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. The solution was discovered also by A. A. Golubinskii and V. V. Sychev in 1979, however, was never applied to flows entrained by jets. The solution is an extension of Taylor's potential flow solution to arbitrary Reynolds number.
Mathematical description
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 α {\displaystyle \alpha } with polar axis along the cone-axis and assume the vertex of the solid cone sits at the origin of the spherical coordinates ( r , θ , ϕ ) {\displaystyle (r,\theta ,\phi )} extending along the negative axis. Now, put the line sink along the positive side of the polar axis. Set this way, α = π / 2 {\displaystyle \alpha =\pi /2} represents the common case of flat wall with jet or plume emerging from the origin. The case α = π {\displaystyle \alpha =\pi } corresponds to jet/plume issuing from a thin injector. The flow is axisymmetric with zero azimuthal motion, i.e., the velocity components are ( v r , v θ , 0 ) {\displaystyle (v_{r},v_{\theta },0)} . The usual technique to study the flow is to introduce the Stokes stream function ψ {\displaystyle \psi } such that
v r = 1 r 2 sin θ ∂ ψ ∂ θ , v θ = − 1 r sin θ ∂ ψ ∂ r . {\displaystyle v_{r}={\frac {1}{r^{2}\sin \theta }}{\frac {\partial \psi }{\partial \theta }},\quad v_{\theta }=-{\frac {1}{r\sin \theta }}{\frac {\partial \psi }{\partial r}}.}
Introducing ξ = cos θ {\displaystyle \xi =\cos \theta } as the replacement for θ {\displaystyle \theta } and introducing the self-similar form ψ = K ν r f ( ξ ) {\displaystyle \psi =K\nu rf(\xi )} into the axisymmetric Navier-Stokes equations, we obtain
K − 1 [ ( 1 − ξ 2 ) f ⁗ − 4 ξ f ‴ ] − f f ‴ − 3 f ′ f ″ = 0. {\displaystyle K^{-1}[(1-\xi ^{2})f''''-4\xi f''']-ff'''-3f'f''=0.}
… excerpt ends here. Continue reading the full article.
