notes/uni/mmme/2047_thermodynamics_and_fluid_dynamics/dimensional_analysis.md

---
author: Akbar Rahman
date: \today
title: MMME2047 // Dimensional Analysis
tags: [ mmme2047, uni, fluid_dynamics, dimensional_analysis ]
uuid: 483f818f-3f67-4cde-bbd6-d9f93c89ca47
---

[Lecture slides](./lecture_slides/T4 - Dimensional analysis - with solutions.pdf)  
[Lecture notes](./lecture_notes/dimensional analysis 2018-2019.pdf)

In lab tests it is not always possible to use the actual scale of the prototype,
actual flow speed, or actual fluid.
In these cases a model is used.

Dimensional analysis allows:

- experiments to be performed on a scaled models or using different fluids
- reduced number of parameters (and therefore tests), and also
- tests to have greater generality


# Introduction Physical Similarity

How to make sure a prototype and a scale model are physically similar

- Geometrical similarity --- all dimensions in all three coordinates have the same
  linear scale ratioi.

  This includes surface roughness (e.g. a 10x smaller model have 10x smaller roughness)
- Kinematic similarity ---velocities at corresponding points in the two flows are in the same direction and related by a constant scale factor in magnitude. Flow regimes (laminar, turbulent, compressible, etc.) must be the same.
- Dynamic similarity --- requires that the magnitude ratio of any two forces in one system must be the same as the magnitude ratio of the corresponding forces in the other system

Kinematic and Dynamic similarity are ensued by equality of the governing
nondimensional parameters.

As a general rule, dynamic and kinematic similarity are ensured if:

- for compressible flow, prototype and model Reynolds and Mach number, and specific heat ratio, are correspondingly equal
- For incompressible flow with no free surface, prototype and model Reynolds numbers are equal
- For incompressible flow with a free surface, Reynolds and Froude numbers are equal (may also require equality
  of Weber and cabiation number as well)

# Dimensions and Units

There are four basic dimensions for fluid dynamics:

- Mass (sometimes replaced by a force)
- Length
- Time
- Temperature

A nondimensional (dimensionless) group does not have any dimension or units, such
as the Reynolds number:

$$\text{Re} = \frac{\rho U x}{\mu}$$

# Example of Dimensional Analysis

![](./images/dim_anal_example.png)

In order to maintain a constant $\omega$ a certain torque $T$ is required to overcome
the shear stress exerted by the fluid on the surface of the rotating cone.

The experiment wants to answer: How are $T$ and $\omega$ related?

Doing this by testing the parameters individually would take too long as there are
several: $\omega$, $D$, $\rho$, $\mu$.
Doing 10 experiments per parameter would take $10^4$ experiments to get a full idea.
This would take ages.

This is where dimensional analysis helps as it means we only need 2 nondimensional groups
to relate the 4 parameters: momentum coefficient, $C_m$, and Reynolds number, $Re$.

$$C_m = \frac{T}{\frac12 \rho \omega^2D^5} \rightarrow C_m = g(\text{Re})$$

This suggests that the parameters can be reduced to a single parameters - the rotating
Reynolds number.
This drastically reduces the number of experiments required.

When you plot the data in a dimensional format, you would see multiple curves,
but when you plot them in a nondimensional format, you will observe that they fall
onto one curve:

![](./images/nondim-format.png)

This verifies the relationship $C_m = g(\text{Re})$ and our dimensional analysis.

You can use this nondimensional graph to find the behaviour for any value of $C_m$ and
$\text{Re}$ within the range of your experiment.
As always, you should be careful when extrapolating.

If nondimensional points do not fall on the same line, there may be a parameters that
are left out:

- Thickness of the gap between stator and rotor
- Cone angle
- Surface roughness
- Fluid heating due to viscous dissipation

# Buckingham Pi Theorem

If a physical process is fully described by $n$ variables and $k$ dimensions, then
$m = n - k$ dimensionless groups are sufficient to describe the process.

For example, say $n = 5$ and $k = 3$ such that $v_1 = f(v_2, v_3, v_4, v_5)$.
The theorem says that $m = 5 -2$ --- two nondimensional groups should be sufficient to
describe the process with a functional relationship: $\Pi_1 = g(\Pi_2)$.

Then pick the two variables that are of the most interest, e.g. $v_1$ and $v_5$, and
use the other three to form the dimensionless groups.

It is important that these three do not themselves form a dimensionless group.

$\Pi_1$, $\Pi_2$ are then formed by the variable we chose and power products of the
three others.:

$$\Pi_1 = v_1 v_2^av_3^bv_4^c$$

$$\Pi_2 = v_5 v_2^dv_3^ev_4^f$$

The exponents are obtained by knowing the groups are dimensionless:

$$[\Pi_1] = [v_1] [v_2]^a[v_3]^b[v_4]^c = [M]^0[L]^0[T]^0$$

$$[\Pi_2] = [v_5] [v_2]^d[v_3]^e[v_4]^f = [M]^0[L]^0[T]^0$$

where $M$, $L$, and $T$ are units of mass, length, and time respectively.

The two equations results in some simple simultaneous equations to solve to find the
coefficients $a$, $b$, $c$, $d$, $e$, $f$.

# Standard Nondimensional Groups in Fluids

This is not an exhaustive list.

## Reynolds Number

$$\text{Re} = \frac{\rho U L}{\mu}$$

Represents ratio of intertial forces over viscous forces.
Important in all viscous flows.

## Froude Number

$$\text{Fr} = \frac{U^2}{gL}$$

Represents ratio of inerital forces over gravitational forces.
Important in flows with interfaces (e.g. gas-liquid).

## Weber Number

$$\text{We} = \frac{rho U^2 L}{\sigma}$$

where $\sigma$ is the surface tension coefficient.

Represents ratio of inertial to capillary forces.
Important to flows with strong surface tension effects (e.g. droplets,
bubbles, jets)

## Strouhal Number

$$\text{St} = \frac{fL}{U}$$

where $f$ is frequency.

Important in flows with velocity oscillations.
$\text{St} \approx 0.21$ for $200 < \text{Re} < 10^5$.

## Mach Number

$$\text{Ma} = \frac U a$$

When $\text{Ma} > 0.3$, the flow should be considered compressible.

# Nondimensional Momentum Equation (From Navier-Stokes)

$$\frac{\delta \pmb{V^*}}{\delta t^*} + \pmb{V^*} \cdot (\nabla^* \pmb{V^*}) = -\nabla^* p^* + \frac{1}{\text{Re}}\nabla^{*2}\pmb{V^*}$$

Therefore for large values of Reynolds number, the viscous forces become negligible.

# Kinematic Similarity

An airfoil model and prototype are geometrically similar, with $1:\alpha$ length ratio:

- $x_m = \frac{x_p}{\alpha}$, $y_m = \frac{y_p}{\alpha}$
- $u_m$ at $(x_m, y_m)$ must have the same direction at $u_p$ at $(x_p, y_p)$
- $\frac{u_p}{u_m} = \beta$ is constant at homologous points

![](./images/vimscrot-2023-02-06T09:46:38,665212789+00:00.png)

# Dynamic Similarity

Dynamic similarity is similarity of forces.
Two systems have dynamic similarity when:

- they are geometrically similar
- identical kind of forces are parallel and related by a constant scale factor at corresponding homologous point

#### Example

Consider two homologous points around two geometrically similar airfoils:

Assume there are 3 forces acting: inertia, friction, and pressure.
These forces must form a closed polygon: $\pmb{F_i} = \pmb{F_f} + \pmb{F_p}$.

![](./images/vimscrot-2023-02-06T10:08:03,569393092+00:00.png)

$F_{f,p}$ and $F_{f,m}$ must be parallel and the same applies for the other two forces.
The force magnitude ratios must be related by a constant scale factor, therefore
this means the triangles must be geometrically similar.

We can estimate $F_i$ and $F_f$ (lecture slides p54-55):

$$F_i ~ \rho L^2U^2$$

$$F_f ~ \mu U L$$

Therefore:

$$\frac{F_i}{F_f} ~ \frac{\rho U L}{\mu} = \text{Re}$$

This means that at homologous points, the magnitude ratio of inertia/viscosity  in one system is the
same as that in the other system if the Reynolds number is the same.

# Consequences of Incomplete Similarity

In example 7 (lecture slides p57), it is not possible to achieve complete dynamic similarity.

- In the case of the example, tests are usually run with water, due to its convenience
- This violates Reynolds number similarity, but this does not matter much as Froude number is
  dominant parameter in free surface flows.
- Reynolds number of the model will be smaller and the model's data will be extrapolated to the desired
  Reynolds number:

  ![](./images/vimscrot-2023-02-06T10:42:58,992360702+00:00.png)

- Extrapolation increases uncertainty and it is left to the engineer to judge the validity of the data