![Wind tunnel x-43](https://upload.wikimedia.org/wikipedia/commons/a/a7/Wind_tunnel_x-43.jpg)
Similitude is a concept applicable to the testing of
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
models
A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
. A model is said to have similitude with the real application if the two share geometric similarity,
kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion of points, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause them to move. Kinematics, as a fie ...
similarity and dynamic similarity. ''Similarity'' 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
Hydraulics (from Greek: Υδραυλική) is a technology and applied science using engineering, chemistry, and other sciences involving the mechanical properties and use of liquids. At a very basic level, hydraulics is the liquid counter ...
and
aerospace engineering
Aerospace engineering is the primary field of engineering concerned with the development of aircraft and spacecraft. It has two major and overlapping branches: aeronautical engineering and astronautical engineering. Avionics engineering is si ...
to test
fluid flow
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
conditions with
scaled models. It is also the primary theory behind many textbook
formula
In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...
s in
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
.
The concept of similitude is strongly tied to
dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
.
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
Scale or scales may refer to:
Mathematics
* Scale (descriptive set theory), an object defined on a set of points
* Scale (ratio), the ratio of a linear dimension of a model to the corresponding dimension of the original
* Scale factor, a number w ...
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 (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
or the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
and type of
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
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)](https://upload.wikimedia.org/wikipedia/commons/c/c8/Similitude_%28model%29.png)
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;
# All parameters required to describe the system are identified using principles from
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
.
#
Dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
is used to express the system with as few independent variables and as many
dimensionless parameters as possible.
# 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. Similitude suggests that the component fatigue life of the two objects will also be similar.
An example
Consider a
submarine
A submarine (or sub) is a watercraft capable of independent operation underwater. It differs from a submersible, which has more limited underwater capability. The term is also sometimes used historically or colloquially to refer to remotely op ...
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
A free body diagram consists of a diagrammatic representation of a single body or a subsystem of bodies isolated from its surroundings showing all the forces acting on it.
In physics and engineering, a free body diagram (FBD; also called a force ...
is constructed and the relevant relationships of force and velocity are formulated using techniques from
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
. The variables which describe the system are:
This example has five independent variables and three
fundamental units. The fundamental units are:
meter
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its prefi ...
,
kilogram
The kilogram (also kilogramme) is the unit of mass in the International System of Units (SI), having the unit symbol kg. It is a widely used measure in science, engineering and commerce worldwide, and is often simply called a kilo colloquially ...
,
second
The second (symbol: s) is the unit of time in the International System of Units (SI), historically defined as of a day – this factor derived from the division of the day first into 24 hours, then to 60 minutes and finally to 60 seconds ...
.
Invoking the
Buckingham π theorem
In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically me ...
shows that the system can be described with two dimensionless numbers and one independent variable.
Dimensional analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as m ...
is used to rearrange the units to form the
Reynolds number
In fluid mechanics, the Reynolds number () is a dimensionless quantity that helps predict fluid flow patterns in different situations by measuring the ratio between inertial and viscous forces. At low Reynolds numbers, flows tend to be domi ...
(
) and
pressure coefficient The pressure coefficient is a dimensionless number which describes the relative pressures throughout a flow field in fluid dynamics. The pressure coefficient is used in aerodynamics and hydrodynamics. Every point in a fluid flow field has its own ...
(
). 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:
:
The pressure (
) is not one of the five variables, but the force (
) is. The pressure difference (Δ
) has thus been replaced with (
) in the pressure coefficient. This gives a required test velocity of:
:
.
A model test is then conducted at that velocity and the force that is measured in the model (
) is then scaled to find the force that can be expected for the real application (
):
:
The power
in watts 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
Fluid mechanics
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 is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
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 (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
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;
Solid mechanics: structural similitude
![Scaledbeams](https://upload.wikimedia.org/wikipedia/commons/9/94/Scaledbeams.jpg)
Similitude analysis is a powerful engineering tool to design the scaled-down structures. Although both dimensional analysis and direct use of the governing equations may be used to derive the scaling laws, the latter results in more specific scaling laws. The design of the scaled-down composite structures can be successfully carried out using the complete and partial similarities. In the design of the scaled structures under complete similarity condition, all the derived scaling laws must be satisfied between the model and prototype which yields the perfect similarity between the two scales. However, the design of a scaled-down structure which is perfectly similar to its prototype has the practical limitation, especially for laminated structures. Relaxing some of the scaling laws may eliminate the limitation of the design under complete similarity condition and yields the scaled models that are partially similar to their prototype. However, the design of the scaled structures under the partial similarity condition must follow a deliberate methodology to ensure the accuracy of the scaled structure in predicting the structural response of the prototype. Scaled models can be designed to replicate the dynamic characteristic (e.g. frequencies, mode shapes and damping ratios) of their full-scale counterparts. However, appropriate response scaling laws need to be derived to predict the dynamic response of the full-scale prototype from the experimental data of the scaled model.
Notes
See also
*
Similitude of ship models
Manned models
Many research workers, hydraulics specialists and engineers have used scale models for over a century, in particular in towing tanks. Manned models are small scale models that can carry and be handled by at least one person on an ...
References
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External links
MIT open courseware lecture notes on Similitude for marine engineering (pdf file)
{{DEFAULTSORT:Similitude (Model)
Dimensional analysis
Conceptual modelling