# Reynolds Stresses and Hemolysis in Turbulent Flow Examined by Threshold Analysis

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Methods

#### 3.1. Modeled Systems

_{w}, defined as ${u}^{*}=\sqrt{\frac{{\mathsf{\tau}}_{w}}{\mathsf{\rho}}}$, while the viscous length scale is defined from the fluid kinematic viscosity, ν and the friction velocity as ${l}^{*}=\frac{v}{{u}^{*}}$). The typical value of the integral length scales in wall turbulence is about 1000 in viscous wall units. The range of the size of our computational domain in the azimuthal direction was changing from 2174 (for the lowest shear stress experiment, 50 Pa) to 6498 (for the highest shear stress experiment, 450 Pa) in viscous wall units. Both values are more than twice the typical value of 1000, justifying the use of 1/32nd of the viscometer in the azimuthal direction for the simulations.

#### 3.2. Computational Procedures

#### 3.2.1. Computational Mesh

^{−3}mm

^{3}. Similar mesh analysis was performed for the capillary tube runs and the mesh includes 1,773,099 cells and 2,023,864 nodes with an average grid cell size of 1 × 10

^{−2}mm

^{3}. The grid independence analysis for velocity is shown in Figure 2b, and it can be seen that velocity values in this capillary system were independent of mesh size.

#### 3.2.2. Flow Simulation

_{t}is turbulent viscosity, G

_{k}is the generation of turbulent kinetic energy due to the mean velocity gradients, and S

_{ij}is the mean strain rate. The standard values of model parameters are C

_{2}= 1.9, C

_{µ}= 0.09 and the turbulent Prandtl numbers for k and ε are σ

_{k}= 1.0, σ

_{ε}= 1.2 [42].

_{k}is the generation of turbulent kinetic energy due to the mean velocity gradients, G

_{ω}is the generation of ω, Y

_{k}and Y

_{ω}are dissipation of k and ω due to turbulence, D

_{ω}is the cross-diffusion term. Calculation of the above terms is shown below.

^{3}for the fluid properties (in accordance with Sutera’s Couette viscometer experiments) [16]. In the case of the capillary tube [4], all simulations employed a Newtonian model with a viscosity of 0.002 Pa·s and a density of 1050 kg/m

^{3}. Because the shear rates that were used in the capillary tube experimental study were much higher than 500 s

^{−1}, Kameneva et al. [4] assumed that blood was a single-phase homogeneous Newtonian fluid. The Newtonian and homogeneous fluid assumptions are valid in both Couette and capillary simulations, since the suspensions used in the experiments contained diluted washed red blood cells. Moreover for turbulent flow, the Schmidt number is to the order of 1 so that the eddy diffusivity can be estimated from the eddy viscosity to gauge mixing. Using Evans distribution of the eddy viscosity and the analysis of Taylor [44,45], we estimated for the capillary tube that, after 10 ms, the standard deviation of displacement corresponds to R at the lowest flow rate.

_{i}, the radius of the inner cylinder, R

_{i}, the gap width, h, and the kinematic viscosity, ν, as $\mathrm{Re}=\frac{{\mathsf{\Omega}}_{i}{R}_{i}h}{\nu}$, giving values of 13,390 (for lowest shear stress, 50 Pa) to 47,382 (for highest shear stress, 450 Pa). Sutera and Mehrjardi [16] stated that when the Taylor number (Ta) is higher than 400, the flow will be turbulent; therefore, the Ta was also determined for the Couette viscometer as $Ta=\frac{{\mathsf{\Omega}}_{i}{{R}_{i}}^{1/2}{h}^{3/2}}{\nu}$. Calculated Taylor number values for the Couette viscometer range from 2725 (for lowest shear stress, 50 Pa) to 9641 (for highest shear stress, 450 Pa) (shown in Table 1), were much higher than the critical Ta. Thus, flow in the Couette viscometer is fully turbulent.

^{−5}. The Reynolds numbers for the capillary experiments ranged from 2783 (for lowest shear stress, 100 Pa) to 6242 (for the highest shear stress, 400 Pa), as seen in Table 2. We assume flow in the capillary tube is turbulent since these Reynolds numbers are higher than the critical Reynolds number for pipe flow of 2100–2300 [46,47].

#### 3.2.3. Turbulence Model

_{τ}, which was defined as ${\mathrm{Re}}_{\mathsf{\tau}}=\frac{R}{{l}^{*}}$ where R is the radius of the pipe and l

^{*}is the viscous length scale in wall turbulence. In the case of Couette viscometer, Re

_{τ}was defined as ${\mathrm{Re}}_{\mathsf{\tau}}=\frac{uh/2}{\nu}$. Couette viscometer simulations were performed with both k-ε and k-ω SST models until obtaining Re

_{τ}value that was given in the DNS data [60]. In the case of capillary tube, simulations were performed at Re

_{τ}matching the DNS data of Chin et al. [59]. After simulations were performed, the mean velocity profiles for the near-wall regions were calculated using the dimensionless wall parameters. The dimensionless distance from the wall, ${y}^{+}$, was calculated based on distance from wall, Y, and viscous length scale, ${l}^{*}$, as ${y}^{+}=\frac{Y}{{l}^{*}}$. The dimensionless velocity, u

^{+}, was calculated as ${u}^{+}=\frac{\langle U\rangle}{{u}^{*}}$, where $\langle U\rangle $ is the mean velocity. The mean velocity profile for the Couette viscometer is plotted with the DNS data in Figure 3a [60]. It can be seen from Figure 3a that using either the k-ε or the k-ω SST model in the computation of the flow domain did not show significant differences. After determination of the root mean square error for both models, the k-ε model was selected for simulation of the Couette experiments (Table 3). The mean velocity profile for the capillary tube is shown in Figure 3b. Based on error calculations, the k-ω SST model was chosen for the capillary tube simulations (Table 3).

#### 3.2.4. Reynolds Stress Calculation

_{Re}>, was calculated by using

_{i}, N

_{bins}, is the number of points in each rake, and r

_{i}is the distance of each point from the center of the capillary tube. For the Couette viscometer, rakes were created in the yellow vertical plane shown in Figure 1a. Calculations of viscous stresses, Reynolds stresses, and area-averaged Reynolds stresses were conducted similarly to the capillary tube, keeping in mind that the total stresses for the Couette viscometer were calculated as

## 4. Results and Discussion

#### 4.1. Reynolds Stress Distributions in Couette Viscometer

^{2}. While these experiments were all run at 4 min duration, we note that the times that cells spent in regions of high shear stress may have varied from one experiment to the other.

#### 4.2. Viscous Stress Distributions in Couette Viscometerr

#### 4.3. Reynolds Stress Distributions in Capillary Tube

#### 4.4. Viscous Stress Distributions in Capillary Tube

#### 4.5. Hemolysis Calculations Using Power Models

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 2.**Determination of cell count for grid independence: (

**a**) Grid independence analysis for Couette viscometer for highest shear stress (450 Pa) experiment by using k-ε model (Bottom row in Table 1); (

**b**) grid independence analysis for capillary tube for the highest shear stress (400 Pa) experiment by using k-ω SST model (Bottom row in Table 2).

**Figure 3.**(

**a**) Couette viscometer mean velocity profiles near wall for the DNS data of Pirro et al. [60]: solid line, k-ε model: circles, k-ω SST model: triangles; (

**b**) capillary tube mean velocity profiles near wall for the DNS data of Chin et al. [59]: solid line, k-ε model: circles, k-ω SST model: triangles.

**Figure 4.**Frequency of Reynolds stresses in Couette viscometer for 450 Pa. The frequency and experimentally observed hemolysis of 85% suggest a threshold of 417 Pa.

**Figure 5.**Relationship of hemolysis to threshold Reynolds stresses in Couette viscometer for all experiments (450 to 50 Pa).

**Figure 6.**Relationship of hemolysis to threshold viscous stresses in Couette viscometer for all experiments (450 to 50 Pa).

**Figure 7.**Frequency of Reynolds stresses in capillary tube for the case whereby ${\mathsf{\tau}}_{w}=400\mathrm{Pa}$.

**Figure 8.**Relationship of hemolysis to threshold Reynolds stresses in capillary tube for all of the experiments (100 to 400 Pa).

**Figure 9.**Relationship of hemolysis to threshold viscous stresses in capillary tube for all experiments (400 to 100 Pa).

**Figure 10.**Changes of area-averaged Reynolds, total, and viscous stress with four different wall shear stresses (first column in Table 2).

**Table 1.**Model conditions for Couette viscometer experiments [16].

Shear Stress (Pa) | Rotation Rate (rad/s) | Experimental Hemolysis (%) | Taylor Number |
---|---|---|---|

50 | 130 | 1.403 | 2725 |

100 | 196 | 1.1364 | 4108 |

150 | 240 | 2.5448 | 5030 |

200 | 300 | 4.2883 | 6288 |

250 | 340 | 11.0547 | 7126 |

350 | 400 | 40.3351 | 8383 |

450 | 460 | 85.3609 | 9641 |

**Table 2.**Model conditions for capillary tube experiments [4].

Shear Stress (Pa) | Flow Rate (L/min) | Experimental Hemolysis (%) | Reynolds Number | Exposure Times (s) |
---|---|---|---|---|

100 | 0.15 | 0.0954 | 2783 | 0.0230 |

200 | 0.23 | 0.1538 | 4253 | 0.0144 |

300 | 0.30 | 0.7625 | 5313 | 0.0102 |

400 | 0.36 | 1.9375 | 6242 | 0.0086 |

Root Mean Square Error | Couette Viscometer | Capillary Tube |
---|---|---|

k-ε Model | 0.39 | 1.32 |

k-ω SST Model | 0.65 | 1.04 |

**Table 4.**Calculated Hemolysis Index (HI) applying four different and commonly used power law models and standard error when calculations are compared to experimental measurements by Kameneva et al. [4]

**.**

^{1}Power Law Models | Type of Stress | Calculated HI for τ_{w} = 100 Pa | Calculated HI for τ_{w} = 200 Pa | Calculated HI for τ_{w} = 300 Pa | Calculated HI for τ_{w} = 400 Pa | Standard Error [24] |
---|---|---|---|---|---|---|

Experimental Hemolysis Data [4] | τ_{w} | 0.0954 | 0.1538 | 0.7625 | 1.9375 | 0 |

Giersiepen et al. [23]$HI(\%)=3.62\times {10}^{-5}{\mathsf{\tau}}^{2.416}{t}^{0.785}$ | τ_{Re} | 0.4106 | 3.9302 | 13.8268 | 32.0492 | 6.6658 |

τ_{t} | 2.0132 | 11.8869 | 34.1216 | 70.8037 | 14.8321 | |

τ_{v} | 0.3455 | 1.0587 | 2.0452 | 3.2602 | 0.2485 | |

τ_{w} | 6.5674 | 38.7766 | 111.3085 | 230.9699 | 49.3428 | |

Heuser et al. [10]$HI(\%)=1.8\times {10}^{-6}{\mathsf{\tau}}^{1.991}{t}^{0.765}$ | τ_{Re} | 0.0051 | 0.0335 | 0.0963 | 0.1944 | 0.3861 |

τ_{t} | 0.0188 | 0.0833 | 0.2028 | 0.3735 | 0.3513 | |

τ_{v} | 0.0044 | 0.0114 | 0.0199 | 0.0296 | 0.4224 | |

τ_{w} | 0.0497 | 0.2208 | 0.5372 | 0.9897 | 0.2126 | |

Zhang et al. [11]$HI(\%)=1.228\times {10}^{-5}{\mathsf{\tau}}^{1.9918}{t}^{0.6606}$ | τ_{Re} | 0.0514 | 0.3568 | 1.065 | 2.1868 | 0.0557 |

τ_{t} | 0.1905 | 0.8886 | 2.2428 | 4.2034 | 0.469 | |

τ_{v} | 0.0445 | 0.121 | 0.2203 | 0.3323 | 0.3686 | |

τ_{w} | 0.5049 | 2.3553 | 5.9447 | 11.1413 | 1.9226 | |

Fraser et al. [18]$HI(\%)=1.745\times {10}^{-6}{\mathsf{\tau}}^{1.963}{t}^{0.7762}$ | τ_{Re} | 0.0043 | 0.0274 | 0.0775 | 0.1546 | 0.3947 |

τ_{t} | 0.0155 | 0.0673 | 0.1614 | 0.2943 | 0.3677 | |

τ_{v} | 0.0037 | 0.0094 | 0.0164 | 0.0241 | 0.4234 | |

τ_{w} | 0.0406 | 0.176 | 0.4217 | 0.7692 | 0.267 |

^{1}In Table 4, HI is the hemolysis index, τ is the shear stress (Pa), t is exposure time (s), τ

_{Re}, τ

_{t}, τ

_{v}, and τ

_{w}are average RS, total, viscous, and wall shear stresses (Pa), to indicate which is used as τ in the power law models.

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**MDPI and ACS Style**

Ozturk, M.; O’Rear, E.A.; Papavassiliou, D.V.
Reynolds Stresses and Hemolysis in Turbulent Flow Examined by Threshold Analysis. *Fluids* **2016**, *1*, 42.
https://doi.org/10.3390/fluids1040042

**AMA Style**

Ozturk M, O’Rear EA, Papavassiliou DV.
Reynolds Stresses and Hemolysis in Turbulent Flow Examined by Threshold Analysis. *Fluids*. 2016; 1(4):42.
https://doi.org/10.3390/fluids1040042

**Chicago/Turabian Style**

Ozturk, Mesude, Edgar A. O’Rear, and Dimitrios V. Papavassiliou.
2016. "Reynolds Stresses and Hemolysis in Turbulent Flow Examined by Threshold Analysis" *Fluids* 1, no. 4: 42.
https://doi.org/10.3390/fluids1040042