Question

A tornado may be simulated as a two-part circulating flow in cylindrical coordinates, with Vr = Vz = 0. Vtheta = ωr if r ≤ R and Vtheta = ωR2 / r if r ≥ R. Determine the vorticity and the strain rates in each part of the flow.

Answer 1

Step-by-step explanation:

`$V_(\theta) = \omega r , r \leq R $`

=
`$\omega (R^2)/(r) $` , r> R

`$V_r = V_z = 0$`

Inner region vorticity

`\Omega _z =(1)/(r).(d)/(dt)(rv_(\theta))`

`\Omega _z =(1)/(r).(d)/(dt)(r\omega r), (e\omega \\eq 0)` (rotational)

Outer region vorticity

`\Omega _z =(1)/(r).(d)/(dt)(\omega e^2/r)`

= 0 (irrotational)

Strain rate of the inner region

`e_(\theta \theta)=(1)/(r).(\delta u_(\theta))/(\delta \theta)+(u_r)/(r)=0`

`e_(z z)=(\delta u_z)/(\delta z)=0`

`e_(r r)=(\delta u_r)/(\delta r)=0`

`e_(r \theta)=e_(\theta r)=(1)/(2)\left \{ r.(\delta)/(\delta r) \left ( (u_(\theta))/(r) \right )+(1)/(r).(\delta u_r)/(\delta \theta)\right \}`

`e_(r \theta)=e_(\theta r)=(1)/(2)\left \{ r.(\delta)/(\delta r) \left ( (\omega r)/(r)+(1)/(r)(0) \right )\right \}`

= 0

Therefore, the strain rate are 0.

`e_(r \theta )= e_(\theta r) =(1)/(2)\left \{ r (\delta)/(\delta r) \left ( (\delta u_(\theta))/(\delta r) \right )+(1)/(r)(\delta v_r)/(\delta \theta) \right \}`

`= (1)/(2)\left \{ r. (\delta)/(\delta r)\left ( (\omega R^2)/(r^2) \right )+(1)/(r). (\delta v_r)/(\delta \theta) \right \}`

`= (1)/(2)\left \{ v-(2)/(v^3) .\omega R^2 \right \}`

`= -(\omega R^2)/(r^2)`