---
title: "Hydrodynamic and thermal boundary layers"
subject: "Heat Transfer"
chapter: 3
key: "semester_5/heat_transfer:hydrodynamic-thermal-boundary-layer"
source: https://www.bctengineer.com/q/semester_5/heat_transfer/hydrodynamic-thermal-boundary-layer
site: BCT Engineering
---

# Hydrodynamic and thermal boundary layers

Heat Transfer · Chapter 3: Convection Heat Transfer

- Asked in: year not given, 6 marks
- Video: [Covection (Hydrodynamic boundary layer & Thermal boundary layer)](https://www.youtube.com/watch?v=x1I3CC-f7nw) — Mechanical Mayank
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/heat_transfer/hydrodynamic-thermal-boundary-layer
- Whole chapter: https://www.bctengineer.com/md/semester_5/heat_transfer/chapter-3

## Question

Define hydrodynamic and thermal boundary layers for flow over a flat plate. Sketch the development of the velocity boundary layer, showing laminar, transition and turbulent regions. State the critical Reynolds number and explain how the relative thickness of the two layers depends on the Prandtl number.

## Short answer

The hydrodynamic boundary layer is the region where velocity rises from zero at the wall to 0.99 u_inf; the thermal boundary layer is where temperature changes from T_s to T_inf. Transition occurs at critical Reynolds number ~5e5. The relative thickness depends on Prandtl number: delta/delta_t ~ Pr^(1/3).

## Answer

#### Hydrodynamic (velocity) boundary layer
When a fluid of free-stream velocity $u_\infty$ flows over a plate, the fluid in contact with the surface is at rest (no-slip) and velocity rises across a thin region to $0.99\,u_\infty$. This region, of thickness $\delta$, is the velocity boundary layer. All the viscous shear is confined to it; outside it the flow is practically inviscid.

#### Thermal boundary layer
If the plate temperature $T_s$ differs from the free-stream temperature $T_\infty$, the fluid temperature changes from $T_s$ at the wall to $T_\infty$. The thickness $\delta_t$ at which $(T-T_s)/(T_\infty - T_s) = 0.99$ is the thermal boundary layer. All convective heat transfer resistance lies within it.

#### Development
```text
 u_inf --> . . . . . . . . . . . . . . . . .
                          _________ turbulent
                  _______/  ~~~~~~~  (buffer,
         _______/  transition        viscous
   _____/ laminar                    sublayer)
 -+--------------+-------------+------------> x
 leading edge    x_cr
```
- **Laminar region** near the leading edge: smooth layers, $\delta \propto \sqrt{x}$.
- **Transition region:** disturbances grow and the flow becomes unstable.
- **Turbulent region:** eddies mix the fluid, the layer grows faster ($\delta\propto x^{4/5}$), with a thin laminar sublayer at the wall. Both $\tau_w$ and $h$ increase sharply.

The transition is fixed by the critical Reynolds number

$$
Re_{x,cr} = \frac{u_\infty x_{cr}}{\nu}\approx 5\times10^5
$$

(it ranges from $10^5$ to $3\times10^6$ with roughness and turbulence level).

#### Effect of Prandtl number
$Pr = \nu/\alpha = \mu c_p/k$ compares momentum diffusion with heat diffusion. For laminar flow:

$$
\frac{\delta}{\delta_t}\approx Pr^{1/3}
$$

| $Pr$ | Fluid | Result |
|---|---|---|
| $\ll 1$ | liquid metals | $\delta_t \gg \delta$ |
| $\approx 1$ | gases | $\delta \approx \delta_t$ |
| $\gg 1$ | oils | $\delta_t \ll \delta$ |

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The velocity boundary layer thickness is where u reaches:
   A. 0.5 u_inf
   B. 0.99 u_inf
   C. u = 0
   D. 1.5 u_inf
   Answer: B. It extends from no-slip at the wall to 0.99 of free-stream velocity. (Key point: Hydrodynamic layer)

2. What is the usual critical Re for a flat plate?
   A. 2300
   B. 10^9
   C. 5 x 10^5
   D. 5 x 10^3
   Answer: C. Transition occurs near Re_x = 5 x 10^5. (Key point: Critical Reynolds number)

3. For Pr >> 1 such as oils:
   A. Thermal layer is much thicker
   B. Both layers are equal
   C. No thermal layer forms
   D. Thermal layer is much thinner than velocity layer
   Answer: D. delta/delta_t ~ Pr^(1/3), so large Pr gives delta_t << delta. (Key point: Prandtl effect)

4. How does the layer grow in the turbulent region?
   A. Faster, delta ~ x^(4/5)
   B. Slower, delta ~ sqrt(x)
   C. Stays constant
   D. It disappears
   Answer: A. Turbulent mixing thickens the layer faster and raises h. (Key point: Turbulent growth)

## Related in Heat Transfer

- [Laminar forced convection over a heated flat plate](https://www.bctengineer.com/question.md?key=semester_5%2Fheat_transfer%3Alaminar-flow-over-heated-plate-air) — Heat Transfer, chapter 3

## Related in other subjects

- [Laminar sub-layer; laminar vs turbulent BL](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Alaminar-sublayer-laminar-vs-turbulent-bl) — Fluid Mechanics, chapter 8
- [BL development along thin flat plate](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Abl-development-flat-plate-features) — Fluid Mechanics, chapter 8
- [Laminar/turbulent flow and friction factor](https://www.bctengineer.com/question.md?key=semester_6%2Fhydro_power%3Alaminar-turbulent-friction-factor) — Hydro Power, chapter 1
- [BL zones sketch and displacement thickness](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Abl-characteristics-sketch-displacement) — Fluid Mechanics, chapter 8