Schneider flow describes the axisymmetric outer flow induced by a laminar or turbulent jet having a large jet Reynolds number or by a laminar plume with a large Grashof number, in the case where the fluid domain is bounded by a wall. When the jet Reynolds number or the plume Grashof number is large, the full flow field constitutes two regions of different extent: a thin boundary-layer flow that may identified as the jet or as the plume and a slowly moving fluid in the large outer region encompassing the jet or the plume. The Schneider flow describing the latter motion is an exact solution of the Navier-Stokes equations, discovered by Wilhelm Schneider in 1981.[1] The solution was discovered also by A. A. Golubinskii and V. V. Sychev in 1979,[2] [3] however, was never applied to flows entrained by jets. The solution is an extension of Taylor's potential flow solution[4] to arbitrary Reynolds number.
For laminar or turbulent jets and for laminar plumes, the volumetric entertainment rate per unit axial length is constant as can be seen from the solution of Schlichting jet and Yih plume. Thus, the jet or plume can be considered as a line sink that drives the motion in the outer region, as was first done by G. I. Taylor. Prior to Schneider, it was assumed that this outer fluid motion is also a large Reynolds number flow, hence the outer fluid motion is assumed to be a potential flow solution, which was solved by G. I. Taylor in 1958. For turbulent plume, the entrainment is not constant, nevertheless, the outer fluid is still governed by Taylors solution.
Though Taylor's solution is still true for turbulent jet, for laminar jet or laminar plume, the effective Reynolds number for outer fluid is found to be of order unity since the entertainment by the sink in these cases is such that the flow is not inviscid. In this case, full Navier-Stokes equations has to be solved for the outer fluid motion and at the same time, since the fluid is bounded from the bottom by a solid wall, the solution has to satisfy the non-slip condition. Schneider obtained a self-similar solution for this outer fluid motion, which naturally reduced to Taylor's potential flow solution as the entrainment rate by the line sink is increased.
Suppose a conical wall of semi-angle
\alpha
(r,\theta,\phi)
\alpha=\pi/2
\alpha=\pi
(vr,v\theta,0)
\psi
vr=
| 1 | |
| r2\sin\theta |
| \partial\psi | |
| \partial\theta |
, v\theta=-
| 1 | |
| r\sin\theta |
| \partial\psi | |
| \partialr |
.
Introducing
\xi=\cos\theta
\theta
\psi=K\nurf(\xi)
K-1[(1-\xi2)f''''-4\xif''']-ff'''-3f'f''=0.
where the constant
K
2\piK\nu
K=4
Pr
Pr=1
K=6
Pr=2
K=4
The above equation can easily be reduced to a Riccati equation by integrating thrice, a procedure that is same as in the Landau–Squire jet (main difference between Landau-Squire jet and the current problem are the boundary conditions). The boundary conditions on the conical wall
\xi=\xiw=\cos\alpha
f(\xiw)=f'(\xiw)=0
and along the line sink
\xi=1
f(1)=1, \lim\xi → (1-\xi)1/2f'' → 0.
The problem has been solved numerically from here. The numerical solution also provides the values
f'(1)
For turbulent jet,
K\gg1
f'(\xiw)=0
f=
| \xi-\xiw | |
| 1-\xiw |
.
In the case of axisymmetric turbulent plumes where the entrainment rate per unit axial length of the plume increases like
r2/3
\psi=CB1/3r5/3g(\xi)
C
B
| g= | \pi |
| \sqrt3 |
| ||||||||||||||||
| \sqrt{1-\xi |
| 1(-\xi)\right] | |
| P | |
| 2/3 |
in which
| 1 | |
| P | |
| 2/3 |
2/3
1
The Schneider flow describes the outer motion driven by the jets or plumes and it becomes invalid in a thin region encompassing the axis where the jet or plume resides. For laminrar jets, the inner solution is described by the Schlichting jet and for laminar plumes, the inner solution is prescribed by Yih plume. A composite solution by stitching the inner thin Schlichting solution and the outer Schneider solution can be constructed by the method of matched asymptotic expansions. For the laminar jet, the composite solution is given by
\psi=4\nur\left[
| (Re\theta)2 | |
| 32/3+(Re\theta)2 |
+f(\cos\theta)-1\right]
in which the first term respresents the Schlichting jet (with a characteristic jet thickness
Re\theta
Re=\nu-1(J/2\pi\rho)1/2
J/\rho
A similar composite solution can be constructed for the laminar plumes.
The exact solution of the Navier-Stokes solutions was verified experimentally by Zauner in 1985.[7] Further analysis[8] [9] showed that the axial momentum flux decays slowly along the axis unlike the Schlichting jet solution and it is found that the Schneider flow becomes invalid when distance from the origin increases to a distance of the order exponential of square of the jet Reynolds number, thus the domain of validity of Schneider solution increases with increasing jet Reynolds number.
The presence of swirling motion, i.e.,
v\phi ≠ 0
\psi=K\nurf(\xi)
K\simO(1)
K
K\simO(1)
2\pi\Gamma
\Gamma=r\sin\thetav\phi
\Gamma=ArλΛ(\xi)
A
λ
Λ(\xi)
K-1[(1-\xi2)Λ''+λ(λ-1)Λ]-fΛ'+λf'Λ=0
subjected to the boundary conditions
Λ(\xiw)=0
(1-\xi)1/2Λ' → 0
\xi → 1