Triple-deck theory

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] boundary layers in swirling flows,^[4]^[5] 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.^[6]^[7] The triple-layer structure itself was independently discovered by Keith Stewartson (1969)^[8] and V. Y. Neiland (1969)^[9] and by A. F. Messiter (1970).^[10] 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 $x$ and $y$ are the streamwise and transverse coordinate with respect to the wall and $Re$ be the Reynolds number, the boundary layer thickness is then $\delta =Re^{-1/2}$ . The boundary layer coordinate is $\eta =yRe^{1/2}$ . Then the thickness of each deck is

${\begin{aligned}{\text{Lower deck}}:&\quad y\sim Re^{-5/8}\\{\text{Middle deck}}:&\quad y\sim Re^{-4/8}\\{\text{Upper deck}}:&\quad y\sim Re^{-3/8}.\end{aligned}}$

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

${\text{Interaction zone}}:\quad x\sim Re^{-3/8}.$

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.^[11]

References

^ Smith, F. T. (1982). "On the high Reynolds number theory of laminar flows". IMA J. Appl. Math. 28 (3): 207–281. doi:10.1093/imamat/28.3.207.
^ Smith, F. T. (1979). "On the non-parallel flow stability of the Blasius boundary layer". Proc. R. Soc. Lond. 366 (1724): 91–109. Bibcode:1979RSPSA.366...91S. doi:10.1098/rspa.1979.0041. S2CID 112228524.
^ Lin, C. C. (1946). "On the stability of two-dimensional parallel flows. III. Stability in a viscous fluid". Quart. Appl. Math. 3 (4): 277–301. doi:10.1090/qam/14894.
^ Burggraf, O. R., Stewartson, K., & Belcher, R. (1971). Boundary layer induced by a potential vortex. The Physics of Fluids, 14(9), 1821-1833.
^ Weiss, A. D., Rajamanickam, P., Coenen, W., Sánchez, A. L., & Williams, F. A. (2020). A model for the constant-density boundary layer surrounding fire whirls. Journal of Fluid Mechanics, 900.
^ Lighthill, Michael James (1953). "On boundary layers and upstream influence II. Supersonic flows without separation". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 217 (1131): 478–507. Bibcode:1953RSPSA.217..478L. doi:10.1098/rspa.1953.0075. S2CID 95497146.
^ E. A. Müller (1953) Dissertation, University of Göttingen.
^ Stewartson, K. (1969). "On the flow near the trailing edge of a flat plate II". Mathematika. 16 (1): 106–121. doi:10.1112/S0025579300004678.
^ Neiland, V. Ya. (1969). "Theory of laminar boundary layer separation in supersonic flow". Fluid Dynamics. 4 (4): 33–35. doi:10.1007/BF01094681.
^ Messiter, A. F. (1970). "Boundary-layer flow near the trailing edge of a flat plate". SIAM Journal on Applied Mathematics. 18 (1): 241–257. doi:10.1137/0118020.
^ Prandtl, L. (1904). "Uber Flussigkeitsbewegung bei sehr kleiner Reibung". Verh. III. Int. Math. Kongr.: 484–491.

[1] Smith, F. T. (1982). "On the high Reynolds number theory of laminar flows". IMA J. Appl. Math. 28 (3): 207–281. doi:10.1093/imamat/28.3.207.

[2] Smith, F. T. (1979). "On the non-parallel flow stability of the Blasius boundary layer". Proc. R. Soc. Lond. 366 (1724): 91–109. Bibcode:1979RSPSA.366...91S. doi:10.1098/rspa.1979.0041. S2CID 112228524.

[3] Lin, C. C. (1946). "On the stability of two-dimensional parallel flows. III. Stability in a viscous fluid". Quart. Appl. Math. 3 (4): 277–301. doi:10.1090/qam/14894.

[4] Burggraf, O. R., Stewartson, K., & Belcher, R. (1971). Boundary layer induced by a potential vortex. The Physics of Fluids, 14(9), 1821-1833.

[5] Weiss, A. D., Rajamanickam, P., Coenen, W., Sánchez, A. L., & Williams, F. A. (2020). A model for the constant-density boundary layer surrounding fire whirls. Journal of Fluid Mechanics, 900.

[6] Lighthill, Michael James (1953). "On boundary layers and upstream influence II. Supersonic flows without separation". Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences. 217 (1131): 478–507. Bibcode:1953RSPSA.217..478L. doi:10.1098/rspa.1953.0075. S2CID 95497146.

[7] E. A. Müller (1953) Dissertation, University of Göttingen.

[8] Stewartson, K. (1969). "On the flow near the trailing edge of a flat plate II". Mathematika. 16 (1): 106–121. doi:10.1112/S0025579300004678.

[9] Neiland, V. Ya. (1969). "Theory of laminar boundary layer separation in supersonic flow". Fluid Dynamics. 4 (4): 33–35. doi:10.1007/BF01094681.

[10] Messiter, A. F. (1970). "Boundary-layer flow near the trailing edge of a flat plate". SIAM Journal on Applied Mathematics. 18 (1): 241–257. doi:10.1137/0118020.

[11] Prandtl, L. (1904). "Uber Flussigkeitsbewegung bei sehr kleiner Reibung". Verh. III. Int. Math. Kongr.: 484–491.

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See also

References