--- title: "BL zones sketch and displacement thickness" subject: "Fluid Mechanics" chapter: 8 key: "semester_3/fluid_mechanics:bl-characteristics-sketch-displacement" source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bl-characteristics-sketch-displacement site: BCT Engineering --- # BL zones sketch and displacement thickness Fluid Mechanics · Chapter 8: Boundary Layer theory - Asked in: 2076 Chaitra, 4 marks - Video: [Boundary Layer Theory - Introduction](https://www.youtube.com/watch?v=54zf68IoPWU) — TutorialsPoint - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/bl-characteristics-sketch-displacement - Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-8 ## Question Flow takes place over a flat plate exposed parallel to free stream. Mention characteristics of flow and draw a neat sketch of the boundary layer development showing, (i) Laminar boundary layer, (ii) Turbulent boundary layer, (iii) Transition zone, (iv) Laminar sub layer. What is displacement thickness? ## Short answer Flow over a flat plate starts laminar, transitions at Re_x ≈ 5×10⁵, and becomes turbulent with a thin laminar sublayer at the wall. Velocity grows from zero at the wall to free-stream U across the layer thickness δ. Displacement thickness δ* = ∫₀ᵟ (1 − u/U) dy, the outward shift of streamlines due to the velocity deficit. ## Answer #### Characteristics of the flow A flat plate held parallel to a free stream develops a boundary layer, in which viscosity slows the fluid near the surface. Velocity is zero at the wall and equals the free-stream velocity $U$ at the edge of the layer ($y = \delta$). The thickness $\delta$ grows with distance $x$. The flow is laminar near the leading edge, changes through a transition region and becomes turbulent downstream, with the change at about $Re_x = Ux/\nu \approx 5\times10^5$. Outside the layer the flow is practically inviscid. #### Sketch ```text U -> --------------------------------------------------- Turbulent BL Laminar BL Transition _________________ U -> ----------____________......___________________ ..... laminar sub-layer ========================================= plate leading edge x ------------------------> (i) laminar (iii) transition (ii) turbulent (iv) laminar sub-layer: thin film at the wall ``` - (i) **Laminar boundary layer**: smooth flow, $\delta = 5x/\sqrt{Re_x}$. - (iii) **Transition zone**: flow changes from laminar to turbulent, $Re_x \approx 3\times10^5$ to $5\times10^5$ or more. - (ii) **Turbulent boundary layer**: eddy mixing, thicker, fuller velocity profile. - (iv) **Laminar sub-layer**: very thin viscous layer next to the wall, even under the turbulent layer. #### Displacement thickness It is the distance by which the outer streamlines are displaced away from the plate because of the reduction of flow in the boundary layer: $$ \delta^* = \int_0^{\delta}\left(1 - \frac uU\right)dy $$ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What are the velocities at the wall and at y = delta? A. U at the wall and zero at the edge B. Zero at both C. Zero at the wall and U (free-stream) at the edge D. U at both Answer: C. No-slip gives u = 0 at the wall; at the edge velocity reaches the free-stream value. (Key point: Velocity at edges) 2. Where does the flow change from laminar to turbulent? A. At the leading edge B. In the transition zone at Re_x of about 5 x 10^5 C. At Re_x = 2000 D. Only at the trailing edge Answer: B. Transition occurs downstream of the laminar region near the critical Reynolds number. (Key point: Transition) 3. Where is the laminar sub-layer? A. A thin film at the wall under the turbulent boundary layer B. In the free stream C. At the outer edge of the laminar layer D. At the leading edge only Answer: A. Even in turbulent flow the wall forces a thin viscous layer. (Key point: Laminar sub-layer) 4. What does displacement thickness represent? A. Distance where u = 0.99U B. Momentum loss in the layer C. Thickness of the laminar sub-layer D. Distance by which outer streamlines are shifted from the plate because of reduced flow in the boundary layer Answer: D. delta* = integral (1 - u/U) dy measures discharge loss. 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