Background

Mixed type of convection occurs in a wide range of turbomachinery applications, and are therefore of great importance. As mentioned earlier, especially high pressure (HP) steam turbine casing is exposed to both natural and mixed convection during shut-down regimes with different operating conditions. The cavity formed by the inner and outer turbine casings could be considered as a horizontal concentric annulus as seen in Figure 1 where various flow regimes taking place in the casing cavity are also presented.

Figure 1.

Variation of flow regimes in casing cavities.

In the case of pure natural convection, thermal plume occurs at the top of the heated inner cylinder, which rises and impinges on the outer casing. This leads to a non-uniform cooling. After impingement, the fluid cools down and moves down the lower portion of the annulus where the flow tends to be stable. When the pure forced convection takes places, a circumferentially uniform boundary layer forms around the inner cylinder as shown in Figure 1. The flow behaviour in the upper portion of the annulus significantly changes when both natural and forced convection take place simultaneously. The thermal plume region is distorted by the cross-flow, and multiple longitudinal structures develop due to the complex flow interactions.

Experimental facility

An experimental facility illustrated in Figure 2 has been designed for the detailed investigations of mixed (natural and forced) convective type of flows in a simplified turbine casing cavity represented by a horizontal concentric annulus. The inner casing temperature is kept constant by a PID controller at a set value, while the incoming air speed is varied by an industrial blower at the outlet for different mixed convection rates. Low conductivity calcium silicate inserts are used at both end of the inner cylinder, which is made of copper, to minimise the heat loss. Self-adhesive ceramic heat shields have been placed around the outer casing cylinder to reduce the heat dissipation. The facility consists of various measurement ports distributed axially and circumferentially. Measurement locations on the test facility are depicted in the facility schematic in Figure 3 and some of the essential dimensions are tabulated in Table 1. The detailed design information regarding the experimental facility can be found in the study of Murat et al. (2021).

Figure 2.

Cross-section of the test facility CAD model.

Figure 3.

Experimental facility schematic and measurement locations.

For pure natural and forced convective flows, Grashof and Reynolds numbers are important non-dimensional parameters to determine the flow regime. Richardson number is used when both natural and forced convection take place simultaneously. The non-dimensional numbers used in the present study are given in Equation 1.

It is important that a test facility is able operate at real engine representative conditions to have realistic conditions. In the present study, the test facility is capable of operating at various mixed convective flow regimes as well as pure natural and forced convective flows. Figure 4 shows the range that the facility can operate in comparison with the real turbine data during a shut-down regime, which indicates that the facility is capable of reproducing engine realistic conditions. In the same figure, mixed type of flow cases that are presented in this paper are also marked, the details of which are tabulated in Table 2.

Figure 4.

Experimental facility operation range.

Linear actuators with accuracy of 0.05 mm have been used to traverse the thermocouple probes, which are distributed in axial and circumferential locations. The thermocouple probe houses two K-type thermocouples and a needle to keep the thermocouple beads at a known distance from the heated copper cylinder, as shown in Figure 5. During the measurements, the thermocouple probes were traversed starting from the heated inner cylinder towards the outer cylinder. Step of 0.1 mm was used near the heated wall to be able to capture the near-wall gradients, while relatively larger steps were used away from the walls. Time averaging of unsteady data acquired by National Instruments data acquisition modules was performed over 120 seconds with 50 Hz sampling rate.

Figure 5.

Thermocouple probe.

Although the heated copper cylinder is covered with a mat black paint, the radiation effect may still be significant. When the heat conduction through the thermocouple sheath tube is ignored, the energy balance applied on a thermocouple bead consists of convective and radiative heat transfer as shown in Equation 2. Measurement or estimation of the effective surrounding temperature, Teff is required to calculate the true fluid temperature, Tf, however it is usually not possible. Therefore, it should somehow be eliminated from the equation. To do so, another thermocouple with different diameter but the same thermophysical properties can be used by relating two energy balance equation for each thermocouple bead (Tagawa and Ohta, 1997; Brohez et al., 2004; Kim et al., 2010).

Boundary conditions

The computational domain used for LES is illustrated in Figure 8. As seen in the figure, CFD domain involve a region at upstream of the bellmouth, referred as “entrance region”, to allow boundary layer development over the bellmouth. Mass flow rate for each case has been calculated using the air speed measured at Section 0 by hot-wire anemometry, which read as m˙1 = 0.46 kg/s and m˙2 = 1.31 kg/s for Case-1 and Case-2, respectively. For the inlet boundary conditions, the ambient conditions have been used. Isothermal wall condition has been applied to inner and outer cylinder walls in the test section, and the details of the boundary conditions are given in Table 6. The remaining walls in the domain have been treated as adiabatic walls.

Figure 8.

Computational domain.

Temperature profiles

Temperature profiles are presented in a non-dimensional form as T∗=((T¯−T∞)/(Ti−T∞)). Figure 13 illustrates the qualitative comparison of the flow fields predicted by LES for buoyancy dominant Case-1 and forced convection dominant Case-2. The most prominent difference between the two cases is the growing thermal plume development at the top. In Case-1, the buoyancy forces are strong enough to cause upward flow movement at the top of the inner cylinder. In Case-2, on the other hand, the thermal plume is not formed due to the strong cross flow, therefore leading to more uniform thermal boundary layer thickness around the inner cylinder than Case-1. The lower portion of the annulus in both cases resembles each other with thin laminar boundary layer as also reported in the study of Murat et al. (2021).

Figure 13.

Time-averaged temperature contours.

Time-averaged non-dimensional temperature profiles predicted by LES are compared against the experimental data in Figure 14 for Case-1 at Section 1, 2 and 3. At Section 1, the thin temperature boundary layer is present around the inner cylinder, indicating that the buoyancy effect is not strongly pronounced at this section. Good agreement between numerical and experimental results has been achieved at Section 1. As the fluid progresses, the development of the boundary layer becomes noticeably distinguishable at the top of the annulus at θ=90∘ at Section 2. The thickening of the boundary is also present at both sides at θ=30∘ and 150∘. The lower portion of the annulus still exhibits a thin boundary layer. Although LES has over-predicted temperature profile at θ=90∘, there has been satisfactory agreement between LES and the experiment. Towards the downstream of the test section, the thermal plume is further developed at Section 3, resulting in the thickest boundary layer at θ=90∘. The buoyancy effect is also present at θ=30∘ and 150∘ with thicker boundary layers compared to those at Section 1 and 2. The lower portion of Section 3 resembles Section 1 and 2 with a thin boundary layer caused by the buoyancy forces forcing the flow move up against the bottom of the inner cylinder. Overall, LES results agree well with the experimental results at Section 3.

Figure 14.

Time-averaged temperature profiles of Case-1 (Ri = 0.246).

Figure 15 shows the temperature profiles from LES and the experiment for Case-2. As opposed to Case-1, Case-2 represents strong momentum of the fluid in the axial direction which reduces the effect of the buoyancy. At Section 1, there is a uniform temperature boundary layer around the inner cylinder which is well predicted by LES. At Section 2, the boundary layer is slightly thickened around the inner cylinder, however the temperature profiles at all measurement locations are very similar, which elucidates the dominance of the forced convection. This flow behaviour can also be seen in Figure 13 where there is no sign of clear plume development. At the upper portion of the annulus at Section 3, the temperature boundary boundary layer is slightly thicker than the lower portion. However, the thermal plume is still not present, therefore the temperature profiles in the annulus are not radically different . Overall, LES has exhibited a good agreement with the experimental data at all sections in Case-2.

Figure 15.

Time-averaged temperature profiles of Case-2 (Ri = 0.032).

Flow field

The flow field in the annulus can be investigated with the iso-surfaces and their footprints on the heated inner cylinder as illustrated in Figure 16 where the iso-surfaces of the second invariant of velocity (Q=0.5(W2−S2)) and the time-averaged wall shear stress on the inner cylinder are shown for each case. In the upstream of the inner cylinder, Case-1 exhibits laminar flow region where the buoyancy is not strongly pronounced in the upper half. As the flow progresses, hairpin-like vortices are formed which causes early transition to turbulence and the the thermal plume lift-off at θ=90∘. This can also be observed in the reduced wall shear stress at the top of the inner cylinder in 16b. On the sides of the inner cylinder between θ=0∘−70∘ and θ=110∘−180∘, some of the low momentum fluid near the heated inner cylinder lifts-off due to buoyancy and is stretched towards the downstream by the cross-flow. After the detachment, the fluid shows tendency to move downwards due to a decrease in the temperature, and then re-join the near-wall low-momentum fluid that moves upwards. Therefore, longitudinal helical structures, which enlarge along the stream-wise direction, form as shown in Figure 16a. These flow structures can be tracked by looking at their footprints on the inner cylinder as illustrated in Figure 16b where the stretched streaks are seen.

Figure 16.

(a) Iso-surfaces of second invariant of velocity, (b) Time-averaged wall shear stress around the inner cylinder.

As opposed to Case-1 which represents a strong buoyancy influence in the flow field, Case-2 exhibits strong cross-flow influence which minimises the buoyancy effect. This phenomenon can be observed by looking at Case-2 iso-surfaces in 16a where the thermal plume is not formed at θ=90∘ due to the strong inertial force. Boundary layer instabilities are present in the upstream of the inner cylinder due to the relatively high speed flow. Towards the downstream, the low momentum buoyant flows near the inner cylinder are convected to the top of the inner cylinder. However, these near-wall fluid movements remain as instabilities as seen in Figure 16 where there is no sign of developed vortices. In the downstream, the turbulent region exists as seen in locally increased/decreased wall shear stress.

Although the lower portion of the annulus remains laminar with both cases, there is a locally increased wall shear stress at the bottom of the inner cylinder in Case-1 due to the buoyant forces pushing the flow upwards against the inner cylinder which suppress the growth of the boundary layer. The shear stress distribution in Case-2 remains relatively uniform until the beginning of the turbulent region.

Heat transfer

Table 8 shows the detailed information on Nusselt number distribution at circumferential and axial measurement locations, which could be used for validation purposes. For Case-1, experimentally and numerically calculated Nusselt numbers as well as the LES errors are tabulated. Since the experimental heat transfer could not be obtained for Case-2 as described earlier, LES results are presented only for Case-2. Figure 17 illustrates the heat transfer distribution around the circumference at Sections 1–3.

Figure 17.

Heat transfer distribution around the inner cylinder.

There is a heat transfer enhancement at the bottom of the inner cylinder compared to the top with Case-1, indicating that the buoyant flow causes a large temperature gradient (thin boundary layer) whereas Case-2 exhibits a uniform heat transfer around the inner cylinder at Section 1. The heat transfer drop phenomenon between the bottom and top in Case-1 is one of the well-known characteristics of the pure natural convection from a horizontal cylinder (Kenjereš and Hanjalić, 1995), which indicates the dominance of the natural convection. There are greater heat transfer values at each sides of the cylinder than that of the top. This could be attributed to the start of formation of the side helical structures. LES results show good agreement with Case-1 experiment with a maximum error of 21.95% at θ=330∘.

At Section 2, the Nusselt number alternates remarkably at the upper half of the cylinder in Case-1 due to developed side structures, footprints of which are shown in Figure 16. Conversely, the bottom exhibits more uniform heat transfer rates which are slightly lower than Section 1 since the boundary layer is thicker. It is also interesting note that the minimum heat transfer has occurred at the top, indicating that the thermal plume lift-off has formed a large boundary layer. Although the highest LES error (-20.19%) has occurred at θ=90∘, the overall trend has been captured at Section 2. The heat transfer distribution around the inner cylinder is still uniform with Case-2 at lower rates than Section 1. This is due to the slightly thicker boundary layer.

Although there are still heat transfer minima/maxima at the upper half of the cylinder at Section 3 in Case-1, the fluctuations are not as strong as those at Section 2 since the flow is more developed at Section 3. Developing helical structures shown in Figure 16a have enhanced the heat transfer rates at both sides of the inner cylinder around 45∘ and 135∘. The heat transfer is further reduced at the bottom as the boundary layer progressively thickens which is also valid for Case-2. Comparable heat transfer behaviour at the bottom in Case-1 and Case-2 implies that the lower half between 210∘ and 330∘ is not significantly affected by the rate at which the mixed convection occurs. On the other hand, the upper half in Case-2 now demonstrates alternating Nusselt numbers since the flow is in the region where the turbulence occurs as shown in Figure 16b. As seen in Table 8, there is a good agreement between LES and experimental results for Case-1 at Section 3.