---
title: "Average velocity from midway velocity reading"
subject: "Hydraulics"
chapter: 1
key: "semester_4/hydraulics:turbulent-pipe-average-velocity-midway"
source: https://www.bctengineer.com/q/semester_4/hydraulics/turbulent-pipe-average-velocity-midway
site: BCT Engineering
---

# Average velocity from midway velocity reading

Hydraulics · Chapter 1: Pipe flow

- Asked in: 2069 Poush, 4 marks
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/hydraulics/turbulent-pipe-average-velocity-midway
- Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-1

## Question

Measurement in a fully developed turbulent flow in pipe indicate that velocity midway between the pipe wall and the pipe centerline is 0.9 times the centerline velocity. Determine the expression for the average velocity in multiples of maximum velocity. What is the value of e/D or K/D (relative roughness) if pipe acts as rough pipe?

## Short answer

Using the velocity defect law, u_max - u = 5.75 u* log(R/y). At y = R/2, u = 0.9 u_max gives u_max = 17.309 u*. Then V = u_max - 3.75 u* = 13.559 u*, so V = 0.783 u_max. For rough pipe, u_max/u* = 5.75 log(R/k) + 8.5 yields k/D = 0.0147.

## Answer

**Use the velocity defect law** (valid for smooth and rough pipes): $\dfrac{u_{max} - u}{u_*} = 5.75\log_{10}\dfrac{R}{y}$, and the mean velocity relation $\dfrac{u_{max} - V}{u_*} = 3.75$.

#### Step 1: use the midway reading
Midway between the wall and the centre line, $y = R/2$ and $u = 0.9\,u_{max}$:
$$
\frac{u_{max} - 0.9u_{max}}{u_*} = 5.75\log_{10}\frac{R}{R/2} = 5.75\log_{10}2 = 1.7309
$$
$$
\frac{0.1\,u_{max}}{u_*} = 1.7309 \;\Rightarrow\; u_{max} = 17.309\,u_*
$$

#### Step 2: average velocity
$$
V = u_{max} - 3.75u_* = 17.309u_* - 3.75u_* = 13.559\,u_*
$$
$$
\frac{V}{u_{max}} = \frac{13.559}{17.309} = 0.7834
$$
$$
\boxed{V = 0.783\,u_{max}}
$$

#### Step 3: relative roughness (rough pipe)
The velocity at the axis of a rough pipe: $\dfrac{u_{max}}{u_*} = 5.75\log_{10}\dfrac{R}{k} + 8.5$ (from $y = R$).
$$
17.309 = 5.75\log_{10}\frac{R}{k} + 8.5 \;\Rightarrow\; \log_{10}\frac{R}{k} = \frac{17.309 - 8.5}{5.75} = 1.532
$$
$$
\frac{R}{k} = 34.04 \;\Rightarrow\; \frac{k}{D} = \frac{1}{2(34.04)} = 0.0147
$$

**Answer:** $V = 0.783\,u_{max}$; relative roughness $k/D \approx 0.0147$ (about 1.5%).

## Check yourself

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

1. At y = R/2 with u = 0.9 u_max, which relation yields u_max in terms of u*?
   A. (u_max - u)/u* = 3.75
   B. u/u* = 5.75 log(y/k) + 8.5 with k unknown
   C. (u_max - u)/u* = 5.75 log(R/y), giving 0.1 u_max/u* = 1.7309
   D. u_max = 2V
   Answer: C. The velocity defect law with R/y = 2 gives 5.75 log 2 = 1.7309, so u_max = 17.309 u*. (Key point: Using the midway reading)

2. How is the average velocity obtained?
   A. V = u_max - 5.75 u*
   B. V = 0.9 u_max
   C. V = u_max/2
   D. V = u_max - 3.75 u*, giving V = 0.783 u_max
   Answer: D. The mean-velocity relation (u_max - V)/u* = 3.75 gives V = 13.559 u*, i.e. 0.783 u_max. (Key point: Average velocity)

3. How is k/D found for the rough pipe?
   A. From u_max/u* = 5.75 log(R/k) + 8.5 with u_max/u* = 17.309, then k/D = k/(2R)
   B. From f = 64/Re
   C. From V/u* = 5.75 log(R/k) + 8.5
   D. From the smooth-pipe law with Re known
   Answer: A. At the axis y = R, so R/k = 34.04 and k/D = 1/(2 x 34.04) = 0.0147. (Key point: Relative roughness)

## Related in Hydraulics

- [Velocity for fully turbulent flow in rough pipe](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Afully-turbulent-velocity-1200mm-pipe) — Hydraulics, chapter 1
- [Roughness height from velocity ratio](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Arough-pipe-velocity-ratio-roughness) — Hydraulics, chapter 1
- [Mean velocity law for rough pipes](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Arough-pipe-log-law-mean-velocity) — Hydraulics, chapter 1
- [Velocity defect law for smooth and rough pipes](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asmooth-rough-pipe-velocity-defect-law) — Hydraulics, chapter 1
- [Inclined smooth pipe with wall shear limit](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Asmooth-pipe-inclined-wall-shear-limit) — Hydraulics, chapter 1
- [Kerosene flow: smooth, rough or transition](https://www.bctengineer.com/question.md?key=semester_4%2Fhydraulics%3Akerosene-pipe-rough-or-smooth) — Hydraulics, chapter 1

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