Radial distribution of the mean streamwise velocity

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Literature review

There have been extensive studies on jet flow through theoretical, experimental, and numerical approaches. Which examined the flow development on the light of several parameters such as Reynolds numbers, inlet conditions, and geometry and surface roughness.
In jet research, jets are classified depending on their persistence into intermittent or continuous injections or according to injection momentum and buoyancy to jet, plumes, and buoyant jet or forced plume. Also, Jet issuing from a nozzle or orifice has been classified depending on the cross-sectional shape of the nozzle (round, rectangular, square, triangular, etc).
The first formulation of a jet problem developed in the 1850s by Helmholtz and Kirchhoff. While the first experiments on plane jets were in 1936 by Forthmann. Corrsin in 1946 carried out a turbulent characteristic measurement for round jets.
Early studies tried to distinguish between laminar and turbulent flow based on Re number. (Todde, Spazzini and Sandberg, 2009) considered Re 1600 as a critical number where the flow changes from laminar to turbulent.
Moreover, the Influence of initial conditions on jet development has been an argument matter for a long time. (George and Arndt, 1989) reported a strong dependency of turbulent motion on initial inflow conditions. And (Bogusławski and Popiel, 1979) found the least turbulence to occur in the core subregion and the highest turbulence occurs at an axial distance of about 6D and radius of (0.7 to 0.8) D. Similar findings were reported by (Deo, Mi and Nathan, 2007) for Re>10000. (Quinn and Militzer, 1989) studied the influence of nozzle geometry on flow development and reported faster spread in the initial region in the contoured nozzle compared to the sharp edge nozzle.
For round jets, numerous studies have been dedicated to study jet development in terms of Re dependency and velocity decay. (Fellouah, Ball, and Pollard, 2009) reported Re dependency in the region (0 ≤x/D ≤25). Moreover, faster velocity decay linked to low Re was found by (Oosthuizen, 1983). Similar findings by (Mi, Xu and Zhou, 2013)
suggested that for Re ˂10,000, the mean flow decay varies with Reynolds number and decay rates are Reynolds-number independent at Re≥10,000. Likewise, (Abdel-Rahman, Al-Fahed and Chakroun, 1996) investigated velocity decay and spread for Reynolds ranged between 1400 and 20000 and reported a significant influence of Reynolds numbers on jet behavior in the near field with faster decay of velocity with Re decrease. (Lemieux and Oosthuizen, 1985) studied experimentally the round jet in the Reynolds numbers ranging between 700 and 4200 by scaling mean velocity and turbulence stresses and found that jet is strongly affected by the characteristics of turbulent jets due to variation in geometry, initial conditions, Reynolds numbers, and upstream disturbances. Also, it has been reported that velocity decay is higher at low Reynold compared to higher Reynold numbers. In contrast, (Kwon and Seo, 2005) investigated non-buoyant water round jet with Reynolds number 177 to 5142 and reported a higher velocity decay and a decrease of length of the zone of flow establishment with Reynolds increase. High Re jets >20000 were found that the mixing is Re independent (Gilbrech, 1991).
Massive studies focused on jet structure and mixing with the reservoir at different initial and boundary conditions.( Dimotakis, Miake‐Lye and Papantoniou, 1983) examined the round jet structure within Re 2500 to 10000 and found that the far-field region of the jet is dominated by large-scale asymmetric or helical vorticial structures. Which their kinematics influences the mixing and entrainment of ambient with jet fluid. (Miller and Dimotakis, 1991) studied turbulent fluctuations in water jet within 3000˂Re˂24000 and found that jets become more homogenous or better mixed when Re increases. Also, (Michalke,1984) found that the instability in the shear layer gets less when Reynolds number gets higher.
Many studies directly linked the entrainment to the initial conditions such as Reynolds numbers, where (Wang and Tan, 2010) revealed that the mixing and bulk entrainment in the near field is influenced by the large vortices which are linked to the initial conditions. Furthermore, (Ricou and Spalding, 1961) studied entrainment into jets in the range 500≤Re≤80000 and estimated the entrainment ratio (entrainment mass flow rate/ jet exit mass flow rate) to be constant for Re≥25000. An experimental study by (Kochesfahani and Dimotakis, 1986) investigated entrainment and mixing at large Schmidt number and with Reynolds numbers 1750 and 23000 and reported better mixing at high Reynolds numbers. (Obot, Graska and Trabold, 1984) studied the flow development from two round nozzles and claimed that entrainment is independent of Reynolds number at the small axial distance. Moreover, (Fondse, Leijdens and Ooms, 1983) suggested 40% less entrainment at 20 D downstream with high turbulence intensity. A PIV study by (Kabanshi and Sandberg, 2019) reported less entrainment with flows with high Reynold numbers and in the near field compared to low Reynold numbers and in the far-field region.
Jet studies have been developing over time. Jet flow characteristics were believed to be washed away as the flow evolves downstream into the far-field. Now, it is known that initial conditions have a direct impact on the entire flow (Ball, Fellouah and Pollard, 2012).

Method

The present work adopted a CFD approach to study the near to intermediate field (NIF) of a free round jet at different exit velocities and two different outlet diameters. Two CFD models (model 1 and model 2) were created and the inlet and boundary conditions were obtained from two previous experiments. The CFD simulations were performed by solving Reynolds Averaged Navier-Stokes (RANS) equations with a standard κ-ω model using the finite volume method. The CFD models were validated against two experimental studies (Kabanshi and Sandberg, 2019) and another unpublished study by the same author, in terms of mean streamwise and spanwise velocity and mean temperature for one mode.

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The Computational models

The geometry was created to capture the NIF flow field. A geometry that considers complexity, computation time, and accuracy was created with the help of Space claim with ANSYS Fluent. And since round jets are considered as axisymmetric jets, geometry that is designed to capture only half of the jet flow, a 3D cylinder quad with inlet face which represents the nozzle exit with radius r = 0.025m inserted in the middle of a bigger face of 20 r and length of 60 r downstream was created (see Figure 2). Four faces at the top, bottom, and outer downstream were defined as pressure outlets. And four internal faces were defined as symmetry faces. For model 2, a similar geometry with radius of 0.0125 m for the inlet face was generated.

The mesh

Two different meshes were generated for the two models 1 (0.05m Diameter) and 2 (0.025 m Diameter). A Hexahedral mesh was generated with a refinement at and close to the inlet. Figure 3 shows X-Z (left) and X-Y (right) sides of the mesh created by ANSYS 2019 for model 1. A similar mesh was created for model 2.
The specifications of the two generated meshes for model 1 (0.05m D) and model 2 (0.025 m D) are shown in more detail in Tables 1 and 2.

Mesh independence

Three meshes with different elements statistics (A, B, and C) were generated for model 1 (0.05 m D). Then, the simulation was done under the same conditions and the decay of two streamwise velocities V was obtained and compared between the three meshes. The same process was done for model 2 (0.025m D). The three meshes for model A had 438295, 658625, and 896775 elements. The average change between meshes with elements number of 658625 and 438295 for 1.93 m/s- Re 6537 and 3.56 m/s- Re 12026 was found to be 7% and 9% respectively. While it decreased to 2% between the two meshes with elements number 658625 and 896775 for the two velocities. Hence, and since the change is small between the two tested parameters and to minimize the computation source and time, the model with 658625 elements was chosen for the model. Likewise, three meshes with elements number 144420, 294420, and 333900 were tested for model 2. The average change between meshes with elements number 144420 and 294420 for 1.39 m/s- Re 2000 and 6.43 m/s- Re 9300 was found to be 9% and 14% respectively. While it decreased to only 0.2% and 0.4% for the same velocities between the two meshes with elements number 294420 and 333900. Hence, and since the change was very small between the mean velocities at the different meshes and to minimize the computation source and time, the model with 294420 elements was chosen for model 2.
Mesh independence for model 1 was tested by comparing the centerline velocity profile at r/D=0 for Re 12026 and Re 6537. A minor difference between the chosen mesh and the refined mesh is seen as discussed in the previous paragraph and as shown by Figure 4 below.
Noticeable change can be seen between the coarse and the working mesh particularly at the region x˂1. The variation deceases between the working and refined mesh.

Model 2

Mesh independence for model 2 was performed by comparing the centerline velocity profile for Re 2000 and Re 9233. A minor difference between the used mesh and the refined mesh is seen in figure 5.

Convergence and iterations

When refining the mesh, better results can be obtained. However, more iterations and more time will be required to obtain a complete convergence. The opposite is true. Hence, balancing the quality of the results, computation time and resources are crucial aspect to be considered when performing a CFD study. Figure 6 shows the number of iterations and the computational time required to obtain the convergence at each mesh.

Table of contents :

1 Introduction
1.1 Motivation of the study
1.2 Aims of the study
1.3 Approach
2 Literature review
3 Method
3.1 The Computational models
3.1.1 The mesh
3.1.2 Mesh independence
3.1.3 Inlet and boundary conditions
3.2 Governing equation
3.3 Turbulence model
3.4 Validation of the CFD Models
3.4.1 Model 1-case A (isothermal flow at 0.05m D)
3.4.2 Model 1-Case B (non-isothermal flow-0.05m D)
3.4.3 Model 2 – Case B (isothermal flow-0.025m D)
4 Results and discussion
4.1 Vector field
4.1.1 The velocity contours
4.1.2 Radial distribution of the mean streamwise velocity
4.1.3 Axial centerline mean velocity
4.1.4 Radial distribution of the spanwise velocity
4.1.5 Axial centerline spanwise velocity
4.2 Turbulence kinetic energy TKE
4.2.1 Radial turbulence kinetic energy
4.2.2 Axial centerline turbulent kinetic energy TKE
4.3 Scalar field
4.3.1 The temperature contours
4.3.2 The radial Temperature
4.3.3 The axial temperature at (r/D=0)
4.4 Jet half width
4.5 Entrainment
5 Conclusion
Future work
References
Appendix A
Appendix B

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