Test case calculated with the finite volume Code ESTET Version 3.2 (Mattéi et al. [1993]) supplied by EDF.
9.3.1. Introduction
This case is related to the isothermal swirling flow developing in a T-junction between a high Reynolds number main pipe and an auxiliary pipe at very small and zero mass flow rates. The motivation is to understand and model the hydraulic behaviour of various auxiliary lines connected to the primary cir-cuit of Pressurised Water Reactors. Experimental data have been obtained on a hydraulic mock-up using flow visualisation and particle image velocimetry (PIV). They show an helicoidal (corkscrew) flow extending along the auxiliary pipe.
This case illustrates the importance of the choice of the turbulence model k-ε eddy viscosity model and a Reynolds stress turbulence model) for computing this type of swirling flow and a grid depend-ency study shows the determining influence of the mesh refinement on the quality of the results.
9.3.2. Geometry and boundary conditions
A sketch of the experimental set-up is shown in figure 1. The cross-section surface is Sm = 1.303 10-2 m2. The dead leg diameter D is 100 mm. Its dimensionless length H/D is 20. The origin is located at the main pipe centre.
VA
VM
1500 mm 2000 mm
500 mm
R = 50 mm
R = 130 mm
100 mm
100 mm Auxiliary pipe (inlet)
Main pipe (inlet) outlet
Figure 1: Sketch of the mock-up.
Main pipe characteristics
Flow-rate (m3/s) 0.120
Bulk velocity (m/s) VM 9.2
Auxiliary pipe characteristics
Bulk velocity (m/s) VA 0.092 0.046 0.023 0.
VA / VM 1 % 0.5 % 0.25 % 0.%
Table 1: Flow rates
At the inlets (main pipe and auxiliary pipe), Dirichlet conditions are used for all variables. In the main pipe, the incoming flow is fully developed. Hence, the mean velocity and turbulent quantities were ob-tained from preliminary periodic pipe computations with the same bulk velocity and the same cross-section. For the auxiliary pipe, due to the very small flow-rates, it is not necessary to describe so
accu-rately the incoming flow. The mean velocity was taken constant and turbulent quantities were esti-mated from usual experimental laws for the friction velocity. In fully developed pipe flows with zero-roughness, and for Reynolds numbers Re (based on the bulk velocity VA and the hydraulic diameter D) ranging between 5,000 and 30,000, the friction velocity uτ can be determined from uτ = VA ((0.3164/Re0.25)/8)0.5. The turbulent kinetic energy k can then be estimated from uτ as k = uτ2/0.3 and the dissipation as ε = uτ3/(0.42 D 0.1). As for Reynolds stresses, the diagonal compo-nents are taken equal to 2/3 k whereas the extra-diagonal compocompo-nents are set to zero (assuming iso-tropic turbulence).
Flow rates and corresponding bulk velocities are given in Table 1. Note that the bulk flow rate remains constant and the auxiliary flow rate varies from zero to 1% of the bulk flow rate. At the outlet (main pipe), zero gradient conditions were applied for all transported variables. Wall functions were used at solid walls.
9.3.3. Grid
The equations are discretized on a 3D semi-staggered grid. Velocity is located at the vertices and pressure is cell-centred. When the k-ε model is used, the turbulent variables are located at the verti-ces, whereas they are cell-centred with the Reynolds stress model. The mesh is single block struc-tured in Cartesian orthogonal coordinates. Curved boundaries are represented by special features of the code which enables the treatment of slanted boundary cells (prisms, tetrahedrons, etc.).
A grid dependence study has been carried out with meshes of 100 000, 400 000, and 1 500 000 nodes for each turbulence model and each auxiliary mass flow rate considered (except for zero mass flow rate, for which only the finest grid has been used). Aspect ratios (ratio of the longest to the small-est edge in each cell) have been kept approximately between 1 and 2.
9.3.4. Features of the simulation
The numerical techniques are based on finite difference and finite volume discretizations. The algo-rithm for the solution of Navier-Stokes equations relies on a segregated velocity-pressure formulation.
The advective terms are treated by a method of characteristics. The trajectory is approximated by a second order Runge Kutta scheme with a third order 3D Hermitian polynomial for interpolation. Diffu-sion with explicit and implicit source terms for dynamic variables is solved implicitly. For the computa-tion of the velocity components, a third step is required in order to prescribe the mass conservacomputa-tion, leading to a Poisson equation for the pressure increment. In order to avoid non physical oscillations of the pressure field and the associated difficulties in obtaining a converged solution, a variant of the Rhie and Chow interpolation is used (Deutsch et al., [1997]).
The turbulence models that have been used are:
- the standard high Reynolds number k-ε model proposed by Launder and Spalding [1974]
- a high Reynolds number Reynolds stress model (second moment closure) with isotropic dissipa-tion, turbulent self-transport modelled by the usual gradient diffusion following Daly and Harlow [1970] and the pressure-strain correlation term recommended by Launder [1989], i.e. consisting of the sum of Rotta's return to isotropy and isotropisation of production terms.
The fluid is incompressible and its viscosity is ν = 1.03 10-6 m2/s. The corresponding Reynolds number based on the main pipe bulk velocity and the diameter of the auxiliary pipe is 895 000.
Default initial conditions implemented in the code have been used for each simulation.
The flow investigated being unsteady, the convergence had to be determined from time averages.
Hence, once the vortical motion was fully established, time average values of the variables were esti-mated in the dead leg. The vortex penetration was then determined by visualising secondary veloci-ties. The calculations were stopped once the value for the mean vortex penetration was stabilised.
9.3.5. Results
Figure 2 compares the length of vortex penetration obtained by numerical simulation with experimental measurements. The length of the vortex penetration has been determined numerically by visualising the time-averaged velocity field obtained in different planes of the dead leg. This method is consistent
with the way the experiments were carried out (visualisation of averaged results from one thousand instantaneous velocity fields obtained using PIV system). It is also important to note that experiments and calculations were conducted by the same team, thus making comparisons easier and less subject to interpretation errors. The results for three different flow rates are presented. Significant improve-ment is obtained when refining the mesh.
The prediction of the vortex penetration in the dead leg using Reynolds stress model on the finest mesh is in very good agreement with the experiments. The differences between medium and fine meshes illustrate the necessity to use very fine grids with the Reynolds stress model. It might be nec-essary to perform yet another simulation on an even finer mesh to achieve full grid independence.
With the k-ε model, on the contrary, the computed vortex penetration is not satisfactory, even on the finest mesh. Moreover, the very small difference between the results obtained on medium and fine meshes indicates that the use of even more refined meshes would be unlikely to bring significant im-provement.
2 4 6 8 10 H/D
0,2 0,4 0,6 0,8 1
k-ε medium mesh SMC medium mesh
k-ε coarse mesh SMC coarse mesh k-ε fine mesh SMC fine mesh Experimental
VA/VM (%)
Figure 2: Length of vortex penetration for different flow rates in the auxiliary pipe.
Comparison between Reynolds stress model (SMC) and k-ε model.
Coarse, medium and fine meshes have been used.
Figure 3 presents the rotation velocity at different sections in the auxiliary pipe obtained by both nu-merical and experimental simulations. According to tests by Robert [1992], the strength of the rotation velocity in the auxiliary line is very sensitive to geometric details of the T-junction. Small differences between the mock-up geometry and the numerical mesh could generate differences on the rotation velocity. To validate independently the vortex decay in the auxiliary line, numerical and experimental results are compared using dimensionless quantities. The rotation velocities obtained from numerical and experimental simulations are scaled by the corresponding rotation velocities near to the T-junction taken at H/D = 3. Using Reynolds stress model, the vortex decay is very close to the experimental re-sult, whereas it is overestimated with the k-ε model.
0.00 0.25 0.50 0.75 1.00
dimensionless rotation velocity
2 4 6 8 10
H/D Experimental k-ε
SMC
Figure 3: Maximum of rotation velocity scaled by the corresponding value at the section H/D=3.
Comparison between Reynolds stress model (SMC), k-ε results and experimental data.
Figures 4 and 5 help to visualise the flow features. They represent the vortex penetration in the dead leg for VA/VM = 0.25 % (k-ε model and RSTM respectively). The positive vertical velocity component is displayed in black and the negative in grey: black trajectories represent fluid particles coming from the main pipe and grey trajectories represent the fluid particles coming from the top of the auxiliary pipe.
Velocity vectors are plotted at three different sections of the auxiliary pipe H/D=2, 5 and 7 from the re-sults of fine, medium and coarse meshes.
H/D= 9
H/D= 7
H/D= 5
H/D=3
Vm
Coarse mesh medium mesh
fine mesh Va
Figure 4: k-ε model prediction of vortex penetration in the dead leg: VA/VM = 0.25 %.
H/D= 9
H/D= 7
H/D= 5
H/D=3
Va
Vm
Coarse mesh medium mesh
fine mesh
Figure 5 : Reynolds stress model prediction of vortex penetration in the dead leg: VA/VM = 0.25 %.
9.3.6. Conclusions
Numerical calculations performed on the T-junction with coarse (100 000), medium (400 000) and fine (1 500 000) meshes were used to test the influence of the grid. The results illustrate that a thorough mesh refinement is necessary before trying to draw conclusions from any numerical simulation. In-deed, on too coarse a mesh, results from the Reynolds stress model might turn out to be significantly worse than those obtained with a k-ε model.
The numerical results obtained with the Reynolds stress turbulence model on the finest mesh confirm the presence of the swirling flow structure observed experimentally in the dead leg. On the other hand, with the k-ε eddy viscosity model, the vortical penetration in the dead leg is not satisfactorily repro-duced. This demonstrates that Reynolds stress modelling might be a very interesting alternative to less sophisticated turbulence models, such as the k-ε model, especially to capture swirling flows.
9.3.7. References
Mattéi J.D., Simonin O., "Logiciel (1993), “Reference manual : ESTET - Manuel théorique de la version 3.1 - Tome 1 - Modélisation physiques" , EDF report HE-44/92/038/A.
Deutsch E., Montanari P., Mallez C. (1997), "Isothermal study of the flow at the junction between an auxiliary line and primary circuit of pressurised water reactors", Journal of hydraulic Research Vol.
35, No. 6, 799-811.
Robert M. (1992), "Corkscrew flow pattern in piping system dead legs", NURETH-5, September, Salt Lake City, USA.