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Research Papers: Techniques and Procedures

# Design and Validation of a Scale-Adaptive Filtering Technique for LRN Turbulence Modeling of Unsteady Flow

[+] Author and Article Information
W. Gyllenram, H. Nilsson

Division of Fluid Dynamics, Department of Applied Mechanics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden

Companies involved: CarlBro, E.ON Vattenkraft Sverige, Fortum Generation, GE Energy (Sweden), Jämtkraft, Jönköping Energi, Mälarenergi, Skellefteå Kraft, Sollefteåforsens, Statoil Lubricants, Sweco VBB, Sweco Energuide, SweMin, Tekniska Verken i Linköping, Vattenfall Research and Development, Vattenfall Vattenkraft, Waplans, and VG Power and Öresundskraft. Universities involved: Chalmers University of Technology, Luleå University of Technology, Royal institute of Technology, and Uppsala University.

J. Fluids Eng 130(5), 051401 (Apr 29, 2008) (10 pages) doi:10.1115/1.2911685 History: Received June 11, 2007; Revised January 11, 2008; Published April 29, 2008

## Abstract

An adaptive low-pass filtering procedure for the modeled turbulent length and time scales is derived and applied to Wilcox’ original low reynolds number $k-ω$ turbulence model. It is shown that the method is suitable for complex industrial unsteady flows in cases where full large eddy simulations (LESs) are unfeasible. During the simulation, the modeled length and time scales are compared to what can potentially be resolved by the computational grid and time step. If the modeled scales are larger than the resolvable scales, the resolvable scales will replace the modeled scales in the formulation of the eddy viscosity. The filtered $k-ω$ model is implemented in an in-house computational fluid dynamics (CFD) code, and numerical simulations have been made of strongly swirling flow through a sudden expansion. The new model surpasses the original model in predicting unsteady effects and producing accurate time-averaged results. It is shown to be superior to the wall-adpating local eddy-viscosity (WALE) model on the computational grids considered here, since the turbulence may not be sufficiently resolved for an accurate LES. Because of the adaptive formulation, the filtered $k-ω$ model has the potential to be successfully used in any engineering case where an LES is unfeasible and a Reynolds (ensemble) averaged Navier–Stokes simulation is insufficient.

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## Figures

Figure 1

Geometry of the test case. The inlet swirl is clockwise in the z direction.

Figure 2

Block structure. Only 10 out of 15 blocks are shown.

Figure 3

Snapshot of the precessing vortex core visualized by isosurfaces of the nondimensional second invariant of the velocity gradient, II∂jUi*=480 (top) and static pressure (bottom). Both methods predict the same location and shape of the vortex core. The results are obtained from a simulation using the filtered k-ω model on the fine grid. Only a part of the computational domain is shown. The helicoidal vortex structure is formed immediately after the expansion and propagates upstream almost all the way up to the inlet. It dominates the flow until a point approximately one and a half diameters downstream of where it is formed. At this point, the flow returns to a quasisymmetric mode.

Figure 4

Isosurfaces of the normalized second invariant of the velocity gradient tensor, II∂jUi*=120, using the filtered k-ω model and the WALE model. Top: filtered k-ω, coarse grid. Center: filtered k-ω, fine grid. Bottom: WALE model, fine grid. The isosurfaces are shaded by the static pressure. A darker shade denotes a lower pressure. As expected, the fine grid resolves a larger part of very small scale turbulence. However, the strongest and largest vortices are well resolved on any of these grids. An even higher density of small-scale turbulence is obtained using the WALE model.

Figure 5

Radial distributions of averaged axial velocity (top row), tangential velocity (center row), and swirl angle (bottom row) at different cross sections. (—) Filtered k-ω model, fine grid. (– – –) Filtered k-ω model, coarse grid. (⋅ –) WALE model, fine grid. (⋅⋅⋅) Standard k-ω model, coarse grid. (◻) Experiment. The scaling between the left and right columns is given by the profiles at z∕D=1. The standard k-ω model fails to predict reasonable results. It converges to a steady solution. The WALE model performs well considering the grid resolution. However, the agreement between the results obtained using the filtered k-ω model and the experimental data is excellent, especially on the fine grid.

Figure 6

Radial distribution of the time-averaged squared filter function, g2 (left), and the (nondimensional) filtered eddy viscosity, ν̂t*=ν̂t∕ν (right), obtained with different grids. (⋅ –) Coarse grid. (—) Fine grid. The filter is inactive near the wall where it reaches the value of 1, and (often) in the strong shear layer near the sudden expansion. The filtered eddy viscosity is much larger on the coarse grid, because the less that is resolved, the more must be modeled. The maximum (time-averaged) value of the filtered eddy viscosity, ν̂t*=75, is found at z∕D=0.5 (coarse grid).

Figure 7

Radial distribution of nondimensional turbulent kinetic energy (TKE) at z∕D=0.75 (left) and z∕D=2 (right) obtained with the filtered k-ω model on two different grids. All quantities marked by superscript asterisks (*) are normalized by Ub2. (– –) Resolved TKE (K*), coarse grid. (—) Resolved TKE (K*), fine grid. (⋅⋅⋅) Filtered modeled TKE (k̂*), coarse grid. (⋅-) Filtered modeled TKE (k̂*), fine grid. (○) Modeled TKE (k*), coarse grid, (+) Modeled TKE (k*), fine grid. The resolved TKE is dominated by large-scale turbulent structures that only weakly depend on grid resolution. The distribution of filtered modeled turbulent kinetic, k̂*, is determined by length and time scales smaller or equal to what can be resolved. Consequently, it depends heavily on grid resolution.

Figure 8

Left: time series of nondimensional wall pressure fluctuations at z∕D=0.5 obtained with the filtered k-ω model. (—) Fine grid. (– –) Coarse grid. Note that the end points of the time series are quite arbitrary, and thus no correlation between the two series is expected. Right: spectral power density of the wall pressure at z∕D=0.5 obtained from using the filtered k-ω model. (—) Fine grid. (– –) Coarse grid. The predicted rotational speed (St=0.6) of the vortex core is not sensitive to the resolution.

Figure 9

Spectral power density of the wall pressure fluctuations at z∕D=0.25 (top left), z∕D=0.5 (top right), z∕D=1 (bottom left), and z∕D=2 (bottom right). (—) Filtered k-ω model, fine grid. (⋅ –) WALE model, fine grid. The most distinct frequency is at St=0.6 and corresponds to the rotational speed of the vortex core. The strong lower frequencies most likely correspond to unsteady structures that are formed in the recirculation zone near the wall, just after the expansion.

Figure 10

Normalized frequency spectra of resolved axial velocity fluctuations at z∕D=2 and r∕D=0.75 using the filtered k-ω model on two grids. (—) Fine grid. (– –) Coarse grid. (⋅ –) Note the higher density of high frequencies obtained when using the fine grid.

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