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Advances in Mechanical Engineering
Volume 2013 (2013), Article ID 797201, 10 pages
Numerical Simulation of the Flow Field in Circumferential Grooved Liquid Seals
School of Energy and Power Engineering, Dalian University of Technology, No. 2, Linggong Road, Dalian, Liaoning 116024, China
Received 21 March 2013; Revised 24 June 2013; Accepted 8 July 2013
Academic Editor: Jaw-Ren Lin
Copyright © 2013 Meng Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Circumferential grooved liquid seals are utilized inside turbomachinery to provide noncontacting control of internal incompressible fluid leakage. Accurate prediction of the flow field is fundamental in producing robust and efficient designs. To validate the capabilities of the computational fluid dynamics FLUENT for incompressible fluid seal flow, comparisons of velocity parameters are made to the published experimental results and other CFD code for a circumferential grooved liquid seal. This work employs a pressure-based CFD code FLUENT to calculate the flow field in the seal, using four different turbulence models respectively. The velocity contours are compared with experimental values. It shows good overall agreement of the axial, radial, and azimuthal velocities in the through-flow jet, shear layer, and recirculation zone. Quantitative comparisons of velocity profiles at the center of the groove are made to experiment. This study verifies the prediction accuracy of three turbulence models. Various structures were considered to obtain a better understanding of the circumferential grooved liquid flow characteristics. The best groove structure to control leakage was also found within the limited designed seal. This study will provide a useful reference for designing the circumferential grooved liquid seal.
Circumferential grooved liquid seals are used extensively in turbomachineries to reduce leakage from high-pressure areas to low-pressure ones. When the leakage is reduced, there will be an accompanying increase in the efficiency of the device. Accurate prediction of the flow field is vital for reliable performance. However, grooved liquid seals have not been investigated thoroughly due to large pressure and velocity gradients near the wall, large recirculation, and high turbulence intensity present in the seals. To better investigate the complex leakage flow characteristics of circumferential grooved liquid seals, numerical simulated approaches have been developed and comparisons to experimental results for the flow field have not been performed by several researchers.
One of the first numerical CFD comparisons to experimental results for circumferential grooved liquid seals was made by Rhode et al. , who compared their prediction with the one-dimensional LDA measurements of Stoff  for swirl velocity and the measured pressure distributions of Morrison et al. . Experimental data was limited to one plane per groove cavity and did not allow a thorough comparison to be made.
Nordmann et al.  calculated the leakage flow of circumferential grooved liquid seals based on Childs  finite length solution, which is derived from Hirs’  bulk-flow theory. Nordmann et al. modified Hirs’ turbulent lubricant equations to account for the different friction factors in the circumferential and axial directions in the grooves.
Demko et al.  compared numerical predictions with experimental hotfilm measurements for the axial and swirl velocity components and turbulent kinetic energy in a grooved liquid annular seal. The results showed reasonable correlation of mean velocity both in the through-flow jet and in the recirculation zone.
Kilgore and Childs’ experiment  provided an additional basis for evaluating the model for grooved liquid seals with different geometries and higher Reynolds numbers. The effects of the combined friction factor model could be seen in the predictions for leakage coefficients. The speed dependency of the friction factor model was particularly noticeable in the predictions for leakage coefficients. At lower pressure differential, leakage predictions and friction factor correlation were both poor. Better leakage predictions were obtained at higher pressure differential. Regardless of the difference in friction factor speed dependency, the predicted friction factors were closer to the measured results.
Iwatsubo et al.  developed a two-control-volume theory, featuring a description of the seal in land and groove sections, in which nonspeed dependent pressure loss coefficients inside the seal were introduced. Their theory gave qualitative agreement with experiment.
Marquette and Childs  provided a new theory, which was derived from Florjancic’s  three-control-volume theory for circumferentially grooved liquid seals. Marquette featured the introduction of diverging flow in groove sections. Validation of the new analysis was achieved through a comparison with existing experimental data taken from Kilgore . Leakage was reasonably well predicted. It provided all numerical parameters that had been set correctly. Each variation of a parameter yielded changes that always have a physical explanation. This showed that the physics of the phenomenon were well captured by three-control-volume theory.
Morrison and Robic  compared a commercial CFD code with Morrison’s  three-dimensional LDA measurements for the whirling, circumferentially grooved liquid seals. The code did a good job predicting the pressure field for circumferentially grooved liquid seals and yielded similar results with LDA measurements. A more recent study by Jeffrey Moore and Palazzolo  employed a pressure-based CFD code SCISEAL developed by Athavale et al.  to calculate the flow field in a whirling, circumferentially grooved liquid seals. Comparisons of basic fluid parameters were made to Morrison’s  three-dimensional LDA measurements for a spinning, circumferentially grooved seal. The results showed good qualitative agreement with experiment for the flow field.
Comparisons were also made between experimental results and predictions from CFD model by Childs et al. . The model included the assumption that a groove was large enough to create separate lands within the seal, creating a zero or negligible pressure perturbation within the groove. Test results showed that even the deepest groove depth tested is not deep enough to satisfy this assumption.
Vijaykumar’s  CFD simulations were performed on 2D grooved liquid seals to compare flows fields with experimental LDA data from Johnson  and to investigate the effects of preswirl, clearance, and Reynolds numbers. The three-dimensional velocity and kinetic energy dissipated were plotted as a function of the tooth with preceding cavity. Vijaykumar’s CFD simulations did a fairly good job in predicting the flow field and the associated flow physics in a 2D grooved liquid seals seal.
The uncertainty of the turbulent model’s accuracy for the estimate of the flow field may result in an inefficient seal design with a larger leakage flow disturbing the main flow at the inlet of the impeller of the pump company, and then the hydraulic efficiency decreases. In this work, predictions are made respectively using the RNG κ-ε turbulence, the standard κ-ε turbulence, the realizable κ-ε turbulence, and the Shear-Stress Transport (SST) κ-ω turbulence models, to validate the capabilities of the computational fluid dynamics (CFD) for incompressible fluid seal flow and find a turbulence model with superior capability in predicting the incompressible flow field of the circumferential grooved seal over others. And it can be helpful in providing an accurate prediction of the flow field for a wide variety of operating conditions and flow parameters.
2. Computational Descriptions
A two-dimensional CFD model is used in this study to calculate the steady incompressible flow within the seal. Water is employed as the sealing medium and the temperature is treated as constant value. The RNG κ-ε turbulence, the standard κ-ε turbulence, the realizable κ-ε turbulence, and the Shear-Stress Transport (SST) κ-ω turbulence models were employed to determine the leakage characteristics. And CFD analysis of the seal is performed by ANSYS FLUENT. A pressure based solver with finite volume method of discretization was used. A second order discretization scheme was used for the pressure, density, and momentum terms. First order upwind scheme was used for the turbulent kinetic energy and turbulent dissipation rate terms.
The turbulence kinetic energy, κ, and its rate of dissipation, ε, are obtained from the following transport equations:
In these equations, represents the generation of turbulence kinetic energy due to the mean velocity gradients, defined as (3); is the generation of turbulence kinetic energy due to buoyancy, defined as (4); represents the contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate, defined as (5). The model constants ,, , and , are the turbulent Prandtl numbers for κ and ε, respectively. and are user-defined source terms. The solution of two separate transport equations allows the turbulent velocity and length scales to be independently determined. It is a semiempirical model, and the derivation of the model equations relies on phenomenological considerations and empiricism .
The main difference among the κ-ε models is ε equation. The modeled transport equation for ε in the RNG κ-ε model is where
One of the weaknesses of the standard κ-ε model lies with the modeled equation for the dissipation rate ε. The round-jet anomaly is considered to be mainly due to the modeled dissipation equation. The RNG model has an additional term in its ε equation that significantly improves the accuracy for rapidly strained flows. It yields a lower turbulent viscosity than the standard κ-ε model .
A new model equation for dissipation ε is based on the dynamic equation of the mean-square vorticity fluctuation. The modeled transport equation for ε in the realizable κ-ε model is
The production term in the ε equation of realizable κ-ε model does not involve the production of κ. The present form better represents the spectral energy transfer. Another feature is that the destruction term (the next to last term on the right-hand side of (8)) does not have any singularity. Its denominator never vanishes, even if κ vanishes. The benefit of the realizable κ-ε model is that it more accurately predicts the spreading rate of round jets. It is likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, separation, and recirculation .
The Shear-Stress Transport (SST) κ-ω turbulence model was developed by Menter  to effectively blend the robust and accurate formulation of the κ-ω model in the near-wall region with the free-stream independence of the κ-ε model in the far field. To achieve this, the κ-ε model is converted into a κ-ω formulation. The definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress .
2.2. Dimensions and Conditions
The geometry of the seal is based on the Morrison’s  experimental model and the simulation uses quad cells to create the entire grid. Figure 1 shows that the geometry consists of a labyrinth seal, a supply nozzle upstream of the seal portion, and the downstream region with increased clearance. The section of the supply expansible nozzle provides an annular flow path leading up to the seal entrance. There is a 0.508 mm gap between the inlet contour assembly and the rotor. The seal portion is sufficiently long to ensure that the flow is fully developed. The flow leaves through the region of large clearance. The geometry of the model will consist of the following dimensions: labyrinth seal length mm, rotor radius mm, and clearance mm.
The geometry of the seal is considered axisymmetric and periodic boundary conditions are assumed in the circumferential direction. Mass flow inlet boundary is maintained at the leakage rate of 0.00486 m3/s, which corresponds to a Reynolds number of 24,000. The flow direction will be normal to the boundary, and there will be no swirl component at the boundary for axisymmetric swirl. Outlet boundary condition is imposed pressure at 413.7 kPa. The rotor was imparted shaft rotation by specifying the rpm at 3600. The turbulence intensity at the core of a fully developed flow can be estimated from the following formula:
The 2D mesh applying the structured cells for the land and groove is created using a commercial mesh generator Gambit as shown in Figure 2. Because the change of the flow near the walls may be large, the mesh distribution was not uniform with a dense mesh in the annular region and a course mesh in the interior of the cavity. When generating a mesh, care is taken to maintain a smaller value , resulting in improved wall shear stress and pressure prediction.
Laws of the wall formulations model the sharp pressure gradients near the wall and are used with standard κ-ε turbulence model: where is the dimensionless velocity, is the dimensionless distance from the wall, and = turbulent viscosity constant (= 0.09), = von Kármán constant (= 0.4187), =empirical constant (= 9.793), = mean velocity of the fluid at the near-wall node P, = turbulence kinetic energy at the near-wall node P, = distance from point P to the wall, and = dynamic viscosity of the fluid.
The laws-of-the-wall for mean velocity and pressure are based on the wall unit, , rather than (). These quantities are approximately equal in equilibrium turbulent boundary layers .
The wall values are ensured less than 10 throughout the seal. And a mesh independence density study is performed to investigate the effect of mesh density and to determine how fine of a mesh is required to capture the important velocity and pressure physics. The mesh is refined in the radial and axial direction dependently. Figure 3 shows the effect of refining the mesh in the clearance and groove regions on the inlet average pressure, which is a representation for pressure loss of the seal. It is seen that increasing the number of nodes from 37000 to 66000 in the seal cross-section led to little change in the inlet average pressure showing that grid independence is established. To obtain a better velocity prediction, maximum density mesh will be used in subsequent calculations.
3. Results and Discussion
3.1. Comparisons of Velocity Amplitude Contour Plots
Figure 4 shows the calculated axial velocity contours of the whole grooved seal. The pattern of vectors at each groove is similar. Figure 5 shows the representative velocity field for groove cross-section for the third seal cavity. The high through-flow jet generated due to a vena contracta effect from the groove to the clearance, which passes through the rotor tooth, remained parallel to and attached to the stator wall. When passing the groove, a smaller part of the core flow impinges obliquely on the front side of the downstream tooth, redirected towards the bottom edge of the rotor groove. And the jet current drives a single large vortex inside the cavity.
For the leakage rate of 0.00486 m3/s, the leakage Reynolds number was 24000, with an average axial mean velocity of 7.4 m/s in the clearance. The velocity contour plots are normalized with the average axial velocity ( m/s). The coordinate represents the radial position () and the axial position () nondimensionalized with the clearance. Mean axial velocity contours for the third seal cavity are compared to the experimental plots  in Figure 6. The mean velocity contours of other seal cavity are essentially identical to those of the third cavity for the distributions of the velocity components. The results show good qualitative agreement in both the through-flow jet and the recirculation zone. The simulated field has similar contour lines in the cavity and the clearance as experimental results. The computed jet current core is observable with axial velocities exceeding the average leakage velocity by 15 percent as the same peak magnitude as the experimental results but slightly underpredicts the radial gradient of the axial velocity in the shear layer above the recirculation zone. Both of the two contour plots show a vena contracta effect with an increase in the radial gradient of the axial velocity as the fluid is accelerated from the large cavity into the small clearance above the downstream tooth and a rapid decrease gradient as the upstream vena contracta effect dissipates widely above the upstream tooth. Both the predicted and experimental jets have the same shear layer width with a rapid increase in the axial velocity near the stator surface.
The predicted recirculation zone has essentially identical flow pattern as the experimental results. Inside the groove, the axial velocities are almost zero near the vertical walls of the rotor teeth and along a horizontal line near the center of the groove. In the top half of the recirculation zone, the computation has the same as the positive value contour lines of measurement but slightly underpredicts the radial gradient of the axial velocity in the shear layer. In the bottom half of the recirculation zone, the computation has negative contour lines; it differs from measurement by a reduction in the magnitude of the maximum negative axial velocity value from to .
Again, the radial velocity is normalized by the average axial velocity as shown in Figure 7. The predicted plots indicate that there is no observable radial velocity contour line above the teeth till the region near the top of the front side of the downstream tooth impinged by a smaller part of the core flow. Comparatively the experimental plots  are evident for the radial velocity contours with a significant radial velocity component present above the recirculation zone.
The predicted recirculation zone has essentially identical radial velocity contour shape as the experimental results. A smaller part of the core flow impinges the top of the front side of the downstream tooth, redirected to radial direction. This corresponds with the larger axial gradient of the radial velocity in the right part of the recirculation zone compared to the left part. In the left part of the recirculation zone, the computation has the positive value contour lines but slightly underpredicts the axial gradient of the radial velocity in the shear layer. The magnitude of the maximum positive radial velocity is much lower in the predicted plots () than in the experimental plots (). In the right part of the recirculation zone, the computation has negative contour lines, which differs from the measurement by a reduction in the magnitude of the maximum negative axial velocity value from to . Additionally the core position of the negative radial velocity contour is nearer to the downstream tooth, slightly deviating from the experiment results. This corresponds with the higher position of the tooth impinged by the small jet current compared to the experimental plots. It indicates that the model does not predict well the divergence angle of jet.
The velocity is normalized by the average azimuthal velocity (tooth tip surface velocity, m/s) as shown in Figure 8. Both the predicted and experimental cavities  have the same magnitude of the azimuthal velocity value () in the central region. The predicted plots have narrower shear layer width near the rotor wall than experiment. The standard near-wall formulations slightly underpredict the turbulence intensity in the shear layer of the groove and model the sharper gradients of azimuthal velocity. Also the azimuthal velocities above the teeth are underpredicted.
The κ-ε model velocity contours are compared with experimental values, showing good general agreement but differences near the cavity wall. It indicates that the model does not perform well in the weak shear layer in the high Reynolds number, because the κ-ε model is based on Boussinesq’s isotropic eddy viscosity assumption, which has weakness in treatment of jet impingement and boundary layers.
3.2. Comparisons of Velocity Profiles
To validate the capabilities of the different turbulence models, quantitative comparisons of velocity profiles at sections in the cavities are made to experiment . Figures 9 and 10 plot the axial and azimuthal velocity profiles at the center of the groove. The coordinate represents the radial position (Y) nondimensionalized with the clearance, while the abscissa for axial velocity is nondimensionalized with the average axial velocity ( m/s). Figure 9 illustrates the axial velocity distribution of the seventh cavity. With the exception of the RNG κ-ε turbulence, the predicted velocity profiles using the standard κ-ε turbulence, the realizable κ-ε turbulence, the SST κ-ω turbulence, and SCISEAL  are essentially identical as the experiment. It validates the capability to predict the axial velocity of the incompressible flow. The CFD result predicts a steeper boundary layer near the stator wall and a through-flow jet with an axial velocity plateau. The RNG κ-ε turbulence and the standard κ-ε turbulence models yield identical peak velocities as the experiment. But another model slightly underpredicts the peak velocities. Various CFD models have a closely predicted through-flow jet as the experiment. It indicates their good capability to calculate the high Reynolds flow.
In the groove, the standard κ-ε turbulence, the realizable κ-ε turbulence, the Shear-Stress Transport (SST) κ-ω turbulence model, and SCISEAL , predict essentially identical recirculation velocities as the experiment. But the RNG κ-ε turbulence predicts extremely different flow pattern with two vortexes. It is evident that the prediction of the recirculation in the groove is unexpectedly poor and shows no advantage for the vortexes. Comparatively other models have higher accuracy of the predicted vortex, with positive velocity in the top half of the groove and negative velocity in the bottom half just appearing to slightly underpredict the recirculation strength.
Near the bottom wall of the rotor groove, the standard near-wall formulations of the standard κ-ε turbulence slightly under-predict the turbulence intensity in the shear layer. It is the standard κ-ε turbulence’s weakness in treatment of boundary layers. The realizable κ-ε model is likely to provide superior performance for flows involving rotation, boundary layers under strong adverse pressure gradients, and recirculation, because a new transport equation for the dissipation rate has been derived from an exact equation for the transport of the mean-square vorticity fluctuation. The Shear-Stress Transport (SST) κ-ω model shows advantage of treating boundary layers because the model effectively blends the robust and accurate formulation of the κ-ω model in the near-wall region. The definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress . SCISEAL employs the two-layer model for a modification to the turbulence diffusion near the wall (inner layer) using an algebraic expression, while the turbulence kinetic energy equation is applied in both the inner and outer layers . And as a consequence, these models’ predictions of the axial velocities in boundary layers are expectedly accurate.
Figure 10 plots the azimuthal velocity profiles at the center of the groove. It was evident that the azimuthal velocity of the standard κ-ε turbulence, the realizable κ-ε turbulence, and the Shear-Stress Transport (SST) κ-ω turbulence model results accord well with the experiment, especially in the recirculation region. It indicates that these models could be used for high accurate azimuthal velocity quantitative analysis, which is fundamental for good rotordynamic characteristics assessment of grooved seals. SCISEAL  curve slightly deviates from the experiment results, under-predicting the azimuthal velocity in the groove and above the teeth. Therefore the standard κ-ε turbulence, the realizable κ-ε turbulence, and the Shear-Stress Transport (SST) κ-ω turbulence model provide higher accurate prediction of the azimuthal velocity than SCISEAL . As can be observed from Figure 10, the RNG κ-ε turbulence has no advantage of the azimuthal velocity, with department forming the experiment in both the recirculation and the through-flow jet. The model grossly overpredicts the azimuthal velocity in the groove but severely underpredicts above the teeth.
3.3. Optimization of Groove Structure of Liquid Seals
This research attempts to validate the accuracy of different turbulence models for predicting the incompressible fluid seal flow. Such predictive ability would greatly aid in the design of grooved seals which are critical pump components. The influence of the groove structure on the resistance of grooved liquid seals in pump is investigated. The friction factor is used for representing the resistance through the seals, which is defined as follows: where is the static pressure, is the axial seal coordinate, is the hydraulic diameter, is the density, and is the average axial velocity.
The friction factors of sixteen different structure seals are predicted over a range of groove widths (Width/1.524 mm = 1, 1.5, 2, 2.5) and depths (Depth/1.016 mm = 1, 2, 3, 4) at the same clearance as the grooved seal (Depth/1.016 mm = 3, Width/1.524 mm = 2) in . Figure 11 illustrates friction factor versus groove width and depth. As can be seen, the grooved seal (Depth/1.016 mm = 1, Width/1.524 mm = 2.5) has a higher resistance to leakage than other seals.
The vector plots of Figure 12 show two different representative flow patterns for comparison of high resistance characteristics, such as Depth/1.016 mm = 1, Width/1.524 mm = 2.5 and Depth/1.016 mm = 4, Width/1.524 mm = 1. As shown in Figure 12(a), there exist two eddies in the groove (Depth/1.016 mm = 4, Width/1.524 mm = 1). The high through-flow jet generates due to a vena contracta effect from the groove to the clearance. When passing the groove, a smaller part of the core flow impinges obliquely on the front side of the downstream tooth, redirected towards the bottom edge of the rotor groove. The groove is deep enough to accommodate two large eddies in the opposite direction. Such a large recirculation is beneficial to energy dissipation because the kinetic energy is altered to the thermal energy in the eddy. It is speculated that the higher resistance of deep groove (Depth/1.016 mm = 4, Width/1.524 mm = 1) than the one (Depth/1.016 mm = 3, Width/1.524 mm = 2) in  is related to the large recirculation zone with two eddies. The other flow pattern of energy dissipation is the high-speed flow impinging on the seal wall. It can be found in Figure 12(b) that the stagnation area exists along the upwind surface of the groove due to the impingement of the high through-flow jet. Then a prolate eddy is formed in the shallow groove. It is consistent with the finding from Figure 13 that pressure distribution of the groove (Depth/1.016 mm = 1, Width/1.524 mm = 2.5) is heterogeneous due to energy dissipation, especially in the stagnation region at the corner. It is considered that the highest resistance to leakage in sixteen different structure seals as illustrated in Figure 11 is benefit from the impingement of the high through-flow jet. Thus the seal (Depth/1.016 mm = 1, Width/1.524 mm = 2.5) has the best performance of enhancing energy dissipation within the limited designed groove. The shallow grooved seal is recommended to be used in the pump in engineering.
This research attempts to validate the prediction accuracy of incompressible fluid seal flow in different CFD models by solving the general Reynolds Averaged Navier-Stokes equations along with the standard κ-ε turbulence, the realizable κ-ε turbulence, the RNG κ-ε turbulence, and the Shear-Stress Transport (SST) κ-ω turbulence model. Comparisons of velocity parameters are made to three-dimensional laser Doppler anemometer (LDA) measurements and the CFD code SCISEAL  for a circumferential grooved liquid seal.
Computed velocity contours are compared with experimental values, showing good general agreement but differences in detail near the wall, while through-flow jet, vortex, rotation, and vena contracta effect are well captured by the model. It validates the capability of the standard κ-ε turbulence for incompressible fluid seal flow.
To compare the predicted accuracy of the different turbulence models, quantitative comparisons of velocity profiles at sections in the cavities are made to experiment. The standard κ-ε turbulence, the realizable κ-ε turbulence, the Shear-Stress Transport (SST) κ-ω turbulence model, and SCISEAL  demonstrate good correlation in the axial velocities but slightly underpredict the recirculation in the groove. The RNG κ-ε turbulence predicts extremely a different flow pattern with two vortexes. The prediction of the recirculation in the groove is unexpectedly poor.
The standard κ-ε turbulence, the realizable κ-ε turbulence, and the Shear-Stress Transport (SST) κ-ω turbulence model really give satisfactory results in terms of the azimuthal velocity. It indicates that these models could be used for high accurate quantitative analysis, which is fundamental for good rotordynamic characteristics assessment of grooved seals. These models give more accurate results, showing substantial improvements over SCISEAL  in the rotational flow.
This study verifies the prediction accuracy of the standard κ-ε turbulence, the realizable κ-ε turbulence, and the Shear-Stress Transport (SST) κ-ω model. These models demonstrate superior capability in predicting the incompressible flow field of the circumferential grooved seal over SCISEAL .
Various structures were considered to obtain a better understanding of the circumferential grooved liquid flow characteristics. The best groove structure to control leakage was also found within the limited designed seal. This study will provide a useful reference in designing the circumferential grooved liquid seal.
|:||Axial Reynolds number|
|:||Static pressure (Pa)|
|:||Hydraulic diameter (m)|
|:||Labyrinth seal length (mm)|
|:||Rotor radius (mm)|
|:||Seal clearance (mm)|
|:||Axial velocity (m/s)|
|:||Average axial velocity (m/s)|
|:||Radial velocity (m/s)|
|:||Azimuthal velocity (m/s)|
|:||Average azimuthal velocity (m/s)|
|:||Axial position (m)|
|:||Radial position (m)|
|:||Friction velocity (m/s)|
|:||Dynamic viscosity (Pas)|
|:||Turbulent kinetic energy (m2/s2)|
|:||Turbulent dissipation rate (m2/s3)|
|:||Turbulent viscosity constant|
|:||Von Kármán constant|
|:||Mean velocity of the fluid at the near-wall node P (m/s)|
|:||Turbulence kinetic energy at the near-wall node P (m2/s2)|
|:||Distance from point P to the wall (m)|
|:||Generation of turbulence kinetic energy due to the mean velocity gradients|
|:||Generation of turbulence kinetic energy due to buoyancy|
|:||Contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate|
|Model constants for ε equations|
|:||Turbulent Prandtl number for κ|
|:||Turbulent Prandtl number for ε|
|:||User-defined source term for κ|
|:||User-defined source term for ε.|
The authors appreciate the financial support from National Basic Research Program of China (973 Program).
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