Corner Vortex Research Articles

Physical mechanisms behind the extreme wind pressures on two tandem square cylinders

The extreme wind pressure might lead to local or overall structural failures. The physical mechanisms behind extreme pressures on two tandem square cylinders are investigated using large-eddy simulation at a Reynolds number of 8 × 104. The spacing ratio P/B ranges from 1.1 to 5 (P: cylinders’ center distance; B: side length). The aerodynamic coefficients, flow patterns, mean and fluctuating pressure coefficients, pressure skewness and kurtosis, and instantaneous flow structures are analyzed. The extreme negative pressures are induced by the intermittent appearance of strong vortices attached to the cylinder surfaces. “Gap vortex” and “wall vortex” are responsible for the extreme pressure at the gap-sides of two cylinders in the single mode (1 < P/B < 2). “Corner vortex” would cause extreme negative pressure near the front corners of the upstream cylinder and the rear corners of both cylinders at most spacings. At the reattached (2 ≤ P/B < 4) and binary (4 ≤ P/B ≤ 5) modes, the Karman vortex released from the rear corner of the downstream cylinders leads to extreme pressure on the rear surface. As the spacing increases, the extreme pressure on non-Gaussian-pressure regions is generally strengthened but the three-dimensionality in vortex structures and instantaneous pressure distributions roughly declines.

LES of HCCI combustion of iso-octane/air in a flat-piston rapid compression machine

Homogeneous Charge Compression Ignition (HCCI) engines promise better efficiency and cleaner emissions than conventional piston engines, but can be challenging to control. Rapid compression machines (RCM) provide a simplified configuration for investigating HCCI combustion behavior, which is necessary for effective control of engine ignition timing and peak pressures. In this study, we assess the utility of large eddy simulations (LES) for predicting HCCI combustion in a 3-D configuration. To this end, LES with finite-rate chemistry employing a 99-species iso-octane/air mechanism of two RCM operating conditions are performed. The RCM configuration under consideration was designed by Strozzi et al. (Combust. Flame, 2019) with a flat piston to introduce large amounts of thermal stratification representative of realistic HCCI engine conditions, through the generation of corner vortices. It is shown that the simulation provides reasonable agreement with temperature fluctuations (7% difference), as well as ignition delay in the short ignition case (1 ms difference), while the long ignition case (35 ms difference) highlights more substantial deficiencies that are still within the expected uncertainty from the employed chemical mechanism. Flame propagation modes predicted by LES agree with experimental observations: spontaneous ignition is seen in the short ignition case, while deflagration is more prominent in the long ignition case. Analysis of global and local quantities classify the short ignition case in a mixed ignition regime, and the long ignition case in the mild ignition regime. These results demonstrate the utility of FRC-LES for investigations of multimode combustion regimes of HCCI combustion in a 3-D configuration.

Numerical Investigation of a Simplified Wing–body Junction Flow

The flow past a simplified wing-body junction configuration is simulated using a shear layer adaptive improved delayed detached-eddy-simulation (SLA-IDDES) approach in which an appropriate subgrid lengthscale is adopted in the boundary region, where the grid may be strongly anisotropic, to prevent the excessive generation of subgrid-scale eddy viscosity. The numerical results show overall good agreement with the experiment carried out by Ölçmen and Simpson. The self-induced chaotic switching of the horseshoe vortex between two flow modes is simulated and analysed, and its effects on the vortex legs and their inherent oscillation are also investigated. It is found that the corner vortex also exhibits an alternating formation and breakdown process with basically the same frequency as the horseshoe vortex. Corner separation is found to be affected also by the upstream horseshoe vortex and the inherent oscillation of the vortex legs.

Effect of compound lean blades on separation structures in high loaded compressor cascades under high subsonic condition

Three-dimensional separation is an inherent flow feature in the blade-end corner of compressors passage, which is a primary source of entropy generation and loss. This paper researches the effect of key geometrical parameters (camber angle, solidity, dihedral angle of compound lean blade) and incoming condition (Mach number) on the evolution of corner separation in a low aspect ratio linear cascade. The underlying flow mechanism is explored in detail. The evolution of typical flow characteristics with the variation of inlet Mach number, camber angle and solidity is interpreted. For different blade loading levels, there are different corner separation forms and vortex structures. The influence of dihedral angle on the flow field structure and flow loss is analyzed, thus the effect mechanism of the compound lean blade in different flow environment is explored. Without the trailing edge shedding vortex, positive dihedral angle alleviates the low momentum fluid accumulation and corner separation at the incidence angle about 0°, but exacerbates the deteriorating corner separation at large positive incidence angle. When the trailing edge shedding vortex and the suction surface separating vortex exist together, positive dihedral angle promotes the upstream migration of the trailing edge shedding vortex, which helps to truncate the suction surface separating vortex. This variation of vortex structures is conducive to weaken the development of corner separation and delay corner stall.

Computational investigation into the effect of constitutive formulations for flow modeling of high elasticity

• Three constitutive formulations of the Leonov model are numerically investigated. • It is verified that keeping positive-definiteness of the conformation tensor is crucial for computational evolution while appropriate interpolation of exponentially convecting domain is essential for accuracy. • Swirling asymmetric corner vortices are described in 3D contraction flow modeling. For the purpose of defeating the high Deborah (or Weissenberg) number problems in viscoelastic flow modeling, the matrix-logarithmic formulation of the constitutive equations has been suggested and proven probably the most effective. This transformation of the conformation tensor endows proper resolution of the boundary layer with exponential convection as well as conservation of its positive definiteness. Herein we make an attempt to evaluate numerical consequences of these two features independently for the Leonov viscoelastic model mainly in the contraction flow. To this end, we from the viewpoint of computational stability and accuracy investigate three formulations written with variables such as the original conformation tensor c (c form), its matrix-logarithm (h form) and additive decomposition (b form). 2D planar 4:1 contraction flow in the limit of vanishing inertia in addition to its 1D toy problem is considered in the framework of finite element method. The h and b forms are designed to preserve the positive definiteness of c , and furthermore the h form grants efficient interpolation of exponentially convecting variables. The 1D analogy expresses the characteristics qualitatively equivalent to the original problem in a simpler fashion. The c form modeling of 2D flow exhibits typical nonevolutionary behavior when the Deborah number ( De ) exceeds a certain limit, and this degradation gets worse with finer mesh. While the h and b forms have always been computationally stable with proper description of steady behavior or elastic instability otherwise, the h form manifests faster mesh convergence. As a complementary study, the h form calculation of full 3D flow along an axisymmetric contraction channel has been performed, which also attests well-posedness without upper bound in De . Its steady result is corroborated with the one under axisymmetric flow approximation. Loss of convergence in the axisymmetric flow regime at high De is verified to result from the occurrence of 3D instability, which is then well portrayed in 3D modeling. In this instability, swirling of asymmetric structure for the corner vortices about the geometrical symmetry axis plausibly mimics experimental observation. Consequently, maintaining the positive definiteness in terms of h or b form is crucial for numerically stable evolution, and the h form interpolation delineating appropriately the exponential convection assures the fastest mesh convergence, hence enhancing the accuracy.

Influence of Urban Road Green Belts on Pedestrian-Level Wind in Height-Asymmetric Street Canyons

This study was conducted to examine the effect on airflow of the shape of an urban road green belt in an asymmetrical street canyon. In this paper, the airflow field at pedestrian height in an asymmetrical street with different building height ratios (ASF) on both sides of the street is modeled and simulated using computational fluid dynamics (CFD) software, ANSYS FLUENT, and the flow rate characteristic distribution index and the average airflow intensity index are used to evaluate and analyze the airflow at the pedestrian level. The study shows that: (1) in an empty street scheme with different building ratios, the static wind area is located on the roof of the downstream building; the closer to the ground in a street with an ASF = 1/3, the lower the airflow rate. However, the situation is the opposite of that in other streets (2/3, 3/1, and 3/2). (2) The position of the green belt makes the windward side flow rate in the step-up street higher than that of the leeward side, and the flow rate of the leeward side in the step-down street is higher than that of the windward side. (3) Compared with other green belt forms, the use of two plates and three belts in the incremental street can increase the circumferential sinking at the roofs of the windward side of the street, thereby improving the wind environment in the entire street. The use of one plate, two-belt and three-plate, four-belt scenarios in the step-down street allows the two ends of the corner vortex to carry more airflow into the interior of the street and reduces both the “wind shadow effect” area in the middle of the street and the “air outlet effect” at both ends.

Large eddy simulation and combined control of corner separation in a compressor cascade

Due to the demand for higher aerodynamic performance of compressors, thoroughly investigating the high-loss flow in the corner region and effectively controlling it are important. In this paper, a novel parameterization method based on the extended free form deformation (EFFD) technique and the constraints for EFFD's control points is proposed. Then, considering the features of typical control techniques and the degrees of freedom of both the blade and hub geometries, the combined control approach is implemented in the corner region of a linear cascade. Furthermore, large eddy simulation is used to simulate the flow, verify the effects of the combined control approach, and explore the underlying physical mechanisms of corner separation. The numerical results show that the combined control can significantly decrease the mean total pressure loss. The loss reduction at the design point reaches 6.05%, while it decreases by almost 2.5% near the stall/blockage operating conditions. The combined control increases the radial pressure gradient at the rear of the blade by depressing the hub and stretching the suction surface. Consequently, although the radial flow slightly increases the mixing loss in the mainstream at large incidences, the accumulation of low-energy flow in the boundary layer and the corresponding development of the corner vortex are significantly restrained. Moreover, by redistributing the static pressure on the hub, the combined control weakens the migration of crossing flow and obstructs the low-velocity flow from the pressure side involved in the separation. Overall, the combined control contributes to reducing the corner separation and improving the aerodynamic performance.

Effects of corner chamfers on the extreme pressures on a square cylinder at incidence to a uniform flow

Wind-induced damages to cladding, glass curtain walls, and windows on surfaces of buildings demand further investigation of extreme pressures on a square prism. The flow physics of extreme wind pressures on a square cylinder has not yet been clarified, especially as they relate to corner modifications and flow incidence. Large-eddy simulation is utilized to investigate uniform flow past a standard square cylinder and a chamfered cylinder with a chamfer ratio of 1/7. The Reynolds number is fixed at 22,000, and the incidence angle ranges from 0° to 45°. The flow mechanism of extreme pressures is studied through analyses of instantaneous flow structures. The influences of corner modifications on extreme pressures are further investigated. The results show that extreme negative pressures are strong in the non-Gaussian regions, which are observed at the rear corners and sides of the two cylinders and change dramatically as the incidence changes. Two types of flow structures are responsible for the extreme pressures, namely, the intermittent rear-corner vortices caused by reverse-flow separation and the Karman vortices near the rear side. The distribution of extreme pressures is highly three-dimensional, owing to 3D vortex structures at a high Reynolds number. By introducing chamfers, the strength and spanwise variations in the corner vortex are reduced, and extreme pressures are weakened.

Transient nature of secondary vortices in an axial compressor stage with a tandem rotor

An efficient and compact design of a compressor continues to be a challenging area of research. The unavoidable streamwise adverse pressure gradient together with passage transverse pressure gradient restricts the maximum allowable blade turning. Tandem blading is an interesting concept for increasing the pressure rise by permitting a higher blade turning angle. The energized flow through the tandem blade nozzle gap helps to minimize the possibility of flow separation over the suction surface of the aft blade. However, a coherent transient analysis of a tandem rotor stage in an axial compressor is yet to be well explored. In the present paper, the complex flow field over a tandem rotor and the succeeding stator passage is analyzed in detail. Although the tandem rotor increases the flow turning and diffusion effect, the presence of separate trailing edge wakes and hub corner vortex causes early onset of flow separation over the stator suction. The flow structures developed within the rotor and the stator passages at different time instances are highlighted with the help of limiting streamlines and iso-surface Q-criterion superimposed with entropy contours. The results indicate that within the rotor passage, transient flow features are observed near the hub corner region close to the aft blade trailing edge rather than the tip region. When the rotor passes the stator leading edge, the rotor trailing edge leakage flow at the hub is entrained into the stator leading edge reverse flow region. This is then further circumferentially dragged into the mid-passage region. The interference of multiple rotor wakes with the stator leading edge leads to the formation of longitudinal and arch like separation vortices at the stator-hub and the stator-casing regions, respectively. These separation vortices grow in size while being convected downstream. Eventually, as time progresses, the vortices split and shed periodically from the stator surface. The present investigation highlights the requirement of a new stator design in a tandem rotor–conventional stator configuration. Such designs could further magnify the significant aerodynamic performance obtained using a tandem configuration.

Effects of Rib Design on the Unsteady Tip Heat Transfer Amplitude for a Turbine Rotor Blade

Abstract By solving the unsteady Reynolds-averaged Navier–Stokes equations and SST k–ω turbulence model, effects of rib design on the unsteady tip heat transfer amplitude for a turbine rotor blade with different rotating speeds are numerically investigated. The results of turbulence model validation are in good agreement with the experimental data. The grid independence verification is also satisfied. The results indicate that the averaged heat transfer coefficient of ribbed tip is 3.1% lower than that of traditional squealer tip and the amplitude of the heat transfer coefficient at the ribbed blade tip is 31.7% lower than that at the traditional squealer tip at standard speed condition. Low-energy passing wake causes the variation of flow field pressure near the tip clearance, which changes the strength of vortex such as pressure corner vortex, scratching vortex, leading edge vortex, and rib vortex inside the cavity, and thus causes the tip heat transfer coefficient fluctuation. At 85% standard speed, the average heat transfer coefficient amplitude of the ribbed tip is reduced by 35.4%. At 115% standard speed, the average heat transfer coefficient amplitude of the ribbed tip is reduced by 44.5%. Ribbed blade tip is suitable to reduce the heat transfer coefficient in turbine design

Effect of blade skew, endplate and casing groove on the aerodynamic performance of Wells turbine for OWC: a review

Abstract The geometry modification of a bidirectional flow Wells turbine used in a wave energy converter can assist in overcoming the mammoth stall triggered by tip leakage, secondary flow on the blade surface, corner vortex, passage vortex, tip scrubbing, tip trailing edge vortex, boundary layer formation on hub and casing, and combination of all the listed making it a complex 3D flow. As the flow varies almost sinusoidally in each wave cycle in the ocean, it is inevitable to get a turbine having a higher operating range and average power output. Skewing a blade reduces the radial flow on the blade suction surface. A casing groove and endplate reduce tip leakage flow. These modifications give a significant improvement in the turbine performance. There are several design alternatives reported in the articles, and they have shown different results. In this article, the modifications and their effects are discussed in detail and concluded for future recommendations.

The axial tip clearance leakage analysis of the winglet and composite blade tip for the liquid-ring vacuum pump

Axial tip clearance leakage of liquid-ring vacuum pump has an important impact on it's hydraulic performance. In order to suppress the axial tip clearance leakage flow and improve the hydraulic performance, the winglet and composite blade tip were introduced to the axial tip clearance of the 2BEA-203 liquid-ring vacuum pump blade. The mechanisms of the winglet and composite blade tip on the clearance leakage flow field were analyzed by numerical simulation method. The analysis results show that the winglet extends the flow channel of axial clearance, and reduces the pressure difference between the two sides of the clearance, thus suppressing the leakage flow. Therefore, the winglet impeller can improve the pump vacuum and efficiency to a certain extent. However, the suction surface winglet will generate the wing corner vortex under the winglet, resulting in additional losses and making the improvement of its hydraulic performance lower than that of the pressure surface winglet. For the composite blade tip, the jet hole directly leads the high-pressure fluid from the pressure surface to the clearance, which weakens the effect of the winglet. So, the suppression effect of the composite blade tip is lower than that of the winglet. Although the winglet reduces the mixing loss of the jet, the suppression effect of the winglet is better than that of the jet. Therefore, as the mechanism of the winglet is different from that of the tip jet, the suppression effect of the composite blade tip is lower than that of the winglet, and greater than that of the tip jet.

Effect of the Axial Casing Groove Geometry on the Production and Distribution of Reynolds Stresses in the Tip Region of an Axial Compressor Rotor

AbstractStereo-PIV data are used for investigating the effect of axial casing groove (ACG) geometry on the distribution, evolution, and production rates of turbulent kinetic energy (TKE) and Reynolds stresses near a rotor tip. The ACGs delay the onset of stall by entraining the tip leakage vortex (TLV) and cause periodic changes to incidence angle. These effects are decoupled using semicircular, U-shaped, and S-shaped grooves that have similar inlets, but different outflow directions. Most TKE distribution trends can be explained by the local turbulence production rates, elucidating the different mechanisms involved and providing a unique database for turbulence modeling. Interaction of the tip flow with the ACGs modifies the highly anisotropic and inhomogeneous passage turbulence. In all cases, the TKE is high in the TLV center and shear layer connecting the TLV to the rotor tip. At prestall flowrate, TLV entrainment reduces the passage turbulence level, but introduces elevated turbulence in the corner vortex formed at the downstream corner of grooves, and in shear layers developing at the exit from grooves. The location of peaks and the dominant components vary among grooves. Near the best efficiency point, interactions of the TLV with the circumferentially negative outflow from the U and semicircular ACGs generate high turbulence levels, which extend deep into the passage. In contrast, interactions with S grooves are limited, resulting in a substantially lower turbulence level. Accordingly, the S groove maintains the untreated endwall efficiency, while the U and semicircular grooves reduce the peak efficiency.

Technical note: Dispersion of cooking-generated aerosols from an urban street canyon

Abstract. The dispersion of cooking-generated aerosols from an urban street canyon is examined with building-resolving computational fluid dynamics (CFD). Using a comprehensive urban CFD model (PALM) with a sectional aerosol module (SALSA), emissions from deep-frying and boiling are considered for near-ground and elevated sources. With representative choices of the source flux, the inclusion of aerosol dynamic processes decreases the mean canyon-averaged number concentration by 15 %–40 % for cooking emissions, whereas the effect is significantly weaker for traffic-generated aerosols. The effects of deposition and coagulation are comparable for boiling, but coagulation dominates for deep-frying. Deposition is maximised inside the leeward corner vortices, while coagulation increases away from the source. The characteristic timescales are invoked to explain the spatial structure of deposition and coagulation. In particular, the relative difference between number concentrations for simulations with and without coagulation is strongly correlated with the ageing of particles along fluid trajectories or the mean tracer age. It is argued that, for a specific emission spectrum, the qualitative nature of the aerosol dynamics within urban canopies is determined by the ratio of the aerosol timescales to the relevant dynamical timescale (e.g. the mean age of air).

Magnetic Field Effects on Backward-Facing Step Flow of Ferrofluids

Abstract Backward-facing step (BFS) flow is a benchmark case study in fluid mechanics. Its control by means of electromagnetic actuation has attracted great interest in recent years. This paper focuses on the effects of a uniform stationary magnetic field on the laminar ferrofluid BFS flows for the Reynolds number range 0.1≤Re≤400 and different expansion ratios. The coupled ferrohydrodynamic equations, including the microscopically derived magnetization equation, for a two-dimensional domain are solved numerically by an openfoam solver after validation and a test of accuracy. The application of a magnetic field causes the corner vortices in the concave corner behind the step to be retracted compared with their positions in the absence of a magnetic field. The maximum percentage of the normalized decrease in length of these eddies reaches 41.23% in our simulations. For small Reynolds numbers (&amp;lt;10), the flow separation points on the convex corner are lowered in the presence of a magnetic field. Furthermore, the dimensionless total pressure drop between the channel inlet and outlet decreases almost linearly with Reynolds number Re, but the drop is greater when a magnetic field is applied. On the whole, the normalized recirculation length of the corner vortex increases nonlinearly with increasing magnetic Reynolds number Rem and Brownian Péclet number Pe, but it tends to constant values in the limits Rem≪1 and Rem≫1.

Production and transport of vorticity in two-dimensional Rayleigh–Bénard convection cell

We present a numerical study of vorticity production and transport in the two-dimensional Rayleigh–Bénard (RB) convection. Direct numerical simulations are carried out in the Rayleigh number (Ra) range 105≤Ra≤106, the Prandtl number (Pr) of 0.71, and the aspect ratio (Γ) of the convection cell range 0.75≤Γ≤6. We found that the flow structure and temperature distribution vary with Γ greatly due to multiple vortices interaction. Further investigation on the vorticity production and transport reveals that, in the RB convection, in addition to the vorticity production due to wall shear stress, buoyancy produces significant vorticity in the bulk region. The produced vorticity is transported via advection and diffusion. An interesting finding is that the main vortices and the corner vortices can be visualized via the contour of buoyancy-produced vorticity. Although a vigorous definition of the vortex is still lacking in the community, our efficient vortex visualization approach in the RB convection may shed light on further research toward vortex identification. We also found that the spatial distribution of vorticity flux along the wall is positively correlated with that of the Nusselt number (Nu), suggesting the amount of vorticity that enters the flow is directly related to the amount of thermal energy that enters the flow.

Flow-induced birefringence study of secondary flow in entrance region of rectangular slit channel for long-chain-branched polyethylene melt

This work presents an analysis of the inner boundary of the secondary flow (i.e., corner vortex flow) in the entrance region of a rectangular slit channel for long-chain-branched polyethylene melt evaluated using particle tracking and stress-induced birefringence measurements. Further, the flow-induced birefringence technique was used to investigate the effects of temperature and shear rate on the size of the secondary entry flow. Obtained results and trends were compared with the laser-Doppler velocimetry data taken from the open literature for the same polymer melt and comparable flow conditions. Interestingly, the results of both methods were very similar. This suggests that the stress-induced birefringence measurements may be a useful tool for investigating secondary entry flows of polymer melts.

The transition to aeration in turbulent two-phase mixing in stirred vessels

We consider the mixing dynamics of an air–liquid system driven by the rotation of a pitched blade turbine (PBT) inside an open, cylindrical tank. To examine the flow and interfacial dynamics, we use a highly parallelised implementation of a hybrid front-tracking/level-set method that employs a domain-decomposition parallelisation strategy. Our numerical technique is designed to capture faithfully complex interfacial deformation, and changes of topology, including interface rupture and dispersed phase coalescence. As shown via transient, a three-dimensional (3-D) LES (large eddy simulation) using a Smagorinsky–Lilly turbulence model, the impeller induces the formation of primary vortices that arise in many idealised rotating flows as well as several secondary vortical structures resembling Kelvin–Helmholtz, vortex breakdown, blade tip vortices and end-wall corner vortices. As the rotation rate increases, a transition to ‘aeration’ is observed when the interface reaches the rotating blades leading to the entrainment of air bubbles into the viscous fluid and the creation of a bubbly, rotating, free surface flow. The mechanisms underlying the aeration transition are probed as are the routes leading to it, which are shown to exhibit a strong dependence on flow history.

The effect of courtyard buildings on the ventilation of street canyons: A wind-tunnel study

Using controlled tests in a wind tunnel, we simulated the pollution of four different street canyons formed by four 3D urban array models. The urban models differed by the geometry of roofs (pitched and flat roofs) and buildings (courtyard and solid buildings). We simulated traffic pollution from a ground-level source positioned in the middle of the street canyons. We show that the courtyard buildings significantly improve (by a factor of 1.3) the ventilation of the street canyons only in the cases with pitched roofs. We explain the differences between the ventilation performances of the street canyons by analysing the dynamics of the coherent structures. The buildings at the roof level shed two main vortex structures into the flow. However, the street canyon with pitched roofs and the courtyard buildings shed more stable structures that collide and penetrate deeper downstream near the wind-facing eaves. Near the pedestrian zone, ventilation is driven by advection, manifested as corner vortices at the street ends and flow convergence from the windward to the leeward side in the middle of the street canyons. The corner vortices are more pronounced in the courtyard buildings regardless of roof shape, resulting in higher concentrations than solid buildings.

Experimental studies of turbulent pulsating flow and heat transfer in a serpentine channel with winglike turbulators

Particle Image Velocimetry and Infrared Thermography measurements are applied to explore the effect of quasi-triangular pulsatile flow inlet on active thermal-fluidic performance in a serpentine channel with innovative winglike turbulators. The novel turbulators are mounted in-line on two sidewalls of the channel with the attack angle (α), thickness to chord line ratio (t/C), and pitch to the channel hydraulic diameter ratio (Pi/DH) respectively fixed at 20°, 0.2, and 0.7. The Strouhal number (St) varies from 0 to 0.67 at a fixed Reynolds number (Re) of 10,000. Relative to the steady-state case (St = 0), the flow pulsation not only strengthens the Fujiwhara co-rotating vortices but also eliminates the corner vortex streams, leading to the overall Nusselt number (Nu¯/Nuo) enhancement by 13%. Moreover, both the Nu¯/Nuo and friction factor (f¯/fo) reveal the tilde-like trends with St, resulting in the highest Nu¯/Nuo of 5.2 and thermal performance factor of 1.39 at St = 0.67, which are higher than the previous best results for 10≤f¯/fo≤100 in the Nu¯/Nuo-f¯/fo diagram. Finally, by combing the present data with our previous steady state results, composite correlations of Nu¯/Nuo and f¯/fo versus Re, α, t/C, Pi/DH, and St are proposed.