Derive Momentum thickness and energy thickness for the given velocity profile. uV=2(yδ)−(yδ)2uV=2(yδ)−(yδ)2

 Derive Momentum thickness and energy thickness for the given velocity profile.

uV=2(yδ)−(yδ)2${\displaystyle \frac{u}{V}}=2\left({\displaystyle \frac{y}{\delta }}\right)-{\left({\displaystyle \frac{y}{\delta }}\right)}^2$

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Solution:-

 Given:

Velocity Distribution, uV=2(yδ)−(yδ)2${\displaystyle \frac{u}{V}}=2\left({\displaystyle \frac{y}{\delta }}\right)-{\left({\displaystyle \frac{y}{\delta }}\right)}^2$

(i) Displacement thickness δ∗∗${\delta }^{\ast \ast }$ is given by

Momentum thickness θ$\theta$ is given by,

θ=∫δ0uV[1−uV]dy$\theta ={\int }_0^{\delta }{\displaystyle \frac{u}{V}}\left[1-{\displaystyle \frac{u}{V}}\right]dy$

θ=∫δ0[2(yδ)−(yδ)2][1−2(yδ)−(yδ)2]dy$\theta ={\int }_0^{\delta }\left[2\left({\displaystyle \frac{y}{\delta }}\right)-{\left({\displaystyle \frac{y}{\delta }}\right)}^2\right]\left[1-2\left({\displaystyle \frac{y}{\delta }}\right)-{\left({\displaystyle \frac{y}{\delta }}\right)}^2\right]dy$

θ=∫δ0[2yδ−y2δ2−4y2δ2+2y3δ3+2y3δ3−y4δ4]dy$\theta ={\int }_0^{\delta }\left[{\displaystyle \frac{2y}{\delta }}-{\displaystyle \frac{y^2}{{\delta }^2}}-{\displaystyle \frac{4y^2}{{\delta }^2}}+{\displaystyle \frac{2y^3}{{\delta }^3}}+{\displaystyle \frac{2y^3}{{\delta }^3}}-{\displaystyle \frac{y^4}{{\delta }^4}}\right]dy$

θ=∫δ0[2yδ−5y2δ2+4y3δ3−y4δ4]dy$\theta ={\int }_0^{\delta }\left[{\displaystyle \frac{2y}{\delta }}-{\displaystyle \frac{5y^2}{{\delta }^2}}+{\displaystyle \frac{4y^3}{{\delta }^3}}-{\displaystyle \frac{y^4}{{\delta }^4}}\right]dy$

θ=[2y22δ−5y33δ2+4y44δ3−y55δ4]δ0$\theta ={\left[{\displaystyle \frac{2y^2}{2\delta }}-{\displaystyle \frac{5y^3}{3{\delta }^2}}+{\displaystyle \frac{4y^4}{4{\delta }^3}}-{\displaystyle \frac{y^5}{5{\delta }^4}}\right]}_0^{\delta }$

θ=[2δ22δ−5δ33δ2+4δ44δ3−δ55δ4]$\theta =\left[{\displaystyle \frac{2{\delta }^2}{2\delta }}-{\displaystyle \frac{5{\delta }^3}{3{\delta }^2}}+{\displaystyle \frac{4{\delta }^4}{4{\delta }^3}}-{\displaystyle \frac{{\delta }^5}{5{\delta }^4}}\right]$

θ=[δ−5δ3+δ−δ5]$\theta =\left[\delta -{\displaystyle \frac{5\delta }{3}}+\delta -{\displaystyle \frac{\delta }{5}}\right]$

θ=[15δ−25δ+15δ−3δ15]$\theta =\left[{\displaystyle \frac{15\delta -25\delta +15\delta -3\delta }{15}}\right]$

θ=[2δ15]$\theta =\left[{\displaystyle \frac{2\delta }{15}}\right]$ - Momentum thickness

(ii) Energy thickness δ∗∗${\delta }^{\ast \ast }$ is given by

δ∗∗=∫δ0uV[1−u2V2]dy${\delta }^{\ast \ast }={\int }_0^{\delta }{\displaystyle \frac{u}{V}}\left[1-{\displaystyle \frac{u^2}{V^2}}\right]dy$

δ∗∗=∫δ0[2(yδ)−(yδ)2][1−(2(yδ)−(yδ)2)2]dy${\delta }^{\ast \ast }={\int }_0^{\delta }\left[2\left({\displaystyle \frac{y}{\delta }}\right)-{\left({\displaystyle \frac{y}{\delta }}\right)}^2\right]\left[1-{\left(2\left({\displaystyle \frac{y}{\delta }}\right)-{\left({\displaystyle \frac{y}{\delta }}\right)}^2\right)}^2\right]dy$

δ∗∗=∫δ0[2yδ−y2δ2][1−4y2δ2+4y3δ3−y4δ4]dy${\delta }^{\ast \ast }={\int }_0^{\delta }\left[{\displaystyle \frac{2y}{\delta }}-{\displaystyle \frac{y^2}{{\delta }^2}}\right]\left[1-{\displaystyle \frac{4y^2}{{\delta }^2}}+{\displaystyle \frac{4y^3}{{\delta }^3}}-{\displaystyle \frac{y^4}{{\delta }^4}}\right]dy$

δ∗∗=∫δ0[2yδ−y2δ2−8y3δ3+4y4δ4−2y5δ5+y6δ6+8y4δ4−4y5δ5]dy${\delta }^{\ast \ast }={\int }_0^{\delta }[{\displaystyle \frac{2y}{\delta }}-{\displaystyle \frac{y^2}{{\delta }^2}}-{\displaystyle \frac{8y^3}{{\delta }^3}}+{\displaystyle \frac{4y^4}{{\delta }^4}}-{\displaystyle \frac{2y^5}{{\delta }^5}}+{\displaystyle \frac{y^6}{{\delta }^6}}$

$\; \; \; \; +{\displaystyle \frac{8y^4}{{\delta }^4}}-{\displaystyle \frac{4y^5}{{\delta }^5}}]dy$

δ∗∗=∫δ0[2yδ−y2δ2−8y3δ3+12y4δ4−6y5δ5+y6δ6]dy${\delta }^{\ast \ast }={\int }_0^{\delta }\left[{\displaystyle \frac{2y}{\delta }}-{\displaystyle \frac{y^2}{{\delta }^2}}-{\displaystyle \frac{8y^3}{{\delta }^3}}+{\displaystyle \frac{12y^4}{{\delta }^4}}-{\displaystyle \frac{6y^5}{{\delta }^5}}+{\displaystyle \frac{y^6}{{\delta }^6}}\right]dy$

δ∗∗=[2y22δ−y33δ2−8y44δ3+12y55δ4−6y66δ5+y77δ6]δ0${\delta }^{\ast \ast }={\left[{\displaystyle \frac{2y^2}{2\delta }}-{\displaystyle \frac{y^3}{3{\delta }^2}}-{\displaystyle \frac{8y^4}{4{\delta }^3}}+{\displaystyle \frac{12y^5}{5{\delta }^4}}-{\displaystyle \frac{6y^6}{6{\delta }^5}}+{\displaystyle \frac{y^7}{7{\delta }^6}}\right]}_0^{\delta }$

δ∗∗=[2δ22δ−δ33δ2−8δ44δ3+12δ55δ4−6δ66δ5+δ77δ6]${\delta }^{\ast \ast }=\left[{\displaystyle \frac{2{\delta }^2}{2\delta }}-{\displaystyle \frac{{\delta }^3}{3{\delta }^2}}-{\displaystyle \frac{8{\delta }^4}{4{\delta }^3}}+{\displaystyle \frac{12{\delta }^5}{5{\delta }^4}}-{\displaystyle \frac{6{\delta }^6}{6{\delta }^5}}+{\displaystyle \frac{{\delta }^7}{7{\delta }^6}}\right]$

δ∗∗=[δ−δ3−2δ+12δ5−δ+δ7]${\delta }^{\ast \ast }=\left[\delta -{\displaystyle \frac{\delta }{3}}-2\delta +{\displaystyle \frac{12\delta }{5}}-\delta +{\displaystyle \frac{\delta }{7}}\right]$

δ∗∗=[−δ3−2δ+12δ5+δ7]${\delta }^{\ast \ast }=\left[-{\displaystyle \frac{\delta }{3}}-2\delta +{\displaystyle \frac{12\delta }{5}}+{\displaystyle \frac{\delta }{7}}\right]$

δ∗∗=[−210δ−35δ+252δ+15δ105]${\delta }^{\ast \ast }=\left[{\displaystyle \frac{-210\delta -35\delta +252\delta +15\delta }{105}}\right]$

δ∗∗=[22δ105]${\delta }^{\ast \ast }=\left[{\displaystyle \frac{22\delta }{105}}\right]$ - energy thickness