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, including the scaling of the lower-branch instability (T-S) of the Blasius flow, 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. The triple-layer structure itself was independently discovered by Keith Stewartson (1969) and V. Y. Neiland (1969) and by A. F. Messiter (1970). 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.
Suppose formula_1 and formula_2 are the streamwise and transverse coordinate with respect to the wall and formula_3 be the Reynolds number, the boundary layer thickness is then formula_4. The boundary layer coordinate is formula_5. Then the thickness of each deck is
formula_6
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
formula_7
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.
Flow near the trailing edge of a flat plate.
Let the length scales be normalized with the plate length formula_8 and the velocity scale by the free-stream velocity formula_9; then the only parameter in the problem is the Reynolds number formula_10. Let the origin formula_11 of the coordinate system be located at the trailing edge of the plate. Further let formula_12 be the non-dimensional velocity components, formula_13 be the non-dimensional pressure field and formula_14 be the non-dimensional stream function such that formula_15 and formula_16. For shortness of notation, let us introduce the small parameter formula_17. The coordinate for horizontal interaction and for the three decks can then be defined by
formula_18
As formula_19 (or formula_20), the solution should approach the asymptotic behaviour of the Blasius solution, which is given by
formula_21
where formula_22 is the Blasisus function which satisfies formula_23 subjected to formula_24. As formula_25 (or formula_26), the solution should approach the asymptotic behaviour of the Goldstein's near wake, which is given by
formula_27
where formula_28 and formula_29. The Goldstein's inner wake solution is not needed here.
Middle deck.
The solution in the middle deck is found to be
formula_30
where formula_31 is referred to as the "displacement function" and formula_32 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 formula_33, although the pressure perturbation is only order formula_34
Upper deck.
In the upper deck, the solution is found to given by
formula_35
where formula_36. Furthermore, the upper deck problem also provides the relation between the displacement and the pressure function as 
formula_37
in which formula_38 stands for Cauchy principal value. One may notice that the pressure function and the derivative of the displacement function (aka transverse velocity) forms a Hilbert transform pair.
Lower deck.
In the lower deck, the solution is given by
formula_39
where formula_40 will satisfy a boundary-layer type equations driven by the pressure gradient formula_41 and the slip-velocity of order formula_42 generated by the middle deck. It is convenient to introduce formula_43 and formula_44, where formula_45 and formula_46 must satisfy
formula_47 
These equations are subjected to the conditions
formula_48
formula_49
where formula_50. The displacement function formula_31 and therefore formula_32 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 formula_53 depends on other locations as well. Because of this, these equations are sometimes referred to as the "interactive boundary-layer" equations. The numerical solution of these equations were obtained by Jobe and Burggraf in 1974. 
References.
 ;templatestyles src="Reflist/styles.css" / ;