# An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids

## Abstract

A new semi-implicit pressure-based Computational Fluid Dynamics (CFD) scheme for simulating a wide range of transient and steady, inviscid and viscous compressible flow on unstructured finite elements is presented here. This new CFD scheme, termed the PCICEFEM (Pressure-Corrected ICE-Finite Element Method) scheme, is composed of three computational phases, an explicit predictor, an elliptic pressure Poisson solution, and a semiimplicit pressure-correction of the flow variables. The PCICE-FEM scheme is capable of second-order temporal accuracy by incorporating a combination of a time-weighted form of the two-step Taylor-Galerkin Finite Element Method scheme as an explicit predictor for the balance of momentum equations and the finite element form of a time-weighted trapezoid rule method for the semi-implicit form of the governing hydrodynamic equations. Second-order spatial accuracy is accomplished by linear unstructured finite element discretization. The PCICE-FEM scheme employs Flux-Corrected Transport as a high-resolution filter for shock capturing. The scheme is capable of simulating flows from the nearly incompressible to the high supersonic flow regimes. The PCICE-FEM scheme represents an advancement in mass-momentum coupled, pressurebased schemes. The governing hydrodynamic equations for this scheme are the conservative form of the balance of momentum equations (Navier-Stokes), mass conservation equation, and total energy equation. An operator splitting processmore »

- Authors:

- Publication Date:

- Research Org.:
- Idaho National Laboratory (INL)

- Sponsoring Org.:
- DOE - SC

- OSTI Identifier:
- 910726

- Report Number(s):
- INEEL/EXT-03-00490

TRN: US200802%%103

- DOE Contract Number:
- DE-AC07-99ID-13727

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 - GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ACCURACY; COMPRESSIBLE FLOW; COMPUTERIZED SIMULATION; FINITE ELEMENT METHOD; FLUID MECHANICS; HYDRODYNAMICS; POISSON EQUATION; SUPERSONIC FLOW; TRANSIENTS; TRANSPORT; Compressed flows; High-Resolution finite element; Inviscid; Semi-Implicit Pressure-Based Scheme; Viscous

### Citation Formats

```
Richard C. Martineau, and Ray A. Berry.
```*An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids*. United States: N. p., 2003.
Web. doi:10.2172/910726.

```
Richard C. Martineau, & Ray A. Berry.
```*An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids*. United States. doi:10.2172/910726.

```
Richard C. Martineau, and Ray A. Berry. Tue .
"An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids". United States. doi:10.2172/910726. https://www.osti.gov/servlets/purl/910726.
```

```
@article{osti_910726,
```

title = {An Efficient, Semi-implicit Pressure-based Scheme Employing a High-resolution Finitie Element Method for Simulating Transient and Steady, Inviscid and Viscous, Compressible Flows on Unstructured Grids},

author = {Richard C. Martineau and Ray A. Berry},

abstractNote = {A new semi-implicit pressure-based Computational Fluid Dynamics (CFD) scheme for simulating a wide range of transient and steady, inviscid and viscous compressible flow on unstructured finite elements is presented here. This new CFD scheme, termed the PCICEFEM (Pressure-Corrected ICE-Finite Element Method) scheme, is composed of three computational phases, an explicit predictor, an elliptic pressure Poisson solution, and a semiimplicit pressure-correction of the flow variables. The PCICE-FEM scheme is capable of second-order temporal accuracy by incorporating a combination of a time-weighted form of the two-step Taylor-Galerkin Finite Element Method scheme as an explicit predictor for the balance of momentum equations and the finite element form of a time-weighted trapezoid rule method for the semi-implicit form of the governing hydrodynamic equations. Second-order spatial accuracy is accomplished by linear unstructured finite element discretization. The PCICE-FEM scheme employs Flux-Corrected Transport as a high-resolution filter for shock capturing. The scheme is capable of simulating flows from the nearly incompressible to the high supersonic flow regimes. The PCICE-FEM scheme represents an advancement in mass-momentum coupled, pressurebased schemes. The governing hydrodynamic equations for this scheme are the conservative form of the balance of momentum equations (Navier-Stokes), mass conservation equation, and total energy equation. An operator splitting process is performed along explicit and implicit operators of the semi-implicit governing equations to render the PCICE-FEM scheme in the class of predictor-corrector schemes. The complete set of semi-implicit governing equations in the PCICE-FEM scheme are cast in this form, an explicit predictor phase and a semi-implicit pressure-correction phase with the elliptic pressure Poisson solution coupling the predictor-corrector phases. The result of this predictor-corrector formulation is that the pressure Poisson equation in the PCICE-FEM scheme is provided with sufficient internal energy information to avoid iteration. The ability of the PCICE-FEM scheme to accurately and efficiently simulate a wide variety of inviscid and viscous compressible flows is demonstrated here.},

doi = {10.2172/910726},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2003},

month = {4}

}