# Differential formulation of the viscous history force on a particle for efficient and accurate computation

## Abstract

It is well known that the computation of the Basset-like history force is very demanding in terms of CPU and memory requirements, since it requires the evaluation of a history integral. We use the recent rational theory of Beylkin & Monzón (Appl. Comput. Harmon. Anal., vol. 19, 2005, pp. 17–48) to approximate the history kernel in the form of exponential sums to reformulate the viscous history force in a differential form. This theory allows us to approximate the history kernel in terms of exponential sums to any desired order of accuracy. This removes the need for long-time storage of the acceleration histories of the particle and the fluid. The proposed differential form approximation is applied to compute the history force on a spherical particle in a synthetic turbulent flow and a wall-bounded turbulent channel flow. Particles of various diameters are considered, and results obtained using the present technique are in reasonable agreement with those achieved using the full history integral.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Florida, Gainesville, FL (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1538910

- DOE Contract Number:
- NA0002378

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Fluid Mechanics

- Additional Journal Information:
- Journal Volume: 844; Journal ID: ISSN 0022-1120

- Publisher:
- Cambridge University Press

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Mechanics; Physics

### Citation Formats

```
Parmar, M., Annamalai, S., Balachandar, S., and Prosperetti, A.
```*Differential formulation of the viscous history force on a particle for efficient and accurate computation*. United States: N. p., 2018.
Web. doi:10.1017/jfm.2018.217.

```
Parmar, M., Annamalai, S., Balachandar, S., & Prosperetti, A.
```*Differential formulation of the viscous history force on a particle for efficient and accurate computation*. United States. doi:10.1017/jfm.2018.217.

```
Parmar, M., Annamalai, S., Balachandar, S., and Prosperetti, A. Mon .
"Differential formulation of the viscous history force on a particle for efficient and accurate computation". United States. doi:10.1017/jfm.2018.217.
```

```
@article{osti_1538910,
```

title = {Differential formulation of the viscous history force on a particle for efficient and accurate computation},

author = {Parmar, M. and Annamalai, S. and Balachandar, S. and Prosperetti, A.},

abstractNote = {It is well known that the computation of the Basset-like history force is very demanding in terms of CPU and memory requirements, since it requires the evaluation of a history integral. We use the recent rational theory of Beylkin & Monzón (Appl. Comput. Harmon. Anal., vol. 19, 2005, pp. 17–48) to approximate the history kernel in the form of exponential sums to reformulate the viscous history force in a differential form. This theory allows us to approximate the history kernel in terms of exponential sums to any desired order of accuracy. This removes the need for long-time storage of the acceleration histories of the particle and the fluid. The proposed differential form approximation is applied to compute the history force on a spherical particle in a synthetic turbulent flow and a wall-bounded turbulent channel flow. Particles of various diameters are considered, and results obtained using the present technique are in reasonable agreement with those achieved using the full history integral.},

doi = {10.1017/jfm.2018.217},

journal = {Journal of Fluid Mechanics},

issn = {0022-1120},

number = ,

volume = 844,

place = {United States},

year = {2018},

month = {4}

}

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