Question 2.5: Force on a Disk Moving into an Axisymmetric Jet A steady uni......
Force on a Disk Moving into an Axisymmetric Jet
A steady uniform round jet impinges upon an approaching conical disk as shown. Find the force exerted on the disk, where \rm v_{jet}, \,A_{jet},\,{v}_{disk}, diameter D, angle θ, and fluid layer thickness t are given.
Assumptions: as stated; constant averaged velocities and properties.
Approach: RTT (mass balance and 1-D force balance)
| Control volume | Sketch |
 |
 |
(a) Mass Conservation
\rm 0=\int\limits_{C.S.}ρ\vec v \cdot d\vec A =-\int\limits_{A_{jet}} ρv_{rel}dA+\int\limits_{A_{exit}}ρwdA (E.2.5.1)
where \rm v_{rel} = v_{jet} –(-v_{disk}) = v_j + v_d ~\text{and}~ A_{exit} ≈ πD\,t.
∴\rm \quad-ρA_j (v_j + v_d) + ρ (πD\,t)w = 0Hence,
\rm w=\frac{(v_j+v_d)A_j}{\pi Dt} (E.2.5.2)
(b) Momentum Conservation
\rm B=(m\vec v )_s;\,\beta =\vec v ;DB\Big/D\,t:=\vec F_{surface}=-R_x.
\Sigma \vec{\mathbf{F}}_{\text {surf }}=\int\limits_{\text {C.S. }} \vec{\mathbf{v}} \rho \vec{\mathbf{v}}_{\text {rel }} \cdot \mathrm{d} \vec{\mathbf{A}} \stackrel{\mathrm{x} \text { – component }}{\Large{\longrightarrow}}-\mathrm{R}_{\mathrm{x}}\rm=\int\limits_{C.S.}{u}ρ\vec{{v}}_ {{rel}}{d}A (E.2.5.3a, b)
Thus,
\rm -R_x = −{v}_{rel}ρ{v}_{rel}A_{jet} + w \sin θ ρ{w}A_{exit}where with Eq. (E.2.5.2):
\rm R_x=\rho A_j\left(v_j+v_d\right)^2\left\lgroup1-\frac{A_j\sin\theta }{\pi tD} \right\rgroup (E.2.5.4)
Comment: The resultant fluid-structure force (E.2.5.4) can be rewritten as:
\rm R_x=(\dot mv_x)_{out}-(\dot mv_x)_{in} (E.2.5.5)
i.e., the result of a change in fluid flow momentum. If the disk would move away with v_d = v_j, i.e., escaping the jet, v_{rel} = 0 and hence R_x = 0.