^{1}and Peter P. Sullivan

^{1}

### Abstract

A point-force model is used to study turbulent momentum transfer in the presence of moderate mass loadings of small (relative to Kolmogorov scales), dense (relative to the carrier phase density) particles. Turbulent Couette flow is simulated via direct numerical simulation, while individual particles are tracked as Lagrangian elements interacting with the carrier phase through a momentum coupling force. This force is computed based on the bulk drag of each particle, computed from its local slip velocity. By inspecting a parameter space consisting of particle Stokes number and mass loading, a general picture of how and under what conditions particles can alter near-wall turbulent flow is developed. In general, it is found that particles which adhere to the requirements for the point-particle approximation attenuate small-scale turbulence levels, as measured by wall-normal and spanwise velocity fluctuations, and decrease turbulent fluxes. Particles tend to weaken near-wall vortical activity, which in turn, through changes in burst/sweep intensities, weakens the ability of the turbulent carrier-phase motion to transfer momentum in the wall-normal direction. Compensating this effect is the often-ignored capacity of the dispersed phase to carry stress, resulting in a total momentum transfer which remains nearly unchanged. The results of this study can be used to interpret physical processes above the ocean surface, where sea spray potentially plays an important role in vertical momentum transfer.

The National Center for Atmospheric Research is supported by the National Science Foundation. The authors would like to thank the Advanced Study Program at NCAR for financial support. The authors would also like to thank Ned Patton for constructive comments and many helpful discussions.

I. INTRODUCTION

II. PROBLEM FORMULATION

A. Carrier phase

B. Dispersed phase

C. Numerical implementation

III. PARAMETER CHOICES

IV. RESULTS

A. Fluctuating velocities

B. Turbulent stress

V. DISCUSSION

A. Fluctuating velocities

B. Near-wall turbulenttransport

VI. CONCLUSIONS

### Key Topics

- Turbulent flows
- 47.0
- Reynolds stress modeling
- 27.0
- Couette flows
- 16.0
- Turbulent channel flow
- 13.0
- Particle fluctuations
- 12.0

## Figures

Schematic of Couette cell geometry. Reference velocity U 0 is the difference in plate velocity and reference length H is the distance between the plates.

Schematic of Couette cell geometry. Reference velocity U 0 is the difference in plate velocity and reference length H is the distance between the plates.

(a) Normalized mean particle concentration ⟨c ⟩ /c 0 versus z +, where c 0 is homogeneous bulk particle concentration. (b) Normalized mean particle feedback force, . Curves represent the following: unladen case (solid); small particle, (dashed); large particle, (dotted); and small particle with the rebound distance specified at z + = 1.25 (dashed-dotted). All particle-laden runs at St K ≈ 11 and ϕ m = 0.25.

(a) Normalized mean particle concentration ⟨c ⟩ /c 0 versus z +, where c 0 is homogeneous bulk particle concentration. (b) Normalized mean particle feedback force, . Curves represent the following: unladen case (solid); small particle, (dashed); large particle, (dotted); and small particle with the rebound distance specified at z + = 1.25 (dashed-dotted). All particle-laden runs at St K ≈ 11 and ϕ m = 0.25.

(a) Fluctuating streamwise velocity profile for varying St K , holding ϕ m = 0.25 constant. (b) Fluctuating streamwise velocity profile for varying ϕ m , holding St K ≈ 11 constant. Fluctuating velocities normalized by plate velocity U 0 and wall-normal position normalized by channel height H.

(a) Fluctuating streamwise velocity profile for varying St K , holding ϕ m = 0.25 constant. (b) Fluctuating streamwise velocity profile for varying ϕ m , holding St K ≈ 11 constant. Fluctuating velocities normalized by plate velocity U 0 and wall-normal position normalized by channel height H.

(a) Fluctuating wall-normal velocity profile for varying St K , holding ϕ m = 0.25 constant. (b) Fluctuating wall-normal velocity profile for varying ϕ m , holding St K ≈ 11 constant. Fluctuating velocities normalized by plate velocity U 0 and wall-normal position normalized by channel height H.

(a) Fluctuating wall-normal velocity profile for varying St K , holding ϕ m = 0.25 constant. (b) Fluctuating wall-normal velocity profile for varying ϕ m , holding St K ≈ 11 constant. Fluctuating velocities normalized by plate velocity U 0 and wall-normal position normalized by channel height H.

(a) Profiles of total, viscous, turbulent, and particle stress for varying St K , holding ϕ m = 0.25 constant. (b) Stress profiles for varying ϕ m , holding St K ≈ 11 constant. Stress normalized by and wall-normal position normalized by channel height H. The curves corresponding to different stress components are indicated on the figures.

(a) Profiles of total, viscous, turbulent, and particle stress for varying St K , holding ϕ m = 0.25 constant. (b) Stress profiles for varying ϕ m , holding St K ≈ 11 constant. Stress normalized by and wall-normal position normalized by channel height H. The curves corresponding to different stress components are indicated on the figures.

Instantaneous snapshots of normalized fluctuating streamwise velocity u′/U 0 with particle locations (black dots) for the following cases: unladen (top row); Run 3: St K = 10.2, ϕ m = 0.5 (middle row); and Run 4: St K = 1.2, ϕ m = 0.25 (bottom row). Left column is at a location of z + = 20 and right column is at the channel midplane: z/H = 0.5. Note that particle sizes have been magnified and thus are not to scale.

Instantaneous snapshots of normalized fluctuating streamwise velocity u′/U 0 with particle locations (black dots) for the following cases: unladen (top row); Run 3: St K = 10.2, ϕ m = 0.5 (middle row); and Run 4: St K = 1.2, ϕ m = 0.25 (bottom row). Left column is at a location of z + = 20 and right column is at the channel midplane: z/H = 0.5. Note that particle sizes have been magnified and thus are not to scale.

Energy spectra E 11 taken in the spanwise direction (k y ) for all cases. Top row is spectra at a height of z + = 20 and bottom row is spectra at the channel midplane: z/H = 0.5. (a) and (c) Comparison of cases holding ϕ m constant at 0.25. (b) and (d) Comparison of cases holding St K ≈ 11 and varying ϕ m

Energy spectra E 11 taken in the spanwise direction (k y ) for all cases. Top row is spectra at a height of z + = 20 and bottom row is spectra at the channel midplane: z/H = 0.5. (a) and (c) Comparison of cases holding ϕ m constant at 0.25. (b) and (d) Comparison of cases holding St K ≈ 11 and varying ϕ m

Probability density function of the fluctuating streamwise velocity, computed at the particle location, for all particles within range 10 ⩽ z + ⩽ 20. Two curves are shown: Run 3 (St K = 10.2, ϕ m = 0.5) and Run 4 (St K = 1.2, ϕ m = 0.25)

Probability density function of the fluctuating streamwise velocity, computed at the particle location, for all particles within range 10 ⩽ z + ⩽ 20. Two curves are shown: Run 3 (St K = 10.2, ϕ m = 0.5) and Run 4 (St K = 1.2, ϕ m = 0.25)

(a) Joint probably distribution function of streamwise (u′) and wall-normal (w′) velocity fluctuations at a height of z + = 40. Panels (b)–(d) show difference between laden and unladen cases: P diff (u′, w′) = P laden (u′, w′) − P unladen (u′, w′). (b) St K = 11.7, ϕ m = 0.25 (Run 2); (c) St K = 10.2, ϕ m = 0.5 (Run 3); and (d) St K = 1.2, ϕ m = 0.25 (Run 4). Note that the u′ and w′ axes use different scales.

(a) Joint probably distribution function of streamwise (u′) and wall-normal (w′) velocity fluctuations at a height of z + = 40. Panels (b)–(d) show difference between laden and unladen cases: P diff (u′, w′) = P laden (u′, w′) − P unladen (u′, w′). (b) St K = 11.7, ϕ m = 0.25 (Run 2); (c) St K = 10.2, ϕ m = 0.5 (Run 3); and (d) St K = 1.2, ϕ m = 0.25 (Run 4). Note that the u′ and w′ axes use different scales.

Probability density function of the nondimensionalized swirling strength, |λ ci |H/U 0. All laden cases are included as well as the unladen case. Vertical lines represent the point where the time scale of the swirling motion (inverse of |λ ci |) is equal to the acceleration time scale of particles from Run 3 (τ p, 3) and Run 4 (τ p, 4).

Probability density function of the nondimensionalized swirling strength, |λ ci |H/U 0. All laden cases are included as well as the unladen case. Vertical lines represent the point where the time scale of the swirling motion (inverse of |λ ci |) is equal to the acceleration time scale of particles from Run 3 (τ p, 3) and Run 4 (τ p, 4).

## Tables

Particle-laden simulation parameters. Runs 1–3 maintain relatively constant St K and vary ϕ m while Runs 4 and 5 vary St K at a constant ϕ m . The particle Stokes number based on wall units, St +, and the mean particle Reynolds number, ⟨Re p ⟩, are included as well for reference. Also indicated are the curve types used for each case in all subsequent plots.

Particle-laden simulation parameters. Runs 1–3 maintain relatively constant St K and vary ϕ m while Runs 4 and 5 vary St K at a constant ϕ m . The particle Stokes number based on wall units, St +, and the mean particle Reynolds number, ⟨Re p ⟩, are included as well for reference. Also indicated are the curve types used for each case in all subsequent plots.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content