Kristian von Bengtson (on twitter, on Wikipedia) of Copenhagen Suborbitals is building a downscaled test item to test the design for their space capsule. He notes that as volume and area scale down differently, he has to make sure that his test results sre still applicable to the full-scale item.

He asks on Wired if the following assumptions he makes on the physical properties of his 1:3 scale model of a space capsule are correct:

  • to test buoyancy behaviour (does it swim, in what position does it come to rest), he has to keep the volume to mass ratio (i. e. the density) constant.
  • to test the forces on impact in the water, he has to keep the ratio of inertia to impact area constant, and therefore add mass to the model (because the volume, and therefore the mass, scales down more quickly than the craft's cross section)

Are those assumptions correct?

  • $\begingroup$ scale the forces 1:3 and scale time 3:1 - that should do it $\endgroup$ – EnergyNumbers Dec 14 '12 at 8:21
  • 1
    $\begingroup$ Shouldn't he adjust pressures and/or fluid viscosities so that the Reynolds numbers agree? $\endgroup$ – RedGrittyBrick Dec 14 '12 at 11:02
  • $\begingroup$ It is very scary that someone trying to build a spaceship to put people in orbit has to ask on public forums about such an elementary engineering topic... $\endgroup$ – Jaime Dec 14 '12 at 16:30
  • $\begingroup$ @Jaime: indeed. With your very nice answer, I expect to see you named a consultant to the project v soon... $\endgroup$ – Art Brown Dec 14 '12 at 17:53

What you are asking about is similitude, the linked wikipedia article does a very good job of explaining what you should aim for. You are also going to want to be familiar with Buckingham's $\pi$ theorem...

Consider the impact force, $F$. It will clearly depend on some parameters of the model cpasule, say its size, $L$, its mass, $M$, and the velocity of impact against the water, $V$. It will also depend on some parameters of the fluid, like its density, $\rho$. We could also consider viscosity, $\mu$, but lets leave it out of the picture for now. So we have 5 variables $(F,L,M,V,\rho)$ that depend on 3 independent physical units (mass, length, time), so we can rewrite their dependency in terms of $5-3=2$ dimensionless quantities. A good option is to use a density ratio, $\alpha = \rho L^3 / M$, and a force ratio, $\beta = F L / MV^2$. So we know that these two quantities will obet some functional relation, $f(\alpha, \beta)=0$, which we can rewite as $\beta = f(\alpha)$, or more precisely for our needs as

$$F = \frac{MV^2}{L}f(\alpha)$$

If you keep alpha unchanged, i.e. scale length by $1/3$ and mass by $1/27$, in order for the force $F$ to be the same in the model and in the full scale capsule, you need to scale the velocity by $3$. This would, incidentally, also keep viscous effects unchanged.

Of course, because you are applying the same force to something 27 times lighter, accelerations are going to be very different. If you wanted to keep those the same, then you would want to see what happens with

$$\frac{F}{M} = \frac{V^2}{L}f(\alpha)$$

and you will now want to scale your velocity by $1/\sqrt{3}$ to get the same acceleration.

On the other hand, if the worry is about structural integrity, this is normally related to stress, so you could go with

$$\frac{F}{L^2} = \frac{MV^2}{L^3}f(\alpha)$$

and now, to keep stresses the same, you will want to keep velocity unchanged from that of the real capsule.


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