Dimensional analysis

Dimensional analysis is a mathematical method to detect the interaction of physical quantities in natural phenomena without having to know the physical process underlying a formula or a precise law. Your application based on applied mathematics, practical observation, implementation and analysis of experiments and on intuitive physical understanding. It has proved itself especially in fluid mechanics.

For realistic problems in engineering and science, the associated mathematical equations are in most cases due to complex boundary conditions, not analytically, but only numerically solvable, ie by computer methods. The application of dimensional analysis to geometrically similar, but the laboratory or numerically more manageable models allowed here often very accurate conclusions as to the solution of highly complex original problem.

Dimensional analysis is used mainly in experimental physics, engineering, but also in medicine and biology application.

• 4.1 What are Π - factors?
• 4.2 How many Π - factors are there?
• 4.3 Formal procedure for dimensional analysis
• 4.4 Control options
• 4.5 conclusions
• 4.6 Locating a fundamental system of Π - factors 4.6.1 Analytical Procedure
• 4.6.2 Method of guessing
• 4.6.3 Evaluation of the methods
• 4.6.4 Education physically useful fundamental systems

• 5.1 Statements of the Π - theorem
• 5.2 conclusions

• 7.1 Full and partial similarity model
• 7.2 Model Laws
• 7.3 Model tests

• 8.1 Galileo's law of falling bodies
• 8.2 Euler's buckling bar
• 8.3 In flow-around body fluids 8.3.1 flow resistance of a sphere
• 8.3.2 Models of ships
• 8.3.3 Models of airplanes and submarines

Areas of application

The problems and possible applications are manifold. Some topics are:

• Aerodynamics and the behavior of bodies in flowing fluids in general. Around the investigation and optimization of the aerodynamic characteristics of aircraft and suspension bridges.
• Flow resistance and pressure drop in flow through tubes.
• Formation of waves and their propagation in various media.
• Diffusion and heat transport.
• Explosion processes
• Material strength tests and crash tests.
• Are geoscientifically interesting effects of earthquakes (eg for high-rise buildings ), Durchsickerungsvorgänge in the soil, bearing capacity of foundations for buildings or landslides and avalanches.
• In hydraulic engineering, the outflow channels and the sediment transport in rivers.
• In medicine and biology, the topic of bionics, the bloodstream or plant growth.

A dimensional analysis of these operations provides useful proportionalities, requirements for calibration of model experiments (see model laws) and concrete evidence for variation studies. Repeats enough that in order to derive functional relationships. In any case, it contributes to a better understanding of the problem.

History and Overview

Already physicists like Ludwig Prandtl, Theodore von Kármán, Albert Shields, Johann Nikuradse and John William Strutt, 3rd Baron Rayleigh, the end of the 19th and beginning of the 20th century as the first deeper in to the properties of currents and moving bodies fluids employed, used the dimensional analysis to close the laboratory experiment with controllable boundary conditions on the behavior of physical problems with geometrically similar bodies or with other fluids toughness and density. This principle of similarity, ie the possibility to investigate physical phenomena at different scales, forms the basis of the similarity theory. Frequently, this theory is also referred to as a model theory.

The similarity of the underlying theory dimensional analysis implies that each dimension bound physical formula can be converted into a dimensionless, ie underlying physical units of shape. To this end, and divided by a product of powers of variables and simultaneously increases the individual in any of magnitude:

So that the left and the right side of the equation is dimensionless. The dimension of purity and thus the accuracy of any physical relationship can be examined in the light of this statement. Satisfies a formula not meet these criteria, then it is not physically accurate. This applies to many approximation formulas, neglect unaware of certain sizes. It is also clear that only variables can be subtracted, so are comparable with each other and added the same dimension. The arguments about trigonometric and other transcendental functions must be dimensionless numbers consequently.

The important, building on the dimensional analysis, and independently of Aimé Vaschy (1890), Dimitri Pavlovitch Riabouchinsky (1911) and Edgar Buckingham (1915 ) demonstrated Π theorem, extended above statement to the effect that the function in the general form

Can be represented. The potency of the products, the so-called Π - factors with, are dimensionless.

By dimensional analysis, it is possible, the functional form of physical formulas to a real-valued constant to "guess " if only a few physical variables influence such as when first formulated by Galileo 's law of fall

As with the dropping path, as the acceleration of gravity, and as the time. The proportionality constant case remains to be determined in the experiment; it is given by.

Dimensions and measurement systems

Base quantities and their units in physics

Measuring a physical quantity is called size types (speed, pressure, ...) compare with something.

For such comparisons, you never need more than seven fundamental variable types, which are called basic variable types. For them to prototype and base units ( meter, second, ...) are defined. Basisgrößenart each represents a dimension of its own, which can not be described on the basis of size remaining species. They are all independent of each other.

→ Table of the basic variables and SI base units

Basic size systems

A basic size system includes all dimensions, in which there is a measurement process. A { M, L, T, Θ, E, I, A} system, which contains all known dimensions, ie, sufficient to cover all processes in nature. In mechanics, the main field of application of dimensional analysis, one can, usually to { M, L, T} - limit system.

The basic size of the system itself, the explicit choice of a base unit is inconsequential. The length [L ] is measured as the basic units of meters, feet, inches, yards, etc.. The base unit is used but only a comparison purpose, it is not to be confused with the dimension.

Basic size systems can be formed not only from those basic variable types, which are also the same base size species, but also with all others. So after Newton is the force equivalent of the mass and independent of all other base sizes. Consequently, there is equivalent to { M, L, T} - system of the base size of species mass [Unit: M], length [L ] and time [T ] a { F, L, T} - System of Grundgrößenart force [ F], which is defined by the mass, and in turn, the length [L ] and time [ T]. The force has here as a concept of dimension has its own independent dimension, which includes the mass term.

Equivalence of basic size systems

All kinds of size { M, L, T} - system can be specified in a { F, L, T} system. A { M, F, L, T} system may not give it because of the dependence of mass and strength. The demand for independent dimensions would be violated.

You can alternatively select basic size systems, in which the pressure, the speed or frequency are basic sizes. Condition is that each basic variable is independent of the other variables used basic dimension for themselves.

They call all the basic size systems, in which the same quantities can be shown equivalent. For the detection of so-called Π - factors the explicit choice of basic variables is trivial. It is only a question of the preferred mode of representation.

In mechanics, common size species in a { M, L, T} system are listed below with their dimension formulas. Your units are products of powers of the base units. Your dimension formulas are products of powers of the dimensions within which these units are described.

In mechanics, common size species in a { M, L, T} system

Formulations, such as "relevant size of the density " or " influence of the parameters speed and acceleration " are colloquial. This use of the term size is not correct in the physical sense. Density, velocity, acceleration, etc. are size kinds. Only in an equation of the type:

Is a size (one can also speak of the measurand) a (dimensional ) unit [ m / s] and a measure described 3. For industrial purposes, but this is not relevant.

Basic size systems and their transformations

Each basic variable system can be transferred using a transition matrix containing the exponents of dimensions in an equivalent thereto. If you want a basic sizing system, for example, the dimension of force, given in the form

,

Expressed by the mass, this is achieved by the simple algebraic conversion

.

Or, presented in a clear format, with the transition matrix of the exponents

The transformation of the basic sizes of the { M, L, T } system for { F, L, T} - basic variable system is given by the matrix multiplication

Possible if the dimension of the matrix contains the exponents of all dimensions of the formula { M, L, T} system. The exponents of this dimension formulas in { F, L, T} system are then found in the dimension matrix.

Since length and time by the transformation remain unaffected, merely the exponents of those variables that are correlated with the dimension of mass change. It can be seen that simplify some fundamental quantities such as the pressure, the dimension formulas. For others, however, such as the right abhängende of the mass density, however, overcomplicate. It is useful to form such a fundamental variable system in which the sizes can be as simple as possible represent the specific problem.

Π - factors

What are Π - factors?

Π - factors called those products that result from a matrix like the above dimensional matrix, if one raises individual sizes in any of magnitude and multiplying it with others occurring in the matrix sizes in such a way that the product is dimensionless and has the dimension 1. The dimension of a size indicated by the brackets. For example, the product of powers

A Π - factor of the matrix of the required dimension

Possesses. The dimension 1, of course, remains the same even if one rises in any of magnitude. It is:

How many Π - factors are there?

One sees that as many representations of a factor once found are possible. The number of Π - factors that can not be written as a power of a factor previously found or collected as a product of factors in powers is, however, limited. About the existence of these Π - factors in a selected dimension matrix can be said that there are exactly linearly independent Π - factors.

Where:

• P: The number of non-dimensional factors Π
• N: The number of dimensionless variables bound
• R: The rank of the matrix.

Formal procedure for dimensional analysis

One finds a row vector with the number of columns for which:

Then you have to:

A Π - factor found.

Control options

The number of linearly independent row vectors that satisfy the equation (2). Their linear independence is proved by showing that the rank of the matrix that you can make out of found row vectors, also is.

Multiplied by zero results in the matrix with the number of selected dimensions ( here: 3) as the column and the number of vectors as rows.

From the matrix algebra it follows that also any linear combination of the row vectors found equation (2) is released and thus is a Π - factor. Accordingly, (5 ) is satisfied also for any matrix which is derived from, multiplied by different lines of arbitrary real numbers of zero and is added to or subtracted from the other rows. At the rank of the matrix, nothing changes. For the number of possible solutions, this means that you can make any number of other factors Π - Π with found - factors:

With their corresponding row vectors were homogeneous solutions of ( 2). However, there are still only just Π - factors that form a fundamental system of dimension matrix.

Conclusions

• With any fundamental system of all existing solutions of ( 2) are determined by (6). Here are as many solutions representable.
• Dimensionless number constants that are often ratio sizes remain dimensionless in this calculation and automatically represent a dimensionless Π - factor

Locating a fundamental system of Π - factors

Analytical Procedure

A first possibility to obtain a fundamental system of Π - factors is to have have any value except zero, the independent variables in the equation system resulting from (2) and to check the rank of the matrix rows according to (4). The number of independent variables is equal to the number of the Π - factors.

Independent or freely selectable are those variables that can be assigned any number of values ​​without causing a contradiction in the solution in the equation system. A clever choice it is, for example, always assign the numerical value of one independent variable and set the other independent variables to zero. The missing dependent variables are obtained by solving the remaining equations.

The disadvantage of this method, however, is that it has very little influence on the appearance of this fundamental system and must solve under certain circumstances a variety of systems of equations.

Method of guessing

A better method is to guess the Π - factors simply from (1). This you have to add " zero " the rows of the variables in the dimension matrix.

Practically, this means:

• If you want to have a size in the numerator must be multiplied their line with " 1 ", otherwise with "-1". ( To multiply the rows with numbers means the sizes in the corresponding powers to raise. )
• If the addition of such lines is zero, one has a product of powers (as previously demonstrated with the matrix ).

This method includes the ability to influence the appearance of Π factors. However, one must acknowledge in retrospect the rank of the resulting matrix rows by finding a non-vanishing subdeterminant, thus shows that (4 ) is satisfied.

Evaluation of the methods

Most guessing the factors with skillful choice of the base system size and clear ratios results significantly faster than a formal approach.

There are in the literature several additional ways to analytical finding the Π - factors demonstrated to the system of equations from (2) as possible sent to solve (eg Gaussian elimination method ). For details, can be found in every math book on Linear Algebra.

Education physically useful fundamental systems

If you arrive at a fundamental set of solutions, it often does not satisfy the desire for a physical significance of each Π - factors. Remedy the use of equation (6).

By skillfully combining the factors with each other and their collection in arbitrary powers can be readily formed a new, physically of richer factor. Should this be available in a new fundamental system is merely to emphasize one of the factors by combining one had the new formed. Thus, the newfound Π factor is linearly independent from the rest. Suppose that there is a fundamental system with a Π -factor, and a new, more meaningful, the shape factor

Would then be a new fundamental system or not, since a linear dependence of the first two.

For model studies, it is useful to have formed such factors, which always contain a characteristic size, which then occurs in only one factor. This is not necessarily possible. Equation (6) allows but to prove that.

Dimension Homogeneous Functions

If there is a dimensionally uniform function with a dimensionally bound function value which is determined by sizes so

Then there is always a product of powers such that can be written:

Any physical formula, and in particular her bound to a unit function value thus can be represented dimensionless power over collection of variables included in the function.

Statements of the Π - theorem

The so-called Π - theorem ( in the literature also often Buckingham theorem), directs a step further. His main message is that each dimension bound equation

In the form of

Can convert and thus only dimensionless power products (and number constants) is constructed. It may be that there are several ways to display the left side of the equation in a dimensionless form. Occasionally, the left side is also referred to as a Π - factor in the literature. This is legitimate, but not consistently, because by the separation into left and right side to give the precise statement

In place of

The significance of the theorem is that a statement about the functional relationship dimension -prone physical quantities can be made that perhaps can not be explicitly specify a formula. This applies to many complex issues in nature (eg, turbulence, Kármán vortex street ). Since variables can occur only in certain relations, the presented Π - factors to each other, can be reached at the same time a useful reduction in function of variables in relation to those in, because it is again.

Conclusions

If the number of Π - factors clear, then:

• When looking for functional relationship is determined up to a constant of proportionality.
• If there are one or more factors (), then a functional relationship, such as from experimental results or by pure intuition, can only guess. Derive explicitly he can not be.

In the second case the product meets Frequently approach to Rayleigh, so that the found Π factors multiplied together and increased in a corresponding, often integer power, deliver the desired end result.

It can still draw two general conclusions:

• Theorem 1: If a size is not required to reach a fundamental system of Π - factors or to make dimensionless, then either hangs not from, or intended functional relationship must be extended by at least another size.

• Theorem 2: If it can be made ​​dimensionless by any product of powers of the, then the dimension of the matrix is incomplete or incorrect.

This means that you can get in every case when dimensionful equations, what physical formulas are always in a beneficial, non-dimensional representation in which the units of the sizes do not matter.

These fundamental principles are important for all of physics.

Vectors and tensors

At its limits the dimensional analysis, if not only scalar variables such as pressure or temperature, or one-dimensional, straight motions are treated, but come vectors and tensors into play.

Since only one physical dimension of length is available for the description of spatial processes but a three-dimensional Cartesian coordinate system is necessary (where vectors come into play ), would have about their time-dependent height and width are examined separately for the two-dimensional parabolic flight of a cannon ball. This does not preclude that one in two formulas and the necessary background knowledge that can derive a valid equation for the flight within a rectangular and non-moving coordinate system about the knowledge of symmetries. The ball is additionally deflected yet by side wind, and the problem with three-dimensional complexity of the to be detected further increases.

The apparent contradiction between the three dimensions of space and one dimension of length available dissolves, if we imagine, is hosting the length dimension in a mitwandernden coordinate system on the trajectory itself. If you follow the path, the curve and the ball speed is one-dimensional. Dimensional analysis is therefore perfectly valid. The web speed along the curve can be one-dimensional, namely capture of the amount of the velocity vector. This is very helpful for a non-moving observer, who wants to gain not only knowledge of the amounts of the speed or the distance traveled by the ball, but also on the direction of the velocity and the ball position in space.

The same is true of three-dimensional stress states (such as for the investigation of material strengths) that need to be detected with a stress tensor.

Transition to model theory

The third important conclusion, which makes the Π theorem important in experimental research equipment, is that:

• Theorem 3: If the dimensionless functional equation

Theorem 3 is crucial for the similarity theory. All boundary conditions which have to be chosen in realistic model tests, go forth from this (see complete and partial similarity model ).

As an example of a significant in model experiments Π - factor, the Reynolds number was called. This is:

Since the Reynolds number is a geometrical length of the flow rate received the density and viscosity, it is possible to analyze to scale smaller models (such as aircraft in the flow channel ), while still obtaining a correct result on the left hand side of the above -dimensional functional equation by adjusting the model for the investigation and / or.

In many problems of the same characteristic Π factors appear repeatedly. So are many, dimensionless using the keyword index, named after their discoverers and explorers.

Full and partial similarity model

If it is possible to keep all Π - factors constant in a physically interesting range of values ​​, we speak of complete model similarity, otherwise of partial similarity model.

Often the complete model similarity, however, does not succeed, and one is forced to estimate the more or less side-effect on the final measurement result. Side effects can also occur elsewhere, namely, when a variable whose influence would be inconsequential to the prototype, the model undesirable strongly influenced (see Froude number in the ship model).

Model laws

About equating the Π - factors of the model and prototype resulting model laws. Varying in the Reynolds number of the model in relation to the prototype length, this can be, as explained above, offset by adjusting the viscosity and / or velocity.

To bring the model laws in an advantageous form, one is always anxious to only those variables to take in the Π - factors that can be varied in the model and not those that would result from the consequences of this variation. The practically most useful form is obtained if it is possible to write these equations in such a way that when inserting the variable values ​​of the prototype is always an unambiguous statement about a single trial setting in the model possible. So in such a way that any change in the initial situation in the prototype always disclose the required test setting in the model.

Model tests

A not to be underestimated advantage is also still in it, not to have to vary all the assumptions used sizes individually in a model experiment, but only those formed from them, and reduced in number to, Π factors. Also the presentation of the subsequent test results this is of crucial importance. By plotting only Π - factors rather than individual, dimensionally -prone, sizes, leads to a much tighter and clearer illustration of the measured variables ( saves dimensions). All diagrams in which the axes are shown dimensionless based on the basis of the dimensional analysis.

When building a model, and the subsequent experimental procedure is necessary to carefully consider all relevant variables in advance. Just got the parameters leads to the right or a complete set of Π - factors and can perform a realistic simulation. If you select too many variables that may have little relevance to the measurement, but the number of attempts increases tremendously. This requires physical expertise.

Maybe it turns out in retrospect that one size, one had intended a meaning much less influence on the outcome has to be accepted. If this variable exists only in a single factor, it is possible to emphasize this. Otherwise, it is recommended that a new set of variables to form the matrix dimension and to find a matching fundamental system.

Examples

To demonstrate the application of the formulas from the previous chapters to follow some calculation examples.

Galileo's law of falling bodies

First, it is mistakenly assumed that in the case law of Galileo Galilei of the fall path depended addition to the acceleration of gravity and time also the mass of the falling body, thus:

The associated dimension matrix is in detailed notation

Or results in mathematical formulation

Since all row vectors are linearly independent of, the ranking results to; there are no Π - factors, because with true. It can only apply:

The approach can not be made ​​dimensionless and thus is physically incorrect. A function of the fall path of the mass only leads to a proper description, when the air is taken into account. Because the person responsible for the braking friction air density contains the dimension of mass.

Galileo was the differential calculus is not available. He was unaware that the rate of fall is the time derivative of the fall path. From time to time he participated in that. Had he makes use of dimensional analysis, would have been clear that the approach to

Leads and this without any knowledge of differential calculus.

Euler Buckling

Vertical bars of a certain length are loaded kink risk, i.e. their failure is often before the actual breaking load of the cross section is reached. The so-called buckling load of such a rod with rectangular cross-section depends on the modulus of elasticity, its length, its cross-sectional height of its cross-sectional thickness and the storage conditions of the ends from:

The dimension matrix for the second case, the adjacent figure is obtained for a { F, L, T} system in detail notation to

And to results in mathematical formulation

The rank of is. The number of Π - factors obtained with and. In both of these are easily guessed Π - factors are the geometrical similarities and so-called. For dimensionless must

Apply, so that the dimensional analysis has shown that you only need to vary the so-called slenderness of the rod and the aspect ratio of the cross section in laboratory experiments to obtain the buckling load for arbitrary moduli of square rods.

According to equation 6 in section existence and number of Π - factors will eventually lead to another Π - factor:

Using this factor, the dimension analysis provides the equivalent relationship

Often, it is natural to be recognized as a product of Π - factors. For this example, we thus come to the equation

The exact, prepared by Leonhard Euler relation

Analog, i.e., the same functional shape. The buckling load can be used in experiments on bars of any length and elasticity, and not just limited to the rectangular shape easily verify and shown in diagram form. Knowledge closed formulas, such as the Euler is not necessary. Noteworthy is the gained knowledge that modulus of elasticity and length of a fixed cross-section for a bend test can in principle be chosen freely. The proportionality between, and is known by dimensional analysis.

In flow-around body fluids

Flow resistance of a sphere

The standard problem in the early days of fluid mechanics was to determine the resistance of a flow around a body of fluid. This can be detected by means of dimensional analysis.

The resistance force of a ball and each of the other body depends on its shape, specified here through the ball diameter, the speed with which it moves in the fluid, the density of the medium and the dynamic viscosity from.

Wanted is the functional relationship.

The dimension matrix into a { M, L, T} system is:

Is the rank of 3 There are Π - factor, the famous Reynolds number, named after the knower of this principle, Osborne Reynolds, and thus:

For is usually to reformulate appears reasonable number of constants, where the convention is that is replaced by the face of the body and the proportionality factor 1/2 is added from the dynamic pressure. Even with this reformulation is considered the relationship

The required resistance is:

Is referred to as a drag coefficient. It can be determined by experiments and is, as can be seen in the non-dimensional diagram, speed dependent and not constant. With the determined by measuring correlation between d and other fluids, and can be converted to spheres of different diameter.

Initially at low applies the analytic difficult herzuleitende linear Stokes law. Then, at higher speeds will vary due to vortex formation on the ball back. Similar diagrams can be determined by experiments for arbitrary geometric shapes and body.

Models of ships

A ship is being investigated as a model on a small scale 1:100.

The prototype, that is the real ship, has the length and the width. Its depth is and it travels at the speed. The water has the density and dynamic viscosity. The process is subject to the acceleration of gravity, because at the water surface incurred by the law of gravity waves unsuccessful. The water is deep enough over.

This experiment investigates the resistance to flow in the direction of travel, as measured by a force. In the dimension matrix only independent variables may take. As about the villains, these three variables are allowed as input variables in only two.

Seeking is the functional relationship

The dimension matrix into a { M, L, T} system presents itself as:

The rank of 3 is valid for the number of Π - factors. With experience in fluid mechanics is hinted at:

And geometric similarities. To scale reproduced curves of the hull form is required. is the Reynolds number and Froude number.

The dimensionless context

Is valid.

Complete model similarity is achieved when all Π - factors can be kept constant in the model and prototype. In and this is trivial. Remain unchanged and in the water. The constancy of the Reynolds number requires to increase the speed by a scale factor of 100, as has been reduced by 100.

• Dilemma in the Froude number of the speed is a square. For the constancy of the acceleration of gravity would adjust, which is not possible without the centrifuge on the earth. Complete model similarity can not be achieved, or can only be constant. Alternatively, the model in another liquid with a suitable density and viscosity can be examined.

• Conclusion: Plays as well as a roll, does not ensure complete model similarity as a rule. Very small models also require large flow velocities. Many models are therefore only realistic if they are sufficiently large.

Models of airplanes and submarines

In flow processes in which the free surface of the fluid does not matter, the Froude number is not relevant surface wave due to lack of formation. Models of submarines or aircraft (below the speed of sound) can be studied in principle with complete model similarity. The decisive factor is the Reynolds number.

To avoid huge, not realizable flow velocities in the wind tunnel aircraft models are often streamed in denser media. If an object moves so fast that the bulk modulus of the fluid is of concern, the Mach number comes into play. Then the relation holds. and are characteristic dimensions. Result is three already known and a new Π - factor:

The denominator of the wave velocity of longitudinal waves in elastic media. In air, the so-called sonic velocity. The Mach number is from values ​​of about of influence. K is in gases strongly pressure and temperature dependent.

Energy of the first atomic bomb test in 1945 in New Mexico

A famous example of the application of dimensional analysis comes from the British physicist Geoffrey Ingram Taylor. After receiving a series of images with precise time intervals of the first atomic bomb exploded in 1945 in New Mexico ( Trinity test ), he was able to determine the free energy of the local laws nuclear explosion. The site measured explosive force had been kept secret by the developers in Los Alamos over the outside Brits.

A number of past thinking on this topic Taylor was clear that the radius of the initially approximately hemispherical explosion largely depends on the time since the ignition of the bomb, the density of the surrounding air and of course the explosion released from the power of the bomb. Other sizes are negligible.

Thus, the following applies:

And:

The rank of 3 and. The functional relationship is determined up to a constant, since it can only apply:

With an estimated temperature at the time of explosion by about 6 clock in the morning in New Mexico by ° is obtained for the density of air.

The radius is the time in the image about.

The proportionality factor could be determined (several kg TNT) from a comparison with conventional explosives explosion. Taylor had enough background knowledge to be able to accept. This is:

1 ton of TNT has an energy of 4.18 billion joules. This leads to the estimate:

Trinity had according to official figures, an energy of approximately 19000-21000 tons of TNT. The deviation to the above explained by the fact that the radius of the 5th power is received. The result is remarkably accurate. Taylor himself calculated approximately 19,000 tons of TNT.