Dimensional analysis means to understand the essentials of a phenomenon in physics and to calculate characteristic numbers – without solving the underlying, often complex, differential equation. The theory of fluid dynamics is full of interesting dimensionless numbers – Reynolds Number is perhaps most famous.

In the previous post on temperature waves I solved the Heat Equation for a very simple case, in order to answer the question *How far does solar energy get into ground in a year? *Reason: I have been working on simulations of our heat pump system since a few years. This also involves heat transport between the water/ice tank and ground. If you set out to simulate a complex phenomenon you have to make lots of assumptions about materials’ parameters, and you have to simplify the system and equations you use for modelling the real world. You need a way of cross-checking if your results sound plausible in terms of orders of magnitude. So my goal has been to find yet another method to confirm assumptions I have made about the thermal properties of ground elsewhere.

Before I am going to revisit heat transport, I’ll try to explain what dimensional analysis is – using the best example I’ve ever seen. I borrow it from theoretical physicist – and awesome lecturer – David Tong:

**How does the speed of a rowing boat depend in the number of rowers?**

References: Tong’s lecture called Dynamics and Relativity (Chapter 3). This is the original paper from 1971 Tong quotes: Rowing: A similarity analysis.

The boat experiences a force of friction in water. As for a car impeded by the friction of the surrounding air, the force of friction depends on velocity.

Force is the change of momentum, momentum is proportional to mass times velocity. Every small ‘parcel’ of water carries a momentum proportional to speed – so force should at least be proportional to one factor of v. But these parcel move at a speed v, so the faster they move the more momentum is exchanged with the boat; so there has to be a second factor of v, and force is proportional to the square of the speed of the boat.

The larger the cross-section of the submerged part of the boat, A, the higher is the number of collisions between parcels of water and the boat, so putting it together:

Rowers need to put in power to compensate for friction. Power is energy per time, and Energy is force times distance. Since distance over time is velocity, thus power is also force times velocity.

So there is one more factor of v to be included in power:

For the same reason wind power harvested by wind turbines is proportional to the third power of wind speed.

A boat does not sink because downward gravity and upward buoyancy just compensate each other; buoyancy is the weight of the volume of water displaced. The heavier the load, the more water needs to be displaced. The submerged volume of the boat V is proportional to the weight of the rowers, and thus to their number N if the mass of the boat itself is negligible:

The volume of something scales with the third power of its linear dimensions – think of a cube or a sphere; so the surface area scales with the square of the length, and the cross-section A scales with V – and thus with N:

Each rower contributes the same share to the total rowing power, so:

Inserting for A in the first expression for P:

Eliminating P as it has been shown to be proportional to N:

… which is in good agreement with measurements according to Tong.

**Heat Transport and Characteristic Lengths
**

In the last post I’ve calculated characteristic lengths, describing how heat is slowly dissipated in ground: 1) The wavelength of the damped oscillation and 2) the run-out length of the enveloping exponential function.

Both are proportional to the square root of a simple number:

… the factor of proportionality being ‘small’ on a logarithmic scale, like π or 2 or their inverse. τ is the period, and D was a number expressing how well the material carries away heat energy.

There is another ‘simple’ scenario that also results in a length scale described by

times a small number: If you deposit a confined ‘lump of heat’, a ‘point heat’ it will peter out and the average width of the lump after some time τ is about this length as well.

Using very short laser pulse to heat solid material is very close to depositing ‘point heat’. Decades ago I worked with pulsed excimer lasers, used for ablation (‘shooting off) material from ceramic targets.This type of lasers is used in eye surgery today:

Heat is deposited in nanosecond pulses, and the run-out length of the heat peak in the material is about with tau being equal to the very short laser’s pulse length of several nanoseconds. As the pulse duration is short, the penetration depth is short as well, and tissue is ‘cut’ precisely without heating much of the underlying material.

**So this type of length is not just a result of a calculation for a specific scenario, but it rather seems to encompass important characteristics of heat conduction as such.**

The unit of D is area over time, m^{2}/s. If you accept the heat equation as a starting point, analysing the dimensions involved by counting x and t you see that D has to contain two powers of x and one of t. Half of applied physics and engineering is about getting units right.

But I pretend I don’t even know the heat equation and ‘visualize’ heat transport in this way: ‘Something’ – like heat energy – is concentrated in space and closely peters out. The spreading out is faster, the more concentrated it is. A thin needle-like peak quickly becomes a rounded hill, and then is flattened gradually. Concentration in space means curvature. The smaller the space occupied by the lump of heat is, the smaller its radius, the higher its curvature as curvature is the inverse of the radius of a tangential circular path.

I want to relate curvature to the change with time. Change in time has to be measured in units including the inverse of time, curvature is related to the second derivative – thus the inverse of the spatial dimension squared. Equating those, you have to come with something including the square of spatial dimension and one temporal dimension – something like D [m^{2}/s].

How to get a characteristic length from this? D has to be multiplied by a characteristic time, and then we need to take a the square root. So we need to put in some characteristic time, that’s a property of the specific system investigated and not of the equation – **like the yearly period or the laser pulse. **And the resulting length is exactly that that shows up in any of of the solutions for specific scenarios.

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Further reading:

The characteristic width of the spreading lump of heat is visible in the so-called Green’s functions. These functions described a system’s response to a ‘source’ which resemble a needle-like peak in time. In this case it is a Gaussian functions with a ‘width’ . See e.g. equation (14) on PDF-page 14 of these lecture notes.

Penetration depth of excimer lasers in human tissue – in this book the square root D times tau formula is used and depths are calculated to be equal to several 10 micrometers.

German version of this postGerman version of this post

Whenever you do a post like this, with various angles of attack, I suspect that people find their own place to fixate. For me it was the graph. You had me at “damped.” It was the first time I’d really considered the concept of damping outside of raw mechanical systems (shock absorbers, for example) or AC electricity. Fascinating. Yes, I know, a physics student should not be saying that :-) It’s got me thinking that I might find opportunities to apply that to non-physical phenomena and so i will be looking :-)

I think you find an exponential decay / damping / attenuation everywhere in ‘nature’ including human behavior – I remember a blog post (by a physicist) about one of his posts that had gone viral: Views over time followed an exponential function. A super-imposed oscillation might be harder to spot :-)

I enjoyed this post, and the one previous that is related to it. I’d fumble a lot to comment on the math that’s going on here, but you’ve made the concepts really accessible. This week one of my classes had a brief introduction into differential equations, which we won’t develop much at this point, but it helped to have your previous post in mind throughout that lecture… actually, to be truthful, a lot of your posts and our conversations have contributed significantly to my understanding and pleasure in my classes this year.

Thanks a lot, Michelle!! A comment like that is maybe the most encouraging feedback a dilettante science blogger can get :-) Much appreciated – and all the best for your explorations into science and math!

OK, that was fun :) I reminded me that engineers just experiment and figure out a rough trend. Mathematicians deduce the formulation from its constituents and physicists are capable of a hybrid like dimensional analysis.

I got a little stuck with the [math]N=Nv^{3}N^{2/3}[/math] which I reduce to [math]v^{3}=N^{-2/9}[/math] but I expect the tilde explains it – or more likely – my maths gets in the way :)

Thanks a lot – of course you have been right about the math. There was one N too much on the right hand side in the first equation in the last block of equations – corrected now (the equations before and after were OK). I can’t believe I did not spot it on proof-reading this N times :-) I guess it shows that I don’t have much experience with LaTex yet, and move and copy around ‘template formulas’ way too often.

I suspected that the result was good, which is what matters most anyway. Great post, loved it. I tend to use an online latex formula editor and then copy and past. I don’t try too hard to learn the language.

Mhhhhh……aha…….aso………

:-)

In my Physics class at school (when was that ?) I was really struck by the derivation of a formula for viscosity using dimensional analysis. Now you have doubled my interest in this stuff. Time for a more thorough read of your post. Thanks.

Thanks, Howard – also a great example! I think I know which one you mean – the two sliding planes with viscuous fluid in between … deriving the velocity gradient etc.? There are so many neat examples in fluid dynamics!

I don’t remember the details, only the feeling. In any case it doesn’t need div, grad and curl, which took me years to get feel for !

Yes, true – it was a simple derivative! A gradient in a 1D scenario :-)

I think the reason why I tend use the term gradient also in these cases is because I find it easier to think of a vector pointing into the direction of maximum change (or against it) then translating it back into postive and negative values of a function that changes along one axis.