^{1,a)}, Luc Vervisch

^{1,b)}and Pascale Domingo

^{1,c)}

### Abstract

Wall-jet interaction is studied with large-eddy simulation(LES) in which a mixed-similarity subgrid scale (SGS) closure is combined with the wall-adapting local eddy-viscosity (WALE) model for the eddy-viscosity term. The macrotemperature and macropressure are introduced to deduce a weakly compressible form of the mixed-similarity model, and the relevant formulation for the energyequation is deduced accordingly. LES prediction capabilities are assessed by comparing flow statistical properties against experiment of an unconfined impinging round jet at Reynolds numbers of and . To quantify the benefit of the proposed WALE-similarity mixed model, the lower Reynolds number simulations are also performed using the standard WALE and Lagrangian dynamic Smagorinsky approaches. The unsteady compressible Navier–Stokes equations are integrated over 2.9 M, 3.5 M, and 5.5 M node Cartesian grids with an explicit fourth-order finite volume solver. Nonreflecting boundary conditions are enforced using a methodology accounting for the three-dimensional character of the turbulent flow at boundaries. A correct wall scaling is achieved from the combination of similarity and WALE approaches; for this wall-jet interaction, the SGS closure terms can be computed in the near-wall region without the necessity of resorting to additional specific treatments. The possible impact of turbulentenergybackscatter in such flow configurations is also addressed. It is found that, for the present configuration, the correct reproduction of reverse energy transfer plays a key role in the estimation of near-wall statistics, especially when the viscous sublayer is not properly resolved.

Support was provided by the project NICE (New Integrated Combustion system for Future Passenger Car Engines), Grant No. TIP3-CT-2004-50620 within the Sixth Framework program of the European Union. Computing resources were provided by IDRIS-CNRS (http://www.idris.fr/) and CRIHAN (http://www.crihan.fr/).

I. INTRODUCTION

II. MODEL PROBLEM AND NUMERICS

A. Filtered balance equations

1. The filtered continuity and momentum equations

2. The filtered energyequation

3. The filtered scalar equation

4. The explicit filtering procedure

B. Flow configuration

C. Numerical formulation

D. Grid spacing

III. RESULTS AND DISCUSSION

A. test-case C1 validation

B. Grid sensitivity assessment

C. test-case C3 validation

D. Energybackscatter

E. Flow field and scalar mixing

IV. CONCLUSIONS

### Key Topics

- Energy transfer
- 32.0
- Reynolds stress modeling
- 32.0
- Backscattering
- 31.0
- Turbulence simulations
- 26.0
- Tensor methods
- 25.0

## Figures

Schematic of the flow configuration and position of the coordinate system.

Schematic of the flow configuration and position of the coordinate system.

test case (coarse grid). Streamwise average velocity (a); streamwise fluctuating velocity (b); wall-normal fluctuating velocity (c); and turbulent shear stress (d); (—) WSM; (– – –) WALE; LDSM; (○) HWA measures (Ref. 23).

test case (coarse grid). Streamwise average velocity (a); streamwise fluctuating velocity (b); wall-normal fluctuating velocity (c); and turbulent shear stress (d); (—) WSM; (– – –) WALE; LDSM; (○) HWA measures (Ref. 23).

test case (coarse grid). Turbulent kinetic energy : (—) WSM; (– – –) WALE; LDSM; (○) HWA measures (Ref. 23).

test case (coarse grid). Turbulent kinetic energy : (—) WSM; (– – –) WALE; LDSM; (○) HWA measures (Ref. 23).

test case (coarse grid). Average wall-normal velocity (a) and average radial velocity (b). (—) WSM; LDSM; PIV measures (Ref. 25); (◻) LDA measures (Ref. 25).

test case (coarse grid). Average wall-normal velocity (a) and average radial velocity (b). (—) WSM; LDSM; PIV measures (Ref. 25); (◻) LDA measures (Ref. 25).

test case (coarse grid). RMS wall-normal velocity (a) and RMS radial velocity (b). (—) WSM; LDSM; PIV measures (Refs. 24 and 25); (◻) LDA measures (Refs. 24 and 25); (○) HWA measures (Ref. 23).

test case (coarse grid). RMS wall-normal velocity (a) and RMS radial velocity (b). (—) WSM; LDSM; PIV measures (Refs. 24 and 25); (◻) LDA measures (Refs. 24 and 25); (○) HWA measures (Ref. 23).

Average value of over horizontal planes: (—) test case C1 (coarse grid); test case C2 (refined grid); (– – –) test case C3.

Average value of over horizontal planes: (—) test case C1 (coarse grid); test case C2 (refined grid); (– – –) test case C3.

test case. Streamwise average velocity (a); streamwise fluctuating velocity (b); wall-normal fluctuating velocity (c); and turbulent shear stress (d); (—) WSM on highly refined grid (C2); (– – –) WSM on coarse grid (C1); WALE on highly refined grid (C2); LDSM on highly refined grid (C2); (○) HWA measures (Ref. 23); (◻) LDA measures (Refs. 24 and 25).

test case. Streamwise average velocity (a); streamwise fluctuating velocity (b); wall-normal fluctuating velocity (c); and turbulent shear stress (d); (—) WSM on highly refined grid (C2); (– – –) WSM on coarse grid (C1); WALE on highly refined grid (C2); LDSM on highly refined grid (C2); (○) HWA measures (Ref. 23); (◻) LDA measures (Refs. 24 and 25).

test case. Streamwise average velocity (a); streamwise fluctuating velocity (b); wall-normal fluctuating velocity (c); and turbulent shear stress (d); (—) resolved fluctuations; resolved fluctuations plus SGS contributions; (○) HWA measures (Ref. 23).

test case. Streamwise average velocity (a); streamwise fluctuating velocity (b); wall-normal fluctuating velocity (c); and turbulent shear stress (d); (—) resolved fluctuations; resolved fluctuations plus SGS contributions; (○) HWA measures (Ref. 23).

test case. Turbulent kinetic energy : (—) resolved fluctuations; resolved fluctuations plus SGS contributions; (○) HWA measures (Ref. 23).

test case. Turbulent kinetic energy : (—) resolved fluctuations; resolved fluctuations plus SGS contributions; (○) HWA measures (Ref. 23).

test-case C1 (coarse grid). Localization of regions (colored in black) of negative [Eq. (93)] over horizontal planes at different heights: (a), (b), (c), (d), (e), and (f). Circles in (a) indicate radial distances in steps of .

test-case C1 (coarse grid). Localization of regions (colored in black) of negative [Eq. (93)] over horizontal planes at different heights: (a), (b), (c), (d), (e), and (f). Circles in (a) indicate radial distances in steps of .

test case C2 (refined grid). Localization of regions (colored in black) of negative [Eq. (93)] over horizontal planes at different heights: (a), (b), (c), (d), (e), and (f). Circles in (a) indicate radial distances in steps of .

test case C2 (refined grid). Localization of regions (colored in black) of negative [Eq. (93)] over horizontal planes at different heights: (a), (b), (c), (d), (e), and (f). Circles in (a) indicate radial distances in steps of .

Maps of normalized SGS energy transfer coefficient over a horizontal plane located at for on coarse grid C1 (a) and for (b). The black contours indicate regions of energy backscatter.

Maps of normalized SGS energy transfer coefficient over a horizontal plane located at for on coarse grid C1 (a) and for (b). The black contours indicate regions of energy backscatter.

test case C2 (refined grid): map of normalized SGS energy transfer coefficient over an horizontal plane located at . The black contours indicate regions of energy backscatter.

test case C2 (refined grid): map of normalized SGS energy transfer coefficient over an horizontal plane located at . The black contours indicate regions of energy backscatter.

test case. Maps of normalized SGS energy transfer coefficient (same scale as in Fig. 12) over a vertical plane in the range , . The black contours: (a) regions of reverse energy transfer, (b) intense strain, and (c) vorticity.

test case. Maps of normalized SGS energy transfer coefficient (same scale as in Fig. 12) over a vertical plane in the range , . The black contours: (a) regions of reverse energy transfer, (b) intense strain, and (c) vorticity.

Tridimensional visualization of the flow at on coarse grid C1 (a) and at (b): isosurfaces of passive scalar (center), isocontours of (left), and passive scalar map (right) over axial planes.

Tridimensional visualization of the flow at on coarse grid C1 (a) and at (b): isosurfaces of passive scalar (center), isocontours of (left), and passive scalar map (right) over axial planes.

test case C1: maps of passive scalar over a horizontal plane at (a) and (b).

test case C1: maps of passive scalar over a horizontal plane at (a) and (b).

test case C3: maps of passive scalar over a horizontal plane at (a) and (b).

test case C3: maps of passive scalar over a horizontal plane at (a) and (b).

Explicit filtering volume in two dimensions.

Explicit filtering volume in two dimensions.

## Tables

Air properties.

Air properties.

Computational grid properties.

Computational grid properties.

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