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Implementation of a parallel ADI algorithm on a finite volume GPU-based elementary porous media flow computation

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Abstract

In this work we present flow simulations in laminar and turbulent regime within a representative elementary volume of a simplified porous media by solving the Navier–Stokes equations and a Low-Re turbulence \(k{-}\epsilon\) model. Numerical solution was achieved with an implementation of the SIMPLE algorithm for pressure velocity coupling of variables, and the solution of the tridiagonal systems of algebraic equations was accomplished by a parallelized ADI scheme based on the Thomas algorithm. Implementation of the numerical solution was done with an in-house C code which combined OMP and CUDA technologies for computations based on CPU and GPU, respectively. Exponential structured grids were employed in the wall vicinity to capture the turbulence behavior. Results indicate that similar profiles for velocity, pressure, turbulent kinetic energy and its dissipation were found. Several CUDA grids were tested and their performances measured over two GPUs: GTX 680 and GTX TITAN. Considerable speedup was achieved by the GPUs over the CPU schemes even without the use of the device shared memory which was not explored due to the nature of the algorithm.

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Abbreviations

a, b :

FVM coefficients

c :

Dimensionless constant

\(\varvec{D}\) :

Deformation tensor (1/s)

f :

Damping function

\(\varvec{g}\) :

Gravitational acceleration (m/s\(^2\))

\(\varvec{I}\) :

Identity tensor

k :

Turbulent kinetic energy (m\(^2\)/s\(^2\))

L :

Length (m)

n :

Coordinate normal to nearest wall (m)

p :

Pressure (Pa)

\(P_{\rm k}\) :

Production of turbulent kinetic energy (m\(^2\)/s\(^3\))

q :

Ratio of rms values \({\text{ rms }}_\phi /({\text{ rms }}_\phi )_{\min }\)

R :

Dimensionless constant

r :

Growth ratio for distance

S :

Generic source term

t :

Time (s)

\(\varvec{u}\) :

Velocity (m/s)

\(\epsilon\) :

Relative to dissipation of turbulent kinetic energy

E:

East coordinate

k :

Relative to turbulent kinetic energy

\(\mu\) :

Relative to turbulent viscosity

N:

North coordinate

n:

Normal

nb:

Neighbors (E, W, N, S)

P:

Central coordinate

p:

Relative to continuity equation

S:

South coordinate

t:

Turbulent

u :

Relative to x velocity component

v :

Relative to y velocity component

W:

West coordinate

\(\alpha\) :

Relaxation factor

\(\beta\) :

Exponential grid parameter

\(\epsilon\) :

Dissipation of turbulent kinetic energy (m\(^2\)/s\(^3\))

\(\Gamma\) :

Generic transport property

\(\gamma\) :

Exponential grid parameter

\(\nu\) :

Kinematic viscosity (m\(^2\)/s)

\(\phi\) :

Generic variable

\(\rho\) :

Density (kg/m\(^3\))

\(\sigma\) :

Dimensionless constant

\(\tau\) :

Mechanistic momentum flux (Pa)

\((\cdot )'\) :

Time fluctuation

\(\overline{(\cdot )}\) :

Time average

T:

Transpose

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Authors and Affiliations

  1. Chemical Engineering Department, Universidad de Santiago, B. O’Higgins #3363, Santiago, Chile

    L. Henríquez-Vargas, E. Villaroel, J. Gutierrez & P. Donoso-García

Authors
  1. L. Henríquez-Vargas
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  2. E. Villaroel
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  3. J. Gutierrez
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  4. P. Donoso-García
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Correspondence to L. Henríquez-Vargas.

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Technical Editor: Jader Barbosa Jr.

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Henríquez-Vargas, L., Villaroel, E., Gutierrez, J. et al. Implementation of a parallel ADI algorithm on a finite volume GPU-based elementary porous media flow computation. J Braz. Soc. Mech. Sci. Eng. 39, 3965–3979 (2017). https://doi.org/10.1007/s40430-017-0882-x

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