Will you get less wet running in the rain?

This is something I did one morning while waiting to take an exam that afternoon on mathematical methods and fluid mechanics. I couldn't bring myself to do any more revision but needed to keep my mind on maths. As it was raining outside I started to wonder whether it was possible to build a simple mathematical model for an object moving in the rain. What follows is a write up of my thinking that morning.
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Common sense would say that one will get less wet running in the rain than walking, but this might not necessarily be true. The slower the speed travelled the more water will land on one's your head, but the faster the speed the greater the water hitting one's front. Given that, for most people, the surface area pointing vertically upwards (top of head, shoulders and chest) is much smaller than that of their surface area pointing forwards (face, front of torso and limbs), it is possible that travelling at greater speeds could in fact make one wetter. To investigate this I build a simple mathematical model for an object moving through rain and start by looking at the case when the rain is falling directly downwards (there is no wind). Later I develop the model to include the effects of wind.

Assumptions

  1. Rain can be modelled by a constant density vector field.
  2. Rain travels at a constant velocity downwards.
  3. A person can be modelled by a regular cuboid.
  4. The person moves at constant velocity through the rain.
  5. Rain that comes in to contact with the person is assumed to be perfectly absorbed.
  6. Other than the person there are no other surface effects that need to be considered.

The first assumption states that rather than modelling the rain as individual drops it is possible to model it spread out evenly throughout space. This appears reasonable as long as the rain droplets are significantly smaller than the object moving through them and there is no clumpiness; i.e. they are evenly spaced out. This the key assumption of the model, without this the following maths is invalid.

The second assumption asserts that rain moves at a constant speed towards the floor and doesn't change direction. While in the real world the wind will affect the direction and speed of rain for the sake of simplicity this model will ignore these effects. This assumption will make the model far simpler than including wind direction, but is an unrealistic assumption, any further models should include wind effect.

The third assumption appears to be fine for out basic initial model, further investigation is needed to confirm this for more advanced models.

The fourth assumption states that we won't be taking the acceleration of the object into account. As the velocities involved are quite small I think this is fair. Again, a more developed model may want to take this into account.

The fifth and sixth assumptions are similar; none of the water hitting the object bounces off and there is no spray onto the object from other surfaces. Both of these are appear to be okay for our simplistic model, but they don't realistically represent a person running in the rain where there will be some spray effects.

The Model

Let the rain be described by the vector field

dRdt=-αk (α>0)
where:
  • R is the rain vector field
  • and α is the downwards speed of the rain

 

The body, O, moving through the rain is a regular cuboid with corners located at (0,0,0), (x,0,0), (0,y,0), (x,y,0), (0,0,z), (x,0,z), (0,y,z) and (x,y,z).

The equation for the velocity of O through the rain is

dsdt=βi (β>0)
where:
  • s is the displacement vector
  • and β is the speed

 

Given the above mathematical definitions it is possible to calculate the flux of the vector field R for each face of O as it moves at velocity dsdt by calculating the surface integrals, giving the volume flow rate through each surface. In other words how wet each surface gets per unit of time. The model has been created so only two surfaces need to be considered; the top and the front faces. That is surface Stop with corners located at (0,0,z), (x,0,z), (0,y,z) and (x,y,z), and the surface Sfront with corners located at (x,0,0), (x,y,0), (x,0,z) and (x,y,z).

The surface integral for a face S is defined as

SF·ndA
where:
  • F is the vector field
  • n is the unit normal vector for the surface
  • and A is the area of the surface

 

For our model it is necessary to calculate how wet each face gets through time. The equation for this is

dwSdt=SF·ndA
where:
  • wS is the wetness of surface S.

 

As the body is moving through a moving field the two vector equations describing velocity need to be combined, hence

F=dsdt+dRdt
=βi-αk

 

For Stop the face normal points directly up, so

n=k

therefore

dwtopdt=Stop(βi-αk)·kdA
=x00y(βi-αk)·kdydx
=x00y-αdydx
=x0-αydx
=αxy

and

wtop=αxydt
=αxyt+C

as wtop=0 when t=0 hence C=0, the surface is dry at the start,
therefore

wtop=αxyt

 

For Sfront the face normal points directly outwards, so

n=i

therefore

dwfrontdt=Sfront(βi-αk)·idA
=0y0z(βi-αk)·idzdy
=0y0zβdzdy
=0yβzdy
=βyz

and

wfront=βyzdt
=βyzt+D

as wfront=0 when t=0 hence D=0, again, the surface is dry at the start,
therefore

wfront=βyzt

 

The total wetness of the body if found by summing the wetness of all the faces.

So

w=wi
=wtop+wfront
=αxyt+βyzt
=(αx+βz)yt(1)

 

The ratio of rain landing on the top surface to rain hitting the front surface is

Wratio=wtopwfront
=αxytβyzt
=αxβz(2)

 

Initial Analysis

The model has produced two equations that will help to shed light on the initial question of how the speed travelled in the rain effects how wet a body gets. These are equation (1), giving the total wetness of the body at a specified time, and equation (2), giving a ratio for the proportion of rain landing on the top compared to the front of the body. I'll start with an analysis of this surface wetness ratio.

Ratio of Surface Wetness, Wratio

Equation (2) gives a ratio for the proportion of rain landing on the top compared to that hitting the front, and depends on 4 independent variables: the speed of the rain, α; the speed of the body, β; the thickness of the body, x; and the height of the body, z. This ratio shows that, according to the model, the two factors affecting how wet the top of the body gets, compared to the front, are the speed of the rain and the thickness of the body. Likewise, the two factors affecting the front compared to the top are the height and the speed of the body. By trying to find standard values for the rain speed, and body width and height it is possible to see how this ratio changes according to the speed the body travels at.

  • According to [1] the speed of falling rain depends on the size of the raindrop. Drizzle falls at 2 m/s, a 2 mm raindrop falls at 6.5 m/s, while a 5 mm raindrop falls at 9 m/s. To compare the effect of different rain speed I'll take a low speed of 3 m/s, a mid speed of 6 m/s, and a high speed of 9 m/s.
  • The mean height of an adult in England in 2008 was 1.68 m [2]. To compare the effects of height I'll take a short person to be 0.3 m lower than this and a tall person 0.3 m higher. Given that people aren't cuboid taking 80% of these values should give a fair value for the height components. This gives the heights 1.10 m, 1.34 m, and 1.58 m for a short, average and tall body, respectively.
  • To estimate people's thickness, finding statistics has been hard. There is data for waist and chest circumferences, but turning these into a reliable figure for thickness is nigh on impossible. The figures I've chosen are arbitrary but hopefully represent a reasonable estimate for the thickness of an individual. I will take the thickness for a slim, average and large person as 0.2 m, 0.35 m and 0.5 m, respectively.
  • The average walking speed is 3 miles per hour [3], or about 1.3 m/s. An Olympic sprinter can run 100m in under 10 s [4], an average speed of under 10 m/s.

An initial look at these figures throws up an interesting, possibly counter intuitive point; rain falls slightly faster than most people run at, while, unsurprisingly, people are taller than thicker. Combined these mean that someone caught in the rain may want to travel slightly faster, but sprinting will === REWRITE ===

Figure 1. Change in rain speed for an average person
Figure 1. Change in rain speed for an average person

Figure 2. Change in height for average thickness and rain speed
Figure 2. Change in height for average thickness and rain speed

Figure 3. Change in thickness for average height and rain speed
Figure 3. Change in thickness for average height and rain speed

Figure 4. Extremes
Figure 4. Extremes

The graphs above show how changes in rain speed, body height or thickness change how the ratio between how wet the top of the body to the front get as the body speeds up. The horizontal axis displays the body speed, β, measured in m/s. The vertical axis displays the wetness ratio of the top to front face, Wratio. The main thing to notice is that when Wratio=1 both faces, top and front, are getting equally wet. When the body is travelling at low speeds the wetness ratio is greater than 1, at this point the top is getting wetter than the front. As the body speeds up the ratio drops below 1, at this point the front is getting wetter than the top. So these graphs demonstrate how, depending on rain speed and body height and thickness. Figure 1 shows .

The Total Wetness of the Body w;;

There is a major flaw to this analysis of the wetness ratio; it doesn't take into account time spent in the rain. It assumes that whether crawling or sprinting the body spends the same time getting wet. This, of course, defeats the object of running in the rain; to get out of it quicker. Equation (1) for the total wetness shows how wetness varies with time.

References

  1. Weather Almanac for April 2003 - The Energy of a Rainshower
  2. NHS Health Survey for England 2008
  3. Wikipedia article on walking
  4. IAAF list of fastest 100 m runners