The purpose of this numerical simulation is to validate the flow velocity profile for a NonNewtonian fluid via the PowerLaw model.
The numerical simulation were carried out using the ReynoldsAveraged 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
$Re=40$.
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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.
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” opensource 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 : InCompressible SteadyState
Turbulence Model : Laminar
NonNewtonian model : PowerLaw
Fluid:
NonNewtonian
based on Re=40
$Re=40$
Boundary Conditions:
For the inlet boundary, a fixed velocity condition was applied, while a pressure boundary condition was applied at the outlet. Noslip 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: 0.5 ms−1
$0.5\text{}m{s}^{1}$

Zero Gradient 
Outlet  Zero Gradient  Fixed Value: 0 Pa
$0\text{}Pa$

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

Zero Gradient 
Custom  2D Empty  2D Empty 
The numerical simulation results for the various Power Index ranging from n=0.5−2
$n=0.52$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 nonNewtonian flows for this case is as follows:
Generalized Reynolds number: Re=ρ V2−nHnK
$Re=\frac{\rho \text{}{V}^{2n}{H}^{n}}{K}$
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
$n=0.5,1.0,2.0$are shown in the figure Fig.4 below.
[1]  (1, 2)

[2]  (1, 2)

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