# Dimensional Analysis

 Question 1
Kinematic viscosity' is dimensionally represented as
 A $\frac{M}{LT}$ B $\frac{M}{L^{2} T}$ C $\frac{T^{2}}{L}$ D $\frac{L^{2}}{T}$
GATE CE 2021 SET-1   Fluid Mechanics and Hydraulics
Question 1 Explanation:
Kinematic viscosity
$v=\frac{\mu }{\rho }=\frac{kg/m\cdot s}{kg/m^3}=m^2/s$
$[v]=\frac{m^2}{s }=\frac{L^2}{T}$
 Question 2
In a laboratory, a flow experiment is performed over a hydraulic structure. The measured values of discharge and velocity are 0.05 $m^{3}$/s and 0.25 m/s, respectively. If the full scale structure (30 times bigger) is subjected to a discharge of 270 $m^{3}$/s, then the time scale (model to full scale) value (up to two decimal places) is ______
 A 0 B 0.18 C 0.55 D 0.75
GATE CE 2018 SET-1   Fluid Mechanics and Hydraulics
Question 2 Explanation:
\begin{aligned} \text { Froude Law }(F r)_{m}&=(F r)_{p} \\ \left(\frac{V}{\sqrt{L \mathrm{g}}}\right)_{m} &=\left(\frac{v}{\sqrt{L g}}\right)_{p} \quad\left(g_{m}=g_{p}\right) \\ v_{r} &=\sqrt{L_{r}} \\ \text{or}\quad \frac{L_{r}}{T_{r}} &=\sqrt{L} \\ T_{r} &=\sqrt{L_{r}} \\ T_{r} &=\sqrt{\frac{1}{30}}=0.1826 \end{aligned}
 Question 3
A 1:50 model of a spillway is to be tested in the laboratory. The discharge in the prototype spillway is 1000 $m^{3}/s$. The corresponding discharge (in $m^{3}/s$, up to two decimal places) to be maintained in the model, neglecting variation in acceleration due to gravity, is ______
 A 0.01 B 0.06 C 0.5 D 0.1
GATE CE 2018 SET-1   Fluid Mechanics and Hydraulics
Question 3 Explanation:
Froude law is valid
\begin{aligned} Q_{r} &=L_{r}^{25} \\ \frac{Q_{m}}{Q_{p}} &=\left(\frac{1}{50}\right)^{25} \\ \frac{Q_{m}}{1000} &=\left(\frac{1}{50}\right)^{25} \\ Q_{m} &=0.0566 \mathrm{m}^{3 / \mathrm{s}} \\ \text{So},\quad Q_{m} & \simeq 0.06 \mathrm{m}^{3 / \mathrm{s}} \end{aligned}
 Question 4
The relationship between the length scale ratio $(L_{r})$ and the velocity scale ratio $(V_{r})$ in hydraulic models, in which Froude dynamic similarity is maintained, is:
 A $V_{r}=L_{r}$ B $L_{r}=\sqrt{V_{r}}$ C $V_{r}=L_{r}^{1.5}$ D $V_{r}=\sqrt{L_{r}}$
GATE CE 2015 SET-2   Fluid Mechanics and Hydraulics
Question 4 Explanation:
As per Froude Law
\begin{aligned} \left(\frac{V}{\sqrt{L g}}\right)_{m}&=\left(\frac{V}{\sqrt{L g}}\right)_{p} \\ \Rightarrow \quad V_{r}&=\sqrt{L_{r}} \end{aligned}
 Question 5
The drag force $F_{D}$, on a sphere kept in a uniform flow field depends on the diameter of the sphere, D; flow velocity, V; fluid density, $\rho$; and dynamic viscosity, $\mu$. Which of the following options represents the non-dimensional parameters which could be used to analyze this problem?
 A $\frac{F_{D}}{VD}\:\: and\: \:\: \: \frac{\mu }{\rho VD}$ B $\frac{F_{D}}{\rho VD^{2}}\:\: and\: \:\: \: \frac{\rho VD }{\mu}$ C $\frac{F_{D}}{\rho V^{2}D^{2}}\:\: and\: \:\: \: \frac{\rho VD }{\mu}$ D $\frac{F_{D}}{\rho V^{3}D^{3}}\:\: and\: \:\: \: \frac{\mu }{\rho VD}$
GATE CE 2015 SET-1   Fluid Mechanics and Hydraulics
Question 5 Explanation:
\begin{aligned} \text{For sphere,}\quad A&=\frac{\pi}{4} D^{2} \\ F_{D} &=C_{D} \times \frac{1}{2} \times p \times A \times V^{2} \\ F_{D} &=C_{D} \times \frac{1}{2} \times \rho \times\left(\frac{\pi}{4} D^{2}\right) \times V^{2} \\ \frac{F_{D}}{\rho D^{2} V^{2}} &=\frac{1}{2} \times\left(\frac{\pi}{4}\right) \times C_{D} \end{aligned}
Hence, right hand side is dimensionless, so $\frac{F_{0}}{\rho D^{2} V^{2}}$ should also be a dimensionless parameter.
Option (C) $\frac{F_{0}}{\rho V^{2} D^{2}}, \frac{\rho V D}{\mu}$
 Question 6
Group-I contains dimensionless parameters and Group- II contains the ratios.

The correct match of dimensionless parameters in Group- I with ratios in Group-II is:
 A P-3, Q-2, R-4, S-1 B P-3, Q-4, R-2, S-1 C P-2, Q-3, R-4, S-1 D P-1, Q-3, R-2, S-4
GATE CE 2013   Fluid Mechanics and Hydraulics
 Question 7
A river reach of 2.0 km long with maximum flood discharge of 10000$m^{3}/s$ is to be physically modeled in the laboratory where maximum available discharge is 0.20 $m^{3}/s$. For a geometrically similar model based on equally of Froude number, the length of the river reach (m) in the model is
 A 26.4 B 25 C 20.5 D 18
GATE CE 2008   Fluid Mechanics and Hydraulics
Question 7 Explanation:
According to Froude model law
\begin{aligned} \therefore \quad \frac{V_{t}}{\sqrt{L_{r} g_{1}}}&=1 \quad\left[\because F_{r}=1\right]\\ \text{Now, we have}\\ \frac{Q_{m}}{Q_{p}}&=L_{1}^{2} \times V_{r}\\ \Rightarrow \quad \frac{Q_{m}}{Q_{p}}&=L_{r}^{2} \times \sqrt{L_{r}} \\ \Rightarrow \quad \frac{Q_{m}}{Q_{0}}&=\left(L_{r}\right)^{5 / 2} \\ \Rightarrow \quad \frac{0.20}{10000}&=\left(\frac{L_{m}}{2 \times 1000}\right)^{5 / 2} \\ \Rightarrow \quad L_{m}&=26.4 \mathrm{m} \end{aligned}
 Question 8
A 1:50 scale model of a spillway is to be tested in the laboratory. The discharge in the prototype is 1000 $m^{3}/s$. The discharge to be maintained in the model test is
 A 0.05$m^{3}/s$ B 0.08$m^{3}/s$ C 0.57$m^{3}/s$ D 5.7$m^{3}/s$
GATE CE 2007   Fluid Mechanics and Hydraulics
Question 8 Explanation:
Froude model law will be applicable in this case
\begin{aligned} \frac{Q_{m}}{Q_{p}} &=\left(\frac{L_{m}}{L_{p}}\right)^{5 / 2} \\ \Rightarrow \quad Q_{m} &=1000 \times\left(\frac{1}{50}\right)^{25} \\ &=0.057 \mathrm{m}^{3} / \mathrm{sec} \end{aligned}
 Question 9
The flow of glycerin (kinematic viscosity $v = 5 \times 10^{-4}m^{2}/s$) in an open channel is to be modeled in a laboratory flume using water ($v = 10^{-6}m^{2}/s$) as the flowing fluid. If both gravity and viscosity are important, what should be the length scale (i.e. ratio of prototype to model dimensions) for maintaining dynamic similarity ?
 A 1 B 22 C 63 D 500
GATE CE 2006   Fluid Mechanics and Hydraulics
Question 9 Explanation:
Equating Reynolds number and Froude number, we get
\begin{aligned} \frac{V_{t} L}{v_{r}} &=\frac{V_{r}}{\sqrt{L}} \\ \therefore\quad L_{r} &=\left(v_{r}\right)^{2 / 3} \\ v_{r} &=\frac{v_{m}}{v_{p}}=\frac{10^{-6}}{5 \times 10^{-4}}=2 \times 10^{-3} \\ \Rightarrow\quad L_{r} &=\left(2 \times 10^{-3}\right)^{2 / 3} \\ \Rightarrow\quad L_{r} &=0.0159 \quad \text { But } \cdot L_{r}=\frac{L_{m}}{L_{p}} \\ \Rightarrow\quad \frac{L_{p}}{L_{m}} &=\frac{1}{L_{r}}=\frac{1}{0.0159}=62.99 \approx 63 \end{aligned}
 Question 10
The height of a hydraulic jump in the stilling pool of 1:25 scale model was observed to be 10 cm. The corresponding prototype height of the jump is
 A not determinable from the data given B 2.5m C 0.5m D 0.1m
GATE CE 2004   Fluid Mechanics and Hydraulics
Question 10 Explanation:
\begin{aligned} L_{r} &=\frac{L_{m}}{L_{p}}=\frac{1}{25} \\ \Rightarrow \quad L_{p} &=L_{m} \times 25 \\ \Rightarrow \quad L_{p} &=10 \times 25 \\ &=250 \mathrm{cm} \\ &=2.5 \mathrm{m} \end{aligned}
There are 10 questions to complete.