Peculiar effects of permeable tabs on the heat transfer behind the backward-facing step

. The effect of square-shaped tabs on the turbulent flow dynamics and heat transfer behind the backward-facing step has been studied experimentally. The Reynolds number, calculated by the hydraulic diameter of the inlet section and the average velocity, calculated based on volumetric flow rate was Re = 42500. The 6x6 mm 2 square-shaped tabs, installed at the edge of the backward-facing step, were of two types: solid and permeable. The permeable tabs had a square hole of 4x4 mm 2 in their center. In the experiments, the spacing between tabs varied from 1.2 to 2.5 calibers. The paper presents the results showing the effect of tabs on pressure distribution, local heat transfer, averaged heat transfer, as well as gives estimates of thermal hydraulic efficiency.


Introduction
In flowing channels with sudden expansion, a flow separation occurs, which increases the hydraulic resistance of the channel, and reduces the heat transfer directly in the separation area behind the step.The review [1] shows the effect of the backward-facing step on the dynamic characteristics of the flow, in particular, on the size of the recirculation area behind the backward-facing step, depending on the flow regime, the ratio of heights (in front of the step and behind the step), as well as the effect of the boundary layer thickness in front of the flow separation.Experimental studies of heat transfer behind the step have been carried out in [2,3].The effect of boundary layer thickness in front of the step on the heat transfer behind the step was studied in [3].Researchers of work [4] studied the effect of backward-facing step height, as well as the degree of turbulence on the heat transfer enhancement behind the backward-facing step.The authors have shown that increasing the step height reduces the maximum heat transfer, while increased degree of turbulence enhances the maximum heat transfer by 10%, and shifts the location of heat transfer maximum towards the step base.
One of the negative features of the separated flow past various vortex generators, i.e. obstacles, is the formation of a low heat transfer stagnant zone behind the separation cross-section.This can generally lead to a significant reduction in heat transfer enhancement effects.Additional intensifying elements, such as transversal ribs, toothing (tabs), vortex generators of various shapes, etc. are used to eliminate this drawback.Despite their significantly smaller size compared to the main separated flow, they can lead to significant flow restructuring, both reducing and expanding the circulation zone.
The effect of the obstacle on the flow dynamics behind the step has been studied in [5][6][7][8][9][10].In an experimental study [11], flow control was conducted using three-dimensional rectangular tabs, arranged at the edge of the step.Studying the irregularity of the separation area revealed that in the wake behind the tabs it lengthens, while between them it approaches the obstacle base.The authors have determined the optimum height and pitch, at which the average recirculation area was minimal.Research work [12] presents the results on effects of height, pitch between tabs, and distance to separation point on heat transfer behind the backward-facing step, as well as the results on flow pattern in the near-wall region.It is shown that by using tabs of different sizes, and varying their arrangement density, one can effectively control the aerodynamic structure of the flow and the heat transfer intensity.
The present study is aimed at experimental investigation of the effect of permeable tabs, installed at the edge of the step, on flow separation and heat transfer behind the backward-facing step.

Experimental setup and procedure
Experiments were carried out in the wind tunnel, which consisted of a medium pressure fan, which was driven by a frequency converter asynchronous motor with a minimum control frequency of 0.01 Hz, and an aerodynamic channel consisting of a prechamber, nozzle, and test channel.The test channel consisted of an initial 20x150 mm section 600 mm long, with a 10 mm high step at the end of the section, and 400 mm long section behind the step.The test section was made of 10 E3S Web of Conferences 459, 03004 (2023) https://doi.org/10.1051/e3sconf/202345903004XXXIX Siberian Thermophysical Seminar mm thick thermally insulating textolite material.To measure the pressure, holes, 0.8 mm in diameter, were drilled in the wall at variable pitch; the pitch near the step was 5 mm, and 40 mm towards the end.The pressure was measured using a multitube pressure gauge; the counterpressure was taken at the wall in the symmetry plane of the model at a distance of 40 mm to the step.
The pressure coefficient Cp = 2(pi-p0)/U 2 was calculated, based on measured pressures.Here pi and p0 are pressure at the wall and counterpressure, respectively; U is the velocity in the center of the initial channel,  is the air density.Measurements were made at Reynolds number Re = 42500, calculated by hydraulic diameter of inlet section and the average velocity calculated based on volumetric flow rate.More details on the installation can be found in [12].
An ohmic heating system with 50 µm thick titanium foil was used for thermal measurements.A 150x300 mm window was made in the opposite wall.The foil was blacked out with black paint with the radiation coefficient of 0.95.The temperature field was recorded using a thermal imaging camera and then the field was verified by means of two thermocouples placed under the heated foil.The Nusselt number was calculated using the formula Nu = /(H), where  is the heat transfer coefficient,  is the heat conductivity coefficient of air, and H is the height of the backward-facing step.Square tabs of 6x6 mm 2 in size were arranged at the edge of the backward-facing step (Fig. 1).The distance between the tab centers was equal to 1.2, 1.8, and 2.5 calibers (step height).Three solid and three perforated tabs were investigated.The flow passage area of the tab was 4x4 mm 2 .

Pressure distribution
Congestion of the channel increases the hydraulic resistance, resulting in increase of pressure coefficient Cp (Fig. 2) in the pressure recovery region, which is located at a distance of about 30 calibers from the step.Thus, for a smooth step Cp = 0.34, and for monolithic tabs with a pitch of 1.2H, Cp = -0.192.As can be seen from the graph, the more sparsely spaced the tabs, the higher the Cp value at the pressure recovery point.Congestion degree of the channel (ratio between the area occupied by tabs and live cross-section area for the smooth step) for tabs with a pitch of 1.2H is 0.156, and for tabs with the pitch of 2.5H is 0.084.Note that the place of largest rarefaction, where Cp is minimum, is observed for the option P/H = 1.2.The denser the installed tabs, the larger Cp representing the difference between pressure recovery values and the minimum Cp value.So, for tabs installed with a pitch P/H = 1.2, the Cp = 0.62, and with a pitch P/H = 2.5, Cp = 0.92.
Installing permeable tabs increases channel crosssection, which results in a decrease in hydraulic resistance of the channel.Thus, pressure build-up for the option P/H = 2.5 occurs at Cp = 0.28, and for the option P/H = 1.2, at Cp = 0.12.The maximum rarefication of permeable tabs also decreases comparing to that of solid ones, and their location coordinate moves away from the step base.Measured transverse Cp profiles at 4.5 and 7 calibers to the step have shown that at close distance, the heterogeneity of Cp did not exceed 50%, and in the far distance it was no more than 25%.

Thermal imaging
Two-dimensional temperature fields of heat transfer surface behind the step are shown in Fig. 3.The following trends in the effect of installed tabs on the temperature fields can be noted.For permeable tabs, the heat transfer near the step decreases and the region of maximum heat transfer shifts away from the step.For both solid and permeable tabs, installed at a pitch P/H = 2.5 and 1.8, similar structures are formed near side walls of the channel.Immediately behind the tabs, there is a more heated longitudinal band, and between the tabs, a cooler band.This is due to the fact that between the tabs, the flow reattachment occurs earlier, while immediately behind the tabs this occurs farther downstream.This phenomenon has been studied in detail in [11].For tabs with a pitch of 1.2, no individual effect of tab on the wall structure of the flow was observed.As can be seen from Figs. 3a and 3b, formed generalized structure enhances the heat transfer forming three large areas near the step.In [11], similar formation of large wall structures was observed in a wide channel for tabs with a pitch equal to twice the height of the step.

Heat transfer
The results of measured convective heat transfer, averaged over the entire channel width, are shown in Fig. 4. When flowing around a smooth backward-faced step, Nu number distribution picture is almost onedimensional, except for the end vortices, where the regions with the lowest local Nu numbers are located in the vicinity of the step base.The cross-averaged profile for the smooth step option is close to the local profile along the symmetry axis of the model.The arrangement of solid tabs with a pitch of P/H = 1.2 leads to an increase in maximum heat transfer; at the same time, two maxima at 3.2 and 9.6 calibers in front of the step appear in the graph, while the minimum value is observed at X/H = 5.9.Note that the Cp profile (Fig. 2a) does not show any significant difference from the distribution pattern for the smooth step between 5 and 11 calibers; only the longitudinal derivative in this section becomes smaller, indicating a slight restructuring of the flow.In the thermogram (Fig. 3a), as noted previously, an area of enhanced heat transfer in the form of three large areas and a smaller extreme area are located near the backward-facing step, while the second peak is observed in similar cooled areas in the vicinity of the step.
With solid tabs set at a pitch P/H = 1.8, the maximum cross-averaged heat transfer is observed at a distance of 2.6 calibers from the step; at this distance, 8 cooled areas are visible in the thermogram, while the second peak degenerates.The option with a pitch P/H = 2.5 is the most efficient in terms of enhancing heat transfer near the wall, where the maximum heat transfer is 33% greater than that for the smooth case, and is by 39% closer to the step.In the thermogram (Fig. 3e), there are 6 cooled areas at this distance from the step.After peak, the heat transfer intensity begins to drop rapidly, because the tab-induced longitudinal vortex decreases in its intensity, while moving away from the step, and, probably, becomes insignificant at a distance of 20 calibers.
In the case of permeable tabs with a pitch P/H = 1.2, the peak near the step disappears; the thermal imaging (Fig. 3b) shows that the heat transfer enhancement region is distant and the maximum longitudinallyaveraged heat transfer is observed at 9.46 calibers.For the permeable tabs with a pitch P/H = 1.8, the maximum heat transfer decreases by 11%, moving away from the step by 69% compared to the solid tabs, as qualitatively seen in the thermogram (Fig. 3d), where a more heated area is formed near the step.The longitudinal structure in the region of maximum heat transfer enhancement is not altered by the solid tabs.Permeable tabs with a pitch P/H = 2.5 also reduce the maximum heat transfer compared to that for solid ones, and the maximum heat transfer shifts away from the step by 1.8 calibers.The integral heat transfer NuI, which characterises the heat transfer averaged over the considered channel length, is of great importance in technical applications.Figure 5 shows the profiles of the average integral Nusselt numbers NuI, calculated by the formula: As can be seen from Fig. 5a, at a distance of 10 calibers, the integral Nusselt number for solid tabs is independent of the tab pitch.For distances between 10 and 30 calibers, the highest NuI is observed for tabs with a pitch of 1.2 calibers, however, this difference from less dense arrangement options is not so significant compared to the option with no tabs.For permeable tabs (Fig. 5b), their arrangement along the entire length with a pitch P/H = 1.8 is more efficient.It should be noted that the effect of heat transfer enhancement is small and does not exceed 20% compared to the separated flowaround a smooth step.
Let's move on to the analysis of the quantities, characteristic of reattached separated flows.In terms of thermal efficiency, these include the maximum heat transfer and its location coordinate.Comparison of such data for solid and permeable tabs is demonstrated in Fig. 6.It should be noted in advance that the coordinates of maximum values of average NuI number (Fig. 5) do not coincide with those for the longitudinally-averaged heat transfer (Fig. 4), while the change in curves in the vicinity of the maximum is smoother.The maximum integral NuI number is located at a distance of 13.6 calibers; this distance is twice the size of the recirculation region.For solid tabs, the distance decreases with increasing pitch, and for a pitch of 2.5H is 58% of the minimum pitch.The maximum NuImax number is not affected by the pitch.For the step with permeable tabs, an optimum value of NuImax is observed at a pitch of 1.8H.

Estimation of hydraulic resistance and thermal hydraulic efficiency
As an estimation of the hydraulic resistance, we use the formula for the total pressure loss.The hydraulic resistance coefficient f is determined by the formula f = 2(Pn-P0)/(U0 2 ), where P0 is the total pressure in the bearing cross-section (4 calibers in front of the step), Pn is the total pressure behind the step, determined in the static pressure recovery area, located 25 calibers behind the step, U0 is the average flow rate in the bearing crosssection.It is clear that the smaller the live cross-section, the greater hydraulic resistance, which is exactly demonstrated in Fig. 7.For solid tabs, this relationship is practically linear, while for permeable tabs there is a slight deviation from linearity.The hydraulic efficiency was estimated using = NuL/NuL0/(f/f0) 0.33 complex, which takes into account the energy input for air pumping.Permeable tabs with large pitch were found to be the most efficient.

Сonclusions
The effect of square tabs on the flow dynamics and heat transfer behind the backward-facing step when flowing around the step by a developed turbulent flow with close-ended up boundary layers at Re=42500, calculated by the hydraulic diameter of the inlet section and the average velocity, calculated based on volumetric flow rate, was studied experimentally.
Tabs, arranged at a pitch of 1.2 calibers, cause intensive flow mixing in the transverse direction, resulting in forming 4 enhanced heat transfer regions.
At the same time, behind the frequently spaced tabs with pitches of 1.8 and 2.5, the individual effect of each tab was observed.Permeable tabs do not qualitatively change the flow pattern compared to solid tabs at the same pitch, however move the maximum heat transfer area farther from the step.
Permeable tabs have less hydraulic resistance.As estimated, the highest thermohydraulic performance is found in the option with permeable tabs at a pitch of 2.5.
This work was financially supported by the Russian Science Foundation (grant No. 21-19-00162).