In these problems, you can use either the Buckingham PI theorem or Ispen method
ID: 2993207 • Letter: I
Question
In these problems, you can use either the Buckingham PI theorem or Ispen methodThe drag force, D, on an oval-shaped submarine moving well below the surface is a function of its maximum diameter, dmax, and its speed as well as the density and viscosity of the seawater. We wish to do a test on a model scaled to one-twentieth of the prototype in a water tunnel. If the speed of the prototype is 5 knots, what speed should the model have in order to dynamically simulate the actual case. Calculate the ratio of model to prototype drag force. Use Table A.3 for the seawater properties at the depth of the prototype and fresh water properties in the water channel.
Explanation / Answer
the question parameters:
D, dmax, v - speed, rho - density, mu - viscosity
D: [ML/T2]
dmax: [L]
v: [L/T]
rho: [M/L3]
mu: [M/LT]
number of parameters - 5, number of dimensions - 3, so there's 2 dimensionless numbers:
we'll select mu, rho and dmax:
D=dmaxarhobmuc
ML/T2=LaMb+c/L3b+cTc
b+c=1
a-3b-c=1
-c=-2 --> a=0,b=-1,c=2
PI1=D*rho/mu2
v=dmaxarhobmuc
L/T=LaMb+c/L3b+cTc
a-3b-c=1
b+c=0
-c=-1 --> a=-1,b=-1,c=1
PI2=v*dmax*rho/mu
we want to keep the PI terms constant so PI2 of sea water/ PI2 os fresh water =1
*Assuming: Fresh water: density:1000 kg/m3, viscosity: 8.96935*10-4 kg/(msec)
Salt water: density:1025 kg/m3, viscosity: 1.369*10-3 kg/(msec)
(v*dmax*rho/mu)_saltwater=(v*dmax*rho/mu)_freshwater
v_salt=v_fresh*(dmax_fresh/dmax_salt)*(rho_fresh/rho_salt)*(mu_salt/mu_fresh)=5*(1/20)*(1000/1025)*(1.369/0.896935)=.3722 knot
// You can use different values for rho and mu based on the table you have to get a bit different result
PI1_salt=PI1_fresh
(D*rho/mu2)_salt=(D*rho/mu2)_fresh
(D_fresh/D_salt)=(rho_salt/rho_fresh)*(mu_fresh/mu_salt)2=(1025/1000)*(0.896935/1.369)2=0.434