“Schneider flow”的意思、由来-开放百科全书

Schneider flow is an axisymmetric flow induced by a laminar or turbulent jet (with large jet Reynolds number or by a laminar plume (with large plume Grashof number), in which the fluid domain is bounded by a wall. The solution is an exact solution of the Navier-Stokes equations, discovered by Wilhelm Schneider in 1981^[1]. The solution is an extension of Taylor's potential flow solution^[2] 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 link sink, as was first done by G. I. Taylor and this sink will be drive the fluid outside the jet or plume. 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

with polar axis along the cone-axis and assume the vertex of the solid cone sits at the origin of the spherical coordinates

extending along the negative axis. Now, put the line sink along the positive side of the polar axis. Set this way,

represents the common case of flat wall with jet or plume emerging from the origin. The case

corresponds to jet/plume issuing from a thin injector. The flow is axisymmetric with zero azimuthal motion, i.e., the velocity components are

. The usual technique to study the flow is to introduce the Stokes stream function

such that

Introducing

as the replacement for

and introducing the self-similar form

into the axisymmetric Navier-Stokes equations, we obtain^[3]

where the constant

is such that the volumetric entrainment rate per unit axial length is equal to

. For laminar jet,

and for laminar plume, it depends on the Prandtl number

, for example with

, we have

and with

, we have

. For turbulent jet, this constant is the order of the jet Reynolds number, which is a large number.

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

become

Taylor's potential flow

For turbulent jet,

, the linear terms in the equation can be neglected everywhere except near a small boundary layer along the wall. Then neglecting the non-slip conditions at the wall, the solution is given by

Other considerations

The exact solution of the Navier-Stokes solutions was verified experimentally by Zauner in 1985^[4]. Further analysis^[5]^[6] 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.

Mathematical description

Taylor's potential flow

Other considerations

See also

References