Similarity Principle
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Similitude is a concept applicable to the testing of engineering models. A model is said to have similitude with the real application if the two share geometric similarity, kinematic 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 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. The concept of similitude is strongly tied to dimensional analysis.


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, temperature or the velocity 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. The following criteria are required to achieve similitude; * Geometric similarity – the model is the same shape as the application, usually scaled. *
Kinematic similarity In fluid mechanics, kinematic similarity is described as “the velocity at any point in the model flow is proportional by a constant scale factor to the velocity at the same point in the prototype flow, while it is maintaining the flow’s streaml ...
– 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 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 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 A base unit of measurement (also referred to as a base unit or fundamental unit) is a unit of measurement adopted for a ''base quantity''. A base quantity is one of a conventionally chosen subset of physical quantities, where no quantity in the ...
. The fundamental units are: meter,
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 shows that the system can be described with two dimensionless numbers and one independent variable. Dimensional analysis 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 ...
( R_e) and pressure coefficient (C_p). 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: : \begin &R_e = \left(\frac\right) &\longrightarrow &V_\text = V_\text \times \left(\frac\right)\times \left(\frac\right) \times \left(\frac\right) \\ &C_p = \left(\frac\right), F=\Delta p L^2 &\longrightarrow &F_\text =F_\text \times \left(\frac\right) \times \left(\frac\right)^2 \times \left(\frac\right)^2. \end The pressure (p) is not one of the five variables, but the force (F) is. The pressure difference (Δp) has thus been replaced with (F/L^2) in the pressure coefficient. This gives a required test velocity of: : V_\text = V_\text \times 21.9 . A model test is then conducted at that velocity and the force that is measured in the model (F_) is then scaled to find the force that can be expected for the real application (F_): : F_\text = F_\text \times 3.44 The power P in watts required by the submarine is then: : P mathrm=F_\text\times V_\text= F_\text mathrm\times 17.2 \ \mathrm 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 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

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


References

* * * * * * *


External links


MIT open courseware lecture notes on Similitude for marine engineering (pdf file)
{{DEFAULTSORT:Similitude (Model) Dimensional analysis Conceptual modelling