^{1}, Simone Camarri

^{1,a)}and Maria Vittoria Salvetti

^{1}

### Abstract

The steady engulfment regime in a fully three-dimensional micro T-mixer is investigated. This regime is of significant interest for applications since it implies high mixing between the flow streams entering the device. Direct numerical simulations are first used to characterize this regime. In particular, the main vortical structures typical of the engulfment regime and their effects on mixing are investigated. Three-dimensional linear stability analysis is successively applied to the characterization of the instability leading to the engulfment regime. The critical Reynolds number and the global unstable mode are first computed for a configuration characterized by fully-developed inlet velocity conditions. The sensitivity of this instability to a generic modification of the base flow is then investigated, thanks to the computation of the mode adjoint to the direct unstable one. Finally, this kind of analysis is specialized to investigate the effect of a perturbation of the velocity distribution at the inlet of the T-mixer. Sensitivity analysis shows that non-fully developed inlet velocity conditions lead to an increase of the critical Reynolds number. More generally, the sensitivity maps can be used for the design of control strategies aimed at promoting or inhibiting the engulfment. An example is provided for a control based on blowing/suction through the mixer walls.

CASPUR (Rome, Italy) and CINECA (Bologna, Italy) computing centers are gratefully acknowledged for allowance of computational resources. The authors wish to thank E. Brunazzi, C. Galletti, and R. Mauri for having stimulated this study.

I. INTRODUCTION

II. PROBLEM DESCRIPTION AND METHODOLOGY

III. VALIDATION

IV. RESULTS

A. DNS investigation

B. Stability and sensitivity analysis

V. CONCLUSIONS

### Key Topics

- Rotating flows
- 49.0
- Flow instabilities
- 32.0
- Reynolds stress modeling
- 21.0
- Eigenvalues
- 20.0
- Boundary value problems
- 10.0

##### B81B

## Figures

Flow configuration and frame of reference; dashed lines indicate the flow direction.

Flow configuration and frame of reference; dashed lines indicate the flow direction.

Adjoint base flow pressure obtained with NEK5000 and FreeFem++ at Re = 240 at section x = −L i in the plane case.

Adjoint base flow pressure obtained with NEK5000 and FreeFem++ at Re = 240 at section x = −L i in the plane case.

Vortices identified according to the λ2-criterion at Re = 140.

Vortices identified according to the λ2-criterion at Re = 140.

(Re = 140) Vorticity normal to the cutting plane and in-plane velocity vectors at sections z = H/2 (a) and y = 0.2 (b); the thick lines indicate the flow region identified as a vortex according to the λ2 criterion.

(Re = 140) Vorticity normal to the cutting plane and in-plane velocity vectors at sections z = H/2 (a) and y = 0.2 (b); the thick lines indicate the flow region identified as a vortex according to the λ2 criterion.

(Re = 140) Velocity in the y direction and in-plane velocity vectors at section y = 0.2; the thick lines indicate the boundary of the vortices.

(Re = 140) Velocity in the y direction and in-plane velocity vectors at section y = 0.2; the thick lines indicate the boundary of the vortices.

(Re = 140) Vorticity component in the y-direction and in-plane velocity vectors at sections y = −1 (a) and y = −3 (b); the thick lines indicate the boundary of the vortices.

(Re = 140) Vorticity component in the y-direction and in-plane velocity vectors at sections y = −1 (a) and y = −3 (b); the thick lines indicate the boundary of the vortices.

Vortices identified according to the λ2-criterion at Re = 160.

Vortices identified according to the λ2-criterion at Re = 160.

Velocity field (iso-contours indicating normal velocity and vectors the planar components) is shown together with a thick line indicating the boundary of the vortices for Re = 140 (a) and Re = 160 (b) at section y = 0.4.

Velocity field (iso-contours indicating normal velocity and vectors the planar components) is shown together with a thick line indicating the boundary of the vortices for Re = 140 (a) and Re = 160 (b) at section y = 0.4.

(Re = 160) Vorticity component in the y-direction and in-plane velocity vectors at sections y = −1 (a) and y = −3 (b); the thick lines indicate the boundary of the vortices.

(Re = 160) Vorticity component in the y-direction and in-plane velocity vectors at sections y = −1 (a) and y = −3 (b); the thick lines indicate the boundary of the vortices.

Vortices identified according to the λ2-criterion at Re = 160 and isocontours of the tracer at different y-sections along the outflow pipe.

Vortices identified according to the λ2-criterion at Re = 160 and isocontours of the tracer at different y-sections along the outflow pipe.

(Re = 160) Passive scalar isocontours and in-plane velocity vectors at plane y = −3; the thick lines localize the vortices.

(Re = 160) Passive scalar isocontours and in-plane velocity vectors at plane y = −3; the thick lines localize the vortices.

Passive scalar on plane z = H/2.

Passive scalar on plane z = H/2.

Vortical structures of the global mode, identified by the λ2-criterion, computed at Re = 140.

Vortical structures of the global mode, identified by the λ2-criterion, computed at Re = 140.

(a)–(d) Global direct mode at four different y section: arrows indicate the in-plane velocity components, whose maximum magnitude is approximately equal to 0.72, and contours represent the velocity component normal to the plane, ranging from −0.6 (dark color) to 0.6 (light color).

(a)–(d) Global direct mode at four different y section: arrows indicate the in-plane velocity components, whose maximum magnitude is approximately equal to 0.72, and contours represent the velocity component normal to the plane, ranging from −0.6 (dark color) to 0.6 (light color).

Isosurface of the magnitude of the adjoint mode velocity u +, computed at Re = 140.

Isosurface of the magnitude of the adjoint mode velocity u +, computed at Re = 140.

(a)–(c) Global adjoint mode at three different x sections: arrows indicate the in-plane velocity components, whose maximum magnitude is approximately equal to 29.2, and contours represent the velocity component normal to the plane, ranging from −15 (dark color) to 15 (light color).

(a)–(c) Global adjoint mode at three different x sections: arrows indicate the in-plane velocity components, whose maximum magnitude is approximately equal to 29.2, and contours represent the velocity component normal to the plane, ranging from −15 (dark color) to 15 (light color).

Instability core , computed at Re = 140. (a) 3D view and (b) slice at y = 0.4.

Instability core , computed at Re = 140. (a) 3D view and (b) slice at y = 0.4.

Sensitivity to base flow modifications at different x sections: arrows indicate the in-plane sensitivity components, whose maximum magnitude is approximately equal to 66, and contours represent the sensitivity component normal to the plane, ranging from −50 (dark color) to 50 (light color); sections at x = 0.2 (a), x = 0.4 (b), and x = 0.6 (c).

Sensitivity to base flow modifications at different x sections: arrows indicate the in-plane sensitivity components, whose maximum magnitude is approximately equal to 66, and contours represent the sensitivity component normal to the plane, ranging from −50 (dark color) to 50 (light color); sections at x = 0.2 (a), x = 0.4 (b), and x = 0.6 (c).

Sensitivity to modifications of the velocity profile normal to the inlet plane (a) and considered velocity perturbation (b), both at x = −L i .

Sensitivity to modifications of the velocity profile normal to the inlet plane (a) and considered velocity perturbation (b), both at x = −L i .

Sensitivity to introduction of a velocity profile normal to wall; sections at y = W (a) and at z = 0 (b).

Sensitivity to introduction of a velocity profile normal to wall; sections at y = W (a) and at z = 0 (b).

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