6 Flow induced by a jet in a confined domain
0 10 20 30 40 50 60 70 80
z/d
0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
Qr/Q0
Data:Qr,exp Theory: Qr,pot Theory: Qr,cont
(a)
0 10 20 30 40 50 60 70 80
z/d
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08
Mr/Q02 /d Data:Mr,expTheory: Mr,pot
Theory: Mr,cont
(b)
Figure 6.15: Experimental and theoretical distributions against non-dimensional height z/d of: (a) the experimental normalized time-averaged volume flux Qr,exp/Q0
(dotted curve) computed using (6.42), the theoretical prediction based on potential the-oryQr,pot/Q0 (solid curve) and computed using (6.43), the theoretical prediction based on continuity Qr,pot/Q0 (dashed curve) and computed using (6.44); (b) the experi-mental normalized time-averaged momentum fluxMr,exp/ Q02/d
(dotted curve) com-puted using (6.45), the theoretical prediction based on potential theoryMr,pot/ Q02/d (solid curve) and computed using (6.46), the theoretical prediction based on continuity Mr,cont/ Q02/d
(dashed curve) and computed using (6.47).
value of < M > / Q02/d
= 0.55. The non-dimensional momentum flux of the return flow increases from Mr/ Q02/d
= 0 at z/d = 0 to approximately 0.06 at z/d = 80. Therefore, in our domain, the momentum flux of the return flow is rather insignificant compared with the jet momentum flux. This finding is completely different from the results for the volume flux of the return flow, which is not insignificant because it has to balance the volume flux of the flow. This crucial difference, which enables us to neglect the influence of the momentum flux of the return flow on the jet flow, is related to the distance between the jet and the lateral boundaries (i.e. to the lateral confinement of the jet) and justifies the assumptions we make in Chapter 2 that the return flow has a weak effect on the dynamics of the evolving jet.
6.4 Conclusion
ratioζ =xj/hi = 3/4 (corresponding to our particular case) between the stream-wise dimension and the cross-stream dimension. The domain is delimited by the jet axis at the right-hand-side boundary (atx=xj), the walls of the experimental apparatus at the left-hand-side and bottom boundaries and the transition height hi between the jet flow region and the impingement region, at the top boundary.
The jet is modelled as a line sink (located on the jet axis) with a lateral flux per unit length varying with height in a similar way to the entrainment velocity due to a quasi-two-dimensional jet. The transition height hi is modelled as a uniform line source, whose total inwards flux matches the total outwards flux of the line sink.
To solve Laplace’s equation in the domain Ds, we decompose the problem into a uniform problem with a uniform line source and a uniform line sink, and a perturbation problem accounting for the varying line sink condition at the jet boundary. We find an analytical solution for the potential field, the stream function and the velocity field in the domain Ds. It appears that in the far field, away from the jet, the results are dominated by the uniform problem with uniform boundary conditions. The influence of the varying line sink (i.e. the entrainment process of the jet) is strong near the source of the jet, because of the singularity at the virtual origin of the jet (located outside the domain below the bottom boundary).
We observe qualitative discrepancies between our analytical solution for the streamlines of the induced flow compared with the solutions of Taylor (1958) or Schneider (1981). The second derivative of the streamlines with respect to the lateral or cross-jet coordinate (x) have a different sign. Our streamlines are convex, whereas the streamlines of Taylor (1958) or Schneider (1981) are concave. This difference is due to the fact that we consider a fully confined domain, which induces a recirculation flow on either side of the jet, whereas Taylor (1958) or Schneider (1981) considered fully unbounded domains or the case of a jet emerging from a wall into a semi-infinite domain, thus ignoring the possibility of recirculation in the ambient flow.
We compare our theoretical flow field with experimental data from quasi-two-dimensional turbulent jets in a confined experimental apparatus of aspect ratio 1 (the ratio between the inner dimensions of the tank). The theoretical streamlines agree with the data in the far-field, away from the boundary of the jet. We find
6 Flow induced by a jet in a confined domain
that the boundary of the jet, defined as the boundary between the turbulent jet flow and the ambient flow (Kotsovinos, 1978), also corresponds to the location x0(z) ≈ 0.22z where the flow is, in average, purely lateral because the time-averaged streamwise velocity vanishes and changes sign at x0(z). In our model, we assume that the jet boundary coincides exactly with the jet axis, instead of being at an angle of approximately 12◦. We find that this assumption is valid in the far-field away from the jet boundary and forz ≤hi.
We find that, to the leading order, the experimental velocity field agrees with the model. Differences are seen near the rigid boundaries, where the experimental time-averaged tangential velocity vanishes at the walls, contrary to the theoretical tangential velocity which is assumed to satisfy a slip boundary condition. Also, near the jet source, the experimental data differ from the model because the flow of the jet is not yet fully established. Finally, the experimental measurements for the volume flux and the momentum flux of the return flow agree to leading order with the model based on potential theory and a model based on volume conservation. In particular, we find that the time-averaged momentum flux of the return flow increases likez2 to approximately 10 % of the jet momentum flux at mid-height in the experimental apparatus.
We believe that a jet emerging from a wall into a fully confined domain is a more realistic case than the case of a jet in an unbounded or semi-infinite domain.
The streamlines of the induced flow are strongly modified by the recirculation cells observed on either side of the jet. This phenomenon is important in mixing problems because the re-entrainment process tends to increase the concentration in the jet of passive tracers injected in the fluid. The momentum flux of the jet can also become negatively affected by the counter-flow after a certain distance. The core and eddy structures also become affected by the confinement at a distance hi approximately equal to 65 % of the depth of the flow, for an experimental apparatus of aspect ratio 1 (i.e. the ratio between the inner dimensions of the tank) or a jet aspect ratioζ =xj/hi = 3/4 (i.e. the ratio of the distance between the jet and the lateral boundary to the transition height of the impingement region). We believe that our model, and in particular the assumption of a uniform flux at the top boundary and the assumption of a jet boundary parallel to the z-axis on the right-hand side of the domain, is valid for a range of jet aspect ratios 2/3< ζ =xj/hi <3. At higher aspect ratios, secondary recirculation cells could
6.4 Conclusion
form on either side of the jet (Jirka & Harleman, 1979), thus affecting the flux at the top boundary. On the other hand, at lower aspect ratios, the expansion of the jet boundary becomes significant compared with the size of the domain, and thus can influence the flux at the top boundary.