--- title: "BL, displacement thickness and ratio, 1/7 profile" subject: "Fluid Mechanics" chapter: 8 key: "semester_3/fluid_mechanics:bl-thickness-displacement-ratio-1-7" source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bl-thickness-displacement-ratio-1-7 site: BCT Engineering --- # BL, displacement thickness and ratio, 1/7 profile Fluid Mechanics · Chapter 8: Boundary Layer theory - Asked in: 2079 Baisakh, 1+2+3 marks - Video: [Lecture 165 Problem Solved on Displacement, Momentum & Energy Thickness in #Fluid #Mechanics](https://www.youtube.com/watch?v=F5r0ZnPKoYc) — PKD Learning Zone - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bl-thickness-displacement-ratio-1-7 - Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-8 ## Question What is boundary layer? Explain boundary layer thickness and displacement thickness. Compute the ratio of these quantities for the boundary layer described by the velocity distribution $\frac{u}{U} = \left(\frac{y}{\delta}\right)^{1/7}$. ## Short answer Boundary layer is the thin viscous region near a wall where velocity rises to free-stream. δ is where u=0.99U; displacement thickness δ* = ∫(1-u/U)dy. For u/U=(y/δ)^{1/7}, integration gives δ* = δ/8, so ratio δ*/δ = 0.125. ## Answer The ratio of displacement thickness to boundary layer thickness for the 1/7 power-law profile is $\delta^*/\delta = 1/8 = 0.125$. #### Boundary layer When a real fluid flows past a solid surface, the fluid layer touching the surface sticks to it (no-slip). Velocity rises from zero at the wall to the free-stream velocity $U$ over a thin region near the surface. This thin region, where viscous shear is important, is the **boundary layer**. Outside it the flow behaves as an ideal (inviscid) fluid. #### Boundary layer thickness ($\delta$) It is the distance from the wall at which the velocity reaches 99% of the free-stream velocity: $u = 0.99\,U$ at $y = \delta$. #### Displacement thickness ($\delta^*$) It is the distance by which the external streamlines are shifted away from the wall because of the slow-moving fluid in the boundary layer. Equivalently, it is the thickness of a layer of free-stream flow that has the same discharge as the loss of discharge caused by the boundary layer. $$ \delta^* = \int_0^{\delta}\left(1 - \frac{u}{U}\right)dy $$ #### Ratio for $u/U = (y/\delta)^{1/7}$ $$ \begin{aligned} \delta^* &= \int_0^{\delta}\left[1 - \left(\frac y\delta\right)^{1/7}\right]dy = \delta - \frac{\delta}{1 + 1/7}\\ &= \delta - \frac{7}{8}\delta = \frac{\delta}{8} \end{aligned} $$ $$ \frac{\delta^*}{\delta} = \frac18 = 0.125 $$ **Answer: $\delta^*/\delta = 1/8$**, so the displacement thickness is 12.5% of the boundary layer thickness. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What is the boundary layer? A. The region far from the wall where the flow is inviscid B. The thin region near a surface where viscous shear is important and velocity rises from zero to U C. The layer where pressure is zero D. The region where flow is always turbulent Answer: B. No-slip makes velocity zero at the wall, and viscous effects are confined to this thin layer. (Key point: Boundary layer definition) 2. How is the boundary layer thickness delta defined? A. Distance from the wall where u = 0.99 U B. Distance where u = 0.5 U C. Distance where u = U/7 D. Distance where shear stress is maximum Answer: A. delta is the distance at which velocity reaches 99% of the free-stream velocity. (Key point: Boundary layer thickness) 3. Which integral gives the displacement thickness? A. Integral of (u/U) dy B. Integral of (u/U)(1 - u/U) dy C. Integral of (1 + u/U) dy D. Integral of (1 - u/U) dy from 0 to delta Answer: D. It measures the loss of discharge due to the boundary layer; the momentum-thickness integral is the other one. (Key point: Displacement thickness formula) 4. What is delta*/delta for u/U = (y/delta)^(1/7)? A. 1/7 B. 7/8 C. 1/8 D. 1/9 Answer: C. delta* = delta - delta/(1+1/7) = delta - 7 delta/8 = delta/8. 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