2.2.20 · D5Fluid Mechanics
Question bank — Boundary layer — Prandtl's concept, growth along flat plate
Reminder of the symbols you'll need (all built in the parent note):
- = free-stream speed far from the wall.
- = boundary-layer thickness, defined at .
- = viscosity; = kinematic viscosity (momentum-diffusion coefficient, units ).
- = local Reynolds number.
- = wall shear stress.
True or false — justify
Is the boundary layer a region where viscosity is large?
False. Viscosity is the same everywhere in the fluid; what is large inside the layer is the velocity gradient , which makes the viscous stress significant there.
At high Reynolds number the boundary layer is thicker.
False. , so higher means a thinner layer — inertia dominates and viscosity is squeezed into a sliver near the wall.
The boundary layer grows linearly with distance down the plate.
False. For a laminar layer ; because momentum diffuses like , the layer fattens but at an ever-decreasing rate.
Velocity reaches exactly at the edge of the boundary layer.
False. The velocity approaches asymptotically and never equals it; the "edge" is a convention placed where .
Wall shear stress is largest far downstream where the layer is fully developed.
False. , so shear is largest at the leading edge where is smallest and the gradient steepest.
Doubling the free-stream speed doubles the boundary-layer thickness.
False. , so doubling multiplies by — a faster stream gives less diffusion time, hence a thinner layer.
A perfectly inviscid ("ideal") fluid still has a boundary layer.
False. With there is no no-slip condition and no viscous stress, so no boundary layer forms — which is exactly why the ideal theory gives zero drag (d'Alembert's paradox).
Outside the boundary layer, treating the fluid as ideal is a good approximation.
True. Away from the wall the gradients are gentle, so is genuinely negligible and the flow behaves as the inviscid outer stream — this is Prandtl's two-region split.
The no-slip condition means the fluid speed at the wall equals the free-stream speed.
False. No-slip means the fluid velocity equals the wall's velocity — for a stationary plate that is zero, not .
At the leading edge the boundary-layer thickness is zero.
True. Fluid arriving at has spent no time over the plate, so viscosity has had no time to diffuse the wall's influence: .
Spot the error
"Since air's viscosity is only , viscous stress is negligible everywhere."
The stress is , not alone. Near the wall is enormous, so a tiny times a huge gradient is not negligible in the layer.
"Because , a more viscous oil gives a thinner boundary layer."
Wrong direction: , so a larger (stickier fluid) gives a thicker layer — viscosity diffuses momentum farther from the wall.
" so as I go downstream falls and the layer thins."
increases with (both and fixed, growing), and even so keeps growing — you must combine the in the numerator with the correctly.
"The viscous term dies at high , so drag on a real plate must be zero."
The viscous term shrinks in the outer region only; inside the thin layer it stays finite and produces Skin friction drag. Ignoring the layer is precisely what caused d'Alembert's false zero-drag prediction.
"Turbulent boundary layers follow the same law."
No — that Blasius result is for a laminar layer only. Once the flow becomes turbulent the mixing is far more vigorous and grows faster (roughly ).
"Since the boundary layer is thin, the pressure changes a lot across it."
The opposite: because the layer is thin, pressure is essentially constant across it (imposed by the outer stream). The strong variation is in velocity, not pressure.
Why questions
Why does Prandtl split the flow into two regions instead of solving the full Navier–Stokes equations everywhere?
The full equations are intractable, but the viscous term matters only in the thin wall layer; splitting lets you keep viscosity where it counts and use simple ideal-flow theory outside.
Why does a tiny distance create such a large velocity gradient?
The fluid must climb from at the wall to nearly across only , so ; making small forces the ratio, and hence the gradient, to be huge.
Why does the layer grow like rather than ?
Viscosity acts as diffusion of momentum, and any diffusion spreads a distance ; with and this gives — a square-root, not linear, law.
Why is kinematic viscosity , not , the quantity that sets ?
has units of , exactly those of a diffusion coefficient, so it directly measures how fast momentum spreads sideways — the physical process that thickens the layer.
Why is wall shear stress a maximum at the leading edge?
There is smallest, so the velocity gradient is steepest, and scales inversely with .
Why does high Reynolds number justify the "almost ideal outer flow" picture?
High makes vanishingly thin, confining all viscous action to a sliver and leaving the rest of the field free of significant friction.
Edge cases
What happens to exactly at ?
It is zero — no exposure time means no diffusion of the wall's influence, so the layer begins with nothing.
What does the boundary-layer concept predict for a truly inviscid fluid ()?
: the layer vanishes, recovering ideal flow everywhere and the (incorrect for real fluids) zero-drag result of d'Alembert.
What happens as with everything else fixed?
, so the layer becomes infinitely thin — the flow looks perfectly ideal outside an ever-shrinking wall film.
What happens to as in the idealised laminar formula?
: the model gives an unbounded shear at the leading edge, a known idealisation that real (finite-thickness, non-sharp) edges soften.
If the plate is very short so never exceeds the transition value, is turbulence possible?
No — with kept below transition the layer stays laminar over its whole length, and the Blasius law applies throughout.
What if the plate itself moves at speed alongside the fluid (no relative motion)?
Then wall and fluid share the same velocity, no-slip creates no gradient, , and no boundary layer forms.
Recall One-line summary to carry away
The trap in almost every question is confusing with , and confusing "more flow / higher " with "thicker layer." Remember: thin layer, huge gradient, growth, scaling.