[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis 1

Similitude

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Dr. Mohammad N. Almasri http://sites.google.com/site/mohammadnablus/Home

Fluid Mechanics

[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Hydraulic Scale Models

Hydraulic projects can be massive, expensive, and complex

Examples of such projects include dams, spillways, culverts, and so on

Even in relatively simple hydraulic projects it is often impossible to predict the exact flow patterns and actual dynamic forces and stresses on a hydraulic structure without conducting a model study

In more complex projects, models are almost mandatory

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Hydraulic Scale Models

A model may be thought of as a prediction tool in which the full-scale “prototype” is reproduced at another scale

A prototype and its model are two similar physical systems of different sizes

The model is smaller in size

Models are used to obtain efficient and satisfactory designs

Hydraulic models are powerful tools in design 3

[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Hydraulic Scale Models

Fluids different from those in prototypes may be used in models

Models may also be subject to different pressures. Hydraulic models allow flexibility in testing prototypes under various conditions to which they might be exposed

The cost of building a model is relatively affordable, especially for the information and benefits they provide

The total cost of a hydraulic project may be increased by two or even five percent, yet better designs can be achieved and many problems can be avoided before the execution of prototype projects

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity

Three types of similarity should exist between a prototype and its model in order to ensure that the model is completely similar to and thus accurately represents the prototype

The main types of similarity are

Geometric

Kinematic

Dynamic

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity

A model should be similar in shape to the prototype (geometric replica)

Any unspecialized person can recognize that the model is just another version of the prototype but with a different scale (generally smaller)

All dimensions in the prototype are reduced by a fixed factor called the scale ratio

A 1/50 model means that each dimension in the model is 1/50 of that in the prototype

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity

Consider, for example, a ship with a length Lp, width Bp, and a total depth of dp

A model is to be built in the laboratory to test the performance and ability of the ship to accommodate lateral dynamic forces due to wave action

The length, width, and total depth of the proposed model are Lm, Bm, and dm, respectively

In order to ensure the geometric similarity, the following conditions should hold:

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity

where L is the length ratio generally less than 1 and may be small as 1/1000 or even less

In the relations to be used in this section all quantities with subscript p will correspond to prototype, those with subscript m to model, and those subscript r to the ratios of model to prototype quantities

From the above equation one can deduce that:

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity

where Ar is the area ratio, and Am and Ap are the areas of model and prototype, respectively

For the horizontal area, Am = LmBm and Ap = LpBp and for the area in the vertical section, Am = Lmdm and Ap = Lpdp

In a similar way, one can write:

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity Example (1)

A model for a huge circular water conduit is constructed. The prototype has a diameter of 5 m and the roughness height of its concrete wall, e, is 3 mm

The model is built to a scale ratio of 1/10; it has a length of 4 m

Estimate the diameter, cross-sectional area, and roughness height of the model and the length of the prototype

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity Example (1)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity Example (2)

A prototype of a volume of 1000 m3 is investigated by a hydraulic model 1 m3 in volume

If the length of the model is 2.25 m, what is the length of the prototype?

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Geometric Similarity Example (2)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Kinematic Similarity

Kinematic similarity is defined as the similarity of motion between two geometrically similar systems (prototype and model)

It requires that the shape of the stream lines at any time and for any two corresponding points is the same in both systems

Geometric similarity does not ensure kinematic similarity, but kinematic similarity does ensure geometric similarity

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Kinematic Similarity

The velocity ratio between any two corresponding points in two kinematically similar systems should be constant

The same is true for discharge, acceleration, time, kinematic viscosity, and all other parameters involving time in their dimensions

For kinematically similar systems, one can write:

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity

Dynamic similarity is defined as the similarity of forces of two kinematically similar systems

The two systems must therefore be geometrically similar

Motion is caused by the action of one or more forces. Forces accelerate or decelerate fluid particles or maintain their constant velocity against viscous and/or frictional forces which tend to bring moving particles to rest

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity

This relation between forces and motion requires that the force action in the model must be identical to that in the prototype

The ratio of any two forces acting in the model must be equal to the ratio of the corresponding two forces acting in the prototype

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity

In general, three types of forces act on a hydraulic system:

(1) external forces such as the gravity force or force due to pressure head in pipes and so on

(2) internal forces involving physical characteristics of the fluid itself such as viscous force and surface tension force

(3) resultant forces, such as drag force on a moving body in a fluid or pressure force on hydraulic structures

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity

The ratio between two different types of forces in two dynamically similar systems (model and prototype) must be constant

Assuming that FI, Fg Fv, Fσ, Fp, and Fe are the inertia force, gravity force, viscous force, surface tension force, pressure force, and elastic force, respectively, then one can write

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity

Regardless of the specific type of the force, its dimensions are equivalent to the dimensions of mass (M) multiplied by acceleration (a)

Therefore,

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity

The work (W) is dimensionally equal to the force multiplied by distance. Therefore

The power (P) is dimensionally equal to the force multiplied by velocity. Therefore,

The pressure (p) is dimensionally equal to the force divided by the area. Therefore,

The specific weight (γ) is dimensionally equal to the force or weight divided by volume. Therefore

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (1)

A 1:10 scale model is constructed for a hydraulic structure. Determine the ratios of force and pressure

If the force and pressure on the model are equal to 1.9 kN and 500 N/m2, respectively, what are the corresponding values in the prototype? Assume the same fluid is used in both the model and prototype

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (1)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (2)

The force required to tow a 1:25 model of a boat at a speed of 2.5 m/s is 0.6 N. What are the corresponding speed and force in the prototype?

If 20 seconds are required by the model to fulfill certain maneuvers, what is the corresponding time in the prototype?

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (2)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (3)

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A model is built for a spillway to a scale of 1:20. The discharge over the spillway is expected to be 1200 m3/s under a head of 3 m

What are the corresponding values of discharge and head in the laboratory?

If the horsepower dissipated by the hydraulic jump in the model is 0.75 HP, what would be the horsepower dissipated in the prototype?

[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (3)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (4)

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A 1:10 model for a submarine is tested in the laboratory. The submarine is required to be tested when it operates 50 m below the seawater level and moves at a velocity of 12 mph

What are the corresponding depth and velocity for the model knowing that Vr between the water used in the model and seawater is 0.95?

[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (4)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (5)

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A model is built to determine the energy loss in a pipeline 1.2 m in diameter and 10 km in length. The pipe conveys a liquid of a density of 0.8 g/cm3 and a dynamic viscosity of 0.02 poise at a rate of 1.8 m3/s.

Tests are conducted on a 12-cm-diameter pipe using freshwater at 20°C (ρ = 0.998 g/cm3, μ = 0.01 poise). The head loss in a length of 20 m of the model is 150 cm

Evaluate the model discharge and the head loss in the entire pipe.

If the pressure at a certain point in the model is 700 kN/m2, what is the corresponding pressure in the actual pipe?

[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (5)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (5)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Types of Similarity Dynamic Similarity Example (5)

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Dynamic Similarity: Common Forces

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[8] Fall – 2012 – Fluid Mechanics Dr. Mohammad N. Almasri [-] Dimensional Analysis

Examples

Question 8.31

Question 8.33

Question 8.45

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