20.8.3 Turbulent flow in an axisymmetric narrowing bend

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20.8.3 Turbulent flow in an axisymmetric $180^\circ$ narrowing bend

(Case Study 3, November 1995)

The problem

This was a benchmark problem investigated at the 1994 annual meeting of the World User Association in Applied Computational Fluid Dynamics. One major question is whether experimentally-observed flow separation after the axisymmetric bend can be predicted by a $k-\epsilon$ model. Numerical turbulence modeling can always predict a recirculation behind sharp corners, such as a step, and the concern in such cases is how large the recirculation should be. Here, however, a balance between pressure, inertia, viscous and turbulent stresses determines whether there is a flow separation and where it occurs. The high normal pressure gradient and the curvature of the walls make some universal wall functions totally unsuitable in this problem.

Computations

The mesh, as shown in Figure 1, contains 5,054 nodes with a higher node concentration near both inner and outer walls. This mesh was created using the meshmap capability in Fastflo. The Reynolds number based on inlet uniform velocity and inlet pipe diameter is 286,000.

Figure 1: Geometry and mesh

Fastflo's operator-splitting turbulence module was used to solve this problem. Three $k-\epsilon$ based turbulence models have been incorporated in the Fastflo module: the standard linear model, the RNG model and the nonlinear model proposed by Speziale [1]. All three models have been applied to this benchmark problem and, provided proper wall functions were used, all gave good results in terms of numerical convergence, flow separation prediction, and comparison with experimental data on wall pressure distribution.

Logarithmic velocity profile wall functions were unsuitable for this problem because of the strong normal pressure gradient and wall curvature [2]. Instead, the van Driest mixing length wall functions were used. The exponential damping function in van Driest mixing length progressively suppresses the mixing length as diminishes.

The solution algorithm is based on operator-splitting timestepping with a pre-conditioned conjugate gradient method [3], in which the nonlinearity and incompressibility in Navier-Stokes equations are separately treated at different fractional time steps.

Results

Flow separation after the bend was predicted by all three models, with the RNG model predicting the shortest recirculation, followed by linear and nonlinear $k-\epsilon$ models. There is very little difference between linear and nonlinear $k-\epsilon$ models in terms of predicted velocity fields, and the nonlinearities mainly affect the distribution of turbulent normal stress and pressure.

Figure 2: Pressure on outer wall

Experimental data for pressure distributions on the inner and outer walls were provided by Daimler-Benz. Figure 2 shows these, together with pressures predicted by Fastflo with the three $k-\epsilon$ models and by Engelman [4]. Figure 3 shows contour plots of streamfunction, pressure, and for the nonlinear $k-\epsilon$ model predictions.

Benefits of using Fastflo for this problem

: Fastflo's versatility enables different models and wall treatments to be programmed
: the mesh mapping function allows evenly graded regular mesh

Figure 3: Non-linear $k-\epsilon$ solutions: clockwise from top left $\psi$ ,

, $\epsilon$ and

References

[1] C G Speziale, "On nonlinear and $k-\epsilon$ models of turbulence", J. Fluid Mech. 178 (1986), 459-475.

[2] X-L Luo, "Operator splitting computation of turbulent flow in an axisymmetric $180^\circ$ narrowing bend using several $k-\epsilon$ models and wall functions", Int. J. Num. Meth. Fluids 22 (1996), 1189-1205.

[3] R Glowinski and O Pironneau "Finite element method for Navier-Stokes equations", Ann. Rev. Fluid Mech. 24 (1992), 167-204.

[4] M Engelman, "Axi-symmetric isothermal turbulent flow in a narrowing bend", Report to WUA-CFD annual meeting, Basel, May (1994).