A spillway model is to be built geometrically similar scale of 1/40 across a flume of 50cm width. The prototype is 20m high and the maximum head on it is expected to be 2m. (a) What height of the model and what head on the model should be used? (b) If the flow over the model at a particular head is 10 lps, what flow per m length of the prototype is expected? (c) If the negative pressure in the model is 150mm, what is the negative pressure in the prototype?

Fluid Mechanics Problem Solution

Problem Statement

A spillway model is to be built geometrically similar scale of 1/40 across a flume of 50cm width. The prototype is 20m high and the maximum head on it is expected to be 2m.

(a) What height of the model and what head on the model should be used?
(b) If the flow over the model at a particular head is 10 lps, what flow per m length of the prototype is expected?
(c) If the negative pressure in the model is 150mm, what is the negative pressure in the prototype?

Given Data

Scale ratio for length (L_r) 1/40

Width of model (B_m) 50 cm = 0.5 m

Height of prototype (H_p) 20 m

Head on prototype (Hd_p) 2 m

Flow through model (Q_m) 10 lps = 0.01 m³/s

Negative pressure in model (P_m) 150 mm = 0.15 m

Solution Approach

To solve this hydraulic model scaling problem, we’ll apply the principles of geometric, kinematic, and dynamic similarity. The solution involves using the length scale ratio to determine model dimensions, and applying appropriate scaling laws for flow and pressure.

Calculations

(a) Height of the Model and Head on the Model

Step 1: Calculate the height of the model using the length scale ratio:

L_r = H_m/H_p

1/40 = H_m/20

H_m = 20 × (1/40) = 0.5 m = 50 cm

Step 2: Calculate the head on the model using the same length scale ratio:

L_r = Hd_m/Hd_p

1/40 = Hd_m/2

Hd_m = 2 × (1/40) = 0.05 m = 5 cm

Height of model (H_m) = 0.5 m = 50 cm
Head on model (Hd_m) = 0.05 m = 5 cm

(b) Flow per Meter Length of Prototype

Step 1: Calculate the total flow through the prototype using Froude similarity (discharge scale is L_r^2.5):

Q_m/Q_p = L_r^2.5

0.01/Q_p = (1/40)^2.5

0.01/Q_p = (1/40)² × (1/40)^0.5

0.01/Q_p = (1/1600) × (1/6.325)

0.01/Q_p = 1/10120

Q_p = 0.01 × 10120 = 101.2 m³/s

Step 2: Calculate the width of the prototype using the length scale ratio:

L_r = B_m/B_p

1/40 = 0.5/B_p

B_p = 0.5 × 40 = 20 m

Step 3: Calculate the discharge per unit width of the prototype:

q_p = Q_p/B_p = 101.2/20 = 5.06 m³/s per meter

Flow per meter length of prototype = 5.06 m³/s/m

(c) Negative Pressure in the Prototype

Step 1: Calculate the negative pressure in the prototype using the length scale ratio:

L_r = P_m/P_p

1/40 = (-0.15)/P_p

P_p = (-0.15) × 40 = -6 m

Negative pressure in prototype = -6 m

Detailed Explanation

Hydraulic Model Scaling

When designing hydraulic structures such as spillways, engineers often build scale models to study their performance before constructing the full-scale prototype. The relationship between the model and prototype is governed by the laws of similarity.

Types of Similarity

Geometric Similarity: All dimensions in the model are related to corresponding dimensions in the prototype by a constant scale ratio (L_r). In this problem, L_r = 1/40.

Kinematic Similarity: Velocities at corresponding points in the model and prototype have the same direction and are related by a velocity scale ratio.

Dynamic Similarity: Forces at corresponding points are related by a constant force scale ratio. In free-surface flows like spillways, Froude number similarity is maintained, meaning the Froude number is the same in both model and prototype.

Scale Ratios Derived from Froude Similarity

For open channel flow problems like spillways, we use Froude number similarity. This gives us the following important scale ratios:

Length ratio: L_r = L_m/L_p = 1/40
Velocity ratio: V_r = V_m/V_p = L_r^1/2 = (1/40)^1/2 = 1/6.325
Discharge ratio: Q_r = Q_m/Q_p = L_r^5/2 = L_r^2.5 = (1/40)^2.5 = 1/10120
Pressure ratio: P_r = P_m/P_p = L_r = 1/40

Importance of Scale Models in Hydraulic Engineering

Scale models provide valuable insights into:

Hydraulic performance of the structure
Flow patterns and potential problems
Pressure distribution, especially negative pressures that could cause cavitation
Energy dissipation
Scouring potential downstream

Negative Pressures in Spillways

The negative pressure of -6 m in the prototype is significant and requires careful consideration. Negative pressures can lead to cavitation, which occurs when local pressure drops below vapor pressure, causing vapor bubbles to form. When these bubbles collapse, they can damage the concrete surface of the spillway. This phenomenon is particularly important in high-velocity flows over spillways.

Practical Considerations

In practice, engineers may modify the spillway design to reduce negative pressures, such as:

Adding aeration devices to introduce air into regions of negative pressure
Modifying the spillway profile to minimize pressure fluctuations
Using specially formulated concrete or protective coatings in areas susceptible to cavitation damage
Implementing stepped spillways to reduce flow velocities

The discharge calculated (5.06 m³/s per meter width) is typical for medium to large spillways. This value helps engineers determine the total width of the spillway needed to pass the design flood safely.