(v) Jet Exit: From the literature review (Catalano et al., 1989, Patankar et al., 1977 and Hwang and Chiang, 1995) it is found that in most of the numerical studies a uniform jet discharge velocity is used. Several researchers, for example Kassimatis et al. (2000) indicated that the assumption of a uniform velocity at the inlet of jet is not appropriate. For accurate prediction and to represent the actual experimental conditions it is better to include the jet slot into the computational domain. In the present study we have considered the jet slot (length 5D) in the computational domain. The conditions at the entry to jet slot are u = 0 and v = vj. The value of turbulence kinetic energy is taken from 6.5% turbulence intensity (k =0.00625v2j) and the value of dissipation rate is taken using the expression proposed by Versteeg and Malalasekera (1996) as,ε =cµ3/4(k1/2/0.5D).
The sides of the non-uniform control volumes along the y-direction are generated using the following relation
−
=
max max
sin j j j
yj j π
π
λ (3.18)
where i and j denote the spatial indices in the x- and y- direction respectively, imax and jmax are their maximum values and λ is the clustering parameter. The value of λ used is 0.5. The grids are clustered near the wall in the y-direction as well as at the jet exit area in the x-direction. The distribution of the control volumes in the computational domain used for the prediction of jet in crossflow with R = 6 is shown in Fig. 3.2. In Fig. 3.2, the distribution of the fine mesh is shown. The fine mesh is fixed based on a grid independence study (Section 3.3.4). 172×143 (= 24596) control volumes (172 along the x-direction and 143 along the y-direction) excluding the jet duct area have been used in the fine mesh. The jet duct area comprises of 7×37 (= 259) control volumes. To avoid the occurrence of pressure oscillations in the domain, a staggered grid arrangement is employed in which the scalar variables (p, k and ε) are positioned at the centre of the control volume and the velocity components are positioned at the cell faces. The positioning of the variables is illustrated in Fig. 3.3.
Fig. 3.2: Computational domain and finite volume grid for the jet with R = 6.
3.3.2 Computational Methodology
A finite volume computer code in FORTRAN 77 based on the SIMPLE algorithm (Patankar, 1980) has been developed to numerically to solve the governing equations.
The momentum and the other transport equations are descritised using the power-law scheme (Patankar, 1980). A good discussion on use of various discretisation schemes
can be found in Patel and Markatos, (1986). Line-by-line iteration with alternating the sweep direction using tri-diagonal matrix algorithm is used to solve the system of linear algebraic equations.
Fig. 3.3: Positioning of variables on the grid.
For any iterative process an initial guess of the unknown variables are made at the starting of the first iteration. In the computation through the curvature modification model, the initial guess is taken from the intermediate solution of the standard k-ε model for better convergence behaviour. All the variables (u, v, k and ε) are under- relaxed during each iteration. The typical values of under-relaxation parameters used are 0.3 for u and v and 0.45 for k and ε. The mass residual which represents the balance of mass in each cell is determined from the continuity equation. The mass residual is defined as
b=(uw∗ −ue∗)∆ ∆ +y z (vb∗−vt∗)∆ ∆ +x z (ws∗−wn∗)∆ ∆x y (3.19) Where the starred quantities (*) are the velocities before correction at each cell and subscript e, w, n, s, t and b denote the directions in east, west, north, south, top and bottom. The maximum mass residual is established as the criterion for assessing the overall convergence of the flow field. The convergence of the computation is assumed to be achieved and therefore iterations are terminated when the sum of the residual mass is less than 10-4 and the relative variation of variables u, v, k, and ε in the successive iteration is less than 10-2. At the residual level mentioned above, the solution is found not to change for several iterations and therefore it is treated as the converged solution. Moreover the exit criterion for various equations has been taken
as 2
max 1
10−
+
− ≤
m i
m i m i
φ φ
φ , where φirepresents the variable for which solution is sought.
The history of the residual fall in the numerical solution for the jet with R = 6 using the standard k- ε method is shown in Fig. 3.4. The mass residual shows some fluctuations initially with high amplitude and gradually it decays. This behaviour has no effect on the final converged solution. The residual fall history is likely to be influenced by the iteration exit criteria chosen for the variables. But to determine the exact nature of this influence, extensive numerical experiments need to be performed.
The code was run on a special Linux server (16 node load clustering machine) with 8 GB RAM and 3G Hz processor speed and it took approximately 72 hours of CPU time for the full convergence of the flow field using the curvature modification model and 40 hours using the standard k- ε model.
Fig. 3.4: Mass residual fall history for R = 6 (standard k-ε model).
3.3.3 Code Validation
The present code is validated against the results reported in the previous numerical studies of similar nature. The prediction of turbulence plane jets in a strong crossflow by Kassimatis et al. (2000) is used as a test case. Using streamline curvature modification model in the present code, the reattachment length and the streamwise velocity at the two different locations are compared for the jet with R = 0.8. Fig 3.5 shows that the present predictions of the streamwise velocity at two different locations at x/D = 2 and 9 agree well with those of Kassimatis et al. (2000).
Fig. 3:5: Comparison of the present predictions using the curvature modification model with those of Kassimatis et al. (2000) for R = 0.8.
The recirculation region predicted by the present code for the value of R = 0.8 is shown in Fig. 3.6 and the comparison of the reattachment length by both the present code and by Kassimatis et al. (2000) is shown in Table 3.2. The difference between the two values is 4% and therefore the present code stands validated.
Table 3.2: Comparison of reattachment length.
Value of R Reattachment length by present prediction (see Fig. 3.6)
Reattachment length by Kassimatis et al. (2000)
0.8 10.8 10.3
Fig. 3.6: Reattachment region predicted by the present code for R = 0.8.
3.3.4 Grid Sensitivity
The grid-independence test of the computations is performed with three different sets of grids, viz., 150×125 (150 along x-axis and 125 along y-axis), 172×143, and 300×240 and by comparing the prediction of the streamwise velocity for each grid for
R = 6. Similarly three different sets of grids, viz., 165×140, 200×165 and 330×250 are used for R = 9. The grids are non-uniform in both the directions and clustered near the wall and the jet source region. The velocity profile at x/D = 2 for R = 6 and 9 using the curvature modification model is shown in Fig. 3.7. It has been observed that the grid refinement in general improves the prediction by reducing the over prediction in the bottom part of the jet and increasing the peak value of the velocity. Moreover the results produced by the fine grid approach the experimental results (not shown in Fig.) than those by the other two grids. The difference between the predictions using the 2nd set and 3rd set grids is only 2%. The results that are presented in the subsequent sections are for the grid size 172×143 for R = 6 and 200×165 for R =9.
Fig. 3.7: Grid sensitivity test: Predicted profile of cross-stream component of velocity.