Triple deck theory is a theory that describes a three-layered boundary-layer structure when sufficiently large disturbances are present in the boundary layer.
This theory is able to successfully explain the phenomenon of boundary layer separation, but it has found applications in many other flow setups as well,[1] including the scaling of the lower-branch instability (T-S) of the Blasius flow,[2][3] etc.
James Lighthill, Lev Landau and others were the first to realize that to explain boundary layer separation, different scales other than the classical boundary-layer scales need to be introduced.
These scales were first introduced independently by James Lighthill and E. A. Müller in 1953.
[4][5] The triple-layer structure itself was independently discovered by Keith Stewartson (1969)[6] and V. Y. Neiland (1969)[7] and by A. F. Messiter (1970).
[8] Stewartson and Messiter considered the separated flow near the trailing edge of a flat plate, whereas Neiland studied the case of a shock impinging on a boundary layer.
are the streamwise and transverse coordinate with respect to the wall and
be the Reynolds number, the boundary layer thickness is then
The lower deck is characterized by viscous, rotational disturbances, whereas the middle deck (same thickness as the boundary-layer thickness) is characterized by inviscid, rotational disturbances.
The upper deck, which extends into the potential flow region, is characterized by inviscid, irrotational disturbances.
The interaction zone identified by Lighthill in the streamwise direction is
The most important aspect of the triple-deck formulation is that pressure is not prescribed, and so it has to be solved as part of the boundary-layer problem.
This coupling between velocity and pressure reintroduces ellipticity to the problem, which is in contrast to the parabolic nature of the classical boundary layer of Prandtl.
; then the only parameter in the problem is the Reynolds number
of the coordinate system be located at the trailing edge of the plate.
For shortness of notation, let us introduce the small parameter
The coordinate for horizontal interaction and for the three decks can then be defined by[10] As
), the solution should approach the asymptotic behaviour of the Goldstein's near wake, which is given by where
The Goldstein's inner wake solution is not needed here.
The solution in the middle deck is found to be where
is referred to as the pressure function, to be determined from the upper and lower deck problems.
Note that the correction to the Blasius stream function is of the order
In the upper deck, the solution is found to given by where
Furthermore, the upper deck problem also provides the relation between the displacement and the pressure function as in which
One may notice that the pressure function and the derivative of the displacement function (aka transverse velocity) forms a Hilbert transform pair.
will satisfy a boundary-layer type equations driven by the pressure gradient
must satisfy These equations are subjected to the conditions where
must be obtained as part of the solution.
The above set of equations may resemble normal boundary-layer equations, however it has an elliptic character since the pressure gradient term now is non-local, i.e., the pressure gradient at a location
The numerical solution of these equations were obtained by Jobe and Burggraf in 1974.