Dimensional analysis and similitude

Wascana Valley Water has chosen Tidewater Pump Works of Moncton, New Brunswick, to supply the new Pump 2 for Northside. The pump is a single-stage, double-suction design with a 375 mm impeller, rated for 0.331 m³/s against 55 m at 1,780 rpm, and it draws about 210 kW. Before cutting patterns for the full-size casting, Tidewater will test a 150 mm geometric model in its test loop. Farah Siddiqui, Tidewater's senior design engineer, has invited Jun and Hana to witness the test, and Hana wants Jun to understand exactly how a model running at 3,560 rpm on a 42 L/s loop can tell the utility what a pump eight times its flow will do. The answer is dimensional analysis: the idea that physical laws do not depend on the units used to express them, and its practical consequence, similitude.

Why dimensional analysis works

Every equation that describes a physical process must be dimensionally homogeneous: each term must have the same dimensions. That requirement constrains how the variables can be combined. A pump's head depends on its flow, speed, diameter, the fluid's density and viscosity, and the roughness of its passages. Without any theory, dimensional analysis can reduce that list of seven variables to four dimensionless groups, and it guarantees that if two geometrically similar pumps have the same values of three of the groups, they will have the same value of the fourth.

The basic dimensions in mechanics are mass M, length L and time T. Velocity has dimensions L/T, force ML/T², pressure M/(LT²) and dynamic viscosity M/(LT). A dimensionless group, also called a Pi group, is a product of variables in which all the dimensions cancel. The Reynolds number \(\rho V D/\mu\) is the best known example.

The Buckingham Pi theorem

The Buckingham Pi theorem, stated by Edgar Buckingham in 1914 after earlier work by others, says that a relation among n dimensional variables that involve k independent dimensions can be rewritten as a relation among n − k independent dimensionless groups. The method of repeating variables builds the groups systematically:

First, list all n variables that affect the problem, including the one to be predicted. Second, write each variable's dimensions and count the independent dimensions k. Third, choose k repeating variables that together contain all the dimensions but do not themselves form a dimensionless group; for fluid problems, a density, a velocity or speed and a length are the usual choice. Fourth, combine each remaining variable with the repeating variables raised to unknown powers and solve for the powers that make the product dimensionless. Fifth, check each group and, if convenient, replace groups with products or powers of themselves to get familiar forms.

Common dimensionless numbers

Many Pi groups appear so often that they have names. Each is a ratio of two effects, and its size tells the engineer which effect dominates.

Writing the Navier-Stokes equations of Lesson 1 in dimensionless form shows where some of these come from. Scaling lengths by L, velocities by V, time by L/V and pressure by \(\rho V^2\) gives:

Any two flows with the same geometry, the same boundary conditions in scaled form and the same Reynolds and Froude numbers have identical dimensionless solutions. This is the formal basis for model testing: a model and its prototype obey the same dimensionless equation, so measurements on one can be scaled to the other.

Similitude and model testing

A model is similar to its prototype when three conditions are met. Geometric similarity requires all lengths to scale by the same ratio, including clearances and, ideally, surface roughness. Kinematic similarity requires the velocity ratios at corresponding points to be the same, which means the flow patterns look alike. Dynamic similarity requires the ratios of all forces to be the same, which is achieved when the governing dimensionless groups match. If dynamic similarity holds and the geometry is similar, kinematic similarity follows.

In practice, it is rarely possible to match every group. A model of an open-channel intake at 1:8 scale run with water cannot match both the Froude number and the Reynolds number: Froude similarity requires velocities to scale as the square root of length, while Reynolds similarity with the same fluid requires them to scale inversely with length. The engineer must decide which effect governs the behaviour being studied and accept a scale effect in the other. Intake and spillway models are run at equal Froude number because gravity controls the free surface and the formation of vortices, and they are built large enough that the Reynolds number stays high and viscous effects remain small.

Scaling the Pump 2 model test

For pumps, the flow and head coefficients are matched, and the Reynolds number is allowed to differ because both model and prototype operate in the fully turbulent range, where its effect is small. Equal coefficients give the pump scaling laws, which are the affinity laws of MEC-270 extended to a change in size:

Tidewater's model is a 0.40-scale copy, run at twice the prototype speed so that its head is large enough to measure accurately and its Reynolds number stays as high as the loop allows. The program applies the laws.

# Similitude: scaling a model pump test and an intake model
import math
g = 9.81
# Prototype: new Pump 2 for Northside (from Lesson 11)
Dp, np_, Qp, Hp, eta_p_guess = 0.375, 1780, 0.331, 55.0, None
# Model built by Tidewater Pump Works for its test loop
Dm, nm = 0.150, 3560
phi = Qp / (np_ / 60 * Dp**3)            # flow coefficient Q/(nD^3), n in rev/s
psi = g * Hp / ((np_ / 60)**2 * Dp**2)   # head coefficient gH/(n^2 D^2)
Qm = phi * (nm / 60) * Dm**3
Hm = psi * (nm / 60)**2 * Dm**2 / g
print(f"Pi groups: flow coefficient {phi:.4f}, head coefficient {psi:.2f}")
print(f"Model at {nm} rpm: Q = {Qm*1000:.1f} L/s, H = {Hm:.1f} m")
# Reynolds numbers based on tip speed and diameter
nu = 1.0e-6
Re_p = math.pi * Dp * np_ / 60 * Dp / nu
Re_m = math.pi * Dm * nm / 60 * Dm / nu
print(f"Reynolds numbers: prototype {Re_p:.2e}, model {Re_m:.2e} (ratio {Re_p/Re_m:.1f})")
# Measured model efficiency and the Moody step-up formula
eta_m = 0.815
eta_p = 1 - (1 - eta_m) * (Dm / Dp) ** 0.25
print(f"Model efficiency {eta_m:.3f} -> prototype estimate {eta_p:.3f} (Moody, exponent 1/4)")
# Power coefficient check: P/(rho n^3 D^5)
rho = 998.0
Pm = rho * g * Qm * Hm / eta_m
Pp = rho * g * Qp * Hp / eta_p
print(f"Model shaft power {Pm/1000:.1f} kW; prototype {Pp/1000:.0f} kW")
print(f"Power coefficients: model {Pm/(rho*(nm/60)**3*Dm**5):.3f}, prototype {Pp/(rho*(np_/60)**3*Dp**5):.3f}")

# Froude-scaled model of the intake sump (1:8) for vortex testing
Lr = 1 / 8
Q_sump = 0.66                            # two pumps, m3/s
Q_model = Q_sump * Lr**2.5
V_ratio = Lr**0.5
print(f"\nIntake sump model 1:8: Q = {Q_model*1000:.2f} L/s, velocities x {V_ratio:.3f}, times x {Lr**0.5:.3f}")
Re_bell_p = 1.1 * 0.6 / nu                # bell velocity about 1.1 m/s, bell diameter 0.6 m
Re_bell_m = 1.1 * V_ratio * 0.6 * Lr / nu
print(f"Bell-mouth Reynolds number: prototype {Re_bell_p:.1e}, model {Re_bell_m:.1e}")

Pi groups: flow coefficient 0.2116, head coefficient 4.36
Model at 3560 rpm: Q = 42.4 L/s, H = 35.2 m
Reynolds numbers: prototype 1.31e+07, model 4.19e+06 (ratio 3.1)
Model efficiency 0.815 -> prototype estimate 0.853 (Moody, exponent 1/4)
Model shaft power 17.9 kW; prototype 209 kW
Power coefficients: model 1.132, prototype 1.081

Intake sump model 1:8: Q = 3.65 L/s, velocities x 0.354, times x 0.354
Bell-mouth Reynolds number: prototype 6.6e+05, model 2.9e+04

To match the prototype's duty point, the model must deliver 42.4 L/s against 35.2 m at 3,560 rpm. Tidewater's loop can do that easily. The Reynolds number of the model is about a third of the prototype's, both above 10⁶, which is why the head curve scales almost perfectly. Efficiency does not scale perfectly. Friction losses in the smaller passages are relatively larger, and the clearances at the wear rings cannot be made 0.4 times as small as the prototype's. Larger pumps are therefore more efficient than their models. Several empirical step-up formulas exist; the oldest and simplest, attributed to Moody, scales the losses with the diameter ratio to the one-quarter power:

With a measured model efficiency of 81.5 percent, the prototype estimate is 85.3 percent. Modern standards for hydraulic machine model tests include more detailed methods that separate friction losses from other losses, and contracts state which method will be used before the test. For Pump 2, the contract with Tidewater uses the full-size factory acceptance test, not the model, to verify the guaranteed efficiency; the model test is a design check.

Froude models of intakes and sumps

The second model in Jun's work term is a hydraulic model of the new raw-water intake sump. Pump intakes can draw air-entraining surface vortices and submerged vortices that disturb the flow into the pump's bell mouth, cause vibration and reduce efficiency. Pump intake design standards, such as ANSI/HI 9.8, give recommended dimensions and say when a physical model study is warranted, typically for larger stations or unusual layouts. A consultant will build a 1:8 model of the sump in a university hydraulics laboratory.

With Froude similarity, velocities scale as the square root of the length ratio, and flows as the length ratio to the 2.5 power. The sump's two pumps take 0.66 m³/s at full scale, so the model needs only 3.65 L/s. Model times are shorter by the same factor as velocities, so an hour of prototype operation takes about 21 minutes in the model. The model's Reynolds number at the bell mouth is lower than the prototype's by a factor of 8^1.5, about 23, which means vortices may appear slightly weaker in the model. Laboratories compensate by also testing at higher flows than the Froude value, sometimes at prototype bell velocity, to check that the conclusions do not change. This is the practical meaning of a scale effect.

Distorted models and the limits of similitude

River models often use a larger scale vertically than horizontally so that depths remain measurable and the flow stays turbulent; these are called distorted models, and they reproduce water levels well but not local details such as scour around a pier. Wind-tunnel models of buildings and structures cannot match the full-scale Reynolds number at all, but for sharp-edged bodies the drag coefficient is nearly independent of Reynolds number once the flow is turbulent, so tests at a lower Reynolds number are still useful. Rounded bodies, such as the elevated water tank in Lesson 6, are harder, because their drag changes sharply with Reynolds number, and engineers roughen the model surface to trip the boundary layer and imitate the full-scale behaviour.

In Canada, the National Research Council's laboratories in Ottawa test hydraulic and coastal models and have a long history of wind-tunnel testing, and several universities run hydraulic laboratories that do intake and spillway studies for utilities. The test standard for the pump's final acceptance test is cited in the purchase order; for Wascana Valley Water, it is an ISO-based pump test standard with tolerances agreed in advance.