23rd Apr, 2021
Menoufia University
Question
Asked 9th Jan, 2013
What is the purpose of dimensionless equations?
Why is it neccesary?
Most recent answer
The main objective of converting the equations to their dimensionless form is to generalize our obtained results, for example, if we study the nonlinear oscillation of systems where these oscillations in the size of mm, cm, and meter. converting these problems (in mm, cm, and meter) to their dimensionless form may be treated as one problem. In the dimensionless forms, we talk about relative quantities that facilitate and generalize the discussion.
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Popular Answers (1)
There are three important reasons for writing complex equations in dimensionless form. These are:
1-It is easier to recognize when to apply familiar mathermatical techniques.
2 It reduces the number of times we might have to solve the equation numerically.
3-It gives us insight into what might be small parameters that could be ignored or treated approximately.
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All Answers (33)
There are three important reasons for writing complex equations in dimensionless form. These are:
1-It is easier to recognize when to apply familiar mathermatical techniques.
2 It reduces the number of times we might have to solve the equation numerically.
3-It gives us insight into what might be small parameters that could be ignored or treated approximately.
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Additionally:
- It facilitates scale up of obtained results to real conditions.
- avoids round-off errors
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As Issam said, it allows you to abstract over a wide variety of phenomena, and gives insight to regimes where some parameters vanish and can be ignored. The Reynolds number in hydrodynamics is a good example of this. At very low Reynolds number, many terms in the equations vanish, and thus systems can be approximated by very simple equations (laminar flow) even though flow at all Reynolds numbers is fundamentally governed by the Navier-Stokes equation.
For another interesting example of this, the mathematician G. I. Taylor used dimensionless forms of certain equations to figure out highly classified and secret information about the atomic bomb in the 40's: http://www.zahniser.net/~physics/index.php?title=Atom%20Bomb%20by%20Dimensional%20Analysis
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There are potentially many ways to create a non-dimensional equation or relationship. Typically there is only one "correct" way to do it, which leads to a simpler, physically meaningful problem. Only when the "correct" scaling is chosen, can you neglect terms where the dimensionless coefficients are small.
Dimensionless analysis done correctly makes the problem simpler. When done wrong, it just makes a mess of new meaningless variables.
Chapter 6 in Lin & Segel's textbook "Mathematics Applied to Deterministic Problems in the Natural Sciences" is an excellent discussion about this with examples.
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The use of a dimensionless model is a very common way to study a wide variety of physics phenomena or engineering problem, even economic task.
A dimensionless equation, algebraic or differential, involves variables without physical dimension.
For example, all consumers prefer to know the price reduction percentage rather than euros (dollars or yen) because it's easier to compare offers.
Consider this example from an electric supply. What to say about the quality comparison of two voltage generators, the first one with an output resistance of 0.1 Ohms and the second with an output resistance of 0.025 Ohms ? You can not conclude about that question and if you try to do, you will probably find that the second one is better.
However, if you know the main parameters of the generators (output voltages Uo & output currents Io), you will be able to perform a more realistic comparison. The first one is a 400V , 40A generator and the second one a 12V , 24A. The dimensionless output resistance is the ratio R / (Uo/Io). Now the comparison is 0.1/10=1% for the first generator (that is really excellent) and 5% for the second one (poor quality).
The same way to analyze voltage drop yields: 4V for the first and 0.6V for the second one. But dimensionless drop are 1% and 5%.
Then the dimensionless output resistance or dimensionless voltage drop give the same result: the quality of the first generator is the best and this conclusion is free of the scaling effect.
Consider now a simple first order differential equation with constant coefficients for the capacitor charge : u+RC*du/dt = E
Knowing that u max = E, voltages will be scaled by supply volatge E: uo=u/E
Knowing that RC product is a time constant, time is scalable by RC: to=t/RC
Then the dimensionless charge equation becomes: uo+duo/dto=1
This equation is free of parameters dependance.
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Dimensionless equations can reduce the complexity of the problem, give you insight into fundamental scales (time-, lenght-, etc.) of the problem. I recommend that you read about the "pi theorem". A good reference is the book by Logan on Applied Mathematics.
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The complex equations where the unknown variables are more and the no. of equations as in case of N-S equations in conservation form, Non dimensionality reduces this complexity by reducing the unknown parameters.
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My answer is from a viewpoint of Control Theory. Dimensionless equations represent the behavior of the dynamical system. For example whether or not I am trying to control an aircraft, or a huge ship or a small servomachine of a robot or a process plant the dimensionless equations may just give me a second order model. Or a simple model for which the theory is valid. However the major impact of dimensionless equations is for non linear phenomena. In the linear case Physical parameters are upto transformations hence solving any one dimensional system can give you an idea about all others of the same behavior. This can be said of non linear systems only when they have same behavior in non dimensional form.
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My answers from Heat transfer view point.In case of convection we use to form group of various parameters known as nondimensional group.No of such groups form nondimensional equation.It is always better to deal with such groups rather than dealing with variables
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In perspective of heat transfer It gives you freedom to analysis for any system irrespective of their material properties.
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Dimensionless form often shows very clearly that not the original variables but rather their ratios (or other combinations) govern the qualitative type of solution.
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From the point of numerical discretization dimensionless form may help to choose appropriate numerical method, e.g. stiff equations are characterized by small coefficient before leading derivative and require specific approaches (A- and L-stable methods etc).
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Dimensionless approach typically generalizes the problem. For fluid mechanics point of view, Solution of dimensional form is the solution of a particular problem. However, dimensionless solution depends on a set of dimensionless parameters, e.g. Rayleigh numbers, Prandtl number, Reynolds number etc. Using one dimensionless solution one can describe many dimensional solution.
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A good and easy example is the 1D-advection-diffusion equation:
a u,x = k u,xx, to be solved in the domain x in [0,L].
where ,x = spatial derivative and ,xx = second spatial derivative.
Now, let us replace u,x by Lu,x. Then we have:
Pe u,x = u,xx, where the well known Péclet number is defined as Pe = aL/k.
Solving this problem and testing it, we don't need to test it for different values of a, k and L! Different values of Pe is sufficient! Generalizing as said before :)
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The purpose of dimensionless equations are:
1. To simplify the equation(s) by reducing the number of variables used.
2. To analyze system behavior regardless of the unit used to measure variables.
3. To rescale parameters and variables so that all computed quantities are of same order (relatively similar magnitudes).
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Along with all the points given above I would like to add that dimensionless numbers helps in standardizing an equation and makes it independent of variable sizes of the reactors used in different labs, as to say, since different chemical labs use different size and shape of equipment, its important to use equations defined in dimensionless quantity.
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1) When you need to solve a problem numerically, dimensionless groups help you to scale your problem - Computers can't deal with wide ranges of numbers especially when adding small numbers to very big numbers. In numerical methods, problems that have certain terms that are much larger than others are said to have a large Condition number and are consequently both difficult to solve and difficult to solve accurately. By scaling the problem to appropriate scales, you can make many terms be of the same order so that it the effects of numerical errors are minimized when calculating the residual.
2)Many useful relationships exist between dimensionless numbers that tell you how specific things influence the system- The classic example of this yet again involves the Reynolds number to predict the onset of turbulence in a system. Critical values for the Reynolds number for many different systems are tabulated and so you can easily predict the onset of turbulence. Other examples include using the Rayleigh number to predict whether a fluid's heat transfer will happen mostly through natural convection or through conduction. Similarly, the Péclet number will tell you whether transport will happen through advection (active convection) or diffusion.
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They usually serve to scale the problems. Sometimes it's done to characterize different physics involved in the problem, for example, Reynolds number comes out of dimensionless Navier-Stokes and we find the important physics involved are inertial forces and viscous forces.
The other times that I can think of, is to solve numerically ill (or ill-posed) problem. For example, you want to to solve fluid flow in a deformable rock (rock has very high viscosity). then the contrast between the fluid and rock viscosity (maybe 20 orders of magnitude) does not let you to solve the linear system easily! You will have what is called "a large condition number".
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Hi,
I just want to add that nondimensionalization shouldn't be confused with normalization. nondimensionalization is related to only the dimensions of the equation and it doesn't necessarily consider scaling, whereas normalization deals with scaling.
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ometimes equations are normalized in order to
• facilitate the scale-up of obtained results to real flow conditions
• avoid round-off due to manipulations with large/small numbers
• assess the relative importance of terms in the model equations
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Hi,
Any one knows how to implement the dimensionless equations in OpenFOAM, as described here: https://www.researchgate.net/post/How_to_use_the_dimensionless_N-S_equations_in_OpenFOAM?_tpcectx=qa_overview_asked&_trid=Nrb2Mj20gvlRA9IrPQPTZvB7_
Thank you in advance.
Best Regards,
Bill
This link will help understand this concept better. https://en.wikipedia.org/wiki/Nondimensionalization
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Non-dimensionalization of a governing equation reveals the governing dimensionless groups and allows the identification of significant and insignificant terms. E.g., the non-dimensional form of the momentum equation reveals the Reynolds number, and it can be seen that for high Reynolds numbers (turbulent flow), the viscous terms are insignificant and inertial terms dominate.
It follows that exactly the same equation will represent the momentum transport about e.g. a full-sized car and a down-scaled model car, as long as the Reynolds numbers are kept identical. This insight can save you a lot of money, since generally small-scale experiments are cheaper to conduct than full-scale. :)
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sometimes equations are normalized in order to
1. facilitate the scale-up of obtained results to real flow conditions
2. avoid round-off due to manipulations with large/small numbers
3. assess the relative importance of terms in the model equations
In the case of confidential research work, you can give nondimensional quantities which will not reveal the actual dimensions/parameters of your research.
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