Relationship between Pulsation Conditions and Pin Fin Spacing for Heat Transfer Enhancement under Pulsating Flow Conditions

1  Introduction

Water-cooling devices are widely used for forced convection cooling in electronic equipment to dissipate high heat loads [1]. Particularly in applications such as cooling power semiconductors for electric/hybrid vehicles [2] and water-cooling systems for AI servers [3,4] and data centers, cooling devices must be designed for environments densely populated with electrical and electronic components. Consequently, extremely stringent conditions are imposed to achieve both miniaturization and high-performance cooling simultaneously [5].

Conventionally, the improvement of heat exchanger performance has largely relied on passive heat transfer enhancement techniques, such as arranging pin fins or rib structures [6–8]. However, relying solely on these passive heat-transfer enhancers in miniaturized, narrow heat exchangers inevitably increases the pressure drop associated with the narrowed flow passages. Therefore, there is a strong need to develop cooling technologies based on a novel concept that does not rely solely on passive heat transfer enhancement structures.

The authors have focused on pulsating flow as a potential solution. The potential for heat transfer enhancement through pulsating flow has been evaluated by various researchers [9–15]. The authors have also been advancing research to develop high-performance cooling devices by inducing flow pulsation. Through both experiments and computational fluid dynamics (CFD) analyses, it has been demonstrated that pulsation can enhance heat transfer by inducing secondary flows in the separation regions behind ribs or fins [16]. Furthermore, previous studies have shown that pulsating flow can achieve a cooling performance index greater than unity, indicating that the enhancement in heat transfer outweighs the accompanying increase in pressure drop [16]. Based on these findings, the present study focuses specifically on the heat transfer characteristics of pulsating flow, with particular emphasis on clarifying the effects of pulsation conditions and pin-fin spacing on heat transfer enhancement.

While the potential for heat transfer enhancement by pulsating flow has been established, its effectiveness remains unclear in scenarios where multiple pin fins are mounted in a continuous array—a configuration commonly found in actual heat exchangers—in which flow separation occurs between the pin fins [17] and the vortices generated by the fins interact with one another. Furthermore, addressing the impending issue of power consumption reduction in AI data centers is a critical social challenge; thus, minimizing energy consumption for cooling is vital. Therefore, discussing the effect of pulsating flow on heat-transfer enhancement structures under conditions that closely resemble actual mounting conditions is considered highly beneficial from an engineering perspective.

Based on the above, this study aims to systematically evaluate the heat transfer enhancement effect of pulsating flow under conditions closer to actual implementation. Specifically, the study investigates pulsating flow across an array of three pin fins mounted in a rectangular channel simulating a cooling device, parameterized by three factors: pin arrangement (fin spacing), time-averaged flow velocity, and pulsation frequency. Using experiments and 3D-CFD analysis, we examined the pulsation-induced heat transfer enhancement and its key flow structures, with the Strouhal number as the primary indicator. This paper reports the specific conditions under which the heat transfer enhancement effect of pulsating flow is maximized.

2  Test Model

To begin with, Fig. 1 shows the test model of the flow passage with pin fins investigated in this paper. The working fluid was water. Three pin fins, each with a width of d, a length of d, and a height of 4d, were mounted in a rectangular cooling channel with a width of 3d and a height of 4d. A heating element was attached to the top of the pin fins, and the generated heat was dissipated into the cooling water flowing through the channel via thermal conduction within the pin fins and forced convection heat transfer from their surfaces. In this paper, the focus was on how pulsation conditions and the secondary flow patterns generated by pulsating flow were affected by fin spacing. The dimensions of the three pin fins were fixed, and the clearance between the fins in the flow direction, c, varied as a parameter. The condition of the c was investigated under three conditions: c = d, c = 2d, and c = 3d. Furthermore, the clearance between the pin fins and the channel walls was fixed at d, equal to the fin width.

Figure 1: Test model of the cooling channel with three pin fins investigated in this paper: (a) overall view; (b) top and front views.

3  Method of CFD Analysis and Experiment

This chapter outlines the CFD and experimental methods used in this study. The section on CFD analysis describes the computational model and the solver employed. The section on the experiments explains the test section and the measurement apparatus used for the heat transfer and flow visualization experiments. Furthermore, the specific analytical and experimental conditions, as well as evaluation methods, are summarized at the end of this chapter.

3.1 CFD Analysis

Fig. 2 shows the schematic of the analytical model. The dimensions of the pin fins and the channel were already explained in Section 2. The lengths of the entrance region and the downstream area, for which the boundary conditions at the inlet and the outlet did not apply to the analytical results around pin fins, were determined.

Figure 2: Schematic of analytical model.

To evaluate the forced convection heat transfer performance of the pin fins and pulsating flow combination, a 3-dimensional transient conjugate flow and heat-transfer analysis was conducted. The water flow in the channel, heat transfer in the flow and on the pin fins’ surfaces, and thermal conduction in the pin fins were calculated. The governing equations were the equation of continuity, the Navier-Stokes equations, and the energy equation. The equations are as follows:

<Liquid region>

(a)   Equation of Continuity

(∇⋅u→)=0,(1)

(b)   Navier-Stokes Equation

∂u→∂t+∇⋅(u→u→)=−∇Pk+∇⋅[ν{∇u→+(∇u→)}T],(2)

(c)   Energy Equation

ρfcp,f∂Tf∂t+u→⋅∇(θ)=∇⋅(λf∇Tf),(3)

<Solid region>

(d)   Energy Equation

ρscp,s∂Ts∂t=∇⋅(λs∇Ts)+q˙s,(4)

(e)   Temperature control at the boundary between solid and fluid

Tf=Ts,(5)

(f)   Energy conservation at the boundary between solid and fluid

−λf∇Tf⋅n→=−λs∇Ts⋅n→,(6)

where u→ [m/s]: velocity vector, t [s]: time, Pk [m2/s2]: kinematic pressure, defined as the static pressure divided by the reference density, ν [m2/s]: kinematic viscosity, ρ [kg/m3]: density, cp [J/(kg K)]: specific heat at constant pressure, T [K]: temperature, λ [W/(m⋅K)]: thermal conductivity, q˙ [W/m3]: volumetric heat flux, n→ [-]: Interfacial normal vector. The subscripts f and s denote fluid and solid, respectively. The CFD code OpenFOAM v. 2306, an open-source CFD toolbox [18], was used for the solver.

Fig. 3 shows the boundary conditions applied to the velocity, pressure, and temperature fields, respectively. For the velocity field, a velocity boundary condition and zero pressure gradient were applied at the inlet of the flow channel, while free outflow and gauge pressure of 0 Pa were applied at the outlet. For the pin fins and other wall surfaces, a no-slip wall boundary condition was applied. Regarding the temperature boundaries, a constant-temperature boundary was applied at the inlet of the flow channel, and a free outflow boundary was applied at the outlet. A constant heat input was applied to the heat transfer surface, and temperature-heat flux coupling conditions based on conjugate heat transfer were applied at the interface between the fin and the fluid. The analysis time was set to the point at which quasi-steady-state conditions were confirmed for each condition. The green plots in Fig. 3 indicate the temperature measurement points used to calculate the heat transfer performance. The temperature at the center of each side, at the top of each pin fin, was treated as the representative temperature, and temperatures on all four sides were evaluated.

Figure 3: Boundary conditions and temperature measurement points: (a) velocity and pressure boundary conditions; (b) temperature boundary conditions and temperature evaluation points.

3.2 Experimental Apparatus for Heat Transfer and Visualization Experiment

Fig. 4 shows a schematic diagram of the experimental apparatus used in the heat transfer experiments. The apparatus consisted of an ultrasonic flow meter (FD-XS20, KEYENCE), a proportional solenoid valve (KFPV300-2-80-FM, KOGANEI), a test section, a water tank, and a pump (RMD-1001, IWAKI). The working fluid was water at room temperature. A target sinusoidal pulsating flow was generated by feeding back the flow rate measured by the flow meter and adjusting the valve drive signal using PID control. DC-regulated power supplies were used to provide power to the proportional solenoid valve, the ultrasonic flow meter, and the valve controller (KFPC1-F07-DN-DC24V, KOGANEI). Additionally, an AC-supplied voltage controller was utilized. The measured temperature data were recorded using data loggers (LR8401, HIOKI; GL900, GRAPHTEC).

Figure 4: Schematic of the experimental apparatus.

Fig. 5 shows the details of the test section. Consistent with the analytical model, the test section is a rectangular flow channel. In the experiment, 180d of the entrance region was prepared in front of 1st pin fin. The total length of the test channel, including the entrance region and the downstream area, was 281d. These were almost the same length as the CFD analysis. The material of the test section was polycarbonate (λs = 0.23 W/(m⋅K), near the heater units) or acrylic (λs = 0.21 W/(m⋅K), in another area) to perform flow visualization and to inhibit heat dissipation through the channel wall. The thickness of the test section materials was 5 mm. The XPS (λs = 0.028 W/(m⋅K)) was additionally installed around the test section to improve its insulation performance.

Figure 5: Schematic of test section and temperature measurement points.

As shown in the right-side figure of Fig. 5, the water inlet temperature, the pin fin surface temperature near the test section, and the temperature distribution between the heater unit and the pin fin surface temperature points were measured. As in the CFD analysis, the temperature at the center of each side, at the top of each pin fin, was treated as the representative temperature, and temperatures on all four sides were evaluated. However, to prevent water leakage from the test section, the clearance between the insert hole in the test section wall and the pin fin must be filled with high-temperature silicone sealant. Therefore, as the representative temperature, the pin fin surface temperature at the point approaching the outside wall. The pin fin surface temperature was measured using K-type thermocouples (0.1 mm in diameter). In addition, to evaluate the heat flux through the pin fins, the temperature distribution inside the pin fin was measured by K-type sheathed thermocouples (0.5 mm in diameter).

Fig. 6 shows a schematic of a flow visualization experiment. For the flow visualization, a test section with the same dimensions as the one used in the heat transfer experiments was employed. However, the pin fins were fabricated from acrylic to ensure optical access. The flow pattern in the xy cross-section at the middle of the channel height was visualized. Specifically, a laser light sheet approximately 1 mm thick was irradiated at the center of the channel in the height direction. Tracer particles (ORGASOL, specific gravity: 1.03) were suspended in the water, and their motion was captured using a high-speed camera.

Figure 6: Schematic of test section for flow visualization.

3.3 Conditions of Pulsating Flow

Table 1 summarizes the conditions of the CFD analysis and the experiments. Because the pulsating flow rate varies over time, the time-averaged Reynolds number shown below was used to determine the flow rate.

Rem=um⋅deν,(7)

where um [m/s] is the time-averaged flow velocity during one cycle, de [m] is the characteristic length of the test channel based on the hydraulic diameter of the channel, and ν [m2/s] is the kinematic viscosity. The time-averaged Reynolds number was set to 50, 100, 200, 300, and 400 in the CFD analysis to evaluate the effects of pulsating flow. In the experiments, the time-averaged Reynolds number was set to 50, 100, 150, 250, and 300, with the maximum value limited to 300 due to the experimental apparatus’s allowable pressure. This specific range was selected because cooling channels are expected to become increasingly narrow as electronic devices continue to miniaturize. The pin fin spacing was nondimensionalized as c/d, where c is the clearance between the pin fins in the flow direction, and d is the pin fin width. To ensure a fair comparison of the heat transfer performance per unit flow rate, the time-averaged supply flow rate per cycle was matched between the steady and pulsating flows. As shown in Fig. 7, the velocity waveform of the pulsating flow was defined as a sine wave. The velocity waveform was defined as follows:

u(t)=um{1+sin⁡(2πft)},(8)

where u(t) [m/s] is the instantaneous flow velocity at time t, um [m/s] is the time-averaged flow velocity over one pulsation cycle, f [Hz] is the pulsation frequency, and t [s] is time. According to Eq. (8), the minimum and maximum velocities are 0 m/s and 2 um [m/s], respectively. At the maximum time-averaged Reynolds number of 400, the maximum and minimum pulsating velocities were 0.032 and 0 m/s, respectively.

Figure 7: Conditions of pulsating waveform.

3.4 Evaluation Methods

First, to comprehensively evaluate the transient flow phenomena induced by the pulsation, a dimensionless number was introduced. In previous papers, Kikuchi et al. [19] have demonstrated that the heat transfer characteristics of pulsating flow around a cylinder depend on the pulsating Strouhal number, which uses the cylinder diameter as the characteristic length. Furthermore, Nagashima and Fukue [20] have shown that the heat transfer enhancement due to pulsating flow on a heat transfer surface in the main flow separation region can be well correlated with an arranged pulsating Strouhal number based on the dimensions of the separation region. Therefore, based on these concepts, in this paper, we attempted to evaluate the characteristics by defining the pulsating Strouhal number around a rectangular pin fin as follows:

St=d⋅fus,(9)

where us [m/s] is the time-averaged flow velocity at the clearance beside the pin fins.

The relationship between pulsating flow conditions and heat transfer performance is evaluated by examining the relationships among the time-averaged heat transfer coefficient, pulsating Strouhal number, and time-averaged Reynolds number. The heat transfer performance of the pin fins was evaluated as the average value of three pin fins. The time-averaged heat transfer coefficient of the pin fins, hm [W/(m2·K)], was defined as Eq. (10).

hm=13∑i=13Qfin,iA⋅(Tfin,i−Tfluid),(10)

where Qfin [W] is the heat value through the pin fin, A [m2] is the surface area of the pin fin, and Tfluid [K] is the inlet temperature of the water. In the experiment, Tfin,i [K] is the average temperature of the pin fins, calculated from 4 evaluation points, as shown in Figs. 3b and 5. About Qfin,i, in the CFD analysis, the input heat value from the thermal boundary at the top of the pin fins was used. In the experiment, the Qfin,i value was estimated from the temperature distribution of the heater unit, measured with sheathed thermocouples, as shown in Fig. 8.

Qfin=12(Qh1+Qh2),(11)

Figure 8: Calculation method of heat value through heater unit by temperature distribution.

As the overall heat transfer performance, the following time-averaged Nusselt number was used. Using this time-averaged local heat transfer coefficient, the time-averaged Nusselt number was calculated by Eq. (12).

Num=hm⋅dλfluid,(12)

where d [m] is the pin-fin width, and λfluid [W/(m·K)] is the thermal conductivity of the water.

4  Results and Discussion

4.1 Relationship between Heat Transfer Performance and c/d

This section evaluates the effect of the dimensionless pin fin spacing, c/d, on the heat transfer enhancement induced by pulsating flow. Fig. 9 shows the relationship between c/d and the time-averaged overall Nusselt number obtained from CFD analysis and experiments. As representative conditions, the results for pulsation frequencies of (a) 0.5 Hz and (b) 3 Hz are presented. The legends indicate the time-averaged Reynolds numbers (Rem).

Figure 9: Relationship between Num and c/d at (a) f = 0.5 Hz and (b) f = 3 Hz. The left and right panels show the analytical and experimental results, respectively.

First, when comparing steady and pulsating flows, the pulsating flow exhibited superior heat-transfer enhancement across all conditions in this study, even when the pin fins were mounted in a continuous array. Conversely, under low pulsation frequency and high Reynolds number, the heat transfer performance was suppressed as c/d decreased. However, this difference became marginal for c/d < 2.0. Furthermore, under high-frequency conditions, the heat transfer performance showed little dependence on c/d. Additionally, the experimental results of average Nusselt number qualitatively confirmed the same trends as those observed in the CFD analysis.

Next, the relationship between c/d and heat transfer characteristics is verified using flow visualization. Figs. 10 and 11 illustrate representative flow fields around the pin fins for c/d = 1, 2, and 3 at Rem = 400, with pulsation frequencies of 0.5 and 3 Hz, respectively. First, across all conditions, during the acceleration period, although some vortices or partial inflows were observed in the clearance between the fins, the flow remained essentially separated from the mainstream. However, during the deceleration period, the formation of a secondary flow attempting to enter the clearance from the mainstream was confirmed. This flow structure is identical to that reported in previous studies [16,19,20] and can enhance heat transfer in the separation region through flow pulsation. These visualization results demonstrated that even in a continuous pin fin array, pulsating flow can suppress deterioration in heat transfer performance between the fins. This phenomenon is considered the primary factor behind the overall enhancement of heat transfer in pulsating flow.

Figure 10: Flow velocity distribution in the case of Rem = 400 and f = 0.5 Hz.

Figure 11: Flow velocity distribution in the case of Rem = 400 and f = 3 Hz.

Next, the results for each pulsation frequency are discussed in detail. Regarding the results at 0.5 Hz, under c/d = 1, the secondary flow generated during the deceleration period failed to penetrate deeply into the clearance and instead recirculated near its entrance. Conversely, as the fin spacing widened, the secondary flow flowed into the entire clearance. Particularly at c/d = 3, a flow attempting to impinge on the rear surface of the fin was clearly observed. Therefore, at low frequencies, a narrow clearance leads to a high pressure drop, and the pressure recovery from mainstream deceleration is insufficient to drive the fluid deep into the clearance, thereby suppressing heat transfer enhancement. In contrast, a wider clearance allows the secondary flow to sufficiently agitate the coolant within the gap, thereby improving heat transfer performance.

On the other hand, focusing on the results at 3 Hz, as shown in Fig. 9, the variation in heat transfer enhancement with respect to c/d is small. Observing the flow field, even when c/d is narrow, it can be confirmed that the flow enters the clearance during the deceleration period and reaches the rear of the fins. In other words, at a high pulsation frequency, the large velocity fluctuation of the mainstream induces a correspondingly large pressure fluctuation. This strong pressure fluctuation allows the fluid to sufficiently flow even into a narrow clearance. Consequently, since the fluid inside the clearance is effectively agitated regardless of spacing, the difference in heat transfer performance due to c/d is considered small.

4.2 Pulsating Strouhal Number

Fig. 12 shows the relationship between the pulsating Strouhal number, St, and the time-averaged Nusselt number obtained from both the numerical analysis and the experiments for the representative conditions of c/d = 1 and 3. The markers indicate the Nusselt numbers for pulsating flow, with different symbol types corresponding to various time-averaged Reynolds numbers (Rem). For comparison, the horizontal lines in each graph represent the Nusselt numbers for the corresponding steady flows. The numerical and experimental results both exhibited a characteristic pulsating Strouhal number at which the Nusselt number reached a maximum. For each Reynolds number condition, the Nusselt number generally increased as St approached the peak condition, whereas it tended to decrease with further increases in St beyond the peak condition. Within the range investigated in this study, the pulsating Strouhal number corresponding to the maximum Nusselt number showed some variation with Rem. Nevertheless, the peak conditions obtained from both the numerical analysis and the experiments were of the same order of magnitude and occurred approximately within the range of St = 0.3–0.7. The absolute Nusselt numbers obtained experimentally tended to be higher than those obtained numerically at comparable Reynolds numbers. This difference is attributed to heat leakage in the experimental apparatus, which led to an apparent overestimation of the heat transfer performance. The effect of heat leakage on the heat transfer performance is discussed in detail in Appendix B. Small deviations in the geometry and installation positions of the experimental pin fins may also have affected the downstream flow structures, as discussed below.

Figure 12: Relationship between Num and St at (a) c/d = 1 and (b) c/d = 3. The left and right panels show the analytical and experimental results, respectively.

To investigate the flow structure associated with the characteristic pulsation condition at which the heat transfer performance was maximized, an evaluation was conducted through flow visualization and visualization of the Q-value (the second invariant of the velocity gradient tensor). Figs. 13–16 compare the analytical Q-value visualizations at Rem = 400 with the experimental flow visualizations at Rem = 300 for each pulsation frequency at c/d = 2. Furthermore, the regions where the asymmetric flow structure is particularly evident are highlighted by red circles in Fig. 15.

Figure 13: Comparison of analytical and experimental flow structures at f = 0.5 Hz and c/d = 2 (Rem = 400 for the analysis and Rem = 300 for the experiment).

Figure 14: Comparison of analytical and experimental flow structures at f = 1 Hz and c/d = 2 (Rem = 400 for the analysis and Rem = 300 for the experiment).

Figure 15: Comparison of analytical and experimental flow structures at f = 2 Hz and c/d = 2 (Rem = 400 for the analysis and Rem = 300 for the experiment).

Figure 16: Comparison of analytical and experimental flow structures at f = 3 Hz and c/d = 2 (Rem = 400 for the analysis and Rem = 300 for the experiment).

The analytical and experimental results correspond to Rem = 400 and Rem = 300, respectively, as indicated at the top of each figure. Among these conditions, pronounced left-right asymmetry developed under the condition corresponding to the maximum heat transfer performance shown in Fig. 15, particularly during the deceleration period, as highlighted by the red circles. For the analytical condition of Rem = 400, the pulsation frequency of 2 Hz corresponds to approximately St = 0.3, whereas for the experimental condition of Rem = 300, the same frequency falls within the experimentally observed peak range of St = 0.5–0.7.

In contrast, under the other pulsation conditions, the vortex structures in the analytical results remained essentially left-right symmetric throughout the pulsation cycle. In the experimental visualizations, slight left-right asymmetry was observed even under these conditions. This slight asymmetry is considered to result from small deviations in the pin-fin geometry and installation positions in the experimental apparatus, which can break the ideal left-right symmetry of the flow passage and promote asymmetric flow development downstream of the pin fins.

Nevertheless, the asymmetry became more pronounced under the condition corresponding to the maximum heat transfer performance. Therefore, the important feature is not the mere presence of flow asymmetry, but its pronounced development under the pulsation condition associated with maximum heat transfer performance.

To quantitatively evaluate the fluid mixing in the downstream regions of the pin fins, the time-averaged volumetric flow rates through the evaluation surfaces shown in Fig. 17 were calculated. Fig. 18 shows the relationship between the pulsating Strouhal number and the total time-averaged volumetric flow rate through these surfaces at Rem = 400 and c/d = 2. The volumetric flow rate reached its maximum near St = 0.3, where pronounced flow asymmetry and the maximum heat transfer performance were also observed. This result indicates that the asymmetric flow structure enhances the exchange of fluid in the downstream regions of the pin fins, thereby promoting fluid mixing. Such enhanced mixing is considered to contribute to the maximization of heat transfer performance.

Figure 17: Schematic of the evaluation surfaces L1–L3 and R1–R3 for the volumetric flow rate in the downstream regions of the pin fins.

Figure 18: Relationship between the pulsating Strouhal number and the total time-averaged volumetric flow rate through the evaluation surfaces at Rem = 400 and c/d = 2.

Although the pulsating Strouhal number corresponding to the maximum heat transfer performance differed quantitatively between the analytical and experimental results, both results indicated that pronounced flow asymmetry developed under their respective maximum-performance conditions. Taken together with the flow visualizations, these results suggest that pronounced asymmetric flow structures develop under characteristic pulsating Strouhal number conditions rather than being determined by the pulsation frequency alone. The enhanced fluid mixing induced by this asymmetric flow structure is considered to contribute to the maximization of heat transfer performance.

4.3 Development of a Correlation for Heat Transfer Performance in the Increasing-St Region

Based on the numerical results discussed in the preceding sections, a correlation was developed to predict the time-averaged Nusselt number by incorporating the effects of the pulsating Strouhal number and the dimensionless pin-fin spacing. In general, the Nusselt number for forced convection is correlated with the Reynolds and Prandtl numbers in the following form:

Nu=C⋅Prm⋅Ren,(13)

where C, m, and n are empirical coefficients determined by the flow configuration and thermal conditions. Based on this conventional form, the present correlation was constructed by additionally incorporating the effects of the pulsating Strouhal number, St, and the dimensionless pin-fin spacing, c/d, which were identified as important parameters governing the heat transfer performance in the present study.

As shown in the preceding sections, the time-averaged Nusselt number increased with increasing St in the low-St region. Therefore, the present correlation was constructed to represent the heat transfer performance in this increasing region. The proposed correlation is expressed as follows:

Num=(110log10⁡St+0.41)⋅(0.186ln⁡cd+1.045)⋅Pr0.25⋅Rem0.47.(14)

The first term represents the effect of the pulsation condition characterized by the pulsating Strouhal number. In the numerical results, Num increased with increasing St in the low-St region; therefore, a logarithmic function was introduced to express this tendency. The second term represents the effect of the dimensionless pin-fin spacing, c/d. As discussed in Section 4.1, the Nusselt number increased approximately logarithmically with increasing c/d, and this trend was therefore incorporated into the correlation. The remaining terms, Pr0.25 and Rem0.47, represent the effects of the thermophysical properties of the working fluid and the time-averaged flow condition, respectively, based on the conventional form of forced-convection heat transfer correlations.

Fig. 19 compares the Nusselt numbers predicted by the proposed correlation with those obtained from the numerical analysis under the representative condition of c/d = 2. The results are shown as a function of the pulsating Strouhal number for different time-averaged Reynolds numbers. As shown in Fig. 19, the predicted Nusselt numbers agree well with the numerical results in the region where Num increases with increasing St. In this region, the average error of the proposed correlation relative to the numerical results was 2.78%. On the other hand, the proposed correlation does not reproduce the decreasing trend of Num observed at higher St. This is because the present correlation was constructed to represent the increasing region of Num with respect to St. The development of a correlation that can also describe the decreasing region at higher St remains a subject for future work.

Figure 19: Comparison of Nusselt numbers predicted by the proposed correlation and obtained from the numerical analysis at c/d = 2.

5  Conclusion

In this study, to establish comprehensive design guidelines for maximizing heat transfer enhancement by combining pulsating flow with pin fin arrays, the heat transfer performance of pulsating flow over a continuous pin fin array was systematically investigated using experiments and 3D-CFD analysis. The following conclusions were obtained.

When comparing steady and pulsating flows, the pulsating flow exhibited superior heat-transfer enhancement under all conditions examined in this study, even when the pin fins were mounted in a continuous array. This indicates that pulsating flow can suppress deterioration in heat transfer performance between the fins, the primary factor contributing to the overall heat transfer enhancement.

The heat transfer performance under pulsating flow depends on the pulsation frequency and the pin-fin spacing. At low pulsation frequencies, the narrow clearance results in a high pressure drop, thereby suppressing heat transfer enhancement. However, as the clearance widens, the secondary flow effectively agitates the coolant within the gap, thereby continuously improving heat transfer performance. At a high pulsation frequency, the large velocity fluctuations of the mainstream induce correspondingly large pressure fluctuations, allowing the fluid to flow into a narrow clearance. Consequently, the heat transfer performance showed almost no dependence on fin spacing, as the fluid within the clearance is effectively agitated regardless of spacing.

Regarding the pulsating Strouhal number, both the analytical and experimental results exhibited characteristic conditions under which the heat transfer performance reached a maximum. Although the pulsating Strouhal number corresponding to the maximum performance showed some variation with the Reynolds number, the peak conditions in both the analytical and experimental results were of the same order of magnitude and occurred approximately within the range of St = 0.3–0.7. Pronounced asymmetric flow structures developed under the respective conditions of maximum heat transfer performance. Furthermore, the analytical results showed that the time-averaged volumetric flow rate through the downstream regions of the pin fins reached its maximum near St = 0.3, indicating enhanced fluid exchange and mixing. These results suggest that the pronounced asymmetric flow structure promotes fluid mixing around and downstream of the pin fins, thereby contributing to the maximization of heat transfer performance.

Based on these findings, a correlation was developed to predict the time-averaged Nusselt number by incorporating the effects of the pulsating Strouhal number, dimensionless pin-fin spacing, time-averaged Reynolds number, and Prandtl number. Within its applicable range up to approximately St = 0.3, the proposed correlation reproduced the numerical results with an average error of 2.78%, demonstrating its capability to quantitatively predict the heat transfer performance under the investigated conditions.

Acknowledgement: This work was partly supported by JSPS KAKENHI Grant Number 25K07625. Part of the CFD analysis in this paper was conducted using the FUJITSU Supercomputer PRIMEHPC FX1000 and FUJITSU Server PRIMERGY GX2570 (Wisteria/BDEC-01) at the Information Technology Center, The University of Tokyo. The authors are deeply grateful to Mizuki Katsumata and Yosuke Shimizu for their kind support of the experiments and CFD analyses conducted in this study.

Funding Statement: This work was supported by JSPS KAKENHI Grant Number 25K07625, awarded to Takashi Fukue (T. F.) (https://www.jsps.go.jp/english/e-grants/).

Author Contributions: The authors confirm contribution to the paper as follows: study conception and design: Jumpei Hatakeyama, Takashi Fukue, Hidemi Shirakawa, and Yasuhiro Sugimoto; experimental investigation, numerical analysis, and data collection: Jumpei Hatakeyama; analysis and interpretation of results: Jumpei Hatakeyama, Takashi Fukue, Hidemi Shirakawa, and Yasuhiro Sugimoto; draft manuscript preparation: Jumpei Hatakeyama; manuscript review, supervision, and revision: Takashi Fukue, Hidemi Shirakawa, and Yasuhiro Sugimoto. All authors reviewed and approved the final version of the manuscript.

Availability of Data and Materials: The data that support the findings of this study are available from the corresponding author, Jumpei Hatakeyama, upon reasonable request.

Ethics Approval: Not applicable.

Conflicts of Interest: The authors declare no conflicts of interest.

Nomenclature

Appendix A

An evaluation of mesh dependency was conducted to determine an appropriate number of computational cells for the CFD model. The number of cells was progressively increased by approximately 1 million, and the heat transfer performance of each computational model was evaluated using the time-averaged Nusselt number, Num. The appropriate mesh resolution was determined based on the convergence behavior of Num with increasing number of computational cells. The evaluation was conducted with Rem = 400, c/d = 2, and f = 2 Hz.

Fig. A1 shows the relationship between the number of computational cells and Num. The value of Num increased with increasing number of cells up to approximately 4.2 million cells. However, further mesh refinement resulted in only a slight increase in Num, with a difference of approximately 0.4% between the model with approximately 4.2 million cells and the finest-mesh model. Therefore, considering the balance between computational accuracy and computational cost, approximately 4.2 million cells were employed for the CFD analyses in this study.

Figure A1: Effect of the number of computational cells on the time-averaged Nusselt number.

Appendix B

As described in the main text, the heat transfer performance obtained experimentally was higher than that obtained from the CFD analysis, particularly under low-Rem conditions. One possible reason for this discrepancy is heat leakage through the test-section walls and thermal insulation. In the CFD analysis, all channel walls other than the fluid-solid interfaces and heating surfaces were treated as perfectly adiabatic boundaries. In contrast, the actual experimental apparatus inevitably allows a certain amount of heat to dissipate through the polycarbonate channel wall and thermal insulation to the surroundings.

To investigate the influence of this heat leakage, a simplified thermal circuit model was constructed using OpenModelica, as shown in Fig. A2. The model consisted of three parallel heat-transfer paths from a constant heat input of 8 W: thermal conduction through the polycarbonate channel wall, thermal conduction through the insulation, and heat dissipation via the copper pin fin to the cooling water. The temperatures on the low-temperature sides of the polycarbonate wall and insulation were fixed at 25°C, corresponding to the coolant and ambient temperatures, respectively. The cooling water temperature was also set to 25°C.

Figure A2: An example of the simplified thermal circuit model constructed in OpenModelica to evaluate the effect of heat leakage.

The time-averaged heat transfer coefficient obtained from the transient CFD analysis of pulsating flow was converted into the convective thermal conductance, Gc, as

Gc=hm⋅A,(A1)

where hm is the time-averaged heat transfer coefficient and A is the heat transfer area of the pin fins. By applying Gc to the convection component of the thermal circuit model, the transient pulsating-flow problem was represented as a virtual steady-state thermal problem, and the time-averaged temperature at the pin-fin root was calculated.

The effect of heat leakage was evaluated for c/d = 1, where a relatively large discrepancy between the CFD and experimental results was observed. As representative low- and high-Reynolds-number conditions, Rem = 50 and Rem = 400 were selected, respectively. Here, Rem = 400 corresponds to the maximum Reynolds number investigated in the CFD analysis.

Fig. A3 compares the time-averaged Nusselt numbers obtained from the CFD analysis, experiments, and OpenModelica simulations with and without heat leakage for c/d = 1 and Rem = 50. The Nusselt numbers predicted by the adiabatic OpenModelica model closely agreed with the CFD results, indicating that the constructed thermal circuit model appropriately reproduced the heat transfer characteristics under the adiabatic condition. When heat leakage through the test-section wall and thermal insulation was included, the predicted Nusselt numbers increased and closely agreed with the experimental results, with an average difference of approximately 3%. These results indicate that heat leakage in the experimental apparatus is a major factor contributing to the higher experimentally evaluated heat transfer performance compared with the CFD results.

Figure A3: Comparison of time-averaged Nusselt numbers obtained from CFD, experiment, and OpenModelica simulations with and without heat leakage at c/d = 1 and Rem = 50.

Fig. A4 shows the relative increase in heat transfer performance caused by heat leakage as a function of the pulsating Strouhal number. The increase in heat transfer performance was greater at Rem = 50 than at Rem = 400, indicating that the relative influence of heat leakage decreases under high-Reynolds-number conditions.

Figure A4: Relative increase in heat transfer performance due to heat leakage as a function of the pulsating Strouhal number at c/d = 1.

This tendency is considered to be related to the heat transfer characteristics of the pin fins and thermal conduction within the copper rods. At higher Reynolds numbers, the enhanced convective heat transfer from the pin fins increases the amount of heat conducted along the copper rods toward the pin fins. Consequently, the one-dimensionality of heat conduction within the copper rods is improved, and the relative amount of heat dissipated through the test-section wall and thermal insulation is reduced. Therefore, the influence of heat leakage on the evaluated heat transfer performance becomes smaller at higher Reynolds numbers. These results demonstrate that heat leakage is one of the main factors responsible for the discrepancy between the experimental and CFD results, particularly under low-Rem conditions.

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