^{1}, Geert Brethouwer

^{1}and Arne V. Johansson

^{1}

### Abstract

In the present investigation, Direct Numerical Simulation (DNS) is used to study a binary irreversible and isothermal reaction in a plane turbulent wall-jet. The flow is compressible and a single-step global reaction between an oxidizer and a fuel species is solved. The inlet based Reynolds, Schmidt, and Mach numbers of the wall-jet are *Re* = 2000, *Sc* = 0.72, and *M* = 0.5, respectively, and a constant coflow velocity is applied above the jet. At the inlet, fuel and oxidizer enter the domain separately in a non-premixed manner. The turbulent structures of the velocity field show the common streaky patterns near the wall, while a somewhat patchy or spotty pattern is observed for the scalars and the reaction rate fluctuations in the near-wall region. The reaction mainly occurs in the upper shear layer in thin highly convoluted reaction zones, but it also takes place close to the wall. Analysis of turbulence and reaction statistics confirms the observations in the instantaneous snapshots, regarding the intermittent character of the reaction rate near the wall. A detailed study of the probability density functions of the reacting scalars and comparison to that of the passive scalar throughout the domain reveals the significance of the reaction influence as well as the wall effects on the scalar distributions. The higher order moments of both the velocities and the scalar concentrations are analyzed and show a satisfactory agreement with experiments. The simulations show that the reaction can both enhance and reduce the dissipation of fuel scalar, since there are two competing effects; on the one hand, the reaction causes sharper scalar gradients and thus a higher dissipation rate, on the other hand, the reaction consumes the fuel scalar thereby reducing the scalar dissipation.

Computer time provided by Swedish National Infrastructure for Computing (SNIC) is gratefully acknowledged. The financial support from the Swedish National Research program of the Centre for Combustion Science and Technology (CECOST) and the Swedish Research Council through Project No. 621-2007-4232 are gratefully acknowledged.

I. INTRODUCTION

II. GOVERNING EQUATIONS

III. NUMERICAL METHOD AND PARAMETERS

IV. DNS OF A REACTING TURBULENT WALL-JET

V. TURBULENCE STATISTICS

VI. REACTANTS STATISTICS

VII. PROBABILITY DENSITY FUNCTIONS

VIII. HIGHER ORDER MOMENTS

IX. DISSIPATION RATES

X. CONCLUSIONS

### Key Topics

- Turbulent flows
- 33.0
- Probability density functions
- 12.0
- Combustion
- 11.0
- Chemical reactions
- 9.0
- Chemically reactive flows
- 8.0

## Figures

(Color online) Schematic of the computational geometry.

(Color online) Schematic of the computational geometry.

(Color online) Instantaneous snapshots of the velocity fluctuations in (a) streamwise and (b) wall-normal directions at a fixed *xz*-plane; 8 < *y* ^{+} < 10. The colorbar at the top of each panel shows the positive and negative fluctuations.

(Color online) Instantaneous snapshots of the velocity fluctuations in (a) streamwise and (b) wall-normal directions at a fixed *xz*-plane; 8 < *y* ^{+} < 10. The colorbar at the top of each panel shows the positive and negative fluctuations.

(Color online) Snapshots of oxidizer concentration (top) and fuel concentration (bottom).

(Color online) Snapshots of oxidizer concentration (top) and fuel concentration (bottom).

(Color online) Snapshots of the reaction rate in (a) *xy*-plane and (b) *xz*-plane at a fixed distance from the wall, *y*/*h* = 1/2, both show the same time instant. The lighter color indicates to the higher reaction rate.

(Color online) Snapshots of the reaction rate in (a) *xy*-plane and (b) *xz*-plane at a fixed distance from the wall, *y*/*h* = 1/2, both show the same time instant. The lighter color indicates to the higher reaction rate.

(Color online) Stoichiometric mixture surface.

(Color online) Stoichiometric mixture surface.

(Color online) Comparison of reacting jet (solid) to non-reacting wall-jet by Ahlman *et al.* (Ref. 23) (dashed), at downstream position *x*/*h* = 25. (a) Inlet normalized mean streamwise velocity and (b) mean streamwise velocity in inner units. Here *y* ^{+} is *yu* _{τ}/ν. (c) Downstream development of friction velocity at the wall and the jet half-height and (d) friction Reynolds number.

(Color online) Comparison of reacting jet (solid) to non-reacting wall-jet by Ahlman *et al.* (Ref. 23) (dashed), at downstream position *x*/*h* = 25. (a) Inlet normalized mean streamwise velocity and (b) mean streamwise velocity in inner units. Here *y* ^{+} is *yu* _{τ}/ν. (c) Downstream development of friction velocity at the wall and the jet half-height and (d) friction Reynolds number.

Comparison of reacting jet (solid) to non-reacting wall-jet by Ahlman *et al.* (Ref. 23) (dashed), at a downstream position *x*/*h* = 25. (a) Streamwise fluctuation intensity in outer scaling, (b) Reynolds shear stress normalized by , (c) cross stream profiles of the conserved scalar fluctuation intensity, and (d) wall-normal flux of the conserved scalar.

Comparison of reacting jet (solid) to non-reacting wall-jet by Ahlman *et al.* (Ref. 23) (dashed), at a downstream position *x*/*h* = 25. (a) Streamwise fluctuation intensity in outer scaling, (b) Reynolds shear stress normalized by , (c) cross stream profiles of the conserved scalar fluctuation intensity, and (d) wall-normal flux of the conserved scalar.

(Color) Statistics of oxidizer (blue), fuel (red), and passive (black) scalars at several downstream positions. Solid: *x*/*h* = 21, dashed: *x*/*h* = 23, dash-dotted: *x*/*h* = 25, dotted: *x*/*h* = 27 (a) reactants mass flux, (b) cross-stream profiles of the reacting scalars, (c) fluctuation intensity of the passive and reacting scalars, and (d) wall-normal fluxes of passive and reacting scalars. Subscript *m* refers to the local maximum value of the corresponding variable.

(Color) Statistics of oxidizer (blue), fuel (red), and passive (black) scalars at several downstream positions. Solid: *x*/*h* = 21, dashed: *x*/*h* = 23, dash-dotted: *x*/*h* = 25, dotted: *x*/*h* = 27 (a) reactants mass flux, (b) cross-stream profiles of the reacting scalars, (c) fluctuation intensity of the passive and reacting scalars, and (d) wall-normal fluxes of passive and reacting scalars. Subscript *m* refers to the local maximum value of the corresponding variable.

Inlet normalized (a) reaction rate and (b) the expected reaction rate corresponding to the product of the mean concentrations at different downstream positions. Line styles are the same as in Figure 8.

Inlet normalized (a) reaction rate and (b) the expected reaction rate corresponding to the product of the mean concentrations at different downstream positions. Line styles are the same as in Figure 8.

Ratio of the maximum reaction rate, peak value of Figure 9(a), to the maximum of the expected corresponding value computed from the mean concentrations, peak value of Figure 9(b).

Ratio of the maximum reaction rate, peak value of Figure 9(a), to the maximum of the expected corresponding value computed from the mean concentrations, peak value of Figure 9(b).

(Color online) Probability density functions of different species at *x*/*h* = 25 in several wall-normal locations. Minimum and maximum concentrations in each plot match the borders of the plot and zero on x-axis points to the mean concentration.

(Color online) Probability density functions of different species at *x*/*h* = 25 in several wall-normal locations. Minimum and maximum concentrations in each plot match the borders of the plot and zero on x-axis points to the mean concentration.

Distributions of third-order moments (skewness) and the fourth-order moments (flatness) of streamwise velocity *u*, wall-normal velocity *v*, spanwise velocity *w*, passive scalar concentration θ, and the fuel species concentration θ_{ f }, in a reacting turbulent wall-jet. Solid: *x*/*h* = 15, dashed: *x*/*h* = 20, and dash-dotted: *x*/*h* = 25.

Distributions of third-order moments (skewness) and the fourth-order moments (flatness) of streamwise velocity *u*, wall-normal velocity *v*, spanwise velocity *w*, passive scalar concentration θ, and the fuel species concentration θ_{ f }, in a reacting turbulent wall-jet. Solid: *x*/*h* = 15, dashed: *x*/*h* = 20, and dash-dotted: *x*/*h* = 25.

The skewness factors for (a) streamwise and (b) wall-normal velocity fluctuations at downstream postion *x*/*h* = 8.7, solid line: present DNS data, star symbols: experimental data of Ref. 41.

The skewness factors for (a) streamwise and (b) wall-normal velocity fluctuations at downstream postion *x*/*h* = 8.7, solid line: present DNS data, star symbols: experimental data of Ref. 41.

(Color online) Turbulence intensity, skewness, and flatness factors of the streamwise velocity at downstream position *x*/*h* = 25. The red lines show the positions of the skewness factor zeros.

(Color online) Turbulence intensity, skewness, and flatness factors of the streamwise velocity at downstream position *x*/*h* = 25. The red lines show the positions of the skewness factor zeros.

Turbulent kinetic energy viscous dissipation rates using (a) inner and (b) outer scaling at different downstream positions; Solid line: *x*/*h* = 21 and dashed line: *x*/*h* = 25.

Turbulent kinetic energy viscous dissipation rates using (a) inner and (b) outer scaling at different downstream positions; Solid line: *x*/*h* = 21 and dashed line: *x*/*h* = 25.

Passive (a) and reacting (b) scalar dissipations using outer scaling at different downstream positions; Solid line: *x*/*h* = 21 and dashed line: *x*/*h* = 25.

Passive (a) and reacting (b) scalar dissipations using outer scaling at different downstream positions; Solid line: *x*/*h* = 21 and dashed line: *x*/*h* = 25.

## Tables

Simulation cases, *L* _{ i } and *N* _{ i } are the domain size and grid points in the i-direction.

Simulation cases, *L* _{ i } and *N* _{ i } are the domain size and grid points in the i-direction.

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