion of the diffuser leads to reduction of the wall friction coefficient. Two-phase flow does not reattach to the wall at angle φ = 5°. Points of flow reattachment and maximum heat transfer are significantly shifted downstream by an increase in the opening angle of the diffuser. The significant heat-transfer suppression (by up to 1.5 times) and turbulence production (by up to two times) are observed for the two-phase mist flow in a diffuser.