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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 modelsModel (physical)

A physical model is used in various contexts to mean a physical representation of some thing....
. A model is said to have similitude with the real application if the two share geometric similarity, kinematic similarity and dynamic similarity. SimilarityFacts About Similarity (geometry)

Geometry Two geometrical objects are called similar if one is congruent 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 aerospaceAerospace

Aerospace comprises air and space travel, manufacturing and associated research....
 engineering to test fluid flow conditions with scaledScaling

The term scaling can have several meanings:...
 models. It is also the primary theory behind many textbook formulaFormula

In mathematics and in the sciences, a formula is a concise way of expressing information symbolically, or a general relation...
s in fluid mechanicsFluid mechanics

Fluid mechanics is the subdiscipline of continuum mechanics that studies fluids, that is, liquids and gases....
.

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 pressurePressure

Pressure is the force per unit area applied on a surface in a direction perpendicular to that surface....
, temperatureTemperature

In thermodynamics, temperature is a measure of the tendency of an object or system to spontaneously give up energy....
 or the velocityVelocity

The velocity of an object is simply its speed in a particular direction....
 and type of fluidFluid

A subset of the phases of matter, fluids include liquids, gases, plasmas and, to some extent, plastic solids....
 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 similaritySimilarity (mathematics)

    In mathematics, there are several notions of 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 mechanicsContinuum mechanics

    Continuum mechanics is a branch of physics that deals with continuous matter, including both solids and fluids ....
    .
  2. Dimensional analysisDimensional analysis

    Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situat...
     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.

An example

Consider a submarineSubmarine

A submarine is a specialized watercraft that can operate underwater....
 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 diagramFree body diagram

Drawing a free body diagram is a method often used by physicists working out kinetics or other mechanics problems to show al...
 is constructed and the relevant relationships of force and velocity are formulated using techniques from continuum mechanicsContinuum mechanics

Continuum mechanics is a branch of physics that deals with continuous matter, including both solids and fluids ....
. The variables which describe the system are;

VariableApplicationScaled modelUnits
L11/40(unit length)
V5Calculate(m/s)
 1028998(kg/m3)
 1.881.00PaPascal (unit)

The pascal is the SI derived unit of pressure or stress ....
·s (N s/m2)
FCalculateTo be measuredNNewton

The newton is the SI unit of force....
     (kg m/s2)


This example has five independent variables and three fundamental units. The fundamental units are MetreMetre

The metre, or meter , is a measure of length....
,KilogramKilogram

The kilogram or kilogramme, is the SI base unit of mass....
,SecondFacts About Second

The second is the name of a unit of time, and today refers to the International System of Units base unit of time....
. (In the SISi

Si, si, or SI may stand for:...
 system of units newtonFacts About Newton

The newton is the SI unit of force....
s can be expressed in terms of kg m/s2.)

Invoking the Buckingham p theoremBuckingham p theorem

The Buckingham π theorem is a key theorem in dimensional analysis....
 shows that the system can be described with two dimensionless numbers and one independent variable (5 variables - 3 fundamental units => 2 dimensionless numbers).

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

In fluid mechanics, the Reynolds number is the ratio of inertial forces to viscous forces and is used for determining whethe...
  and Pressure coefficientPressure coefficient

The pressure coefficient is a dimensionless number used in aerodynamics and fluid mechanics, most often in the design and an...
 ). 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 simulationComputer simulation

A computer simulation or a computer model is a computer program that attempts to simulate an abstract model of a parti...
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 heliumHelium

|-| 3He || 0.000137%* || colspan="4" | He is stable with 1 neutron...
 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 numberReynolds number

In fluid mechanics, the Reynolds number is the ratio of inertial forces to viscous forces and is used for determining whethe...
, Pressure coefficientPressure coefficient

The pressure coefficient is a dimensionless number used in aerodynamics and fluid mechanics, most often in the design and an...
,
Compressible flows - Reynolds numberReynolds number

In fluid mechanics, the Reynolds number is the ratio of inertial forces to viscous forces and is used for determining whethe...
, Mach numberFacts About Mach number

Mach number is defined as a ratio of the speed of an object or flow relative to the speed of sound in the medium through w...
, Prandtl numberPrandtl number

The Prandtl Number is a dimensionless number approximating the ratio of momentum diffusivity and thermal diffusivity....
, Specific heat ratio
Flow excited vibration Strouhal numberStrouhal number

In dimensional analysis, the Strouhal number is a dimensionless number describing oscillating flow mechanisms....
Centrifugal compressors - Reynolds numberReynolds number Overview

In fluid mechanics, the Reynolds number is the ratio of inertial forces to viscous forces and is used for determining whethe...
, Mach numberMach number

Mach number is defined as a ratio of the speed of an object or flow relative to the speed of sound in the medium through w...
, Pressure coefficientPressure coefficient

The pressure coefficient is a dimensionless number used in aerodynamics and fluid mechanics, most often in the design and an...
, Velocity ratio
 

See also

  • Dimensionless number
  • Buckingham p theoremBuckingham p theorem

    The Buckingham π theorem is a key theorem in dimensional analysis....
  • Dimensional analysisDimensional analysis

    Dimensional analysis is a conceptual tool often applied in physics, chemistry, and engineering to understand physical situat...
  • MKS system of fundamental unitsSi

    Si, si, or SI may stand for:...