The purpose of this numerical simulation is to validate the flow velocity profile for a Non-Newtonian fluid via the Power-Law model.

The numerical simulation were carried out using the Reynolds-Averaged Navier–Stokes (RANS) approach at laminar flow conditions. The results of SimScale simulation runs were compared to the analytical results shown in [1][2]. The flow regime selected for the study has a Reynolds number of Re=40

The geometry of the study is a 2 dimensional channel with a 3:1 expansion (see Fig.1.). A brief description of the dimensions are provided by the table below.

with respect to expansion step.

Domain and Analysis type

The domain is the internal region of the geometry with the domain extents same as the geometrical dimensions. For this study a full hexahedral structured mesh was created with the “blockMesh” open-source tool (see Fig.2.). The Mesh was refined at the expansion step in both horizontal and vertical directions. Two meshes, mesh M_2 (intermediate) and mesh M_3 (fine) were used for the study to check for mesh independence of results. The details of the mesh are listed in the following table:

Mesh and Element types :

Mesh type

Number of cells

Type

blockMesh

30.7 – 77.6 thousand

3D hexahedral

The numerical analysis performed is detailed as follows:

Tool Type : OPENFOAM®

Analysis Type : In-Compressible Steady-State

Turbulence Model : Laminar

Non-Newtonian model : Power-Law

Simulation Setup

Fluid:

Non-Newtonian

normalized Consistancy Index (by density): k[m2/s] $k[{m}^{2}/s]$

based on Re=40

$Re=40$

Flow/Power Index n=0.5−2 $n=0.5-2\text{}$

Boundary Conditions:

For the inlet boundary, a fixed velocity condition was applied, while a pressure boundary condition was applied at the outlet. No-slip condition was applied for the walls and empty condition for the side faces. The following table provides the further details.

Boundary type

Velocity

Pressure

Inlet

laminar Fixed Value: 0.5ms−1$0.5\text{}m{s}^{-1}$

Zero Gradient

Outlet

Zero Gradient

Fixed Value: 0Pa$0\text{}Pa$

Wall no-slip

Fixed Value: 0.0ms−1$0.0\text{}m{s}^{-1}$

Zero Gradient

Custom

2D Empty

2D Empty

Results

The numerical simulation results for the various Power Index ranging from n=0.5−2

$n=0.5-2$

for a Re=40

$Re=40$

are compared with analytical formulation given by Tanner [1] and shown by [2]. The formulations for the Generalized Reynolds number for non-Newtonian flows for this case is as follows:

Generalized Reynolds number: Re=ρV2−nHnK

$Re=\frac{\rho \text{}{V}^{2-n}{H}^{n}}{K}$

where, ρ

$\rho $

is the density, V

$V$

is the flow inlet velocity, H

$H$

is the inlet height, n

$n$

is the power index and K

$K$

is the consistency factor. All quantities here are in standard S.I units.

Two meshes M_2 and M_3 were analyzed. As no further improvement was observed for mesh M_3 the results presented are for mesh M_2. A comparison of the velocity profile in the fully developed region downstream of the expansion at X=28

$X=28$

is shown in fig.3. The figure shows the normalized velocity profile along the normalized channel height for n>1

$n>1$

shear thickening or dilatant fluid (e.g corn starch water), n=1

$n=1$

Newtonian fluid and n<1

$n<1$

Shear thinning or pseudoplastic fluids (e.g ketchup, paint, blood).

The Velocity contours with streamlines and corresponding profiles for Power Index n=0.5,1.0,2.0

MANICA, A.L. de BORTOLI, “Simulation of Incompressible Non-Newtonian Flows Through Channels with Sudden Expansion Using the Power-Law Model”.

Disclaimer

This offering is not approved or endorsed by OpenCFD Limited, producer and distributor of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trade marks. OPENFOAM® is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software.

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