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Similitude (model)

 
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Similitude (model)
 
 
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Similitude is a concept used in the testing of engineering models
Model (physical)

A physical model is a smaller or larger physical copy of an object. The object being modelled may be small or large .The geometry of the model and the object it represents are often similar in the sense that one is a rescaling of the other; in such cases the Scale is an important characteristic....
. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity. Similarity
Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. Equivalently and more precisely, one is congruence to the result of a uniform Scaling of the other....
 and similitude are interchangeable in this context.

The term dynamic similitude is often used as a catch-all because it implies that geometric and kinematic similitude have already been met.

Similitude's main application is in hydraulic and aerospace
Aerospace

Aerospace comprises the atmosphere of Earth and surrounding outer space. Typically the term is used to refer to the industry that researches, designs, manufactures, operates, and maintains vehicles moving through Aircraft and Space exploration....
 engineering to test fluid flow conditions with scaled
Scaling

Scaling may refer to:* Scaling , a linear transformation that enlarges or diminishes objects* Scaling , a network's ability to function as the number of people or computers on the network increases....
 models.






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Wind Tunnel X 43
Similitude is a concept used in the testing of engineering models
Model (physical)

A physical model is a smaller or larger physical copy of an object. The object being modelled may be small or large .The geometry of the model and the object it represents are often similar in the sense that one is a rescaling of the other; in such cases the Scale is an important characteristic....
. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity. Similarity
Similarity (geometry)

Two geometrical objects are called similar if they both have the same shape. Equivalently and more precisely, one is congruence to the result of a uniform Scaling of the other....
 and similitude are interchangeable in this context.

The term dynamic similitude is often used as a catch-all because it implies that geometric and kinematic similitude have already been met.

Similitude's main application is in hydraulic and aerospace
Aerospace

Aerospace comprises the atmosphere of Earth and surrounding outer space. Typically the term is used to refer to the industry that researches, designs, manufactures, operates, and maintains vehicles moving through Aircraft and Space exploration....
 engineering to test fluid flow conditions with scaled
Scaling

Scaling may refer to:* Scaling , a linear transformation that enlarges or diminishes objects* Scaling , a network's ability to function as the number of people or computers on the network increases....
 models. It is also the primary theory behind many textbook formula
Formula

In mathematics and in the sciences, a formula is a concise way of expressing information symbolically , or a general relationship between quantities....
s in fluid mechanics
Fluid mechanics

Fluid mechanics is the study of how fluids move and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion....
.

Overview

Engineering models are used to study complex fluid dynamics problems where calculations and computer simulations aren't reliable. Models are usually smaller than the final design, but not always. Scale models allow testing of a design prior to building, and in many cases are a critical step in the development process.

Construction of a scale model, however, must be accompanied by an analysis to determine what conditions it is tested under. While the geometry may be simply scaled, other parameters, such as pressure
Pressure

Pressure is the force per unit area applied to an object in a direction surface normal to the surface. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure....
, temperature
Temperature

In physics, temperature is a physical property of a Physical system that underlies the common notions of hot and cold; something that feels hotter generally has the greater temperature....
 or the velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
 and type of fluid
Fluid

A fluid is defined as a substance that continually deforms under an applied shear stress. All liquids and all gases are fluids. Fluids are a subset of the Phase and include liquids, gas, Plasma physics and, to some extent, plasticity ....
 may need to be altered. Similitude is achieved when testing conditions are created such that the test results are applicable to the real design.

Similitude (model)
The following criteria are required to achieve similitude;
  • Geometric similarity - The model is the same shape as the application, usually scaled.
  • Kinematic similarity - Fluid flow of both the model and real application must undergo similar time rates of change motions. (fluid streamlines are similar)
  • Dynamic similarity - Ratios of all forces acting on corresponding fluid particles and boundary surfaces in the two systems are constant.


To satisfy the above conditions the application is analyzed;
  1. All parameters required to describe the system are identified using principles from Continuum mechanics
    Continuum mechanics

    Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids ....
    .
  2. Dimensional analysis
    Dimensional analysis

    Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving certain physical quantities....
     is used to express the system with as few independent variables and as many dimensionless parameters as possible.
  3. The values of the dimensionless parameters are held to be the same for both the scale model and application. This can be done because they are dimensionless and will ensure dynamic similitude between the model and the application. The resulting equations are used to derive scaling laws which dictate model testing conditions.


It is often impossible to achieve strict similitude during a model test. The greater the departure from the application's operating conditions, the more difficult achieving similitude is. In these cases some aspects of similitude may be neglected, focusing on only the most important parameters.

The design of marine vessels remains more of an art than a science in large part because dynamic similitude is especially difficult to attain for a vessel that is partially submerged: a ship is affected by wind forces in the air above it, by hydrodynamic forces within the water under it, and especially by wave motions at the interface between the water and the air. The scaling requirements for each of these phenomena differ, so models cannot replicate what happens to a full sized vessel nearly so well as can be done for an aircraft or submarine -- each of which operates entirely within one medium.

Similitude is a term used widely in fracture mechanics relating to the strain life approach. Under given loading conditions the fatigue damage in an un-notched specimen is comparable to that of a notched specimen. Smilitude suggests that the component fatigue life of the two objects will also be similar.

An example

Consider a submarine
Submarine

A submarine is a watercraft capable of independent operation below water. It differs from a submersible, which has only limited underwater capability....
 modeled at 1/40th scale. The application operates in sea water at 0.5 °C, moving at 5 m/s. The model will be tested in fresh water at 20 °C. Find the power required for the submarine to operate at the stated speed.

A free body diagram
Free body diagram

A free body diagram is a pictorial representation often used by physicists and engineers to analyze the forces acting on a free body. A free body diagram shows all contact force and non-contact force forces acting on the body....
 is constructed and the relevant relationships of force and velocity are formulated using techniques from continuum mechanics
Continuum mechanics

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and mechanical behavior of materials modeled as a continuum, e.g., solids and fluids ....
. The variables which describe the system are:

VariableApplicationScaled modeUnits
L (diameter
Length

Length is the long dimension of any object. The length of a thing is the distance between its ends, its linear extent as measured from end to end....
 of submarine)		11/40(m)
V (speed
Speed

Speed is the rate of Motion , or equivalently the rate of change of distance.Speed is a Scalar quantity with dimensions length/time; the equivalent Vector quantity to speed is velocity....
)		5calculate(m/s)
(density
Density

The density of a material is defined as its mass per unit volume. The symbol of density is ....
)		1028998(kg/m3)
(dynamic viscosity
Viscosity

Viscosity is a measure of the Drag of a fluid which is being deformed by either shear stress or extensional stress. In everyday terms , viscosity is "thickness"....
)		1.88x10-31.00x10-3Pa
Pascal (unit)

The pascal is the SI derived unit of pressure, stress , Young's modulus and tensile strength. It is a measure of force per unit area i.e. equivalent to one newton per square meter or one joule per cubic meter....
·s (N s/m2)
F (force
Force

In physics, a force is that which can cause an object with mass to change its velocity. Force has both Euclidean_vector#Length of a vector and Direction , making it a Vector quantity....
)		calculateto be measuredN
Newton

The newton is the International System of Units SI derived unit of force, named after Isaac Newton in recognition of his work on classical mechanics....
   (kg m/s2)


This example has five independent variables and three fundamental units. The fundamental units are: metre
Metre

The metre or meter is a Unit of measurement of length. It is the SI base unit of length in the metric system and in the International System of Units , used around the world for general and scientific purposes....
, kilogram
Kilogram

The kilogram or kilogrammeThe spelling kilogram is used by the International Committee for Weights and Measures and the U.S....
, second
Second

The second , sometimes abbreviated sec., is the name of a units of measurement of time, and is the International System of Units SI base unit of time....
.

Invoking the Buckingham p theorem
Buckingham p theorem

The Buckingham p theorem is a key theorem in dimensional analysis. The theorem loosely states that if we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k  independent Fundamental unit, then the original expression is equivalent to an equa...
 shows that the system can be described with two dimensionless numbers and one independent variable.

Dimensional analysis is used to re-arrange the units to form the Reynolds number
Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
  and Pressure coefficient
Pressure coefficient

The pressure coefficient is a dimensionless number less than one which describes the relative pressures throughout a flow field in fluid dynamics....
 . These dimensionless numbers account for all the variables listed above except F, which will be the test measurement. Since the dimensionless parameters will stay constant for both the test and the real application, they will be used to formulate scaling laws for the test.

Scaling Laws:
   
   


This gives a required test velocity of:

.

The force measured from the model at that velocity is then scaled to find the force that can be expected for the real application:

The power required by the submarine is then:

Note that even though the model is scaled smaller, the water velocity needs to be increased for testing. This remarkable result shows how similitude in nature is often counterintuitive.

Typical applications

Similitude has been well documented for a large number of engineering problems and is the basis of many textbook formulas and dimensionless quantities. These formulas and quantities are easy to use without having to repeat the laborious task of dimensional analysis and formula derivation. Simplification of the formulas (by neglecting some aspects of similitude) is common, and needs to be reviewed by the engineer for each application.

Similitude can be used to predict the performance of a new design based on data from an existing, similar design. In this case, the model is the existing design. Another use of similitude and models is in validation of computer simulation
Computer simulation

A computer simulation, a computer model or a computational model is a computer program, or network of computers, that attempts to simulation an abstract model of a particular system....
s with the ultimate goal of eliminating the need for physical models altogether.

Another application of similitude is to replace the operating fluid with a different test fluid. Wind tunnels, for example, have trouble with air liquefying in certain conditions so helium
Helium

Helium is a colorless, odorless, tasteless, non-toxic, inert monatomic chemical element that heads the noble gas group in the periodic table and whose atomic number is 2....
 is sometimes used. Other applications may operate in dangerous or expensive fluids so the testing is carried out in a more convenient substitute.

Some common applications of similitude and associated dimensionless numbers;
Incompressible flow (see example above)- Reynolds number
Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
, Pressure coefficient
Pressure coefficient

The pressure coefficient is a dimensionless number less than one which describes the relative pressures throughout a flow field in fluid dynamics....
, (Froude number
Froude number

The Froude number is a dimensionless number comparing inertial and gravitational forces. It may be used to quantify the resistance of an object moving through water, and compare objects of different sizes....
 and Weber number
Weber number

The Weber number is a dimensionless number in fluid mechanics that is often useful in analysing fluid flows where there is an interface between two different fluids, especially for multiphase flows with strongly curved surfaces....
 for open channel hydraulics)
Compressible flows - Reynolds number
Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
, Mach number
Mach number

Mach number is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance. It is commonly used to represent an object's speed, when it is travelling at the speed of sound....
, Prandtl number
Prandtl number

The Prandtl number is a dimensionless number approximating the ratio of momentum diffusivity and thermal diffusivity. It is named after the German physicist Ludwig Prandtl....
, Specific heat ratio
Flow excited vibration Strouhal number
Strouhal number

In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms. The parameter is named after Vincenc Strouhal, a German physicist who experimented in 1878 with wires experiencing vortex shedding and singing in the wind....
Centrifugal compressors - Reynolds number
Reynolds number

In fluid mechanics and heat transfer, the Reynolds number is a dimensionless number that gives a measure of the ratio of inertial forces to viscosity forces and, consequently, it quantifies the relative importance of these two types of forces for given flow conditions....
, Mach number
Mach number

Mach number is the speed of an object moving through air, or any fluid substance, divided by the speed of sound as it is in that substance. It is commonly used to represent an object's speed, when it is travelling at the speed of sound....
, Pressure coefficient
Pressure coefficient

The pressure coefficient is a dimensionless number less than one which describes the relative pressures throughout a flow field in fluid dynamics....
, Velocity ratio
 


See also

  • Dimensionless number
  • Buckingham p theorem
    Buckingham p theorem

    The Buckingham p theorem is a key theorem in dimensional analysis. The theorem loosely states that if we have a physically meaningful equation involving a certain number, n, of physical variables, and these variables are expressible in terms of k  independent Fundamental unit, then the original expression is equivalent to an equa...
  • Dimensional analysis
    Dimensional analysis

    Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situations involving certain physical quantities....
  • MKS system of fundamental units
    Si

    Si, si, or SI may refer to :...


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