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Triple deck theory
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<h2 id="flow-near-the-trailing-edge-of-a-flat-plate">Flow near the trailing edge of a flat plate</h2> <p>Let the length scales be normalized with the plate length L {\displaystyle L} and the velocity scale by the free-stream velocity U {\displaystyle U} ; then the only parameter in the problem is the Reynolds number R e = U L / ν {\displaystyle Re=UL/\nu } . Let the origin ( x , y ) = ( 0 , 0 ) {\displaystyle (x,y)=(0,0)} of the coordinate system be located at the trailing edge of the plate. Further let ( u , v ) {\displaystyle (u,v)} be the non-dimensional velocity components, p {\displaystyle p} be the non-dimensional pressure field and ψ {\displaystyle \psi } be the non-dimensional <a href="/facts/Stream_function/nEoYTGAl">stream function</a> such that u = ∂ ψ / ∂ y {\displaystyle u=\partial \psi /\partial y} and v = − ∂ ψ / ∂ x {\displaystyle v=-\partial \psi /\partial x} . For shortness of notation, let us introduce the small parameter ε = 1 / R e 1 / 8 {\displaystyle \varepsilon =1/Re^{1/8}} . The coordinate for horizontal interaction and for the three decks can then be defined by<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> </p> χ = x / ε 3 , ξ = y / ε 5 , η = y / ε 4 , ζ = y / ε 3 . {\displaystyle \chi =x/\varepsilon ^{3},\quad \xi =y/\varepsilon ^{5},\quad \eta =y/\varepsilon ^{4},\quad \zeta =y/\varepsilon ^{3}.} <p>As χ → − ∞ {\displaystyle \chi \to -\infty } (or x → 0 − {\displaystyle x\to 0^{-}} ), the solution should approach the asymptotic behaviour of the <a href="/facts/Blasius_boundary_layer/wBJanmIS">Blasius solution</a>, which is given by </p> ψ ε 4 = 2 f B ( η 2 ) + x 2 [ f B ( η 2 ) − η 2 f B ′ ( η 2 ) ] + ⋯ {\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\sqrt {2}}f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)+{\frac {x}{\sqrt {2}}}\left[f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)-{\frac {\eta }{\sqrt {2}}}f_{B}'\left({\frac {\eta }{\sqrt {2}}}\right)\right]+\cdots } <p>where f B ( η ) {\displaystyle f_{B}(\eta )} is the Blasisus function which satisfies f B ‴ + f B f B ″ = 0 {\displaystyle f_{B}'''+f_{B}f_{B}''=0} subjected to f B ( 0 ) = f B ′ ( 0 ) = f B ′ ( ∞ ) − 1 = 0 {\displaystyle f_{B}(0)=f_{B}'(0)=f_{B}'(\infty )-1=0} . As χ → + ∞ {\displaystyle \chi \to +\infty } (or x → 0 + {\displaystyle x\to 0^{+}} ), the solution should approach the asymptotic behaviour of the Goldstein's near wake, which is given by </p> Outer wake : ψ ε 4 = 2 f B ( η 2 ) + μ x 1 / 3 λ f B ′ ( η 2 ) + ⋯ {\displaystyle {\text{Outer wake}}:{\frac {\psi }{\varepsilon ^{4}}}={\sqrt {2}}f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)+{\frac {\mu x^{1/3}}{\lambda }}f_{B}'\left({\frac {\eta }{\sqrt {2}}}\right)+\cdots } <p>where μ = 1.1321 {\displaystyle \mu =1.1321} and λ = 0.8789 {\displaystyle \lambda =0.8789} . The Goldstein's inner wake solution is not needed here. </p> <h3>Middle deck</h3> <p>The solution in the middle deck is found to be </p> ψ ε 4 = 2 f B ( η 2 ) + ε A ( χ ) f B ′ ( η 2 ) + ε 2 Φ ( χ , η ) + ⋯ , p = ε 2 P ( χ ) + ⋯ {\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\sqrt {2}}f_{B}\left({\frac {\eta }{\sqrt {2}}}\right)+\varepsilon A(\chi )f_{B}'\left({\frac {\eta }{\sqrt {2}}}\right)+\varepsilon ^{2}\Phi (\chi ,\eta )+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots } <p>where A ( χ ) {\displaystyle A(\chi )} is referred to as the <i>displacement function</i> and P ( χ ) {\displaystyle P(\chi )} is referred to as the <i>pressure function</i>, to be determined from the upper and lower deck problems. Note that the correction to the Blasius stream function is of the order ε {\displaystyle \varepsilon } , although the pressure perturbation is only order ε 2 . {\displaystyle \varepsilon ^{2}.} </p> <h3>Upper deck</h3> <p>In the upper deck, the solution is found to given by </p> ψ ε 4 = ζ ε − 2 β − ε π ∫ − ∞ + ∞ A ′ ( χ ) [ tan − 1 ( χ − χ ^ ζ ) + π 2 ] d χ ^ + ⋯ , p = ε 2 P ( χ ) + ⋯ {\displaystyle {\frac {\psi }{\varepsilon ^{4}}}={\frac {\zeta }{\varepsilon }}-{\sqrt {2}}\beta -{\frac {\varepsilon }{\pi }}\int _{-\infty }^{+\infty }A'(\chi )\left[\tan ^{-1}\left({\frac {\chi -{\hat {\chi }}}{\zeta }}\right)+{\frac {\pi }{2}}\right]d{\hat {\chi }}+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots } <p>where β = 1.2168 {\displaystyle \beta =1.2168} . Furthermore, the upper deck problem also provides the relation between the displacement and the pressure function as </p> P ( χ ) = p . v . 1 π ∫ − ∞ + ∞ A ′ ( χ ^ ) χ − χ ^ d χ ^ and A ′ ( χ ) = − p . v . 1 π ∫ − ∞ + ∞ P ( χ ^ ) χ − χ ^ d χ ^ . {\displaystyle P(\chi )=\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {A'({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}\quad {\text{and}}\quad A'(\chi )=-\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {P({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}.} <p>in which p . v . {\displaystyle \mathrm {p.v.} } stands for <a href="/facts/Cauchy_principal_value/Gm33Qevf">Cauchy principal value</a>. One may notice that the pressure function and the derivative of the displacement function (aka transverse velocity) forms a <a href="/facts/Hilbert_transform/RoQaC8MS">Hilbert transform</a> pair. </p> <h3>Lower deck</h3> <p>In the lower deck, the solution is given by </p> ψ ε 4 = ε 2 Ψ ( χ , ξ ) + ⋯ , p = ε 2 P ( χ ) + ⋯ {\displaystyle {\frac {\psi }{\varepsilon ^{4}}}=\varepsilon ^{2}\Psi (\chi ,\xi )+\cdots ,\quad p=\varepsilon ^{2}P(\chi )+\cdots } <p>where Ψ ( χ , ξ ) {\displaystyle \Psi (\chi ,\xi )} will satisfy a boundary-layer type equations driven by the pressure gradient d P / d χ {\displaystyle dP/d\chi } and the slip-velocity of order ε 2 {\displaystyle \varepsilon ^{2}} generated by the middle deck. It is convenient to introduce u ^ = ∂ Ψ / ∂ ξ {\displaystyle {\hat {u}}=\partial \Psi /\partial \xi } and v ^ = − ∂ Ψ / ∂ χ {\displaystyle {\hat {v}}=-\partial \Psi /\partial \chi } , where u ^ {\displaystyle {\hat {u}}} and v ^ {\displaystyle {\hat {v}}} must satisfy </p> ∂ u ^ ∂ χ + ∂ v ^ ∂ ξ = 0 , u ^ ∂ u ^ ∂ χ + v ^ ∂ u ^ ∂ ξ = − p . v . 1 π ∫ − ∞ + ∞ A ″ ( χ ^ ) χ − χ ^ d χ ^ + ∂ 2 u ^ ∂ ξ 2 . {\displaystyle {\begin{aligned}{\frac {\partial {\hat {u}}}{\partial \chi }}+{\frac {\partial {\hat {v}}}{\partial \xi }}&=0,\\{\hat {u}}{\frac {\partial {\hat {u}}}{\partial \chi }}+{\hat {v}}{\frac {\partial {\hat {u}}}{\partial \xi }}&=-\mathrm {p.v.} {\frac {1}{\pi }}\int _{-\infty }^{+\infty }{\frac {A''({\hat {\chi }})}{\chi -{\hat {\chi }}}}d{\hat {\chi }}+{\frac {\partial ^{2}{\hat {u}}}{\partial \xi ^{2}}}.\end{aligned}}} <p>These equations are subjected to the conditions </p> ξ = 0 : { u ^ = v ^ = 0 , χ ≤ 0 , ∂ u ^ ∂ ξ = v ^ = 0 , χ > 0 , and ξ → ∞ : u ^ → ξ + A ( χ ) 2 α 3 , {\displaystyle \xi =0:\,\,\,{\begin{cases}{\hat {u}}={\hat {v}}=0,\,\,\chi \leq 0,\\{\frac {\partial {\hat {u}}}{\partial \xi }}={\hat {v}}=0,\,\,\chi >0,\end{cases}}\quad {\text{and}}\quad \xi \to \infty :{\hat {u}}\to {\frac {\xi +A(\chi )}{\sqrt {2\alpha ^{3}}}},} χ → − ∞ : A → 0 , and χ → + ∞ : A → μ λ χ 1 / 3 {\displaystyle \chi \to -\infty :\,\,\,A\to 0,\quad {\text{and}}\quad \chi \to +\infty :\,\,\,A\to {\frac {\mu }{\lambda }}\chi ^{1/3}} <p>where α = 1.6552 {\displaystyle \alpha =1.6552} . The displacement function A ( χ ) {\displaystyle A(\chi )} and therefore P ( χ ) {\displaystyle P(\chi )} 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 χ {\displaystyle \chi } depends on other locations as well. Because of this, these equations are sometimes referred to as the <i>interactive boundary-layer</i> equations. The numerical solution of these equations were obtained by Jobe and Burggraf in 1974.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Flow_separation/Lnvdqo4p">Flow separation</a></li> <li><a href="/facts/Boundary_layer/go8gqWM1">Boundary layer</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="https://academic.oup.com/imamat/article-abstract/28/3/207/660853" target="_blank">https://academic.oup.com/imamat/article-abstract/28/3/207/660853</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="https://doi.org/10.1090%2Fqam%2F14894" target="_blank">https://doi.org/10.1090%2Fqam%2F14894</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="/wiki/James_Lighthill" target="_blank">/wiki/James_Lighthill</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>E. A. Müller (1953) Dissertation, University of Göttingen. <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Stewartson, K. (1969). "On the flow near the trailing edge of a flat plate II". Mathematika. 16 (1): 106–121. doi:10.1112/S0025579300004678. <a href="/wiki/Mathematika" target="_blank">/wiki/Mathematika</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Neiland, V. Ya. (1969). "Theory of laminar boundary layer separation in supersonic flow". Fluid Dynamics. 4 (4): 33–35. Bibcode:1969FlDy....4d..33N. doi:10.1007/BF01094681. <a href="https://doi.org/10.1007%2FBF01094681" target="_blank">https://doi.org/10.1007%2FBF01094681</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>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. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Prandtl, L. (1904). "Uber Flussigkeitsbewegung bei sehr kleiner Reibung". Verh. III. Int. Math. Kongr.: 484–491. <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Sobey, I. J. (2000). Introduction to interactive boundary layer theory (Vol. 3). Oxford Texts in Applied and En. <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Jobe, C. E., & Burggraf, O. R. (1974). The numerical solution of the asymptotic equations of trailing edge flow. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 340(1620), 91-111. <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> </ol>