# A Graphics Process Unit-Based Multiple-Relaxation-Time Lattice Boltzmann Simulation of Non-Newtonian Fluid Flows in a Backward Facing Step

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## Abstract

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## 1. Introduction

## 2. Governing Equations on the Macro Scale

## 3. Formulation of the Problem in LBM

#### Multiple-Relaxation-Time Lattice Boltzmann Method

## 4. Non-Newtonian Viscosity Model for the Lattice Boltzmann Method

#### 4.1. Power-Law Viscosity Model (PL)

#### 4.2. Modified Power-Law Viscosity Model (MPL)

## 5. Boundary Conditions for the Flow in a Backward-Facing Step

## 6. GPU Implementation for Lattice Boltzmann Method

#### CUDA Programming Interface

**blockIdx**variable. Block size and grid size can also be determined by using

**block_size**and

**grid_size**variables consecutively in one, two, or three dimensions. For defining the thread ID in three-dimensional block of size $({D}_{i},{D}_{j})$, the thread ID of a thread of index $(i,j)$ is $(i+j{D}_{i})$. Additionally, threads are executed in groups of 32 threads defined as

**warps**by [62], which are created by the thread execution unit to ensure instruction oriented data parallelism inside SM. In CUDA implementation for the GPU acceleration in CPU, the shared memory is accessible within a thread block. The special function unit in each block is the processing unit for the solution of different transcendental functions in single precision. For that reason, the CUDA programming environment implements a hybrid approach for executing serial and parallel code together by using both the parent hardware as the host system and CUDA capable hardware as the device. The environment activates the host to perform sequential programs, including parallel programs of the

**kernel**functions and programs for visual data representations. The CUDA comprises three types of memory organization, thread’s local memory, shared memory, and global memory, managed by CUDA run-time for different types of data transfers between the host and device. The global memory is the constant and texture memory of the DRAM of the GPU hardware, which allocates all the memory allocation for the blocks within the grid and the threads for enabling data parallelism. Thread’s local memory is the sole register memory only used for instruction processing in each thread. In contrast, the shared memory is the cache memory of each block for inter-communication between threads.

**cudaMalloc()**function is used for CUDA memory allocation of DRAM constant and texture variables defined in the host program while

**cudaMemcpy()**performs device-to-host data transfers after executing the parallel computation in the GPU. Shared memory can be defined using

**_shared_**qualifier and can be used in kernel functions. Since data transaction between the host and device is time expensive, the optimal amount of global memory usage can achieve good computational performance for any parallel program in the CUDA C programming environment. The optimized algorithm for the implementation of the latice Boltzmann method is described as in [63]. NVIDIA’s GPU Tesla k40 with 2880 CUDA cores and 12GB RAM and Geforce RTX 2080 Ti with 4352 CUDA cores and 11GB RAM were used in the present research work. Table 1 shows the performance of the two NVIDIA GPUs, and it is clear that Geforce RTX 2080 Ti is more efficient than the Tesla k40 GPU. From this performance table, it can be concluded that performance depends on the higher data transfer bandwidth and the number of CUDA cores.

## 7. Results and Discussion

#### 7.1. Grid Sensitivity Test and Code Validation with Lid-Driven Cavity Flow

#### 7.2. Validation with Channel Flow

#### 7.3. Flow in a Backward-Facing Step

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**The schematic model and coordinates systems: (

**a**) flow in a lid-driven cavity, (

**b**) flow in a channel, and (

**c**) flow in a backward facing step.

**Figure 2.**Grid sensitivity test interms of the horizontal velocity profiles, $u/U$, for the lid driven cavity (

**a**) $n=0.5$, (

**b**) $n=1.0$ and (

**c**) $n=1.5$ at $x/H=0.5$$\text{}Re=500$.

**Figure 3.**Velocity profiles for the lid driven cavity (

**a**) $u/U$ at $x/H=0.5$ and (

**b**) $v/U$ at $y/H=0.5$ while $Re=100$.

**Figure 4.**Velocity profiles for the lid driven cavity (

**a**) $u/U$ at $x/H=0.5$ and (

**b**) $v/U$ at $y/H=0.5$ while $Re=500$.

**Figure 5.**Streamlines for the lid driven cavity flow: (

**a**) $n=0.5$, (

**b**) $n=1.0$ (Newtonian), and (

**c**) $n=1.5$ while $Re=100$.

**Figure 6.**Qualitative comparison of the the streamlines with the results of Li et al. [32] for the lid driven cavity flow: (

**a**) $n=0.5$ (shear-thinning), (

**b**) $n=1.0$ (Newtonian), and (

**c**) $n=1.5$ (shear-thickening) while $Re=500$.

**Figure 7.**Velocity profile $u/U$ for the channel flow at (

**a**) $n=0.5$, (

**b**) $n=0.8$, (

**c**) $n=1.0$, (

**d**) $n=1.2$, and (

**e**) $n=1.5$.

**Figure 8.**Streamlines for the power-law non-Newtonian fluids flow in a sudden expansion channel with aspect ratio ($H/d=3.0$): (

**i**) BGK-LBM results of Ilio et al. [65] and (

**ii**) present MRT-LBM for (

**a**) $n=0.6$, (

**b**) $n=1.0$ and (

**c**) $n=1.4$.

**Figure 9.**Streamlines for different power-law indexes: (

**a**) $n=0.5$, (

**b**) $n=0.8$, (

**c**) $n=1.0$, (

**d**) $n=1.2$, (

**e**) $n=1.5$ while $Re=100$ (left) and (

**f**) $n=0.5$, (

**g**) $n=0.8$, (

**h**) $n=1.0$, (

**i**) $n=1.2$, (

**j**) $n=1.5$ while $Re=200$ (right).

**Figure 10.**Streamlines for different power-law indexes: (

**a**) $n=1.5$, (

**b**) $n=1.2$, (

**c**) $n=1.0$, (

**d**) $n=0.8$, (

**e**) $n=0.5$ while $Re=300$.

**Figure 11.**Streamlines for different power-law indexes: (

**a**) $n=1.5$, (

**b**) $n=1.2$, (

**c**) $n=1.0$, (

**d**) $n=0.8$, (

**e**) $n=0.5$ while $Re=400$.

**Figure 12.**Velocity $u/U$ distribution at the different axial locations of the downstream region while $Re=100$ and n (=0.5, 1.0, 1.5).

**Figure 13.**Velocity $u/U$ distribution at the different axial locations of the downstream region while $Re=200$ and n (=0.5, 1.0, 1.5).

**Figure 14.**Skin-friction coefficient, ${C}_{f}={\tau}_{w}/{\textstyle \frac{1}{2}}{\rho}_{0}{U}^{2}$ at the lower wall for the different values of power-law index n at (

**a**) $Re=100$ and (

**b**) $Re=200$.

**Figure 15.**Skin-friction coefficient, ${C}_{f}={\tau}_{w}/{\textstyle \frac{1}{2}}{\rho}_{0}{U}^{2}$ at the lower wall for the different values of power-law index n at (

**a**) $Re=300$ and (

**b**) $Re=400$.

**Table 1.**Performance of the Double precession MRT-LBM simulation time in GPU1:Tesla k40 and GPU2: Geforce RTX 2080 Ti for the different mesh arrangements while $Re=400$ for the backward facing step problem.

Lattice Size | Average Time (s)/Step: | ${\mathit{t}}_{\mathbf{GPU}1}$ | ${\mathit{t}}_{\mathbf{GPU}2}$ | Speed up = ${\mathit{t}}_{\mathbf{GPU}1}/{\mathit{t}}_{\mathbf{GPU}2}$ | ||
---|---|---|---|---|---|---|

512 × 32 | $1.68\times {10}^{-4}$ | $3.89\times {10}^{-5}$ | 4.3 | |||

1024 × 64 | $4.45\times {10}^{-4}$ | $1.37\times {10}^{-4}$ | 3.3 | |||

2048 × 128 | $7.93\times {10}^{-4}$ | $2.56\times {10}^{-4}$ | 3.1 |

**Table 2.**Comparison of the center of the primary vortex for the different power-law indexes n while $Re=500$.

Center of Primary Vortex | $\mathit{n}=0.5$ | $\mathit{n}=1.0$ | $\mathit{n}=1.5$ | |
---|---|---|---|---|

Neoftou [3] | 0.5731 | 0.5494 | 0.5495 | |

${x}_{c}$ | Li et al. [32] | 0.5793 | 0.5467 | 0.5495 |

Present | 0.5762 | 0.5449 | 0.5469 | |

Neoftou [3] | 0.5490 | 0.5935 | 0.6380 | |

${y}_{c}$ | Li et al. [32] | 0.5497 | 0.5947 | 0.6378 |

Present | 0.5449 | 0.5938 | 0.6367 |

**Table 3.**Grid sensitivity test for the flow in a backward-facing step in terms of the length of reattachment point ${x}_{r}/h$ of the primary recirculation zone at the bottom wall for the different power-law index n while $Re=400$.

Grid Size | $\mathit{n}=0.5$ | $\mathit{n}=1.0$ | $\mathit{n}=1.5$ |
---|---|---|---|

512 × 32 | 11.774 (time-mean) | 8.182 | 5.181 |

1024 × 64 | 9.601 (time-mean) | 8.219 | 5.667 |

2048 × 128 | 10.258 (time-mean) | 8.231 | 5.412 |

**Table 4.**Length of reattachment point ${x}_{r}/h$ of the primary recirculation zone at the bottom wall for the different power-law indexes n and for different Reynolds numbers $Re$.

$\mathbf{Re}$ | $\mathit{n}=0.5$ | $\mathit{n}=0.8$ | $\mathit{n}=1.0$ (Erturk [48]) | $\mathit{n}=1.2$ | $\mathit{n}=1.5$ |
---|---|---|---|---|---|

100 | 4.501 | 3.512 | 2.984 (2.922) | 2.539 | 1.943 |

200 | 8.171 | 6.032 | 5.032 (4.921) | 4.213 | 3.229 |

300 | – | 7.332 | 6.951 (6.751) | 5.853 | 4.530 |

400 | – | – | 8.231 (8.237) | 6.974 | 5.412 |

$\mathit{y}/\mathit{h}$ | n = 0.5 | n = 1.0 | n = 1.5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{x}/\mathit{h}=\mathbf{1}$ | $\mathit{x}/\mathit{h}=\mathbf{2}$ | $\mathit{x}/\mathit{h}=\mathbf{3}$ | $\mathit{x}/\mathit{h}=\mathbf{4}$ | $\mathit{x}/\mathit{h}=\mathbf{1}$ | $\mathit{x}/\mathit{h}=\mathbf{2}$ | $\mathit{x}/\mathit{h}=\mathbf{3}$ | $\mathit{x}/\mathit{h}=\mathbf{4}$ | $\mathit{x}/\mathit{h}=\mathbf{1}$ | $\mathit{x}/\mathit{h}=\mathbf{2}$ | $\mathit{x}/\mathit{h}=\mathbf{3}$ | $\mathit{x}/\mathit{h}=\mathbf{4}$ | |

0.0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

0.2 | $-2.99\times {10}^{-2}$ | $-2.46\times {10}^{-2}$ | $1.55\times {10}^{-2}$ | −0.3055 | $-7.09\times {10}^{-2}$ | −0.1299 | −0.1668 | −0.1786 | −0.1441 | −0.2448 | −0.1785 | $8.195\times {10}^{-3}$ |

0.4 | $-3.7\times {10}^{-2}$ | $-2.46\times {10}^{-2}$ | $-2.90\times {10}^{-2}$ | −0.3067 | −0.1003 | −0.1467 | −0.1481 | −0.1008 | −0.1870 | −0.1544 | 2.3625 | 0.2547 |

0.6 | $-2.78\times {10}^{-2}$ | $-2.67\times {10}^{-2}$ | $2.03\times {10}^{-2}$ | $-9.93\times {10}^{-2}$ | $-6.35\times {10}^{-2}$ | $-3.48\times {10}^{-2}$ | $3.93\times {10}^{-2}$ | 0.1626 | $-4.12\times {10}^{-2}$ | 0.1388 | 0.3697 | 0.5906 |

0.8 | $2.25\times {10}^{-2}$ | $6.69\times {10}^{-2}$ | 0.1593 | 0.3514 | $8.45\times {10}^{-2}$ | 0.2331 | 0.3718 | 0.3712 | 0.5431 | 0.5349 | 0.7686 | 0.9284 |

1.0 | 0.3123 | 0.4578 | 0.4463 | 0.80259 | 0.4771 | 0.6674 | 0.8231 | 0.9627 | 0.7183 | 0.9881 | 1.1310 | 1.1668 |

1.2 | 1.0666 | 1.1125 | 1.0396 | 1.1888 | 1.1104 | 1.1846 | 1.2351 | 1.2488 | 1.2863 | 1.3904 | 1.3391 | 1.1122 |

1.4 | 1.3821 | 1.3420 | 1.2854 | 1.3895 | 1.4705 | 1.4295 | 1.3542 | 1.2265 | 1.6628 | 1.4538 | 1.1298 | 0.8389 |

1.6 | 1.3583 | 1.3033 | 1.2564 | 1.2903 | 1.3764 | 1.2506 | 1.0873 | 0.8813 | 1.3926 | 1.0624 | 0.7338 | 0.4911 |

1.8 | 0.9275 | 0.8582 | 0.8651 | 0.7826 | 0.8378 | 0.6930 | 0.5435 | 0.3868 | 0.7577 | 0.5172 | 0.3165 | 0.1888 |

2.0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

$\mathit{y}/\mathit{h}$ | n = 0.5 | n = 1.0 | n = 1.5 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{x}/\mathit{h}=\mathit{1}$ | $\mathit{x}/\mathit{h}=\mathit{2}$ | $\mathit{x}/\mathit{h}=\mathit{3}$ | $\mathit{x}/\mathit{h}=\mathit{4}$ | $\mathit{x}/\mathit{h}=\mathit{1}$ | $\mathit{x}/\mathit{h}=\mathit{2}$ | $\mathit{x}/\mathit{h}=\mathit{3}$ | $\mathit{x}/\mathit{h}=\mathit{4}$ | $\mathit{x}/\mathit{h}=\mathit{1}$ | $\mathit{x}/\mathit{h}=\mathit{2}$ | $\mathit{x}/\mathit{h}=\mathit{3}$ | $\mathit{x}/\mathit{h}=\mathit{4}$ | |

0.0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

0.2 | $1.0\times {10}^{-2}$ | −0.1114 | −0.4377 | $2.24\times {10}^{-2}$ | $-5.64\times {10}^{-2}$ | −0.1073 | −0.1440 | −0.1697 | −0.1066 | −0.2209 | −0.2415 | −0.1359 |

0.4 | $-1.24\times {10}^{-2}$ | −0.1399 | −0.4377 | $-7.92\times {10}^{-2}$ | $-8.53\times {10}^{-2}$ | −0.1334 | −0.1525 | −0.1464 | −0.1626 | −0.1908 | $-9.24\times {10}^{-2}$ | $9.41\times {10}^{-2}$ |

0.6 | $-9.73\times {10}^{-6}$ | $-6.24\times {10}^{-2}$ | −0.1411 | 0.1766 | $-6.33\times {10}^{-2}$ | $-5.53\times {10}^{-2}$ | $-1.74\times {10}^{-2}$ | $4.88\times {10}^{-2}$ | $-7.04\times {10}^{-2}$ | $5.22\times {10}^{-2}$ | 0.2302 | 0.4469 |

0.8 | $5.43\times {10}^{-2}$ | 0.1036 | 0.3798 | 0.6877 | $5.4756\times {10}^{-2}$ | 0.1586 | 0.2653 | 0.3859 | 0.1778 | 0.4136 | 0.6304 | 0.8303 |

1.0 | 0.2437 | 0.3805 | 0.9326 | 1.0827 | 0.4214 | 1.1430 | 0.7124 | 0.8338 | 0.6241 | 0.8702 | 1.0475 | 1.1564 |

1.2 | 0.8616 | 0.8558 | 1.3894 | 1.2368 | 1.0801 | 1.1846 | 1.1938 | 1.2318 | 1.2100 | 1.3259 | 1.3617 | 1.2435 |

1.4 | 1.3162 | 1.3480 | 1.4705 | 1.0493 | 1.4635 | 1.4403 | 1.3998 | 1.3348 | 1.6266 | 1.5137 | 1.2763 | 0.9784 |

1.6 | 1.3205 | 1.3520 | 1.1135 | 0.5684 | 1.3900 | 1.3028 | 1.1941 | 1.0572 | 1.4076 | 1.1568 | 0.8707 | 0.5929 |

1.8 | 0.8676 | 0.8775 | 0.4677 | $3.71\times {10}^{-2}$ | 0.8634 | 0.7518 | 0.6393 | 0.5169 | 0.7793 | 0.5822 | 0.3914 | 0.2297 |

2.0 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 |

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## Share and Cite

**MDPI and ACS Style**

Molla, M.M.; Nag, P.; Thohura, S.; Khan, A.
A Graphics Process Unit-Based Multiple-Relaxation-Time Lattice Boltzmann Simulation of Non-Newtonian Fluid Flows in a Backward Facing Step. *Computation* **2020**, *8*, 83.
https://doi.org/10.3390/computation8030083

**AMA Style**

Molla MM, Nag P, Thohura S, Khan A.
A Graphics Process Unit-Based Multiple-Relaxation-Time Lattice Boltzmann Simulation of Non-Newtonian Fluid Flows in a Backward Facing Step. *Computation*. 2020; 8(3):83.
https://doi.org/10.3390/computation8030083

**Chicago/Turabian Style**

Molla, Md. Mamun, Preetom Nag, Sharaban Thohura, and Amirul Khan.
2020. "A Graphics Process Unit-Based Multiple-Relaxation-Time Lattice Boltzmann Simulation of Non-Newtonian Fluid Flows in a Backward Facing Step" *Computation* 8, no. 3: 83.
https://doi.org/10.3390/computation8030083