Similitude

Last updated November 29, 2023

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.

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 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 – 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.
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 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. The variables which describe the system are:

Variable	Application	Scaled model	Units
L (diameter of submarine)	1	1/40	(m)
V (speed)	5	calculate	(m/s)
$\rho$ (density)	1028	998	(kg/m³)
$\mu$ (dynamic viscosity)	1.88x10⁻³	1.00x10⁻³	Pa·s (N s/m²)
F (force)	calculate	to be measured	N (kg m/s²)

This example has five independent variables and three fundamental units. The fundamental units are: meter, kilogram, second.^[1]

Invoking the Buckingham π theorem shows that the system can be described with two dimensionless numbers and one independent variable.^[2]

Dimensional analysis is used to rearrange the units to form the Reynolds number ( $R_{e}$ ) 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{aligned}&R_{e}=\left({\frac {\rho VL}{\mu }}\right)&\longrightarrow &V_{\text{model}}=V_{\text{application}}\times \left({\frac {\rho _{a}}{\rho _{m}}}\right)\times \left({\frac {L_{a}}{L_{m}}}\right)\times \left({\frac {\mu _{m}}{\mu _{a}}}\right)\\&C_{p}=\left({\frac {2\Delta p}{\rho V^{2}}}\right),F=\Delta pL^{2}&\longrightarrow &F_{\text{application}}=F_{\text{model}}\times \left({\frac {\rho _{a}}{\rho _{m}}}\right)\times \left({\frac {V_{a}}{V_{m}}}\right)^{2}\times \left({\frac {L_{a}}{L_{m}}}\right)^{2}.\end{aligned}}

The pressure ( $p$ ) is not one of the five variables, but the force ( $F$ ) is. The pressure difference (Δ $p$ ) has thus been replaced with ( $F/L^{2}$ ) in the pressure coefficient. This gives a required test velocity of:

V_{\text{model}}=V_{\text{application}}\times 21.9

.

A model test is then conducted at that velocity and the force that is measured in the model ( $F_{model}$ ) is then scaled to find the force that can be expected for the real application ( $F_{application}$ ):

F_{\text{application}}=F_{\text{model}}\times 3.44

The power $P$ in watts required by the submarine is then:

P[\mathrm {W} ]=F_{\text{application}}\times V_{\text{application}}=F_{\text{model}}[\mathrm {N} ]\times 17.2\ \mathrm {m/s}

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 simulations 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;

Incompressible flow (see example above)	Reynolds number, pressure coefficient, (Froude number and Weber number for open channel hydraulics)
Compressible flows	Reynolds number, Mach number, Prandtl number, specific heat ratio
Flow-excited vibration	Strouhal number
Centrifugal compressors	Reynolds number, Mach number, pressure coefficient, velocity ratio
Boundary layer thickness	Reynolds number, Womersley number, dynamic similarity

Solid mechanics: structural similitude

Schematic of scaled composite laminated I-beams: prototype (top) and models with different scales and layups (bottom) Scaledbeams.jpg — Schematic of scaled composite laminated I-beams: prototype (top) and models with different scales and layups (bottom)

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.^[3] The design of the scaled-down composite structures can be successfully carried out using the complete and partial similarities.^[4] 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.^[5] 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.^[6]

Related Research Articles

In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities and units of measurement and tracking these dimensions as calculations or comparisons are performed. The term dimensional analysis is also used to refer to conversion of units from one dimensional unit to another, which can be used to evaluate scientific formulae.

The Mach number, often only Mach, is a dimensionless quantity in fluid dynamics representing the ratio of flow velocity past a boundary to the local speed of sound. It is named after the Austrian physicist and philosopher Ernst Mach.

Buckingham <span class="texhtml mvar" style="font-style:italic;">π</span> theorem Theorem in dimensional analysis

In engineering, applied mathematics, and physics, the Buckingham $π$ 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 meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters $π$ ₁, $π$ ₂, ..., $π$ _p constructed from the original variables, where k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.

In fluid mechanics, the Grashof number is a dimensionless number which approximates the ratio of the buoyancy to viscous forces acting on a fluid. It frequently arises in the study of situations involving natural convection and is analogous to the Reynolds number.

In fluid dynamics, the drag equation is a formula used to calculate the force of drag experienced by an object due to movement through a fully enclosing fluid. The equation is:

In fluid dynamics, the Darcy–Weisbach equation is an empirical equation that relates the head loss, or pressure loss, due to friction along a given length of pipe to the average velocity of the fluid flow for an incompressible fluid. The equation is named after Henry Darcy and Julius Weisbach. Currently, there is no formula more accurate or universally applicable than the Darcy-Weisbach supplemented by the Moody diagram or Colebrook equation.

In viscous fluid dynamics, the Archimedes number (Ar), is a dimensionless number used to determine the motion of fluids due to density differences, named after the ancient Greek scientist and mathematician Archimedes.

In continuum mechanics, the Froude number is a dimensionless number defined as the ratio of the flow inertia to the external field. The Froude number is based on the speed–length ratio which he defined as:

The Knudsen number (Kn) is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. The number is named after Danish physicist Martin Knudsen (1871–1949).

In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality.

Large eddy simulation (LES) is a mathematical model for turbulence used in computational fluid dynamics. It was initially proposed in 1963 by Joseph Smagorinsky to simulate atmospheric air currents, and first explored by Deardorff (1970). LES is currently applied in a wide variety of engineering applications, including combustion, acoustics, and simulations of the atmospheric boundary layer.

There are two different Bejan numbers (Be) used in the scientific domains of thermodynamics and fluid mechanics. Bejan numbers are named after Adrian Bejan.

The Womersley number is a dimensionless number in biofluid mechanics and biofluid dynamics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. It is named after John R. Womersley (1907–1958) for his work with blood flow in arteries. The Womersley number is important in keeping dynamic similarity when scaling an experiment. An example of this is scaling up the vascular system for experimental study. The Womersley number is also important in determining the thickness of the boundary layer to see if entrance effects can be ignored.

The Dean number (De) is a dimensionless group in fluid mechanics, which occurs in the study of flow in curved pipes and channels. It is named after the British scientist W. R. Dean, who was the first to provide a theoretical solution of the fluid motion through curved pipes for laminar flow by using a perturbation procedure from a Poiseuille flow in a straight pipe to a flow in a pipe with very small curvature.

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 dominated by laminar (sheet-like) flow, while at high Reynolds numbers, flows tend to be turbulent. The turbulence results from differences in the fluid's speed and direction, which may sometimes intersect or even move counter to the overall direction of the flow. These eddy currents begin to churn the flow, using up energy in the process, which for liquids increases the chances of cavitation.

In a nozzle or other constriction, the discharge coefficient is the ratio of the actual discharge to the ideal discharge, i.e., the ratio of the mass flow rate at the discharge end of the nozzle to that of an ideal nozzle which expands an identical working fluid from the same initial conditions to the same exit pressures.

In fluid mechanics, dynamic similarity is the phenomenon that when there are two geometrically similar vessels with the same boundary conditions and the same Reynolds and Womersley numbers, then the fluid flows will be identical. This can be seen from inspection of the underlying Navier-Stokes equation, with geometrically similar bodies, equal Reynolds and Womersley Numbers the functions of velocity (u’,v’,w’) and pressure (P’) for any variation of flow.

Geotechnical centrifuge modeling is a technique for testing physical scale models of geotechnical engineering systems such as natural and man-made slopes and earth retaining structures and building or bridge foundations.

In fluid mechanics, non-dimensionalization of the Navier–Stokes equations is the conversion of the Navier–Stokes equation to a nondimensional form. This technique can ease the analysis of the problem at hand, and reduce the number of free parameters. Small or large sizes of certain dimensionless parameters indicate the importance of certain terms in the equations for the studied flow. This may provide possibilities to neglect terms in the considered flow. Further, non-dimensionalized Navier–Stokes equations can be beneficial if one is posed with similar physical situations – that is problems where the only changes are those of the basic dimensions of the system.

Morris Muskat et al. developed the governing equations for multiphase flow in porous media as a generalisation of Darcy's equation for water flow in porous media. The porous media are usually sedimentary rocks such as clastic rocks or carbonate rocks.

References

↑ In the SI system of units, newtons can be expressed in terms of kg·m/s².
↑ 5 variables - 3 fundamental units ⇒ 2 dimensionless numbers.
↑ Rezaeepazhand, J.; Simitses, G.J.; Starnes, J.H. (1996). "Scale models for laminated cylindrical shells subjected to axial compression". Composite Structures. 34 (4): 371–9. doi:10.1016/0263-8223(95)00154-9.
↑ Asl, M.E.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2016). "Similitude Analysis of Composite I-Beams with Application to Subcomponent Testing of Wind Turbine Blades". Experimental and Applied Mechanics. Conference Proceedings of the Society for Experimental Mechanics Series. Vol. 4. Springer. pp. 115–126. doi:10.1007/978-3-319-22449-7_14. ISBN 978-3-319-22449-7.
↑ Asl, M.E.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2017). "Vibration prediction of thin-walled composite I-beams using scaled models". Thin-Walled Structures. 113: 151–161. doi: 10.1016/j.tws.2017.01.020 .
↑ Eydani Asl, M.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2015). "Predicting the Vibration Response in Subcomponent Testing of Wind Turbine Blades". Special Topics in Structural Dynamics. Conference Proceedings of the Society for Experimental Mechanics Series. Vol. 6. Springer. pp. 115–123. doi:10.1007/978-3-319-15048-2_11. ISBN 978-3-319-15048-2.

External links

MIT open courseware lecture notes on Similitude for marine engineering

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[1] In the SI system of units, newtons can be expressed in terms of kg·m/s².

[2] 5 variables - 3 fundamental units ⇒ 2 dimensionless numbers.

[3] Rezaeepazhand, J.; Simitses, G.J.; Starnes, J.H. (1996). "Scale models for laminated cylindrical shells subjected to axial compression". Composite Structures. 34 (4): 371–9. doi:10.1016/0263-8223(95)00154-9.

[4] Asl, M.E.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2016). "Similitude Analysis of Composite I-Beams with Application to Subcomponent Testing of Wind Turbine Blades". Experimental and Applied Mechanics. Conference Proceedings of the Society for Experimental Mechanics Series. Vol. 4. Springer. pp. 115–126. doi:10.1007/978-3-319-22449-7_14. ISBN 978-3-319-22449-7.

[5] Asl, M.E.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2017). "Vibration prediction of thin-walled composite I-beams using scaled models". Thin-Walled Structures. 113: 151–161. doi: 10.1016/j.tws.2017.01.020 .

[6] Eydani Asl, M.; Niezrecki, C.; Sherwood, J.; Avitabile, P. (2015). "Predicting the Vibration Response in Subcomponent Testing of Wind Turbine Blades". Special Topics in Structural Dynamics. Conference Proceedings of the Society for Experimental Mechanics Series. Vol. 6. Springer. pp. 115–123. doi:10.1007/978-3-319-15048-2_11. ISBN 978-3-319-15048-2.

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