Validation Case: Non-Newtonian Flow Through Expansion Channel

The non-Newtonian flow through an expansion channel validation case belongs to fluid dynamics. This test case aims to validate the following parameter:

Non-Newtonian fluid with the power law model

The simulation results of SimScale were compared to the analytical results presented in [Tanner, 1992 apud Manica]\(^1\).

The geometry consists of a pseudo-2D channel, with a 3:1 expansion ratio, as seen in Figure 1:

The dimensions of the channel are given in Table 1:

A

B

C

D

E

F

G

H

x \([m]\)

0

0

5

5

30

30

5

5

y \([m]\)

-0.5

0.5

0.5

1.5

1.5

-1.5

-1.5

-0.5

z \([m]\)

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

Table 1: Dimensions of the expansion channel

The corresponding nodes marked with an apostrophe (‘) are translated 0.5 m along the negative z-direction.

Note

SimScale requires a domain with volume to perform simulations. Therefore, we are going to use a pseudo-2D approach for this validation case.

The meshes used in this project contain a single cell along the z-direction. An empty 2D boundary condition will be applied to both sides of the domain, so the z-direction won’t be resolved.

Mesh and Element Types: Both meshes used in this validation case are hexahedral meshes created locally and imported to SimScale. Cases A through D were run with an intermediate fineness mesh (30700 cells) and with a fine mesh (77600 cells).

In Table 2, an outline of the cases is presented, including information regarding the mesh and the power-law model parameters.

Case

Mesh Type

Cells

Element Type

\(K\ [m^2/s]\)

\(N\)

\(\nu_{min}\ [m^2/s]\)

\(\nu_{max}\ [m^2/s]\)

(A)

blockMesh

30700 – 77600

3D hexahedral

0.008838835

0.5

0.000001

0.5

(B)

blockMesh

30700 – 77600

3D hexahedral

0.0125

1

0.000001

0.5

(C)

blockMesh

30700 – 77600

3D hexahedral

0.01767767

1.5

0.000001

0.5

(D)

blockMesh

30700 – 77600

3D hexahedral

0.025

2

0.000001

0.5

Table 2: Overview of the mesh and the power law material characteristics for each case

Find below the 30700 cells intermediate fineness hexahedral mesh. It contains a single cell in the z-direction.

Simulation Setup

Material:

Viscosity model: Power law non-Newtonian model, with coefficients as described in Table 2;

\((\rho)\) Density: 1 \(kg/m^3\).

Note

In the non-Newtonian model, fluids with \(N\) < 1 are called shear thinning or pseudoplastic fluids. As examples, we have paint, blood, and ketchup.

Fluids with \(N\) > 1 are referred to as shear thickening or dilatant fluids. Oobleck (mixture of cornstarch and water) is an example of shear thickening fluid.

Lastly, for \(N\) = 1, we have a Newtonian fluid. Therefore, three types of fluids are evaluated in this validation case.

Boundary Conditions:

Before defining the boundary conditions, the current nomenclature will be used for the rest of this documentation:

In the table below, the configuration for both velocity and pressure are given at each of the boundaries:

Boundary type

Velocity \([m/s]\)

Pressure \([Pa]\)

Inlet

Velocity inlet

Fixed value: 0.5 in the x-direction

Zero gradient

Outlet

Pressure outlet

Zero gradient

Fixed value: 0

Sides

Empty 2D

Empty 2D

Empty 2D

Top and bottom

Wall

Fixed value: 0

Zero gradient

Table 3: Summary of the boundary conditions used for all cases

Reference Solution

A dimensionless analytical solution for the developed velocity profiles within a channel is presented by Tanner [Tanner, 1992 apud Manica]\(^1\).

Result Comparison

The numerical simulation results for the various power indexes \(N\), ranging from 0.5 to 2, are compared with the analytical solution. For all cases, the Reynolds number was kept at 40.

The generalized Reynolds number formulation for non-Newtonian flows is as follows (SI units apply for all parameters):

$$Re=\frac{\rho\ V^{2-N} H^{N} }{K} \tag{1}$$

Where:

\(\rho\) is the density;

\(V\) is the flow inlet velocity;

\(H\) is the inlet height;

\(N\) is the power index from the power law formulation;

\(K\) is the consistency factor from the power law formulation. The consistency factor was adjusted for each case, to maintain \(Re\) = 40.

Cases A through D were run with both intermediate and fine meshes. The presented results are those from the intermediate mesh, as no further improvement was observed with the finer discretization.

The figure below shows the normalized velocity profiles in the fully developed region downstream of the expansion, at \(x\) = 28 \(m\). The \(y\) coordinates, represented in the horizontal axis of Figure 4, are normalized from 0 (center of the channel) to 1 (top of the channel).

Note that, in the figure above, ‘n’ represents the same power index \(N\) used within the workbench.

SimScale’s results show very good agreement with the analytical solution for all configurations.

In Figure 5, we can see how the velocity profile is developing along the expansion channel for case A (\(N\) = 0.5) with the intermediate fineness mesh.

This website uses cookies so that we can provide you with the best user experience possible. Cookie information is stored in your browser and performs functions such as recognising you when you return to our website and helping our team to understand which sections of the website you find most interesting and useful.

Strictly Necessary Cookies

Strictly Necessary Cookie should be enabled at all times so that we can save your preferences for cookie settings.

If you disable this cookie, we will not be able to save your preferences. This means that every time you visit this website you will need to enable or disable cookies again.