Dimensional Analysis and Dynamic Similitude

Buckingham's pi theorem

If a physical phenomenon involves $n$ variables and these contain $m$ fundamental dimensions (M, L, T), then the relation among them can be expressed in terms of $(n - m)$ independent dimensionless groups ($\pi$-terms).

Step 1: Variables

$\Delta p = f(D, L, V, \rho, \mu, \varepsilon)$, so $n = 7$.

Step 2: Number of pi terms

$m = 3$ (M, L, T), so number of $\pi$ terms $= 7 - 3 = 4$.

Step 3: Repeating variables

Choose 3 repeating variables that together contain M, L and T and are not themselves dimensionless in a group: $D$ (geometric), $V$ (kinematic) and $\rho$ (dynamic).

Step 4: Form the pi terms

$\pi_1$ with $\Delta p$: $\pi_1 = \Delta p\, D^a V^b \rho^c$

• M: $1 + c = 0 \Rightarrow c = -1$
• T: $-2 - b = 0 \Rightarrow b = -2$
• L: $-1 + a + b - 3c = 0 \Rightarrow a = 0$

$$\pi_1 = \frac{\Delta p}{\rho V^2}$$

$\pi_2$ with $L$: length has the same dimension as $D$, so $\pi_2 = L/D$.

$\pi_3$ with $\mu$: $\pi_3 = \mu D^a V^b \rho^c$

• M: $1 + c = 0 \Rightarrow c = -1$
• T: $-1 - b = 0 \Rightarrow b = -1$
• L: $-1 + a + b - 3c = 0 \Rightarrow a = -1$

$$\pi_3 = \frac{\mu}{\rho V D} = \frac{1}{Re}$$

$\pi_4$ with $\varepsilon$: $\pi_4 = \varepsilon/D$ (relative roughness).

Step 5: Final relation

$$\frac{\Delta p}{\rho V^2} = \phi\left(\frac{L}{D}, Re, \frac{\varepsilon}{D}\right)$$

Experiment shows that $\Delta p$ is directly proportional to $L$. Therefore

$$\Delta p = \frac{L}{D}\,\rho V^2\, \phi_1\left(Re, \frac{\varepsilon}{D}\right) = f\,\frac{L}{D}\,\frac{\rho V^2}{2}$$

where $f = 2\phi_1(Re, \varepsilon/D)$ is the Darcy friction factor. This is the Darcy-Weisbach equation, $h_f = f L V^2/(2 g D)$.

Dimensionless numbers

Each is the ratio of the inertia force to another force.

Types of similarity

• Geometric similarity: the model and prototype have the same shape; all linear dimensions have the same scale ratio $L_r = L_p/L_m$. Angles are equal.
• Kinematic similarity: velocities at corresponding points are in the same direction and have the same ratio, so streamline patterns are similar. This needs geometric similarity and a fixed time scale.
• Dynamic similarity: forces at corresponding points have the same ratio (including direction). It requires the relevant dimensionless numbers of the model and prototype to be equal. Dynamic similarity implies kinematic and geometric similarity.

Incomplete similarity

In practice more than one force may matter, e.g. both gravity and viscosity for a ship. To keep both $Re$ and $Fr$ equal, the model fluid would need an unrealistic viscosity: for $Fr_m = Fr_p$, $V_r = \sqrt{L_r}$, and for $Re_m = Re_p$ with the same fluid, $V_r = 1/L_r$. Both cannot be satisfied with the same fluid, unless $L_r = 1$.

Therefore only the dominant force is matched:

• Gravity dominant (spillway, ship wave resistance, open channel): equate $Fr$.
• Viscosity dominant (pipe flow, submarine, aircraft at low speed): equate $Re$.

The other effects are estimated by correction factors or by running tests at several scales. A model is then said to have incomplete (partial) similarity.

Criterion of similarity

Flow over a spillway is a free-surface flow dominated by gravity, so the Froude number is the same in the model and prototype. The scale ratio $L_r = L_p/L_m = 25$. The same fluid (water) and the same $g$ are used.

Ratios from Froude law

$$\begin{aligned} V_r &= \sqrt{L_r} = \sqrt{25} = 5 \\ T_r &= \frac{L_r}{V_r} = \sqrt{L_r} = 5 \\ Q_r &= V_r L_r^2 = L_r^{5/2} = 25^{2.5} = 3125 \\ F_r &= \frac{\rho_p}{\rho_m} L_r^3 = L_r^3 = 15\,625 \end{aligned}$$

(a) Model discharge

$$Q_m = \frac{Q_p}{Q_r} = \frac{1200}{3125} = 0.384\ \text{m}^3/\text{s}$$

(b) Velocity and time ratios

$V_p/V_m = 5$ and $T_p/T_m = 5$.

(c) Force on prototype gate

$$F_p = 40 \times 15\,625 = 625\,000\ \text{N} = 625\ \text{kN}$$

(d) Prototype time

$$T_p = 5 \times 2 = 10\ \text{minutes}$$

Comment

The Reynolds number is not matched in the model ($Re_r = V_r L_r = 125$), so viscous effects and surface tension are not scaled correctly. For a large model with a turbulent flow this error is small. This is incomplete similarity.

Answer: $Q_m = 0.384\ \text{m}^3/\text{s}$; $V_r = T_r = 5$; $F_p = 625\ \text{kN}$; $T_p = 10\ \text{min}$ (Froude law)