4.4 Results
4.4.2 Turbulence Kinetic Energy and Turbulence
The prediction of the turbulence kinetic energy at different transverse planes and at different downstream locations are shown in Figs. 4.40 to 4.43 for the velocity ratio R
= 6. The variation of the turbulence kinetic energy with the vertical distance (y/D) at different downstream locations in the jet central plane (z/D = 0) is shown in Fig. 4.40.
The development of the turbulence kinetic energy can be described in three regions.
The first region is the jet exit region (x/D = 0), where its distribution is governed by an interaction of different mechanisms. The kinetic energy is transported from the crossflow boundary layer and the jet flow. It is produced and destroyed by various mean velocity gradients. Though the generation of the kinetic energy depends upon different velocity gradients, among them the cross-stream velocity gradient (∂u/∂y) is the dominant factor. Thus the maximum value of the kinetic energy is observed at the location of the maximum velocity gradient.
The second region is at the downstream of the jet exit (x/D = 2 and 5). In this region the velocity gradient (∂u/∂y) becomes significantly large in the jet-shear layer and due to this, the production of the kinetic energy is high in this region.
Fig. 4.40: Predictions of turbulence kinetic energy at different downstream locations (z/D = 0) for R = 6.
Due to a sharp velocity gradient in the wall-jet structure near the wall a steep increase of turbulence kinetic energy is reported near the wall. The third region is the further downstream regions (x/D = 10 and 20), where strong gradients of the velocity disappear thus producing less turbulence kinetic energy.
Fig. 4.41: Predictions of turbulence kinetic energy at different downstream locations (z/D = 3) for R = 6.
Fig. 4.42: Predictions of turbulence kinetic energy at different downstream locations (z/D = 6) for R = 6.
Fig. 4.43: Predictions of turbulence kinetic energy at different downstream location (z/D = -3) for R = 6.
The variation of the kinetic energy at the plane z/D = 3 is shown in Fig. 4.41. In this plane although (∂u/∂y) is the predominant gradient in producing turbulence kinetic energy k, the role of the spanwise velocity gradient ∂w/∂yshould not be underestimated. In the wake-like region where there is entrainment of the fluid towards the jet central plane, the streamlines converge (∂w/∂y< 0) and the production of k is reduced. The higher value of kinetic energy at the wall-jet region is observed in this plane due to a strong velocity gradient of the w-component. Fig. 4.42 shows the variation of the kinetic energy at the plane z/D = 6 for R = 6. The value of the k at the wall-jet region is higher in this case compared to the earlier two cases due to a high value of∂u/∂y. However the value is smaller at the upward position of the flow field due to lesser gradients of all components of mean velocity than that in earlier two cases.
Fig. 4.44: Predictions of turbulence kinetic energy at different downstream locations (z/D = 0) for R = 9.
The observation of the symmetry of the profile of turbulence kinetic about the jet centre plane is made by comparing the profiles at z/D = - 3 (Fig. 4.43) with the profile at z/D = 3 (Fig. 4.41).
The prediction of the turbulence kinetic energy at different transverse planes and at different downstream locations are shown in Figs. 4.44 to 4.47 for the velocity ratio R
= 9. Fig. 4.44 shows the variation of turbulence kinetic energy at z/D = 0. The trend of the kinetic energy profile is same as that in the case with R = 6. However the value of the kinetic energy is more in this case due to a stronger effect of the jet. A high value of turbulence kinetic energy is transported from the jet flow. At other two transverse planes (z/D = 3 and 6), the kinetic energy profiles show some qualitative difference.
Fig. 4.45: Predictions of turbulence kinetic energy at different downstream locations (z/D = 3) for R = 9.
Fig. 4.46: Predictions of turbulence kinetic energy at different downstream locations (z/D = 6) for R = 9.
Fig. 4.47: Predictions of turbulence kinetic energy at different downstream locations (z/D = -3) for R = 9.
Two peak values of the turbulence kinetic energy profiles are more distinguished in this case (Figs. 4.45 and 4.46) compared to the case of R = 6 (Figs. 4.41 and 4.42).
Moreover the value of turbulence kinetic energy at the wall-jet region is less compared to the case of R = 6 due to a weak velocity gradient ∂u/∂y in that region.
The turbulence kinetic energy profile is also symmetric for R = 9 about jet centre plane (z/D = 0), which can be observed by comparing Figs. 4.45 and 4.47.
The prediction of turbulence shear-stress profile at two different planes (z/D = 0 and 5) and at different downstream locations are shown in Figs. 4.48 and 4.49 for the velocity ratio R = 6. Fig. 4.48 shows the development of turbulence shear stress at jet centre plane (z/D = 0) and at different downstream positions. In general the shear stress −u′v′ and the velocity gradient (∂u/∂y) have the same sign and therefore the eddy viscosity νt =−u′v′/(∂u/∂y)is positive in all regions of the flow field. The maxima of the shear stress −u′v′ profile correspond closely to the position of maxima of the velocity gradient (∂u/∂y).
Fig. 4.48: Predictions of turbulence shear stress at different downstream locations (z/D
= 0) for R = 6.
In the crossflow direction initially the maximum value of shear stress increases as the shear layer with a higher velocity gradient which is developed over the wake-like region. Further downstream the maximum of shear stress decreases as the velocity gradient becomes smaller. Near the jet exit region (x/D = 0 and 2) there is no wake-like region and therefore the shear stress shown represents only the jet-shear layer.
Fig. 4.49: Predictions of turbulence shear stress at different downstream locations (z/D
= 5) for R = 6.
Downstream of the jet exit region the shear stress distinctly shows the wake-like region, where the value of the shear stress is low. In this region the shear stress shows a trend of increasing the value due to the wall-jet structures, further downstream it decreases in the wake-like region, reaches a minimum value and again increases in the jet-shear layer. After reaching the maximum value in the jet-shear layer, the value of the shear stress again decreases where the jet and the crossflow does not have much interaction.
The prediction of the shear stress at the jet edge plane (z/D = 5) at different downstream locations is shown in Fig. 4.49 for R = 6. The profiles follow the trends of the velocity gradient (∂u/∂y) shown in Fig. 4.8. In this plane the value of the shear stress is significantly higher in the wall-jet layer compared to that in the jet central plane (z/D = 0). The maximum value of the shear stress in the jet-shear layer is lower compared to that at z/D = 0 due to lesser velocity gradient.
Fig. 4.50: Variation of turbulence shear stress in spanwise direction at different downstream locations for R = 6, y/D = 5.
The shear stress v′w′represents the transverse turbulence mixing and is associated with the transverse spreading of the jet in the vertical (y-z) plane. The variations of the shear stress v′w′ in the spanwise directions at a vertical distance of y/D = 5 from the bottom wall and at different downstream locations are shown in Fig. 4.50 for the velocity ratio R = 6. At the jet exit region (x/D = 0) the value of the shear stress is quite high near both the side edges of the slot (z/D = 5 and -5). The high velocity jet after leaving the slot interacts with the slower moving crossflow thus producing a shear layer near both the edges where the shear stress is quite high. Downstream of the slot the shear stress is governed by the CRVP formation and the value of the shear stress reduces due to a reduced velocity gradients. The value of the shear stress at the plane of symmetry (z/D
= 0) is zero.
Fig. 4.51: Predictions of turbulence shear stress at different downstream locations (z/D
= 0) for R = 9.
Fig. 4.52: Predictions of turbulence shear stress at different downstream locations (z/D
= 5) for R = 9.
The predictions of the turbulence shear-stress profiles at two different planes (z/D = 0 and 5) and at different downstream locations are shown in Figs. 4.51 and 4.52 for the velocity ratio R = 9. The trend of the profiles is similar to the case with R = 6, but the value of the shear stress is higher compared to the case with R = 6. Moreover the shear stress predictions in the three regions of wall-jet, wake-like and jet-shear layer are also observed more prominent in this case.
Fig. 4.53: Variation of turbulence shear stress in spanwise direction at different downstream locations for R = 9, y/D = 5.
The value of shear stress v′w′is higher at the jet exit region x/D = 0 near both the edges of the jet compared to the case with R = 6. At the downstream positions the value of the stresses are the similar to the case with R = 6.