Physical Law and Properties

The channel flows are characterized by free surface, maximum velocity below the free surface and condition walls with no-slip.  The boundary walls together with the no-slip condition allow the wall shear stress and the development of the velocity gradient to depend upon the walls roughness and the fluid viscosity.  While observing the fluid flow along the x- axis and the chancels depth along the y-axis a velocity result as shown in figure 3b is established (Ranade, 2013). The fact that the average velocity along the flow depends on the stream distance x, the velocity ends up becoming one-dimensional variable. The trait enables the longitudinal section analysis as shown by the figure 3.a in the form of two dimensions since the domain depth is smaller to have any impact. Once the configuration is attained, the fluid flow is normally considered to be two dimensional because the boundary layers developed on the walls are consistent with the ducts and the pipes.

The no-slip condition alongside the walls affects the flow of the fluid due to the fluid’s viscosity effects.  The boundary layer normally affects the region along the entrance (Steiner, 2014). The entry length is where the flow develop the velocity profile until it reaches the hydrodynamic developed region where the profile of the velocity remain unchanged as illustrated by the figure 4.

When the liquid flows along a pipe with a constant diameter, the viscous impacts normally causes the pressure to reduce because the fluid flow velocity is associated to the pressure gradient along the pipe. The result of the fluid flow is referred to as poiseuille flow or parabolic flow. The Poiseuille or parabolic equation is normally applicable to the laminar flows but is never suitable to the turbulent flows. The equation for solving the laminar plane flow velocity is usually as:

Assumptions and Approximations

The CFD model development depends on the fundamental approximation and assumptions which leads into two dimension domain. The assumptions under all conditions are based on the water entering at a constant velocity, the walls being smooth while the flow is incompressible. For the velocity of 0.1m/s, the fluid flow is considered to be laminar, while for the fluid velocity of 0.2m/s and 0.5m/s the flow is considered to be turbulent (Wendt, 2012).  In order to develop CFD simulation in the geometrical model, the prior mentioned dimensional slice of the longitudinal section are usually converted into 3 dimensional domains without mainly affecting the flow. The relative depth of the experiment must be about 0 to 20 times less than the base and the height of approximately 0.02m. Moreover, the effective depth for the calculated value is 0.002m.

Theoretical Calculations

Reynold’s number

The Reynold’s number in equation one was used to establish whether each flow was turbulent, transitional or laminar. The figures of the velocity in table 2 reveal that the flow is laminar, turbulent and transitional respectively. As the simulations were being conducted, the flow regimes were calculated in order to show that they were reviewed and indicative.

Laminar Poisueille Flow solution

The laminar Poiseuille flow equation was employed in solving the axial velocity for the laminar flow in order to attain are fined solution, the y-axis values were calculated from 0 to 0.02m in additions of 0.001m thus resulting into 20 points of data. Excel was used to conduct the calculation while the results were indicated in Appendix 1.  Once the data is obtained, the results were plotted were plotted on the y axis while the observed height of the x-axis as illustrated by figure 5 to 7. Despite the fact that the poiseuille flow equation was intended for the laminar flow the other flow conditions were solved using the reference.

The three graphs indicates that the velocity profiles for the entire flow condictions share similar parabolic traits which is expected when U∞=0.1m/s and possible when u∞=0.2m/s when calculated to have a Reynold’s number transitional. Nevertheless, U∞=0.5m/s was calculated as a turbulent. Hence, the calculation was most likely to be inapplicable for the flow. Give the laminar flow regime is for the analytical solution, the initial validation and grid convergence will be made depending on the solution of the laminar flow of 0.1m/s.

Modeling

Process Overview

The Computational Fluid Dynamics, (ANSYS Academic R16.2. Fluid Flow) was used to model the flow of water in the channel. The approach involved creation of the geometry from the subsequently and physical model for determining the sufficient mesh in establishment of the converged solution (Wendt, 2012). The initial and boundary conditions must be established in the set up prior to the performance processing. The CFX modeling overview is present in figure 8. The CFX solution manager is used to obtain the solution where the convergence may be verified or observed. As illustrated in the diagram below, the refinement mesh may be performed using the quality and accuracy of the result. The moment the expected solution is attained the results may be exported and viewed in CFD-Post for processing.

 

Geometry

The geometry was established using the built-in modeling tool of the Ansys CFX which is the design modeler. The 2cm*1m rectangle created the workspace in the fluid travelled in the x to y plane. The symmetry in the z direction was about 0.002 in order to enable discretization of the flow domain.

Mesh

In order to analyze the multiple nodes, the Fluid Flow (CFX) was used to create grid pattern mesh. By default, due to the low number of nodes and lack of nodes in the free stream automatic mesh was used (Wendt, 2012). In addition, there were inadequate data points in the mesh which would generate the solution. The Mesh refinements were performed by increasing the accuracy and quality of the grid. Figure 10 to 13 represent the development of the grid refinements.

 

Using the inlet velocity of 0.1m/s, a simulation was performed in order to obtain to obtain a solution for comparison with the laminar profile. The grid resolution was later transformed from 20 to 50 divisions with each simulation results being compared against the laminar flow solution analytical profiles (Wendt, 2012). Upon analyzing the mesh with the 50 sweep divisions sufficient to perform the other simulations were identified.

The velocities of 0.2m/s and 0.5m/s inlet input were transformed in order to be aligned with the flow conditions. At U∞=0.2m/s , the simulations for both turbulent and laminar regimes were performed because the transitional region could change between turbulent and laminar. At U∞=0.5m/s where the Reynolds’s number was considered to be higher, the CFD setup was set at k-epsilon turbulence model.

Boundary and Initial Conditions

As specified by the project description, each boundary face was established in order to correspond to the conditions. The inlet speed was set at 0.1m/s while the outlet pressure was decreased at 0 pa. The boundary faces minimum and maximum in the y-axis direction were modeled so that the non-slip conditions on the walls were applied. The Nominal roughness was assumed to be 0 in the model (Wendt, 2012).The material was fixed in the water of approximately 10⁰C, dynamic velocity of 1.307 *10³pa/s and density of 1000kg/m³.  For the case of 0.1m/s flow of the laminar model and k-epsilon turbulence model applied both 0.2m/s and 0.5m/s. the maximum initials were set at 500while the convergence criteria was set at 1*10^-8.

Processing and Convergence

After mesh, geometry and setup was done, each model was conducted the quality of each was performed as illustrated by the converged presentation in figures15 to 17.

 

Post Processing

When the post processing vectors were being established, data visualized the flow velocity throughout the channel length. The data velocity was used to prepare graphs points of 0.1m, 0.5m and 0.9m respectively along the channel. The information collected was compared and analyzed  the theoretical calculations.

Data and Model Analysis

Mesh Refinement

The laminar flow of 0.1m/s flow condition was the validated model which was used in the simulations. The diagram below validated the model.

 

 

 

The generated automatic model mesh is displayed by figure 10, it displays the inadequate number of divisions which produce accurate fluid flow representation through 1m*0.02m*0.0001m rectangular pipe.

 

Comparison of results to theoretical

Discuss the effect mesh refinement has on the results

The comparison and analysis of figure 18 and 19 assists in determination of the impact of grid refinement on the results accuracy. Figure 18 illustrate sharp points at for both 0.5 and 0.9m measurement points.  These results illustrate that the mesh usually have inadequate refinement (Xiao, 2016). On the other hand, figure 19 demonstrates de-void sharp points sufficient for the correct fluid model.  Figure 15 and 16 illustrate convergence and RMS error values for the respective meshes. Therefore, 50 sweep meshes have the potential to offer fundamental improvement to the axial velocity comparisons while minimizing improvement to the convergence.

Discuss any observations about the boundary layer

After observing the figures, 15, 16, 17, 19, 20 and 20 the effect of the inlet velocity upon the layer boundary can be determined.  Figure 19 demonstrates laminar traits as the velocity of both 0m abd 0.02m in the y direction at 0m/s. in addition, the axial velocity at 0.9 when distinguish against figure 4 emphasizes the match between the models.  Figure 18 illustrates that the velocity of the fluid on the boundary becomes stagnant hence forming boundary thickness of increased thickness (Xiao, 2016). The increasing boundary layer thickness is as a result of viscous interactions between the second and boundary layer. Therefore, both the layers interact efficiently with the third layer. The internal fluid core normally increases the velocity in accordance with the boundary layer thickness in order to satisfy mass conservation.  Eventually, the velocity returns the average profile across the channel which is no longer influenced by the flow type and boundary layer are accurately established.

Establishment of figure 18 resulted into representation of the laminar flow as shown in table 2.  The display assisted in the calculation of the Reynolds number which falls within the specified range of the laminar flow.  The figure 20 and 21 respectively displayed the velocity of 0.1, 0.5 and 0.9m which was indicative of turbulent flow at the inlet as the fluid particles proceeding along the channel (Terline, 2009). In order to establish the flow profile equation 2 and 3 must be used to explain how the channel with similar specifications and inlet velocity of 0.2m/s and 0.5m/s must have laminar of o.2 and 0.25 and have a length of 2.92m and 7.30m if the turbulent before the flow is fully established while the boundary layer  remain stable and accurate. The Reynolds number for figure 23 and 24 respectively and the state figure 16 and figure 24 is believed to be transitional and turbulent respectively. It is therefore recommended that length of the model to be analyzed increased in order to allow a fully developed analysis of boundary conditions and velocity profiles to be undertaken.