Exercise 7

A pressure peak $p$ can often be heard when closing a valve abruptly, stopping a fluid flow in a pipe.  We can  therefore assume it depends on propagation of pressure waves in the fluid, i.e. the speed  of sound $v_{sound}$ of the fluid (340m/s for air, about 4-5 times faster in water). Our intuition might also say that it depends on the density $\rho$ $[kg/m^3]$of the fluid and the fluid velocity $v$ [m/s] before closing the valve.

Find two  dimensionless combinations $\Pi_1$ and $\Pi_2$ of $p$,$v_{sound}$,$\rho$,$v$. Can you from this guess the expression for the maximum $p$ (which is obtained when the closing of the valve is infinitely fast) as a function of the other 3 variables ?

Check your result against the Joukowski formula, described in https://www.youtube.com/watch?v=GfCCsdHKrF8 Links to an external site.

a) Construct the dimension matrix A of size 3*7 when we consider the variables

Drag force $F$ [N], Viscosity $\mu$ [kg/(m s)], Speed of pressure waves in fluid $v_{sound}$ [m/s],  gravitational constant $g$[m/s^2] speed of object $v$ [m/s], characteristic length of object $d$ [m], fluid density $\rho$[kg/m^3].

b) Show that you can construct the four dimensionless quantities mentioned in the book,

Pressure coefficient=$P_c = \frac{F}{\rho v^2d^2}$, Reynolds number = $R_e = \frac{\rho vd}{\mu}$, Mach number = $M_m = \frac{v}{v_{sound}}$,  Froude number= $\tilde F_r = \frac{v^2}{gd}$

(literature usually defines "Froude number $F_r$" as the square root  of this expression...)

b) What is approximately the Reynolds number when you use the water hose ?

c) Same question when you drink water through a straw ?

d) What about when you breathe air through a straw ?

Hint: You might find these links useful

Googling gives that max exhale pressure is around 50mbar (female) about double for men. Links to an external site.

• Heating 1kg water 100 degrees
• Melting 1kg ice (0degree ice -> 0 degree water)
• Standard 9V battery
• Energy in 2mF capacitor charged to 230V
• 1000kg car moving at 50km/h
• 1 liter of car petrol
• 1kg chocolate used as food $\sim$ 5000kcal
• Your daily electric energy consumption
• A fully charged Tesla electric car
• Daily consumption of food
• 1 second usage of ESS

Also construct some rules of of thumb for heat energy:

• For water, how many meters upwards movement does the energy in a 1 degree temperature increase correspond to ?
• How many meters up of a 100kg body does daily consumption of food correspond to ?

SI Units: 1 Pa = 1N/m^2 and 1atm $\approx$1bar $\approx$ normal air pressure $\approx$ 10m water column$\approx$$10^5$Pa

Find out (Google) What is the typical water pressure at your home? Is it smaller, larger, or about the same as the air pressure in a bike tire? Roughly what pressure can you generate by your lungs? What is approximately the pressure difference inside a balloon compared to surrounding. What is the pressure under your foot if you stand on one leg ?

A rule of thumb for sailing boats is that a longer boat can sail faster and that speed $v_{hull}$ is proportional to

$\sqrt{L}$. Have a look at https://en.wikipedia.org/wiki/Hull_speed Links to an external site. and formulate the

expression $v_{hull}={\sqrt {L_{WL}\cdot g \over 2\pi }}$ as a condition on the Froude number. What is the intuitive explanation to this limit speed ?

Some bonus material:

Does the Froude number F_r=1 roughly predict the radius $r$ for where the hydraulic jump occurs in the water in your water sink at home? Here $d$ would be the (low)  water depth [m] in the supercritical flow region near the center, and $v\left(r\right)$  its (radially outwards) water speed [m/s]. Use that flow rate Q at radius $r$would be $Q = d 2\pi r v(r)$ [$m^3/s$] and guesstimate Q, d, r

Some related material: